Problem 55185. Ridge regularized linear regression

Created by Yuvraj

Solve Later

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margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93px 8px; transform-origin: 93px 8px; unicode-bidi: normal; ">Given a predictor data matrix X<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; "> of size <img src="data:image/png;base64,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" style="width: 47.5px; height: 18.5px;" width="47.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73px 8px; transform-origin: 73px 8px; unicode-bidi: normal; ">, target variable vector y<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; "> of size <img src="data:image/png;base64,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" style="width: 46px; height: 18.5px;" width="46" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75px 8px; transform-origin: 75px 8px; unicode-bidi: normal; "> and a shrinkage factor λ<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 8px; transform-origin: 28px 8px; unicode-bidi: normal; "> (scalar) (ridge regularization), write the function to compute linear regression model coefficients β<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 47.5px; height: 18.5px;" width="47.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.5px 8px; transform-origin: 75.5px 8px; unicode-bidi: normal; "> to model the data. The data has n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46.5px 8px; transform-origin: 46.5px 8px; unicode-bidi: normal; "> observations, p<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82px 8px; transform-origin: 82px 8px; unicode-bidi: normal; "> predictor variables in the X<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 22.5px 8px; transform-origin: 22.5px 8px; unicode-bidi: normal; "> matrix </div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79px 8px; transform-origin: 79px 8px; unicode-bidi: normal; ">The model is defines as: <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJAAAAAkCAYAAABmHbPbAAAFpklEQVR4Xu2bTehVRRjGdZ8EtUqiFrkwCqxFH9BOUEGECNG0Ni2ij0VEi4JciQsFXbRMxaBVWKAQiVBIiyLoY1NR1EIXRdSqWtS+np+cV16HmTNz75nzP+d658CD997/nPl432ferxk3b2pPk8AACWwe8G57tUlgUyNQI8EgCTQCDRJfe7kRqHFgkAQagQaJr73cCFSfA3eoy4PCU8Ltwi/CO8In9Yeavsd1I9BuiXy/8IxwWyD+D/T9W+FE9/t9+veNSNuv9NsrwjcR9T2i3z4UPhJOCn8LXwjbhbPCi9OrvO4M1o1AJj0U/bUTJeR5OiFaiPC68K/wvPB+oh19fipcDvp6Sd/f7t7Zpn+v1VXhtL2tK4GQOkTA1fBgVR6PqAJ39IOwRcBypdwQ7bA0dwsPBSTB6n3c9X2oh4DTMmHJ0deZQF6xiC9mHVA87XKKNysVc1N+nEfVV8z1lagPS/ac8JmAa53Fs84EQgG/C3d1mjgVKOaMvr8gHBEsLoopDetDoExMtUcIrdSb+u248IfwoPDXkpo3ktI/48ziWXcCmXJRBjEOrorH4hZio5czSre2KYL8pPcJounn9ACtNwINEN5Yr2I9/nSd287GdbHTDxdYDCNILBA391VCxNwaV4pAlu7u0KqoZbCDYnGACW+VU1SLc1DgzwKB8G/CPiGXMZHqX+00bxaGbMzc1Of6DHJWLEce/j42gdhMlBmeFB4QwjKHzfEmF5pzYRDpQtcZwr0/WKkRiJ/vLNittPtSeKxEYj1tvLsZ2NX19P2864S+dwolwa53gQThZHXEO/YgsxIilqxhTALRNyRPkcbP76aYMEcgXrSJ8znMImAtdQ+s1BMrSiDW1RdM9ynXrJffXGw6LBfWqWb6PgaB0B81KitnsHneE6hnnet0SxXdP6zthmUuIZA302GmYgS7Vx9ShbiS3TV1G28VCYa3Fk7ov65dyoXb3xfpMzX0GATytTA2wauCZZFmXXuz0BICsSBzVbGCG3/zAxfKfjbNfKXYJlVSr/GuL5a+05cRiM85Fx/WpZYVUK7sYP2GGWhYAKXdPwJuLdlnKYG8G/MFNxZ9TNgrLFvfWFZQNd7zWRJW1GKzvqMNG9d2L2afd8P1++MSdnfOxW80gYwcrCdVKDXLnIw5SwmUKsczwFtC6nyohpLH6sPOrsi4UO4uwQfTOdlY3JQq7PkdXkLI3DprujBvdfEqKQPgXXvUguaE5BcV+vtlrc8csjA7u2J9PkvywXRf4c9bl5R5932VuMSNJJBV2Rmzz+WZlUpa0EUIZIq3jAMBvbaE9ZkDgZgDtY7wgNS76tQBK0L31iVGDr/Da1gfxqxpgbwOUjcEfIw3OAbyC8AfkvpRZJzNmUxu+7q/W+wSC3x9xskrKcvhN1MY29gJPmdsJbFP6dTHIlAquLc19rm4hf5Xho+DIFEsai8VxlTtjDx9lXNfHI2188cfMeviyVWriFjbAoUF0LDibmTNFkIXcWFecKWp4lREiY3r/X7fxS7fjn7CHerdk5cD1uuSYLcPUULuKGQR+dS0QBYDhjcl+Z2bB9xCYPNk17AIgVgsgXSvSVtEIhvUFsv5rmDXNhiWncW1U3+vBmI8LMSuuyJMjnTIuHz6/qO+cz8Hd36P8L1wsWtTe3k1CcTcrApNBkaw/KvAiQLroQpdcpSzkAtjh3FneBVdV01lWmbSF2TXHM/6sgtl3+mH2dytLrVAZvKOjrS7xhD4GH2WpO9jjDvbPksJhNnm1t1srlJOJNFURX6i6Uw/bIxA7LIDwhXBfD4zrXGnZfoVD5uBZWg10/NhM5r47RiBwpPpWheiJl7q4OFztxIGD7CKHcQIRLDGtQ2CRQ5Kh9zjXUWZpOZM7YTbejzPCjVT9JWVU2kMtLILbBMfVwKNQOPK95bvvRHollfxuAv8H/xaYjSm2MJCAAAAAElFTkSuQmCC" style="width: 72px; height: 18px;" width="72" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 100px 8px; transform-origin: 100px 8px; unicode-bidi: normal; "> where sigma is gaussian noise.</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 248px 8px; transform-origin: 248px 8px; unicode-bidi: normal; ">(Hint: search on google for closed form solution of a linear regression problem)</div></div></div>

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15 Solutions
7 Solvers

Last Solution submitted on Nov 12, 2022

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2 Comments

2 Comments

ChrisR on 14 Jul 2022

Yuvraj, this problem looks interesting, and I was looking forward to learning about ridge regularization. I suggest that you remove the code from the function template: you are giving us the answer! Also, several Cody players have recommended at least four tests to discourage lookup table solutions and other cheats.

Christian Schröder on 12 Nov 2022 at 10:16

The test suite's incorrect: the matrix inversion in the ridge estimator is missing.

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