{"componentChunkName":"component---src-templates-post-tsx","path":"/machine_learning_models/elastic_net_regression/","result":{"data":{"logo":{"childImageSharp":{"gatsbyImageData":{"layout":"fixed","backgroundColor":"#f8f8f8","images":{"fallback":{"src":"/static/e43c6530b7b2b056de6bfee951c9e393/b181d/compasslogo.png","srcSet":"/static/e43c6530b7b2b056de6bfee951c9e393/b181d/compasslogo.png 1920w","sizes":"1920px"},"sources":[{"srcSet":"/static/e43c6530b7b2b056de6bfee951c9e393/e02c9/compasslogo.webp 1920w","type":"image/webp","sizes":"1920px"}]},"width":1920,"height":472}}},"mdx":{"body":"const _excluded = [\"components\"];\n\nfunction _extends() { _extends = Object.assign || function (target) { for (var i = 1; i < arguments.length; i++) { var source = arguments[i]; for (var key in source) { if (Object.prototype.hasOwnProperty.call(source, key)) { target[key] = source[key]; } } } return target; }; return _extends.apply(this, arguments); }\n\nfunction _objectWithoutProperties(source, excluded) { if (source == null) return {}; var target = _objectWithoutPropertiesLoose(source, excluded); var key, i; if (Object.getOwnPropertySymbols) { var sourceSymbolKeys = Object.getOwnPropertySymbols(source); for (i = 0; i < sourceSymbolKeys.length; i++) { key = sourceSymbolKeys[i]; if (excluded.indexOf(key) >= 0) continue; if (!Object.prototype.propertyIsEnumerable.call(source, key)) continue; target[key] = source[key]; } } return target; }\n\nfunction _objectWithoutPropertiesLoose(source, excluded) { if (source == null) return {}; var target = {}; var sourceKeys = Object.keys(source); var key, i; for (i = 0; i < sourceKeys.length; i++) { key = sourceKeys[i]; if (excluded.indexOf(key) >= 0) continue; target[key] = source[key]; } return target; }\n\n/* @jsxRuntime classic */\n\n/* @jsx mdx */\nconst _frontmatter = {\n  \"layout\": \"post\",\n  \"title\": \"Elastic Net Regression Explained, Step by Step\",\n  \"author\": [\"Lari Giba\"],\n  \"tags\": [\"Machine Learning Models\"],\n  \"image\": \"img/ElasticNet/ElasticNetRegression_Thumbnail.png\",\n  \"date\": \"2021-06-26T08:28:02.234Z\",\n  \"dateModified\": \"2021-06-26T08:28:02.234Z\",\n  \"changelog\": [],\n  \"draft\": false,\n  \"imgCaption\": \"Background image by Pawel Czerwinski\",\n  \"imgSource\": \"https://unsplash.com/photos/BPrk2cOoCq8\",\n  \"timeToRead\": 7,\n  \"uniquePostId\": 11,\n  \"excerpt\": \"Elastic net is a combination of the two most popular regularized variants of linear regression: ridge and lasso.  Ridge utilizes an L2 penalty and lasso uses an L1 penalty. With elastic net, you don't have to choose between these two models, because elastic net uses both the L2 and the L1 penalty! In practice, you will almost always want to use elastic net over ridge or lasso, and in this article you will learn everything you need to know to do so, successfully!\",\n  \"dependencies\": [],\n  \"proficiencyLevel\": \"Beginner\"\n};\n\nconst makeShortcode = name => function MDXDefaultShortcode(props) {\n  console.warn(\"Component \" + name + \" was not imported, exported, or provided by MDXProvider as global scope\");\n  return mdx(\"div\", props);\n};\n\nconst ConditionalLazyPlot = makeShortcode(\"ConditionalLazyPlot\");\nconst ToolTip = makeShortcode(\"ToolTip\");\nconst layoutProps = {\n  _frontmatter\n};\nconst MDXLayout = \"wrapper\";\nreturn function MDXContent(_ref) {\n  let {\n    components\n  } = _ref,\n      props = _objectWithoutProperties(_ref, _excluded);\n\n  return mdx(MDXLayout, _extends({}, layoutProps, props, {\n    components: components,\n    mdxType: \"MDXLayout\"\n  }), mdx(\"h2\", {\n    \"id\": \"outline\"\n  }, `Outline`), mdx(\"p\", null, `You have probably heard about linear regression. Most likely you have also heard\nabout ridge and lasso. Maybe you have even read\nsome articles about `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/ridge_regression\"\n  }, `ridge`), ` and `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/lasso_regression\"\n  }, `lasso`), `.\nRidge and lasso are the two most popular variations of\nlinear regression which try to make it a bit more `, mdx(\"em\", {\n    parentName: \"p\"\n  }, `robust`), `. Nowadays it is actually very uncommon\nto use regular linear regression, and not one of its variations like ridge or lasso.\nIn previous articles we have seen how `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/ridge_regression\"\n  }, `ridge`), ` and `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/lasso_regression\"\n  }, `lasso`), `\noperate, what their differences, as well as strengths and weaknesses are,\nand how you can implement them in practice.\nBut what should you use? Ridge or lasso?\nThe good news is that you don’t have to choose!\nWith elastic net, you can use both the ridge penalty as well as the lasso penalty at once.\nAnd in this article, you will learn how!`), mdx(\"p\", null, `This article is the third article in a series where we take a deep dive into ridge and lasso regression.\nLet’s start!`), mdx(\"h2\", {\n    \"id\": \"prerequisites\"\n  }, `Prerequisites`), mdx(\"p\", null, `This article is a direct follow up to the articles\nabout `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/ridge_regression\"\n  }, `ridge`), ` and `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/lasso_regression\"\n  }, `lasso`), `,\nso ideally you should read the articles about ridge and lasso before reading this article.\nElastic net is based on ridge and lasso, so it’s important to understand\nthose models first.\nWith that being said, let’s take a look at elastic net regression!`), mdx(\"h2\", {\n    \"id\": \"the-problem\"\n  }, `The Problem`), mdx(\"p\", null, `So what is wrong with linear regression? Why do we need more machine learning algorithms\nthat do the same thing? And why are there two of them? We’ve explored this question in the\narticles about ridge and lasso. Here’s a lightning-quick recap:`), mdx(\"p\", null, `We had a dataset of figure prices, where each entry in the dataset contained the age of the figure as well as its price for that age in € (or any other currency).\nWe then wanted to predict the price of a figure given its age using linear regression, to see how much the figures depreciate over time.\nThe dataset looked like this:`), mdx(ConditionalLazyPlot, _extends({}, olsVsRidgeData, {\n    mdxType: \"ConditionalLazyPlot\"\n  }), mdx(\"p\", null, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"gatsby-resp-image-wrapper\",\n    \"style\": {\n      \"position\": \"relative\",\n      \"display\": \"block\",\n      \"marginLeft\": \"auto\",\n      \"marginRight\": \"auto\",\n      \"maxWidth\": \"1106px\"\n    }\n  }, `\n      `, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"gatsby-resp-image-background-image\",\n    \"style\": {\n      \"paddingBottom\": \"63%\",\n      \"position\": \"relative\",\n      \"bottom\": \"0\",\n      \"left\": \"0\",\n      \"backgroundImage\": \"url('data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAANCAYAAACpUE5eAAAACXBIWXMAABYlAAAWJQFJUiTwAAABS0lEQVQ4y52Tf3OCMAyGPUEFJy2gKIityG+9nXfb9/9u7y6ZRep2N90fOULTPEmaZDKLjnBDjVXaQuY95L6H2HeYxwWfk/0ZMXcnpEyFwjKpGLhKGwRZawHp+5eYe5PF+oQgbeCnPdyogCMVy7OZ/ciQgCJrsUx7LNYlZpF+GWYBqbQw7+HIg2V0X8xyAIb5GUHajgwKXlLD33UvQQfg+vjOwPvDGmD7PyCPS9ZCqCsWm2rUuW8YPYk8ftwdKQg5hxretkGgrlZgbkp0OGMWF+z8GHnO31ujQs3Nowoem2c1RWQdH9jjoq2LFHAa5AiLT4Y6IrdmdJhDU563OSFWFzZQ1t6m5Atvu5r/SZdZx7q/rXjUKAF6/+W2hmMyHGdBzkY3m0JrSJvjJxULnZHNT0oGUmDvFnBYPQM1JY/LoBmN1IWhY7sj9a8lfwFWWzy/hBs7RgAAAABJRU5ErkJggg==')\",\n      \"backgroundSize\": \"cover\",\n      \"display\": \"block\"\n    }\n  }), `\n  `, mdx(\"picture\", {\n    parentName: \"span\"\n  }, `\n          `, mdx(\"source\", {\n    parentName: \"picture\",\n    \"srcSet\": [\"/static/040ab5967b5b4f588dc8db80d21003fb/601b1/figurePricesData.webp 500w\", \"/static/040ab5967b5b4f588dc8db80d21003fb/80c60/figurePricesData.webp 1000w\", \"/static/040ab5967b5b4f588dc8db80d21003fb/e6a9b/figurePricesData.webp 1106w\"],\n    \"sizes\": \"(max-width: 1106px) 100vw, 1106px\",\n    \"type\": \"image/webp\"\n  }), `\n          `, mdx(\"source\", {\n    parentName: \"picture\",\n    \"srcSet\": [\"/static/040ab5967b5b4f588dc8db80d21003fb/0eb09/figurePricesData.png 500w\", \"/static/040ab5967b5b4f588dc8db80d21003fb/1263b/figurePricesData.png 1000w\", \"/static/040ab5967b5b4f588dc8db80d21003fb/22593/figurePricesData.png 1106w\"],\n    \"sizes\": \"(max-width: 1106px) 100vw, 1106px\",\n    \"type\": \"image/png\"\n  }), `\n          `, mdx(\"img\", {\n    parentName: \"picture\",\n    \"className\": \"gatsby-resp-image-image\",\n    \"src\": \"/static/040ab5967b5b4f588dc8db80d21003fb/22593/figurePricesData.png\",\n    \"alt\": \"Figure prices dataset\",\n    \"title\": \"Figure prices dataset\",\n    \"loading\": \"lazy\",\n    \"decoding\": \"async\",\n    \"style\": {\n      \"width\": \"100%\",\n      \"height\": \"100%\",\n      \"margin\": \"0\",\n      \"verticalAlign\": \"middle\",\n      \"position\": \"absolute\",\n      \"top\": \"0\",\n      \"left\": \"0\"\n    }\n  }), `\n        `), `\n    `))), mdx(\"p\", null, `We then split our dataset into a train set and a test set, and trained our linear regression (OLS regression) model\non the training data. Here’s how that looked like:`), mdx(ConditionalLazyPlot, _extends({}, olsVsRidgeDataTrainTestOls, {\n    mdxType: \"ConditionalLazyPlot\"\n  }), mdx(\"p\", null, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"gatsby-resp-image-wrapper\",\n    \"style\": {\n      \"position\": \"relative\",\n      \"display\": \"block\",\n      \"marginLeft\": \"auto\",\n      \"marginRight\": \"auto\",\n      \"maxWidth\": \"1103px\"\n    }\n  }, `\n      `, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"gatsby-resp-image-background-image\",\n    \"style\": {\n      \"paddingBottom\": \"63%\",\n      \"position\": \"relative\",\n      \"bottom\": \"0\",\n      \"left\": \"0\",\n      \"backgroundImage\": \"url('data:image/png;base64,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')\",\n      \"backgroundSize\": \"cover\",\n      \"display\": \"block\"\n    }\n  }), `\n  `, mdx(\"picture\", {\n    parentName: \"span\"\n  }, `\n          `, mdx(\"source\", {\n    parentName: \"picture\",\n    \"srcSet\": [\"/static/026e7e9074d90660410322b8ef8dda0a/601b1/figurePricesDataOLS.webp 500w\", \"/static/026e7e9074d90660410322b8ef8dda0a/80c60/figurePricesDataOLS.webp 1000w\", \"/static/026e7e9074d90660410322b8ef8dda0a/8a7c0/figurePricesDataOLS.webp 1103w\"],\n    \"sizes\": \"(max-width: 1103px) 100vw, 1103px\",\n    \"type\": \"image/webp\"\n  }), `\n          `, mdx(\"source\", {\n    parentName: \"picture\",\n    \"srcSet\": [\"/static/026e7e9074d90660410322b8ef8dda0a/0eb09/figurePricesDataOLS.png 500w\", \"/static/026e7e9074d90660410322b8ef8dda0a/1263b/figurePricesDataOLS.png 1000w\", \"/static/026e7e9074d90660410322b8ef8dda0a/890d8/figurePricesDataOLS.png 1103w\"],\n    \"sizes\": \"(max-width: 1103px) 100vw, 1103px\",\n    \"type\": \"image/png\"\n  }), `\n          `, mdx(\"img\", {\n    parentName: \"picture\",\n    \"className\": \"gatsby-resp-image-image\",\n    \"src\": \"/static/026e7e9074d90660410322b8ef8dda0a/890d8/figurePricesDataOLS.png\",\n    \"alt\": \"OLS for predicting figure prices\",\n    \"title\": \"OLS for predicting figure prices\",\n    \"loading\": \"lazy\",\n    \"decoding\": \"async\",\n    \"style\": {\n      \"width\": \"100%\",\n      \"height\": \"100%\",\n      \"margin\": \"0\",\n      \"verticalAlign\": \"middle\",\n      \"position\": \"absolute\",\n      \"top\": \"0\",\n      \"left\": \"0\"\n    }\n  }), `\n        `), `\n    `))), mdx(\"p\", null, `We then noticed that this model had a very low training error but a rather high testing error\nand thus we concluded that our linear regression model is overfit.\nWe then tried to come up with an imaginary, better model that was less overfit and looked more like this:`), mdx(ConditionalLazyPlot, _extends({}, olsVsRidgeDataTrainTestOlsRidge, {\n    mdxType: \"ConditionalLazyPlot\"\n  }), mdx(\"p\", null, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"gatsby-resp-image-wrapper\",\n    \"style\": {\n      \"position\": \"relative\",\n      \"display\": \"block\",\n      \"marginLeft\": \"auto\",\n      \"marginRight\": \"auto\",\n      \"maxWidth\": \"1106px\"\n    }\n  }, `\n      `, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"gatsby-resp-image-background-image\",\n    \"style\": {\n      \"paddingBottom\": \"63.6%\",\n      \"position\": \"relative\",\n      \"bottom\": \"0\",\n      \"left\": \"0\",\n      \"backgroundImage\": \"url('data:image/png;base64,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')\",\n      \"backgroundSize\": \"cover\",\n      \"display\": \"block\"\n    }\n  }), `\n  `, mdx(\"picture\", {\n    parentName: \"span\"\n  }, `\n          `, mdx(\"source\", {\n    parentName: \"picture\",\n    \"srcSet\": [\"/static/bcd8acf2452585219b152cfe41dc8a87/601b1/figurePricesDataOLSRidge.webp 500w\", \"/static/bcd8acf2452585219b152cfe41dc8a87/80c60/figurePricesDataOLSRidge.webp 1000w\", \"/static/bcd8acf2452585219b152cfe41dc8a87/e6a9b/figurePricesDataOLSRidge.webp 1106w\"],\n    \"sizes\": \"(max-width: 1106px) 100vw, 1106px\",\n    \"type\": \"image/webp\"\n  }), `\n          `, mdx(\"source\", {\n    parentName: \"picture\",\n    \"srcSet\": [\"/static/bcd8acf2452585219b152cfe41dc8a87/0eb09/figurePricesDataOLSRidge.png 500w\", \"/static/bcd8acf2452585219b152cfe41dc8a87/1263b/figurePricesDataOLSRidge.png 1000w\", \"/static/bcd8acf2452585219b152cfe41dc8a87/22593/figurePricesDataOLSRidge.png 1106w\"],\n    \"sizes\": \"(max-width: 1106px) 100vw, 1106px\",\n    \"type\": \"image/png\"\n  }), `\n          `, mdx(\"img\", {\n    parentName: \"picture\",\n    \"className\": \"gatsby-resp-image-image\",\n    \"src\": \"/static/bcd8acf2452585219b152cfe41dc8a87/22593/figurePricesDataOLSRidge.png\",\n    \"alt\": \"OLS vs Ridge for predicting figure prices\",\n    \"title\": \"OLS vs Ridge for predicting figure prices\",\n    \"loading\": \"lazy\",\n    \"decoding\": \"async\",\n    \"style\": {\n      \"width\": \"100%\",\n      \"height\": \"100%\",\n      \"margin\": \"0\",\n      \"verticalAlign\": \"middle\",\n      \"position\": \"absolute\",\n      \"top\": \"0\",\n      \"left\": \"0\"\n    }\n  }), `\n        `), `\n    `))), mdx(\"p\", null, `This imaginary model turned out to be ridge regression. We analyzed what exactly lead to our\nlinear regression model overfitting and we noticed that the main cause of overfitting were\nlarge model parameters.\nAfter discovering this insight, we developed a new loss function that penalizes large model parameters\nby adding a `, mdx(\"em\", {\n    parentName: \"p\"\n  }, `penalty term`), ` to our mean squared error.\nIt looked like this (where `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `m`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `m`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `m`))))), ` is the number of model parameters):`), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `N`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `w`), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, ` `), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `L`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `o`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `s`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `s`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `)`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `E`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: 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\"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.036108em\",\n      \"verticalAlign\": \"-0.286108em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `N`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.02691em\"\n    }\n  }, `w`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\"\n  }, ` `), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `L`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `o`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `s`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `s`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.036108em\",\n      \"verticalAlign\": \"-0.286108em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.929066em\",\n      \"verticalAlign\": \"-1.277669em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-limits\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.6513970000000002em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.872331em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord 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}, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `m`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.277669em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `θ`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.31166399999999994em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.02778em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `i`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))))), mdx(\"p\", null, `There’s just one problem with this loss function. Since our model parameters can be negative, adding them might `, mdx(\"em\", {\n    parentName: \"p\"\n  }, `decrease`), `\nour loss instead of increasing it.\nIn order to circumvent this, we can either square our model parameters or take their absolute values:`), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mtable\", {\n    parentName: \"mrow\",\n    \"rowspacing\": \"0.24999999999999992em\",\n    \"columnalign\": \"right left\",\n    \"columnspacing\": \"0em\"\n  }, mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `R`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `i`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `d`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `g`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `E`)), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `)`)))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `E`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `)`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `+`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`), mdx(\"munderover\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"munderover\"\n  }, `∑`), mdx(\"mrow\", {\n    parentName: \"munderover\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `i`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `1`)), mdx(\"mi\", {\n    parentName: 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`S`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `E`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `)`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `+`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`), mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"bold-italic\"\n  }, `θ`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"msubsup\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mi\", {\n    parentName: \"msubsup\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, `2`), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, `2`)))))))), mdx(\"mspace\", {\n    parentName: \"mrow\",\n    \"linebreak\": \"newline\"\n  }), mdx(\"mtable\", {\n    parentName: \"mrow\",\n    \"rowspacing\": \"0.24999999999999992em\",\n    \"columnalign\": \"right left\",\n    \"columnspacing\": \"0em\"\n  }, mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `L`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `a`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `s`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `s`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `o`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `E`)), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `)`)))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `E`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `)`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `+`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`), mdx(\"munderover\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"munderover\"\n  }, `∑`), mdx(\"mrow\", {\n    parentName: \"munderover\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `i`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `1`)), mdx(\"mi\", {\n    parentName: \"munderover\"\n  }, `m`)), mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"msub\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `θ`), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `i`)), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`)))))), mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `E`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `)`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `+`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`), mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"bold-italic\"\n  }, `θ`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"msub\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, `1`))))))))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\begin{aligned}\n{\\\\color{#26a6ed}RidgeMSE}(y,y_{pred}) &= MSE(y,y_{pred}) + \\\\alpha \\\\sum_{i=1}^m{{\\\\color{#26a6ed}\\\\theta_i^2}} \\\\\\\\\n&= MSE(y,y_{pred}) + \\\\alpha {\\\\color{#26a6ed}||\\\\boldsymbol{\\\\theta}||_2^2} \\\\\\\\\n\\\\end{aligned}\n\n\\\\\\\\\n\n\\\\begin{aligned}\n{\\\\color{#26a6ed}LassoMSE}(y,y_{pred}) &= MSE(y,y_{pred}) + \\\\alpha \\\\sum_{i=1}^m{{\\\\color{#26a6ed}|\\\\theta_i|}} \\\\\\\\\n&= MSE(y,y_{pred}) + \\\\alpha {\\\\color{#26a6ed}||\\\\boldsymbol{\\\\theta}||_1} \\\\\\\\\n\\\\end{aligned}`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"4.7531740000000005em\",\n      \"verticalAlign\": \"-2.126587em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mtable\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.6265870000000007em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.626587000000001em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.00773em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `R`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `d`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `g`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `E`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.1848100000000006em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.126587em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-l\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.6265870000000007em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.626587000000001em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-limits\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.6513970000000002em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.872331em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel mtight\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.050005em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\"\n  }, `∑`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.3000050000000005em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `m`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.277669em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `θ`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.4530000000000003em\",\n      \"marginLeft\": \"-0.02778em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `i`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `2`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.247em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.1848100000000006em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord boldsymbol\",\n    \"style\": {\n      \"marginRight\": \"0.03194em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `θ`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.4530000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `2`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `2`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.247em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.126587em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace newline\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"4.729066em\",\n      \"verticalAlign\": \"-2.1145329999999998em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mtable\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.6145330000000007em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.614533000000001em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `L`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `a`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `s`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `s`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `o`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `E`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.1968640000000006em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.1145329999999998em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-l\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.6145330000000007em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.614533000000001em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-limits\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.6513970000000002em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.872331em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel mtight\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.050005em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\"\n  }, `∑`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.3000050000000005em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `m`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.277669em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `θ`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.31166399999999994em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.02778em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `i`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.1968640000000006em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord boldsymbol\",\n    \"style\": {\n      \"marginRight\": \"0.03194em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `θ`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.1145329999999998em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))))), mdx(\"p\", null, `The first function is the loss function of ridge regression, while the second one is the loss function of lasso regression.`), mdx(\"h2\", {\n    \"id\": \"elastic-net\"\n  }, `Elastic Net`), mdx(\"p\", null, `What we can do now is combine the two penalties, and we get the loss function of elastic net:`), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mtable\", {\n    parentName: \"semantics\",\n    \"rowspacing\": \"0.24999999999999992em\",\n    \"columnalign\": \"right left\",\n    \"columnspacing\": \"0em\"\n  }, mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mstyle\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `E`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `l`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `a`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `s`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `t`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `i`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `c`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `N`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `t`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `E`)))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `E`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `)`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `+`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `α`), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, `1`)), mdx(\"munderover\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"munderover\"\n  }, `∑`), mdx(\"mrow\", {\n    parentName: \"munderover\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `i`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `1`)), mdx(\"mi\", {\n    parentName: \"munderover\"\n  }, `m`)), mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"msub\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `θ`), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `i`)), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`)), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `+`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `α`), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, `2`)), mdx(\"munderover\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"munderover\"\n  }, `∑`), mdx(\"mrow\", {\n    parentName: \"munderover\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `i`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mn\", {\n   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parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `E`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `)`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `+`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `α`), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, `1`)), mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"bold-italic\"\n  }, `θ`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"msub\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, `1`))), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `+`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `α`), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, `2`)), mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"bold-italic\"\n  }, `θ`), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"msubsup\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mi\", {\n    parentName: \"msubsup\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, `2`), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, `2`)))))))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\begin{aligned}\n{\\\\color{#26a6ed}ElasticNetMSE} &= MSE(y,y_{pred}) + \\\\alpha_1 \\\\sum_{i=1}^m{{\\\\color{#26a6ed}|\\\\theta_i|}} +  \\\\alpha_2 \\\\sum_{i=1}^m{{\\\\color{#26a6ed}|\\\\theta_i|}} \\\\\\\\\n&= MSE(y,y_{pred}) + \\\\alpha_1 {\\\\color{#26a6ed}||\\\\boldsymbol{\\\\theta}||_1} + \\\\alpha_2 {\\\\color{#26a6ed}||\\\\boldsymbol{\\\\theta}||_2^2}  \\\\\\\\\n\\\\end{aligned}`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"4.7531740000000005em\",\n      \"verticalAlign\": \"-2.126587em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mtable\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.6265870000000007em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.626587000000001em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.01968em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `l`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `a`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `s`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `t`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `c`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `N`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `t`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `E`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.1848100000000006em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.126587em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-l\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.6265870000000007em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.626587000000001em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.0037em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-limits\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.6513970000000002em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.872331em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel mtight\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.050005em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\"\n  }, `∑`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.3000050000000005em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `m`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.277669em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `θ`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.31166399999999994em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.02778em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `i`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.0037em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `2`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-limits\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.6513970000000002em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.872331em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel mtight\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.050005em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\"\n  }, `∑`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.3000050000000005em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `m`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.277669em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `θ`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.31166399999999994em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.02778em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `i`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.1848100000000006em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.6513970000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.0037em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord boldsymbol\",\n    \"style\": {\n      \"marginRight\": \"0.03194em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `θ`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.0037em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `2`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord boldsymbol\",\n    \"style\": {\n      \"marginRight\": \"0.03194em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `θ`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.4530000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `2`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `2`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.247em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"2.126587em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))))), mdx(\"p\", null, `And that’s pretty much it! Instead of one regularization parameter `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`))))), ` we now use\ntwo parameters, one for each penalty. `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `α`), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, `1`))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha_1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.58056em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.0037em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), ` controls the L1 penalty and `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `α`), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, `2`))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha_2`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.58056em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.0037em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `2`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), ` controls\nthe L2 penalty.\nWe can now use elastic net in the same way that we can use ridge or lasso.\nIf `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `α`), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, `1`)), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `0`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha_1 = 0`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.58056em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.0037em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.64444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, `0`))))), `, then we have ridge regression. If `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `α`), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, `2`)), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `0`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha_2 = 0`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.58056em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.0037em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `2`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.64444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, `0`))))), `, we have lasso.\nAlternatively, instead of using two `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`))))), `-parameters, we can also use just one `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`))))), `\nand one `, mdx(\"em\", {\n    parentName: \"p\"\n  }, `L1-ratio`), `-parameter, which determines the percentage of our L1 penalty with regard to `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`))))), `.\nSo if `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `1`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha = 1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.64444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, `1`))))), ` and L1-ratio = 0.4, our L1 penalty will be multiplied with 0.4 and our L2\npenalty will be multiplied with `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `1`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `−`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `L`), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `1`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `−`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `a`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `t`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `i`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `o`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `0.6`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `1 - L1-ratio = 0.6`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.72777em\",\n      \"verticalAlign\": \"-0.08333em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, `1`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `−`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.76666em\",\n      \"verticalAlign\": \"-0.08333em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `L`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, `1`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `−`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.65952em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `a`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `t`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `o`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.64444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, `0`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, `.`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, `6`))))), `.\nHere’s the equation:`), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `E`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `l`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `a`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `s`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `t`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `i`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `c`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `N`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `t`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `E`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `M`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `S`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `E`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `y`), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, `,`), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `y`), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `p`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `r`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `e`), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `d`))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, `)`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `+`), mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `α`), mdx(\"mo\", {\n    parentName: \"mstyle\"\n  }, `⋅`), mdx(\"mo\", {\n    parentName: \"mstyle\",\n    \"stretchy\": \"false\"\n  }, `(`), mdx(\"mn\", {\n    parentName: \"mstyle\"\n  }, `1`), mdx(\"mo\", {\n    parentName: \"mstyle\"\n  }, `−`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `L`), mdx(\"mn\", {\n    parentName: \"mstyle\"\n  }, `1`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `R`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `a`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `t`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `i`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `o`), mdx(\"mo\", {\n    parentName: \"mstyle\",\n    \"stretchy\": \"false\"\n  }, `)`)), mdx(\"munderover\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"munderover\"\n  }, `∑`), mdx(\"mrow\", {\n    parentName: \"munderover\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `i`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `1`)), mdx(\"mi\", {\n    parentName: \"munderover\"\n  }, `m`)), mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"msub\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `θ`), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `i`)), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`)), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `+`), mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `α`), mdx(\"mo\", {\n    parentName: \"mstyle\"\n  }, `⋅`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `L`), mdx(\"mn\", {\n    parentName: \"mstyle\"\n  }, `1`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `R`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `a`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `t`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `i`), mdx(\"mi\", {\n    parentName: \"mstyle\"\n  }, `o`)), mdx(\"munderover\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"munderover\"\n  }, `∑`), mdx(\"mrow\", {\n    parentName: \"munderover\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `i`), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, `=`), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, `1`)), mdx(\"mi\", {\n    parentName: \"munderover\"\n  }, `m`)), mdx(\"mstyle\", {\n    parentName: \"mrow\",\n    \"mathcolor\": \"#26a6ed\"\n  }, mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`), mdx(\"msub\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `θ`), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, `i`)), mdx(\"mi\", {\n    parentName: \"mstyle\",\n    \"mathvariant\": \"normal\"\n  }, `∣`))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\nElasticNetMSE = MSE(y,y_{pred}) + {\\\\color{#26a6ed}\\\\alpha \\\\cdot (1 - L1Ratio)} \\\\sum_{i=1}^m{{\\\\color{#26a6ed}|\\\\theta_i|}} + {\\\\color{#26a6ed}\\\\alpha \\\\cdot L1Ratio} \\\\sum_{i=1}^m{{\\\\color{#26a6ed}|\\\\theta_i|}}\n\n`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.69444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.01968em\"\n    }\n  }, `l`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `a`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `s`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `t`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `c`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `N`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, `t`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.036108em\",\n      \"verticalAlign\": \"-0.286108em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }, `M`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `S`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }, `E`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, `,`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, `y`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.3361079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `p`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, `r`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `e`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `d`))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.286108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, `)`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.929066em\",\n      \"verticalAlign\": \"-1.277669em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `⋅`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `(`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `1`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `−`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `L`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `1`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.00773em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `R`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `a`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `t`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `o`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `)`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-limits\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.6513970000000002em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.872331em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel mtight\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.050005em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\"\n  }, `∑`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.3000050000000005em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `m`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.277669em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `θ`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.31166399999999994em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.02778em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `i`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, `+`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.929066em\",\n      \"verticalAlign\": \"-1.277669em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `α`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `⋅`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `L`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `1`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.00773em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `R`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `a`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `t`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `o`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-limits\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.6513970000000002em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.872331em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `i`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel mtight\"\n  }, `=`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, `1`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.050005em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\"\n  }, `∑`))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.3000050000000005em\",\n      \"marginLeft\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.05em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, `m`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.277669em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\",\n      \"color\": \"#26a6ed\"\n    }\n  }, `θ`), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.31166399999999994em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.02778em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `i`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, `​`)), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"color\": \"#26a6ed\"\n    }\n  }, `∣`)))))))), mdx(\"p\", null, `Ok, looks good!\nIn this case we have ridge regression if L1-ratio = 0 and lasso regression if L1-ratio = 1.\nIn most cases, unless you already have some information about the importance\nof your features, you should use elastic net instead of lasso or ridge.\nYou can then use cross-validation to determine the best ratio between L1 and L2 penalty strength.\nNow let’s look at how we determine the optimal model parameters `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold-italic\"\n  }, `θ`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\boldsymbol{\\\\theta}`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.69444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord boldsymbol\",\n    \"style\": {\n      \"marginRight\": \"0.03194em\"\n    }\n  }, `θ`))))))), ` for our elastic net model.`), mdx(\"h2\", {\n    \"id\": \"solving-elastic-net\"\n  }, `Solving Elastic Net`), mdx(\"p\", null, `If L1-ratio = 0, we have ridge regression. This means that we can treat our model\nas a ridge regression model, and solve it in the `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/ridge_regression/#solving-ridge-regression\"\n  }, `same ways we would solve ridge regression`), `.\nNamely, we can use the normal equation for ridge regression to solve our model directly,\nor we can use gradient descent to solve it iteratively.`), mdx(\"p\", null, `If L1-ratio = 1, we have lasso regression. Then we can solve it with the `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/lasso_regression/#solving-lasso-regression\"\n  }, `same ways we would use to solve lasso regression`), `.\nSince our model contains absolute values, we can’t construct a normal equation,\nand neither can we use (regular) gradient descent. Instead,\nwe can use an adaptation of gradient descent like `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_math/subgradient_descent\"\n  }, `subgradient descent`), ` or `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_math/coordinate_descent\"\n  }, `coordinate descent`), `.`), mdx(\"p\", null, `If we are using both the L1 and the L2-penalty, then we also have absolute values,\nso we can use the same techniques as the ones we would use for lasso regression,\nlike subgradient descent or coordinate descent.`), mdx(\"h3\", {\n    \"id\": \"implementing-elastic-net-regression\"\n  }, `Implementing Elastic Net Regression`), mdx(\"p\", null, `If you’re interested in implementing elastic net from scratch,\nthen I recommend that you take a look at the articles about `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_math/subgradient_descent\"\n  }, `subgradient descent`), ` or `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_math/coordinate_descent\"\n  }, `coordinate descent`), `,\nwhere we do exactly that! In this article, we will use scikit-learn to help us out.\nScikit-learn provides a `, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `ElasticNet`), `-class, which implements coordinate descent under the hood.\nWe can use it like this:`), mdx(\"pre\", null, mdx(\"code\", {\n    parentName: \"pre\",\n    \"className\": \"language-python\"\n  }, `elastic_pipeline = make_pipeline(StandardScaler(),\n                                  ElasticNet(alpha=1, l1_ratio=0.1))\nelastic_pipeline.fit(X_train, y_train)\nprint(elastic_pipeline[1].intercept_, elastic_pipeline[1].coef_)\n# output: 41.0 [-1.2127174]\n`)), mdx(\"p\", null, `Just like with `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/lasso_regression/#lassos-lesser-known-twin-sgdregressor\"\n  }, `lasso`), `,\nwe can also use scikit-learn’s `, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `SGDRegressor`), `-class, which uses `, mdx(\"em\", {\n    parentName: \"p\"\n  }, `truncated`), ` gradients instead of regular ones.\nHere’s the code:`), mdx(\"pre\", null, mdx(\"code\", {\n    parentName: \"pre\",\n    \"className\": \"language-python\"\n  }, `elastic_sgd_pipeline = make_pipeline(StandardScaler(), SGDRegressor(alpha=1, l1_ratio=0.1, penalty = \"elasticnet\"))\n\nelastic_sgd_pipeline.fit(X_train, y_train)\n\nprint(elastic_sgd_pipeline[1].intercept_, elastic_sgd_pipeline[1].coef_)\n# output: [40.69570804] [-1.21309447]\n`)), mdx(\"div\", null, mdx(\"p\", null, `Cool! In practice, you should probably stick to `, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `ElasticNet`), ` instead of `, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `SGDRegressor`), ` since\ncoordinate descent converges more quickly than the truncated SGD in this scenario.`, mdx(ToolTip, {\n    width: \"20em\",\n    mdxType: \"ToolTip\"\n  }, `Coordinate descent for lasso in particular is extremely efficient. The `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_math/coordinate_descent\"\n  }, `article about coordinate descent`), `\ngoes into more depth as to why this is, but in general coordinate descent is the preferred way to train lasso or elastic net models.`))), mdx(\"p\", null, `Since we’re using regularized models like lasso or elastic net it is important to first standardize our data before feeding it into our regularized model!\nIf you’re interested in what happens when we `, mdx(\"em\", {\n    parentName: \"p\"\n  }, `don’t`), ` standardize our data, check out `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/dataset_optimization/standardization\"\n  }, `When You Should Standardize Your Data`), `.\nThere you will learn all about standardization as well as pipelines in scikit-learn, which is what we’ve\nused in the above code to make our lives a bit easier.`), mdx(\"h3\", {\n    \"id\": \"parameter-sparsity-testing-for-elastic-net\"\n  }, `Parameter Sparsity Testing for Elastic Net`), mdx(\"p\", null, `The most important property of lasso is that lasso produces sparse model weights,\nmeaning weights can be set all the way to 0. Whenever you are presented with an implementation\nof lasso (or any model that incorporates an L1-penalty, like elastic net),\nyou should verify that this property actually holds.\nThe easiest way to do so is to generate a randomized dataset, fit the model on it,\nand see whether or not all of the parameters are zeroed-out. Here it goes:`), mdx(\"pre\", null, mdx(\"code\", {\n    parentName: \"pre\",\n    \"className\": \"language-python\"\n  }, `elastic_rand_pipeline = make_pipeline(StandardScaler(),\n                                  ElasticNet(alpha=1, l1_ratio=0.1))\nelastic_rand_pipeline.fit(X_rand, y_rand)\nprint(elastic_rand_pipeline[1].intercept_, elastic_rand_pipeline[1].coef_)\n# output:\n# 0.4881255425051216 [-0.  0. -0. -0. -0. -0. -0. -0. -0.  0. -0.  0. -0. -0.  0.  0. -0. -0.\n#  -0.  0. -0.  0. -0. -0.  0. -0.  0.  0.  0. -0. -0.  0. -0. -0.  0.  0.\n# -0.  0. -0. -0.  0.  0.  0.  0. -0.  0.  0.  0. -0. -0.]\n`)), mdx(\"p\", null, `Nice, the weights are all zeroed out!\nWe can perform the same test for `, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `SGDRegressor`), `:`), mdx(\"pre\", null, mdx(\"code\", {\n    parentName: \"pre\",\n    \"className\": \"language-python\"\n  }, `elastic_sgd_rand_pipeline = make_pipeline(StandardScaler(), SGDRegressor(alpha=1, l1_ratio=0.1, penalty = \"elasticnet\"))\n\nelastic_sgd_rand_pipeline.fit(X_rand, y_rand)\n\nprint(elastic_sgd_rand_pipeline[1].intercept_, elastic_sgd_rand_pipeline[1].coef_)\n# output:\n# [0.46150165] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n#  0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n#  0. 0.]\n`)), mdx(\"p\", null, `Nice!`), mdx(\"h3\", {\n    \"id\": \"finding-the-optimal-value-for-alpha-and-the-l1-ratio\"\n  }, `Finding the optimal value for `, mdx(\"span\", {\n    parentName: \"h3\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`))))), ` and the L1-ratio`), mdx(\"p\", null, `Here, we can use the power of cross-validation to compute the most optimal\nparameters for our model. Scikit-learn even provides a special class for this\ncalled `, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `ElasticNetCV`), `. It takes in an array of `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`))))), `-values to compare and select\nthe best of. If no array of `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`))))), `-values is provided, scikit-learn will automatically\ndetermine the optimal value of `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`))))), `. We can use it like so:`), mdx(\"pre\", null, mdx(\"code\", {\n    parentName: \"pre\",\n    \"className\": \"language-python\"\n  }, `elastic_cv_pipeline = make_pipeline(StandardScaler(),\n                                  ElasticNetCV(l1_ratio=0.1))\nelastic_cv_pipeline.fit(X_train, y_train)\nprint(elastic_cv_pipeline[1].alpha_)\n# output: 0.6385668121344372\n`)), mdx(\"p\", null, `Ok that’s nice, but how can you find an optimal value for the L1-ratio?\n`, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `ElasticNetCV`), ` only determines the optimal value for `, mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, `α`)), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, `\\\\alpha`)))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.0037em\"\n    }\n  }, `α`))))), `, so if we want to\ndetermine the optimal value for the L1-ratio as well, we’ll have to do an additional round\nof cross-validation. For this, we can use techniques such as grid or random search,\nwhich you can learn more about by reading the article `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/model_optimization/grid_and_random_search\"\n  }, `Grid and Random Search Explained, Step by Step`), `.`), mdx(\"h3\", {\n    \"id\": \"a-summary-of-scikit-learn-classes\"\n  }, `A Summary of Scikit-Learn-Classes`), mdx(\"p\", null, `We’ve looked at quite a few models so far.\nTo make it easier to remember when you should use which scikit-learn-class,\nI’ve created this little table. Here you can find the corresponding scikit-learn\nclass for every model and every solver. This should make it a bit more organized.`), mdx(\"div\", {\n    className: \"largeTable\"\n  }, mdx(\"table\", null, mdx(\"thead\", {\n    parentName: \"table\"\n  }, mdx(\"tr\", {\n    parentName: \"thead\"\n  }, mdx(\"th\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, `Model / Solver`), mdx(\"th\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, `Normal Equation`), mdx(\"th\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, `Gradient Descent variant`))), mdx(\"tbody\", {\n    parentName: \"table\"\n  }, mdx(\"tr\", {\n    parentName: \"tbody\"\n  }, mdx(\"td\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, `OLS Regression`), mdx(\"td\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, mdx(\"a\", {\n    parentName: \"td\",\n    \"href\": \"https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html?highlight=linearregression\",\n    \"target\": \"_blank\",\n    \"rel\": \"nofollow noopener noreferrer\"\n  }, `LinearRegression`)), mdx(\"td\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, mdx(\"a\", {\n    parentName: \"td\",\n    \"href\": \"https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.SGDRegressor.html\",\n    \"target\": \"_blank\",\n    \"rel\": \"nofollow noopener noreferrer\"\n  }, `SGDRegressor`), ` `, mdx(\"br\", null), ` `, mdx(\"span\", {\n    style: {\n      \"color\": \"#26a6ed\"\n    }\n  }, `[SGD]`))), mdx(\"tr\", {\n    parentName: \"tbody\"\n  }, mdx(\"td\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, `Ridge`), mdx(\"td\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, mdx(\"a\", {\n    parentName: \"td\",\n    \"href\": \"https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.Ridge.html\",\n    \"target\": \"_blank\",\n    \"rel\": \"nofollow noopener noreferrer\"\n  }, `Ridge`)), mdx(\"td\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, mdx(\"a\", {\n    parentName: \"td\",\n    \"href\": 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\"href\": \"https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.SGDRegressor.html\",\n    \"target\": \"_blank\",\n    \"rel\": \"nofollow noopener noreferrer\"\n  }, `SGDRegressor`), ` with penalty=“l1” `, mdx(\"span\", {\n    style: {\n      \"color\": \"#26a6ed\"\n    }\n  }, `[Truncated SGD]`))), mdx(\"tr\", {\n    parentName: \"tbody\"\n  }, mdx(\"td\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, `Elastic Net`), mdx(\"td\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, `/`), mdx(\"td\", {\n    parentName: \"tr\",\n    \"align\": null\n  }, mdx(\"a\", {\n    parentName: \"td\",\n    \"href\": \"https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.ElasticNet.html\",\n    \"target\": \"_blank\",\n    \"rel\": \"nofollow noopener noreferrer\"\n  }, `ElasticNet`), ` `, mdx(\"span\", {\n    style: {\n      \"color\": \"#26a6ed\"\n    }\n  }, `[Coordinate Descent]`), ` or `, mdx(\"br\", null), ` `, mdx(\"a\", {\n    parentName: \"td\",\n    \"href\": \"https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.SGDRegressor.html\",\n    \"target\": \"_blank\",\n    \"rel\": \"nofollow noopener noreferrer\"\n  }, `SGDRegressor`), ` with penalty=“elasticnet” `, mdx(\"span\", {\n    style: {\n      \"color\": \"#26a6ed\"\n    }\n  }, `[Truncated SGD]`)))))), mdx(\"h2\", {\n    \"id\": \"further-reading\"\n  }, `Further Reading`), mdx(\"p\", null, `We’ve looked at ridge, lasso, and elastic net in the context of regression,\nbut we can also take the corresponding penalties and apply them to other models,\nlike logistic regression or polynomial regression, as well.\nIf you’re interested in these regularized models,\nI recommend you take a look at the articles `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/logistic_regression\"\n  }, `Logistic Regression Explained, Step by Step`), `\nand `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/machine_learning_models/polynomial_regression\"\n  }, `Polynomial Regression Explained, Step by Step`), ` respectively.\nIn those articles you will learn everything about the named models as well as their regularized variants!`), mdx(\"p\", null, `Classes like `, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `RidgeCV`), `, `, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `LassoCV`), ` or `, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `ElasticNetCV`), ` are handy,\nbut not every model has a `, mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, `CV`), `-variant. But if you know how cross-validation works,\nyou can make these models yourself! Cross-validation is an extremely important method\nto train machine learning models effectively and to optimize hyperparameters.\nYou can read more about this technique in the article `, mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/model_optimization/cross_validation\"\n  }, `Cross-Validation Explained, Step by Step`), `,\nwhere you will learn everything you need to know to start using cross-validation in your own projects!`));\n}\n;\nMDXContent.isMDXComponent = true;","excerpt":"Elastic net is a combination of the two most popular regularized variants of linear regression: ridge and lasso.  Ridge utilizes an L2 penalty and lasso uses an L1 penalty. With elastic net, you don't have to choose between these two models, because elastic net uses both the L2 and the L1 penalty! In practice, you will almost always want to use elastic net over ridge or lasso, and in this article you will learn everything you need to know to do so, successfully!","tableOfContents":{"items":[{"url":"#outline","title":"Outline"},{"url":"#prerequisites","title":"Prerequisites"},{"url":"#the-problem","title":"The Problem"},{"url":"#elastic-net","title":"Elastic Net"},{"url":"#solving-elastic-net","title":"Solving Elastic Net","items":[{"url":"#implementing-elastic-net-regression","title":"Implementing Elastic Net Regression"},{"url":"#parameter-sparsity-testing-for-elastic-net","title":"Parameter Sparsity Testing for Elastic Net"},{"url":"#finding-the-optimal-value-for-alpha-and-the-l1-ratio","title":"Finding the optimal value for \\alpha and the L1-ratio"},{"url":"#a-summary-of-scikit-learn-classes","title":"A Summary of Scikit-Learn-Classes"}]},{"url":"#further-reading","title":"Further Reading"}]},"frontmatter":{"title":"Elastic Net Regression Explained, Step by Step","userDate":"26 June 2021","date":"2021-06-26T08:28:02.234Z","dateModified":"2021-06-26T08:28:02.234Z","changelog":[],"tags":["Machine Learning Models"],"timeToRead":7,"excerpt":"Elastic net is a combination of the two most popular regularized variants of linear regression: ridge and lasso.  Ridge utilizes an L2 penalty and lasso uses an L1 penalty. With elastic net, you don't have to choose between these two models, because elastic net uses both the L2 and the L1 penalty! In practice, you will almost always want to use elastic net over ridge or lasso, and in this article you will learn everything you need to know to do so, successfully!","image":{"childImageSharp":{"gatsbyImageData":{"layout":"fullWidth","placeholder":{"fallback":"data:image/png;base64,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"},"images":{"fallback":{"src":"/static/73e4a37e19eb2668d1fddc5ebf3bd7c1/87706/ElasticNetRegression_Thumbnail.png","srcSet":"/static/73e4a37e19eb2668d1fddc5ebf3bd7c1/0dee1/ElasticNetRegression_Thumbnail.png 750w,\n/static/73e4a37e19eb2668d1fddc5ebf3bd7c1/8beaa/ElasticNetRegression_Thumbnail.png 1080w,\n/static/73e4a37e19eb2668d1fddc5ebf3bd7c1/87706/ElasticNetRegression_Thumbnail.png 1280w","sizes":"100vw"},"sources":[{"srcSet":"/static/73e4a37e19eb2668d1fddc5ebf3bd7c1/a66aa/ElasticNetRegression_Thumbnail.webp 750w,\n/static/73e4a37e19eb2668d1fddc5ebf3bd7c1/65dd5/ElasticNetRegression_Thumbnail.webp 1080w,\n/static/73e4a37e19eb2668d1fddc5ebf3bd7c1/71d4d/ElasticNetRegression_Thumbnail.webp 1280w","type":"image/webp","sizes":"100vw"}]},"width":1,"height":0.5625}}},"author":[{"id":"Lari Giba","bio":"Hi! 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You can find more information in the \"About\"-tab.","avatar":{"children":[{"fluid":{"base64":"data:image/jpeg;base64,/9j/2wBDABALDA4MChAODQ4SERATGCgaGBYWGDEjJR0oOjM9PDkzODdASFxOQERXRTc4UG1RV19iZ2hnPk1xeXBkeFxlZ2P/2wBDARESEhgVGC8aGi9jQjhCY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2P/wgARCAAbABQDASIAAhEBAxEB/8QAGAAAAwEBAAAAAAAAAAAAAAAAAAMEAgH/xAAWAQEBAQAAAAAAAAAAAAAAAAABAgP/2gAMAwEAAhADEAAAAeDIop5sDKn0aCSgzv8A/8QAHhABAAICAQUAAAAAAAAAAAAAAQACAxEEEyEiMjP/2gAIAQEAAQUC3ZAE6ZGW8HtMw1M2O+tTlfI9Z//EABURAQEAAAAAAAAAAAAAAAAAABAR/9oACAEDAQE/AYU//8QAFxEBAAMAAAAAAAAAAAAAAAAAABAREv/aAAgBAgEBPwFbMf/EABwQAAIBBQEAAAAAAAAAAAAAAAABMQIQESFxkf/aAAgBAQAGPwLfpJvBhuRVIgXRNwpsxct//8QAHRABAAMAAgMBAAAAAAAAAAAAAQARIUFRMWGBcf/aAAgBAQABPyG78vaUcDUeYJrJrCUcHAhzK6pYRVinqYAdKT7Y0/Yi+8UGE//aAAwDAQACAAMAAAAQsO1C/8QAGBEBAQEBAQAAAAAAAAAAAAAAAREAECH/2gAIAQMBAT8QPF4uTBS7/8QAGBEBAQEBAQAAAAAAAAAAAAAAAQARIVH/2gAIAQIBAT8QO2vYEgOZf//EAB8QAQEAAgIBBQAAAAAAAAAAAAERACExQVFhcZGxwf/aAAgBAQABPxDlRMDhDreGljrJ8uK5DiW/WJE4rXSx7wgAOy8sEKuzsxERAF59XxirS0k0dz03g4T3iY5jH9MRu2FvejEgEAAM/9k=","aspectRatio":0.7407407407407407,"src":"/static/9eab840ebd4087faf31c48702c92af36/2f1b1/lari.jpg","srcSet":"/static/9eab840ebd4087faf31c48702c92af36/b3d27/lari.jpg 40w,\n/static/9eab840ebd4087faf31c48702c92af36/5a735/lari.jpg 80w,\n/static/9eab840ebd4087faf31c48702c92af36/6e81a/lari.jpg 120w,\n/static/9eab840ebd4087faf31c48702c92af36/2f1b1/lari.jpg 800w,\n/static/9eab840ebd4087faf31c48702c92af36/75b83/lari.jpg 1520w","sizes":"(max-width: 800px) 100vw, 800px"}}]}}],"imgCaption":"Background image by Pawel Czerwinski","imgSource":"https://unsplash.com/photos/BPrk2cOoCq8","uniquePostId":11,"tutorialId":null,"tutorialEntryIdx":null,"dependencies":[],"proficiencyLevel":"Beginner"}},"relatedPosts":{"totalCount":4,"edges":[{"node":{"id":"8da3beae-1c9f-5fae-a6b9-3fd103f104ef","excerpt":"Elastic net is a combination of the two most popular regularized variants of linear regression: ridge and lasso.  Ridge utilizes an L2 penalty and lasso uses an L1 penalty. With elastic net, you don't have to choose between these two models, because elastic net uses both the L2 and the L1 penalty! In practice, you will almost always want to use elastic net over ridge or lasso, and in this article you will learn everything you need to know to do so, successfully!","frontmatter":{"title":"Elastic Net Regression Explained, Step by Step","timeToRead":7,"date":"2021-06-26T08:28:02.234Z"},"fields":{"slug":"/machine_learning_models/elastic_net_regression/"}}},{"node":{"id":"f6288123-30a0-5918-8d58-1607badab75e","excerpt":"Lasso regression is an adaptation of the popular and widely used linear regression algorithm. It enhances regular linear regression by slightly changing its cost function, which results in less overfit models. Lasso regression is very similar to ridge regression, but there are some key differences between the two that you will have to understand if you want to use them effectively. In this article, you will learn everything you need to know about lasso regression, the differences between lasso and ridge, as well as how you can start using lasso regression in your own machine learning projects.","frontmatter":{"title":"Lasso Regression Explained, Step by Step","timeToRead":16,"date":"2021-06-20T16:35:36.458Z"},"fields":{"slug":"/machine_learning_models/lasso_regression/"}}},{"node":{"id":"a9afab9b-71b2-5906-a280-fb2dd2afa180","excerpt":"Ridge Regression is an adaptation of the popular and widely used linear regression algorithm. It enhances regular linear regression by slightly changing its cost function, which results in less overfit models. In this article, you will learn everything you need to know about Ridge Regression, and how you can start using it in your own machine learning projects.","frontmatter":{"title":"Ridge Regression Explained, Step by Step","timeToRead":26,"date":"2021-05-23T19:13:45.907Z"},"fields":{"slug":"/machine_learning_models/ridge_regression/"}}},{"node":{"id":"6be475fd-5ce4-55fb-a78f-49f61e026a2e","excerpt":"Linear regression is one of the most famous algorithms in statistics and machine learning. In this post you will learn how linear regression works on a fundamental level. You will also implement linear regression both from scratch as well as with the popular library scikit-learn in Python. You will learn when and how to best use linear regression in your machine learning projects. You do not need any knowledge prior to reading this article.","frontmatter":{"title":"Linear Regression Explained, Step by Step","timeToRead":23,"date":"2021-03-20T18:17:30.352Z"},"fields":{"slug":"/machine_learning_models/linear_regression/"}}}]}},"pageContext":{"slug":"/machine_learning_models/elastic_net_regression/","prev":{"excerpt":"Lasso regression is an adaptation of the popular and widely used linear regression algorithm. It enhances regular linear regression by slightly changing its cost function, which results in less overfit models. Lasso regression is very similar to ridge regression, but there are some key differences between the two that you will have to understand if you want to use them effectively. In this article, you will learn everything you need to know about lasso regression, the differences between lasso and ridge, as well as how you can start using lasso regression in your own machine learning projects.","frontmatter":{"title":"Lasso Regression Explained, Step by Step","tags":["Machine Learning Models"],"date":"2021-06-20T16:35:36.458Z","dateModified":"2021-06-26T08:28:02.234Z","changelog":[["2021-06-26T08:28:02.234Z","Added section about SGDRegressor."],["2022-01-18T09:43:47.766Z","Fixed small typo in section about qualitative difference between Ridge and Lasso. Thanks to Tianyu for pointing this out!"]],"draft":false,"excerpt":"Lasso regression is an adaptation of the popular and widely used linear regression algorithm. It enhances regular linear regression by slightly changing its cost function, which results in less overfit models. Lasso regression is very similar to ridge regression, but there are some key differences between the two that you will have to understand if you want to use them effectively. In this article, you will learn everything you need to know about lasso regression, the differences between lasso and ridge, as well as how you can start using lasso regression in your own machine learning projects.","uniquePostId":10,"timeToRead":16,"tutorialId":null,"tutorialEntryIdx":null,"dependencies":[],"proficiencyLevel":"Beginner","image":{"childImageSharp":{"gatsbyImageData":{"layout":"fullWidth","placeholder":{"fallback":"data:image/png;base64,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"},"images":{"fallback":{"src":"/static/1cbee7448495e7e7fa3771995e853206/87706/LassoRegression_Thumbnail.png","srcSet":"/static/1cbee7448495e7e7fa3771995e853206/0dee1/LassoRegression_Thumbnail.png 750w,\n/static/1cbee7448495e7e7fa3771995e853206/8beaa/LassoRegression_Thumbnail.png 1080w,\n/static/1cbee7448495e7e7fa3771995e853206/87706/LassoRegression_Thumbnail.png 1280w","sizes":"100vw"},"sources":[{"srcSet":"/static/1cbee7448495e7e7fa3771995e853206/a66aa/LassoRegression_Thumbnail.webp 750w,\n/static/1cbee7448495e7e7fa3771995e853206/65dd5/LassoRegression_Thumbnail.webp 1080w,\n/static/1cbee7448495e7e7fa3771995e853206/71d4d/LassoRegression_Thumbnail.webp 1280w","type":"image/webp","sizes":"100vw"}]},"width":1,"height":0.5625}}},"author":[{"id":"Lari Giba","bio":"Hi! I'm Lari and I run this website. 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What is even more important is that many models, especially regularized ones, require the data to be standardized in order to function properly. In this article, you will learn everything you need to know about standardization. You will learn why it works, when you should use it, and how you can do so with just a few lines of code.","frontmatter":{"title":"When, Why, And How You Should Standardize Your Data","tags":["Dataset Optimization"],"date":"2021-07-06T11:47:10.172Z","dateModified":"2021-07-06T11:47:10.172Z","changelog":[],"draft":false,"excerpt":"Standardization is one of the most useful transformations you can apply to your dataset. What is even more important is that many models, especially regularized ones, require the data to be standardized in order to function properly. In this article, you will learn everything you need to know about standardization. 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