{"id":463,"date":"2018-09-23T23:04:25","date_gmt":"2018-09-23T14:04:25","guid":{"rendered":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/how-to-conduct-ridge-lasso-elastic-net-in-r\/"},"modified":"2024-10-12T19:20:52","modified_gmt":"2024-10-12T10:20:52","slug":"how-to-conduct-ridge-lasso-elastic-net-in-r","status":"publish","type":"post","link":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/how-to-conduct-ridge-lasso-elastic-net-in-r\/","title":{"rendered":"R \u3067\u30ea\u30c3\u30b8\u56de\u5e30\u30fb\u30e9\u30c3\u30bd\u56de\u5e30\u30fb\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u3092\u5b9f\u884c\u3059\u308b\u65b9\u6cd5"},"content":{"rendered":"\n

\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u3092\u7c21\u5358\u306b\u89e3\u8aac<\/p>\n\n\n\n

R \u3067\u5b9f\u884c\u3059\u308b\u65b9\u6cd5\u3082\u89e3\u8aac<\/p>\n\n\n\n\n\n\n\n

\u30ea\u30c3\u30b8\u30fb\u30e9\u30c3\u30bd\u30fb\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u3068\u306f\uff1f<\/h2>\n\n\n\n

\u7dda\u5f62\u56de\u5e30\u30e2\u30c7\u30eb\u306f\u3001\u4fc2\u6570 \u03b2\uff08\u30d1\u30e9\u30e1\u30fc\u30bf\uff09\u3092\u63a8\u5b9a\u3059\u308b\u3068\u304d\u306b\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3092\u7528\u3044\u308b\u3002<\/p>\n\n\n\n

\u901a\u5e38\u306e\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306f\u3001\u5f93\u5c5e\u5909\u6570\u306e\u5b9f\u6e2c\u5024\u3068\u30e2\u30c7\u30eb\u304b\u3089\u8a08\u7b97\u3055\u308c\u305f\u5024\u3068\u306e\u5dee\uff1d\u6b8b\u5dee\u3092\u4e8c\u4e57\u3057\u3066\u8db3\u3057\u5408\u308f\u305b\u308b\u3002<\/p>\n\n\n\n

\u3053\u306e\u5408\u8a08\u304c\u6700\u5c0f\u306b\u306a\u308b\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u30bb\u30c3\u30c8\u3092\u898b\u3064\u3051\u308b\u3002<\/p>\n\n\n\n

\u30ea\u30c3\u30b8\u3001\u30e9\u30c3\u30bd\uff08LASSO\uff09\u3001\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u306f\u6b8b\u5dee\u306e\u4e8c\u4e57\u548c\u306b\u4ee5\u4e0b\u306e\u5f0f\u3092\u8db3\u3057\u305f\u3082\u306e\u3092\u6700\u5c0f\u306b\u3059\u308b\u65b9\u6cd5\u3092\u53d6\u308b\u3002\u300c\u7f70\u5247\u300d\u3068\u547c\u3076\u3002<\/p>\n\n\n\n

$$ \\lambda \\left( \\frac{1 – \\alpha}{2} || \\beta ||_2^2 + \\alpha || \\beta ||_1 \\right) $$<\/p>\n\n\n\n\n\n\n\n

$$ = \\lambda \\sum_{j=1}^p \\left( \\frac{1 – \\alpha}{2} \\beta_j^2 + \\alpha |\\beta_j| \\right) $$<\/p>\n\n\n\n\n\n\n\n

\u30ea\u30c3\u30b8\u306f\u6700\u521d\u306e\u9805\u3060\u3051\u3001\u30e9\u30c3\u30bd\u306f\u5f8c\u306e\u9805\u3060\u3051\u3001\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u306f\u3069\u3061\u3089\u306e\u9805\u3082\u4f7f\u3046\u65b9\u6cd5\u3060\u3002<\/p>\n\n\n\n

\u03b1 \u304c0\u306e\u6642\u306f\u30ea\u30c3\u30b8\u3001\u03b1 \u304c1\u306e\u6642\u306f\u30e9\u30c3\u30bd\u30010\u30681\u306e\u9593\u306e\u3068\u304d\u306f\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

\u6700\u521d\u306e\u9805\u306f$ \\beta^2 $ \u306e\u548c\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

\u5f8c\u306e\u9805\u306f \u03b2 \u306e\u7d76\u5bfe\u5024\u306e\u548c\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

\u3044\u305a\u308c\u3082 \u03b2 \u304c\u591a\u304f\u306a\u308b\u3001\u5927\u304d\u304f\u306a\u308b\u3068\u3001\u5168\u4f53\u304c\u5927\u304d\u304f\u306a\u308b\u3002<\/p>\n\n\n\n

\u3064\u307e\u308a\u3001\u3069\u3093\u306a\u306b\u6b8b\u5dee\u4e8c\u4e57\u548c\u304c\u5c0f\u3055\u304f\u306a\u308b\u3088\u3046\u306b\u3057\u3066\u3082\u3001\u3053\u306e\u9805\u305f\u3061\u304c\u5927\u304d\u304f\u306a\u3063\u305f\u306e\u3067\u306f\u3001\u76ee\u7684\u304c\u9054\u6210\u3067\u304d\u306a\u3044\u3002<\/p>\n\n\n\n

\u300c\u7f70\u5247\u300d\u3068\u547c\u3070\u308c\u308b\u3086\u3048\u3093\u3060\u3002<\/p>\n\n\n\n

\u03b2 \u3092\u5c0f\u3055\u304f\u3059\u308b\u3001\u3044\u304f\u3064\u304b\u30920\u306b\u3059\u308b\u306a\u3069\u3057\u3066\u3001\u300c\u7f70\u5247\u300d\u3092\u5c0f\u3055\u304f\u3059\u308b\u3002<\/p>\n\n\n\n

\u7d50\u679c\u3068\u3057\u3066\u3001\u5f53\u3066\u306f\u307e\u308a\u3059\u304e\uff08\u904e\u5b66\u7fd2\uff09\u3092\u9632\u3050\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

\u3061\u306a\u307f\u306b $ || \\beta ||_2^2 $ \u3068\u304b\u3001$ || \\beta ||_1 $ \u306f\u30ce\u30eb\u30e0\u3068\u547c\u3070\u308c\u308b\u3002<\/p>\n\n\n\n

$$ || \\beta ||_2^2 = \\sqrt{| \\beta |_1^2 + |\\beta|_2^2 + \\cdots + |\\beta|_n^2} $$<\/p>\n\n\n\n\n\n\n\n

$$ ||\\beta||_1 = |\\beta|_1 + |\\beta|_2 + \\cdots + |\\beta|_n $$ <\/p>\n\n\n\n\n\n\n\n

\u30ea\u30c3\u30b8\u30fb\u30e9\u30c3\u30bd\u30fb\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u306e\u540d\u524d\u306e\u7531\u6765<\/h2>\n\n\n\n

\u30ea\u30c3\u30b8\uff08Ridge\uff09\u306f\u3001\u4e00\u822c\u7684\u306a\u610f\u5473\u306f\u76db\u308a\u4e0a\u304c\u3063\u305f\u5834\u6240\u3001\u4f8b\u3048\u3070\u5c71\u306e\u80cc\u3001\u5c3e\u6839\u306a\u3069\u3002<\/p>\n\n\n\n

\u4e8c\u4e57\u548c\u306e\u9805\u3092\u5165\u308c\u308b\u3053\u3068\u3067\u3001\u4e0a\u306b\u51f8\u578b\u306e\u4e8c\u6b21\u53cd\u5fdc\u95a2\u6570\u306b\u3001\u6570\u5b66\u7684\u985e\u4f3c\u6027\u304c\u3042\u308b\u3068\u3053\u308d\u304b\u3089\u540d\u3065\u3051\u3089\u308c\u305f\u3002<\/p>\n\n\n\n

\u30e9\u30c3\u30bd\uff08LASSO\uff09\u306f\u3001Least Absolute Shrinkage and Selection Operator \u306e\u982d\u6587\u5b57\u8a9e\u3002<\/p>\n\n\n\n

\u4fc2\u6570\u306e\u7d76\u5bfe\u5024\uff08Absolute\uff09\u3092\u7528\u3044\u308b\u65b9\u6cd5\u3067\u3001\u4fc2\u6570\u3092\u5c0f\u3055\u304f\u3057\u3066\uff08Shrinkage\uff09\u7f70\u5247\u3092\u5c0f\u3055\u304f\u3057\u3001\u8981\u3089\u306a\u3044\u5909\u6570\u306e\u4fc2\u6570\u3092\u30bc\u30ed\u306b\u3057\u3066\u5909\u6570\u3092\u9078\u629e\u3059\u308b\uff08Selection\uff09\u65b9\u6cd5\u3060\u3002<\/p>\n\n\n\n

\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\uff08Elastic net\uff09\u306f\u3001\u307e\u308b\u3067\u4f38\u7e2e\u6027\u306e\u6295\u7db2\u306e\u3088\u3046\u306b\u300c\u5927\u304d\u306a\u9b5a\uff08\u91cd\u8981\u306a\u4fc2\u6570\uff09\u5168\u90e8\u300d\u3092\u6b8b\u3059\u30a4\u30e1\u30fc\u30b8\u304b\u3089\u3064\u3051\u3089\u308c\u305f\u540d\u524d\u3060\u3002<\/p>\n\n\n\n

\uff1e\uff1e\u3082\u3046\u7d71\u8a08\u3067\u60a9\u3080\u306e\u306f\u7d42\u308f\u308a\u306b\u3057\u307e\u305b\u3093\u304b\uff1f\u00a0<\/a><\/strong><\/span><\/p>\r\n\"\"<\/a>\r\n

\u21911\u4e07\u4eba\u4ee5\u4e0a\u306e\u533b\u7642\u5f93\u4e8b\u8005\u304c\u8cfc\u8aad\u4e2d<\/span><\/strong><\/span><\/p><\/div>

\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u3092 R \u3067\u5b9f\u884c\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u306e\u6e96\u5099<\/h2>\n\n\n\n

\u6700\u521d\u306b\u4e00\u56de\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3002<\/p>\n\n\n\n

glmnet\u30d1\u30c3\u30b1\u30fc\u30b8\u3068ISLR\u30d1\u30c3\u30b1\u30fc\u30b8\u3002<\/p>\n\n\n\n

install.packages<\/span>(<\/span>\"glmnet\"<\/span>)<\/span>\ninstall.packages<\/span>(<\/span>\"ISLR\"<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
\u3068\u3082\u306b\u547c\u3073\u51fa\u3057\u3066\u304a\u304f\u3002<\/p>\n\n\n\n
library<\/span>(<\/span>glmnet)<\/span>\nlibrary<\/span>(<\/span>ISLR)<\/span>\n<\/code><\/pre>\n\n\n\n
\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u3092 R \u3067\u884c\u3063\u3066\u307f\u308b\u4f8b\u984c\u306e\u30c7\u30fc\u30bf\u3092\u6e96\u5099<\/h2>\n\n\n\n
ISLR\u30d1\u30c3\u30b1\u30fc\u30b8\u4e2d\u306eCollege\u30c7\u30fc\u30bf\u3092\u4f7f\u3046\u3002<\/p>\n\n\n\n
\u7c73\u56fd\u306e777\u5927\u5b66\u306e\u51fa\u9858\u8005\u3084\u5b66\u751f\u6570\u3001\u66f8\u7c4d\u8cfc\u5165\u306b\u304b\u304b\u308b\u30b3\u30b9\u30c8\u306a\u3069\u3068\u3044\u304618\u500b\u306e\u30d7\u30ed\u30d5\u30a1\u30a4\u30eb\u30c7\u30fc\u30bf\u3002<\/p>\n\n\n\n
\u5b66\u7fd2\u30c7\u30fc\u30bf\u3068\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u306b\u5206\u3051\u308b\u3002<\/p>\n\n\n\n
5\u5206\u306e4\u3092\u5b66\u7fd2\u30c7\u30fc\u30bf\u30025\u5206\u306e1\u3092\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u306b\u3059\u308b\u3002<\/p>\n\n\n\n
set.seed<\/span>(<\/span>20180921<\/span>)<\/span>\nsub <-<\/span> sample<\/span>(<\/span>1<\/span>:<\/span>777<\/span>,<\/span> round<\/span>(<\/span>777<\/span>*<\/span>4<\/span>\/<\/span>5<\/span>))<\/span>\n<\/code><\/pre>\n\n\n\n
\u4f8b\u984c\u306f\u3001\u767b\u9332\u3059\u308b\u5b66\u751f\u6570\uff08Enroll\uff09\u3092\u4e88\u6e2c\u3059\u308b\u30e2\u30c7\u30eb\u3092\u4f5c\u308b\u3053\u3068\u3002<\/p>\n\n\n\n
\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u306e\u524d\u306b\u5f93\u6765\u306e\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3092\u5b9f\u884c\u3057\u3066\u307f\u308b<\/h2>\n\n\n\n
Enroll\u3092\u305d\u306e\u4ed6\u306e\u5909\u6570\u5168\u90e8\u3067\u4e88\u6e2c\u3059\u308b\u56de\u5e30\u5f0f\u3092\u4f5c\u308b\u3002<\/p>\n\n\n\n
lm.res <-<\/span> lm<\/span>(<\/span>Enroll ~<\/span> .,<\/span> data=<\/span>College[<\/span>sub,])<\/span>\nsummary<\/span>(<\/span>lm.res)<\/span>\n<\/code><\/pre>\n\n\n\n
\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u7d50\u679c\u304c\u8868\u793a\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
><\/span> summary<\/span>(<\/span>lm.res)<\/span>\nCall:<\/span>\nlm<\/span>(<\/span>formula =<\/span> Enroll ~<\/span> .,<\/span> data =<\/span> College[<\/span>sub,<\/span> ])<\/span>\nResiduals:<\/span>\nMin       1Q   Median       3Q      Max\n-<\/span>1577.18<\/span>   -<\/span>63.47<\/span>   -<\/span>12.88<\/span>    45.64<\/span>  1581.09<\/span>\nCoefficients:<\/span>\nEstimate Std. Error t value Pr<\/span>(<\/span>>|<\/span>t|<\/span>)<\/span>\n(<\/span>Intercept)<\/span> 219.946039<\/span>  87.176801<\/span>   2.523<\/span> 0.011892<\/span> *<\/span>\nPrivateYes  -<\/span>20.716365<\/span>  29.254284<\/span>  -<\/span>0.708<\/span> 0.479126<\/span>\nApps         -<\/span>0.040325<\/span>   0.007908<\/span>  -<\/span>5.099<\/span> 4.57e-07<\/span> ***<\/span>\nAccept        0.162698<\/span>   0.014738<\/span>  11.040<\/span>  <<\/span> 2e-16<\/span> ***<\/span>\nTop10perc     4.381100<\/span>   1.270779<\/span>   3.448<\/span> 0.000605<\/span> ***<\/span>\nTop25perc    -<\/span>2.206551<\/span>   0.943228<\/span>  -<\/span>2.339<\/span> 0.019642<\/span> *<\/span>\nF.Undergrad   0.142490<\/span>   0.004467<\/span>  31.901<\/span>  <<\/span> 2e-16<\/span> ***<\/span>\nP.Undergrad  -<\/span>0.029804<\/span>   0.008131<\/span>  -<\/span>3.666<\/span> 0.000269<\/span> ***<\/span>\nOutstate     -<\/span>0.004381<\/span>   0.004120<\/span>  -<\/span>1.063<\/span> 0.288023<\/span>\nRoom.Board   -<\/span>0.018011<\/span>   0.010351<\/span>  -<\/span>1.740<\/span> 0.082361<\/span> .\nBooks         0.018743<\/span>   0.049424<\/span>   0.379<\/span> 0.704654<\/span>\nPersonal     -<\/span>0.004377<\/span>   0.014736<\/span>  -<\/span>0.297<\/span> 0.766533<\/span>\nPhD           0.353353<\/span>   1.024941<\/span>   0.345<\/span> 0.730399<\/span>\nTerminal     -<\/span>1.275895<\/span>   1.139472<\/span>  -<\/span>1.120<\/span> 0.263276<\/span>\nS.F.Ratio    -<\/span>0.842953<\/span>   2.686361<\/span>  -<\/span>0.314<\/span> 0.753789<\/span>\nperc.alumni   1.920409<\/span>   0.879343<\/span>   2.184<\/span> 0.029353<\/span> *<\/span>\nExpend        0.001368<\/span>   0.002644<\/span>   0.517<\/span> 0.605075<\/span>\nGrad.Rate     0.277165<\/span>   0.639241<\/span>   0.434<\/span> 0.664745<\/span>\n---<\/span>\nSignif. codes:<\/span>  0<\/span> \u2018***<\/span>\u2019 0.001<\/span> \u2018**<\/span>\u2019 0.01<\/span> \u2018*<\/span>\u2019 0.05<\/span> \u2018.\u2019 0.1<\/span> \u2018 \u2019 1<\/span>\nResidual standard error:<\/span> 199.2<\/span> on 604<\/span> degrees of freedom\nMultiple R-<\/span>squared:<\/span>  0.9529<\/span>,<\/span>    Adjusted R-<\/span>squared:<\/span>  0.9516<\/span>\nF<\/span>-<\/span>statistic:<\/span> 718.9<\/span> on 17<\/span> and 604<\/span> DF,<\/span>  p-<\/span>value:<\/span> <<\/span> 2.2e-16<\/span>\n<\/code><\/pre>\n\n\n\n
\u30e2\u30c7\u30eb\u304b\u3089\u5f97\u3089\u308c\u308b\u4e88\u6e2c\u5024\u3068\u5b9f\u6e2c\u5024\u306e\u304b\u3044\u96e2\u3092 mean squared error of prediction (MSEP) \u3067\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n\n\n\n
mean<\/span>((<\/span>predict<\/span>(<\/span>lm.res)<\/span>-<\/span>College[<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n><\/span> mean<\/span>((<\/span>predict<\/span>(<\/span>lm.res)<\/span>-<\/span>College[<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n[<\/span>1<\/span>]<\/span> 38530.83<\/span>\n<\/code><\/pre>\n\n\n\n
\u5b66\u7fd2\u30c7\u30fc\u30bf\u304b\u3089\u5f97\u3089\u308c\u305f\u30e2\u30c7\u30eb\u3092\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u306b\u9069\u7528\u3057\u3066\u3001\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u3067\u5f97\u3089\u308c\u308b\u4e88\u6e2c\u5024\u3068\u5b9f\u6e2c\u5024\u306e\u304b\u3044\u96e2\u3092 MSEP \u3067\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n\n\n\n
lm.test <-<\/span> predict<\/span>(<\/span>lm.res,<\/span> newdata=<\/span>College[<\/span>-<\/span>sub,])<\/span>\nmean<\/span>((<\/span>lm.test -<\/span> College[<\/span>-<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n><\/span> mean<\/span>((<\/span>lm.test -<\/span> College[<\/span>-<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n[<\/span>1<\/span>]<\/span> 47213.26<\/span>\n<\/code><\/pre>\n\n\n\n
MSEP\u306f\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u3067\u5927\u304d\u304f\u306a\u3063\u305f\u3002<\/p>\n\n\n\n
\u3053\u308c\u306f\u5b66\u7fd2\u30c7\u30fc\u30bf\u3067\u306e\u4e88\u6e2c\u30e2\u30c7\u30eb\u306e\u6027\u80fd\u304c\u3042\u307e\u308a\u9ad8\u304f\u306a\u3044\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
\u30ea\u30c3\u30b8\u3067\u89e3\u6790\u3057\u305f\u7d50\u679c<\/h2>\n\n\n\n
glmnet\u30d1\u30c3\u30b1\u30fc\u30b8\u306eglmnet()\u3092\u4f7f\u3046\u3002<\/p>\n\n\n\n
\u30c7\u30fc\u30bf\u306f\u884c\u5217\u3067\u4e0e\u3048\u308b\u3002<\/p>\n\n\n\n
data.matrix()\u3092\u4f7f\u3046\u3002<\/p>\n\n\n\n
\u8aac\u660e\u5909\u6570, \u5f93\u5c5e\u5909\u6570\u306e\u9806\u756a\u306b\u6307\u5b9a\u3057\u3001alpha=0\u3092\u6307\u5b9a\u3059\u308b\u3002<\/p>\n\n\n\n
ridge.res <-<\/span> glmnet<\/span>(<\/span>data.matrix<\/span>(<\/span>College[<\/span>sub,<\/span>-<\/span>4<\/span>]),<\/span> College[<\/span>sub,]$<\/span>Enroll,<\/span> alpha=<\/span>0<\/span>)<\/span>\nmean<\/span>((<\/span>predict<\/span>(<\/span>ridge.res,<\/span> newx=<\/span>data.matrix<\/span>(<\/span>College[<\/span>sub,<\/span> -<\/span>4<\/span>]))<\/span>-<\/span>College[<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\nridge.test <-<\/span> predict<\/span>(<\/span>ridge.res,<\/span> newx=<\/span>data.matrix<\/span>(<\/span>College[<\/span>-<\/span>sub,<\/span>-<\/span>4<\/span>]))<\/span>\nmean<\/span>((<\/span>ridge.test -<\/span> College[<\/span>-<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
\u5b66\u7fd2\u30c7\u30fc\u30bf\u3067\u4f5c\u3089\u308c\u305f\u30e2\u30c7\u30eb\u3092\u3001\u5b66\u7fd2\u30c7\u30fc\u30bf\u3068\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u306e MSEP \u3067\u898b\u3066\u307f\u308b\u3068\u3001\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u3067\u306f\u306f\u308b\u304b\u306b MSEP \u304c\u5927\u304d\u304f\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
\u3042\u307e\u308a\u4e88\u6e2c\u6027\u80fd\u306f\u3088\u304f\u306a\u3044\u3002<\/p>\n\n\n\n
><\/span> mean<\/span>((<\/span>predict<\/span>(<\/span>ridge.res,<\/span> newx=<\/span>data.matrix<\/span>(<\/span>College[<\/span>sub,<\/span> -<\/span>4<\/span>]))<\/span>-<\/span>College[<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n[<\/span>1<\/span>]<\/span> 442744.8<\/span>\n><\/span> mean<\/span>((<\/span>ridge.test -<\/span> College[<\/span>-<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n[<\/span>1<\/span>]<\/span> 558366.1<\/span>\n<\/code><\/pre>\n\n\n\n
\u4e88\u6e2c\u6027\u80fd\u3092\u4e0a\u3052\u308b\u65b9\u6cd5\u3068\u3057\u3066\u3001\u30af\u30ed\u30b9\u30d0\u30ea\u30c7\u30fc\u30b7\u30e7\u30f3\uff08cross-validation\uff09\u3068\u3044\u3046\u65b9\u6cd5\u304c\u3042\u308b\u3002<\/p>\n\n\n\n
\u5b66\u7fd2\u30c7\u30fc\u30bf\u3092\u3055\u3089\u306b\u5206\u5272\u3057\u3066\u3001\u30c6\u30b9\u30c8\u30bb\u30c3\u30c81\u3064\u3068\u5b66\u7fd2\u30bb\u30c3\u30c8\u3068\u3057\u3066\u7e70\u308a\u8fd4\u3057\u3001\u6700\u9069\u306a \u03bb \u3092\u898b\u3064\u3051\u308b\u65b9\u6cd5\u3002<\/p>\n\n\n\n
\u5b66\u7fd2\u30c7\u30fc\u30bf\u306bcross-validation\u4ed8\u304d\u306eglmnet\u3092\u9069\u7528\u3057\u3066\u3001\u7d50\u679c\u3092\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3002<\/p>\n\n\n\n
ridge.res.cv <-<\/span> cv.glmnet<\/span>(<\/span>data.matrix<\/span>(<\/span>College[<\/span>sub,<\/span>-<\/span>4<\/span>]),<\/span> College[<\/span>sub,]$<\/span>Enroll,<\/span> alpha=<\/span>0<\/span>)<\/span>\nplot<\/span>(<\/span>ridge.res.cv)<\/span>\n<\/code><\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
MSEP\uff08Y\u8ef8\uff09\u304c\u6700\u3082\u5c0f\u3055\u3044\u5024\u3092\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n\n\n\n
\u5de6\u5074\u306e\u7e26\u7834\u7dda\u304c\u901a\u3063\u3066\u3044\u308b\u8d64\u3044\u30c9\u30c3\u30c8\u306eY\u8ef8\u5024\u3002<\/p>\n\n\n\n
><\/span> min<\/span>(<\/span>ridge.res.cv$<\/span>cvm)<\/span>\n[<\/span>1<\/span>]<\/span> 61345.74<\/span>\n<\/code><\/pre>\n\n\n\n
MSEP \u304c\u6700\u3082\u5c0f\u3055\u3044\u3068\u304d\u306e \u03bb \u3068 $\\log (\\lambda) $ \u3092\u78ba\u8a8d\u3059\u308b\uff08\u5de6\u5074\u306e\u7e26\u7834\u7dda\u306eX\u8ef8\u5024\uff09\u3002<\/p>\n\n\n\n
><\/span> ridge.res.cv$<\/span>lambda.min\n[<\/span>1<\/span>]<\/span> 95.77041<\/span>\n><\/span> log<\/span>(<\/span>ridge.res.cv$<\/span>lambda.min)<\/span>\n[<\/span>1<\/span>]<\/span> 4.561954<\/span>\n<\/code><\/pre>\n\n\n\n
MSEP \u304c\u6700\u3082\u5c0f\u3055\u3044\u3068\u304d\u306e \u03bb\u306b\u5bfe\u5fdc\u3057\u305f\u56de\u5e30\u4fc2\u6570\u3092\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n\n\n\n
><\/span> coef<\/span>(<\/span>ridge.res.cv,<\/span> s=<\/span>\"lambda.min\"<\/span>)<\/span>\n18<\/span> x 1<\/span> sparse Matrix of class \"dgCMatrix\"<\/span>\n1<\/span>\n(<\/span>Intercept)<\/span>  2.234662e+02<\/span>\nPrivate     -<\/span>7.114514e+01<\/span>\nApps         1.738649e-02<\/span>\nAccept       1.106883e-01<\/span>\nTop10perc    2.040518e+00<\/span>\nTop25perc   -<\/span>3.542996e-01<\/span>\nF.Undergrad  1.035560e-01<\/span>\nP.Undergrad  8.384248e-03<\/span>\nOutstate    -<\/span>4.248129e-03<\/span>\nRoom.Board  -<\/span>3.121038e-02<\/span>\nBooks        2.823821e-02<\/span>\nPersonal     1.564955e-02<\/span>\nPhD          5.052552e-01<\/span>\nTerminal     2.447818e-02<\/span>\nS.F.Ratio    2.254079e+00<\/span>\nperc.alumni  1.854235e+00<\/span>\nExpend      -<\/span>2.158011e-04<\/span>\nGrad.Rate    1.225929e-01<\/span>\n<\/code><\/pre>\n\n\n\n
\u5b66\u7fd2\u30c7\u30fc\u30bf\u3067MSEP\u304c\u6700\u3082\u5c0f\u3055\u3044\u3068\u304d\u306e \u03bb \u306e\u30e2\u30c7\u30eb\u3092\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u306b\u9069\u7528\u3059\u308b\u3002<\/p>\n\n\n\n
ridge.test.cv <-<\/span> predict<\/span>(<\/span>ridge.res.cv,<\/span> s=<\/span>\"lambda.min\"<\/span>,<\/span> newx=<\/span>data.matrix<\/span>(<\/span>College[<\/span>-<\/span>sub,<\/span>-<\/span>4<\/span>]))<\/span>\nmean<\/span>((<\/span>ridge.test.cv-<\/span>College[<\/span>-<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u306e\u4e88\u6e2c\u5024\u3068\u5b9f\u6e2c\u5024\u306e\u304b\u3044\u96e2\u3092 MSEP \u3067\u6c42\u3081\u308b\u3002<\/p>\n\n\n\n
><\/span> mean<\/span>((<\/span>ridge.test.cv-<\/span>College[<\/span>-<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n[<\/span>1<\/span>]<\/span> 44719.1<\/span>\n<\/code><\/pre>\n\n\n\n
\u5b66\u7fd2\u30c7\u30fc\u30bf\u306e MSEP \u306f\u3001\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
><\/span> min<\/span>(<\/span>ridge.res.cv$<\/span>cvm)<\/span>\n[<\/span>1<\/span>]<\/span> 61345.74<\/span>\n<\/code><\/pre>\n\n\n\n
\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u306e MSEP \u306e\u307b\u3046\u304c\u5c0f\u3055\u304f\u306a\u3063\u3066\u3044\u308b\u306e\u304c\u308f\u304b\u308b\u3002<\/p>\n\n\n\n
\u4e88\u6e2c\u6027\u80fd\u304c\u3088\u3044\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
\u30e9\u30c3\u30bd\u3067\u89e3\u6790\u3057\u305f\u7d50\u679c<\/h2>\n\n\n\n
glmnet()\u3067alpha=1\u3068\u6307\u5b9a\u3059\u308b\u3002<\/p>\n\n\n\n
\u6700\u521d\u304b\u3089\u30af\u30ed\u30b9\u30d0\u30ea\u30c7\u30fc\u30b7\u30e7\u30f3\u3092\u4f7f\u3046\u3002<\/p>\n\n\n\n
lasso.res.cv <-<\/span> cv.glmnet<\/span>(<\/span>data.matrix<\/span>(<\/span>College[<\/span>sub,<\/span>-<\/span>4<\/span>]),<\/span> College[<\/span>sub,]$<\/span>Enroll,<\/span> alpha=<\/span>1<\/span>)<\/span>\nplot<\/span>(<\/span>lasso.res.cv)<\/span>\n<\/code><\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
MSEP \u304c\u6700\u5c0f\u5024\u3092\u8868\u793a\u3055\u305b\u308b\uff08Y\u8ef8\uff09\u3002<\/p>\n\n\n\n
MSEP \u304c\u6700\u5c0f\u5024\u306e $ \\log (\\lambda) $ \u3092\u8868\u793a\u3055\u305b\u308b\uff08X\u8ef8\uff09\u3002<\/p>\n\n\n\n
MSEP \u6700\u5c0f\u5024\u306e \u03bb \u306e\u3068\u304d\u306e\u56de\u5e30\u4fc2\u6570\u3092\u8868\u793a\u3055\u305b\u308b\u3002<\/p>\n\n\n\n
><\/span> min<\/span>(<\/span>lasso.res.cv$<\/span>cvm)<\/span>\n[<\/span>1<\/span>]<\/span> 46512.83<\/span>\n><\/span> log<\/span>(<\/span>lasso.res.cv$<\/span>lambda.min)<\/span>\n[<\/span>1<\/span>]<\/span> -<\/span>0.5781604<\/span>\n><\/span> coef<\/span>(<\/span>lasso.res.cv,<\/span> s=<\/span>\"lambda.min\"<\/span>)<\/span>\n18<\/span> x 1<\/span> sparse Matrix of class \"dgCMatrix\"<\/span>\n1<\/span>\n(<\/span>Intercept)<\/span> 223.249334574<\/span>\nPrivate     -<\/span>17.323492035<\/span>\nApps         -<\/span>0.034361012<\/span>\nAccept        0.152731795<\/span>\nTop10perc     3.758399461<\/span>\nTop25perc    -<\/span>1.812982422<\/span>\nF.Undergrad   0.142965370<\/span>\nP.Undergrad  -<\/span>0.029337386<\/span>\nOutstate     -<\/span>0.003177355<\/span>\nRoom.Board   -<\/span>0.018717120<\/span>\nBooks         0.012830680<\/span>\nPersonal     -<\/span>0.003259145<\/span>\nPhD           0.191660710<\/span>\nTerminal     -<\/span>1.050251664<\/span>\nS.F.Ratio    -<\/span>0.547854820<\/span>\nperc.alumni   1.834686143<\/span>\nExpend        0.001059255<\/span>\nGrad.Rate     0.177922346<\/span>\n<\/code><\/pre>\n\n\n\n
\u5b66\u7fd2\u30c7\u30fc\u30bf\u3067\u3082\u3063\u3068\u3082\u3059\u3050\u308c\u305f\u30e2\u30c7\u30eb\u3067\u3001\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u4e88\u6e2c\u3092\u3059\u308b\u3002<\/p>\n\n\n\n
MSEP \u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n
lasso.test.cv <-<\/span> predict<\/span>(<\/span>lasso.res.cv,<\/span> s=<\/span>\"lambda.min\"<\/span>,<\/span> newx=<\/span>data.matrix<\/span>(<\/span>College[<\/span>-<\/span>sub,<\/span>-<\/span>4<\/span>]))<\/span>\nmean<\/span>((<\/span>lasso.test.cv-<\/span>College[<\/span>-<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
><\/span> mean<\/span>((<\/span>lasso.test.cv-<\/span>College[<\/span>-<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n[<\/span>1<\/span>]<\/span> 46711.87<\/span>\n<\/code><\/pre>\n\n\n\n
\u5b66\u7fd2\u30c7\u30fc\u30bf\u3067\u306e MSEP \u3068\u307b\u307c\u540c\u3058\u5024\u3002<\/p>\n\n\n\n
\u4ee5\u4e0b\u304c\u3001\u5b66\u7fd2\u30c7\u30fc\u30bf\u3067\u306e MSEP \u306e\u5024\u3002<\/p>\n\n\n\n
><\/span> min<\/span>(<\/span>lasso.res.cv$<\/span>cvm)<\/span>\n[<\/span>1<\/span>]<\/span> 46512.83<\/span>\n<\/code><\/pre>\n\n\n\n
\u5927\u304d\u304f\u5897\u52a0\u3057\u3066\u3044\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u306e\u3067\u3001\u4e88\u6e2c\u306b\u4f7f\u3048\u306a\u304f\u306f\u306a\u3044\u3002<\/p>\n\n\n\n
\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u306e\u7d50\u679c<\/h2>\n\n\n\n
0 < \u03b1< 1 \u304c\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u3060\u3002<\/p>\n\n\n\n
\u30ea\u30c3\u30b8\u3068\u30e9\u30c3\u30bd\u306e\u4e2d\u9593\u3067\u3001\u3044\u3044\u3068\u3053\u3069\u308a\u306e\u65b9\u6cd5\u3002<\/p>\n\n\n\n
\u03b1 \u3092\u632f\u3063\u3066\u8a66\u3057\u3066\u304f\u308c\u308b\u95a2\u6570\u304c glmnetUtils \u30d1\u30c3\u30b1\u30fc\u30b8\u306e cva.glmnet() \u3060\u3002<\/p>\n\n\n\n
\u6700\u521d\u4e00\u56de\u3060\u3051\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3059\u308b\u3002<\/p>\n\n\n\n
\u4f7f\u3046\u3068\u304d\u306flibrary()\u3067\u547c\u3073\u51fa\u3059\u3002<\/p>\n\n\n\n
install.packages<\/span>(<\/span>\"glmnetUtils\"<\/span>)<\/span>\nlibrary<\/span>(<\/span>glmnetUtils)<\/span>\n<\/code><\/pre>\n\n\n\n
cva.glmnet() \u306f cv.glmnet() \u3068\u66f8\u304d\u65b9\u306f\u4e00\u7dd2\u3002<\/p>\n\n\n\n
\u03b1 \u306f\u30c7\u30d5\u30a9\u30eb\u30c8\u306711\u500b\u306b\u632f\u3063\u3066\u304f\u308c\u308b\u3002<\/p>\n\n\n\n
elnet.res.cv <-<\/span> cva.glmnet<\/span>(<\/span>data.matrix<\/span>(<\/span>College[<\/span>sub,<\/span>-<\/span>4<\/span>]),<\/span> College[<\/span>sub,]$<\/span>Enroll)<\/span>\nplot<\/span>(<\/span>elnet.res.cv)<\/span>\n<\/code><\/pre>\n\n\n\n
$ \\log (\\lambda) $ \u3068MSEP\u306e\u30d7\u30ed\u30c3\u30c8\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
\u03b1\u304c11\u500b\u306b\u632f\u3089\u308c\u3066\u3044\u308b\u306e\u3067\u300111\u672c\u306e\u30b0\u30e9\u30d5\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
\u03b1\u3092\u78ba\u8a8d\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
><\/span> elnet.res.cv$<\/span>alpha\n[<\/span>1<\/span>]<\/span> 0.000<\/span> 0.001<\/span> 0.008<\/span> 0.027<\/span> 0.064<\/span> 0.125<\/span> 0.216<\/span> 0.343<\/span> 0.512<\/span> 0.729<\/span> 1.000<\/span>\n<\/code><\/pre>\n\n\n\n
\u6700\u9069\u306a \u03b1 \u3092\u9078\u3076\u305f\u3081 MSEP \u304c\u3082\u3063\u3068\u3082\u5c0f\u3055\u3044 \u03b1 \u3092\u63a2\u3059\u3002<\/p>\n\n\n\n
\u5404 \u03b1 \u306e\u3046\u3061\u6700\u3082\u5c0f\u3055\u3044 MSEP \u3092\u53d6\u308a\u3060\u3057\u3001\u305d\u306e\u4e2d\u3067\u3082\u3063\u3068\u3082\u5c0f\u3055\u3044MSEP\u3092\u53d6\u308a\u51fa\u3057\u305f\u3068\u3053\u308d\u3001\u03b1 = 1 \u304c\u3082\u3063\u3068\u3082 MSEP \u304c\u5c0f\u3055\u304b\u3063\u305f\u3002<\/p>\n\n\n\n
\u3064\u307e\u308a\u3001\u30e9\u30c3\u30bd\u304c\u3082\u3063\u3068\u3082\u5f53\u3066\u306f\u307e\u308a\u304c\u3088\u3044\u30e2\u30c7\u30eb\u3067\u3042\u3063\u305f\u3002<\/p>\n\n\n\n
><\/span> sapply<\/span>(<\/span>sapply<\/span>(<\/span>elnet.res.cv$<\/span>modlist,<\/span> \"[[\"<\/span>,<\/span> \"cvm\"<\/span>),<\/span>min)<\/span>\n[<\/span>1<\/span>]<\/span> 61274.57<\/span> 61208.48<\/span> 47948.64<\/span> 47276.72<\/span> 47268.60<\/span> 47292.46<\/span> 47324.64<\/span> 47347.99<\/span> 47177.38<\/span>\n[<\/span>10<\/span>]<\/span> 47006.23<\/span> 46913.66<\/span>\n><\/span> min<\/span>(<\/span>sapply<\/span>(<\/span>sapply<\/span>(<\/span>elnet.res.cv$<\/span>modlist,<\/span> \"[[\"<\/span>,<\/span> \"cvm\"<\/span>),<\/span>min))<\/span>\n[<\/span>1<\/span>]<\/span> 46913.66<\/span>\n<\/code><\/pre>\n\n\n\n
\u03b1 = 1 \u306e\u3068\u304d\u306e\u3001MSEP \u304c\u6700\u5c0f\u306b\u306a\u308b \u03bb \u306e\u30e2\u30c7\u30eb\u3092\u4f7f\u3063\u3066\u3001\u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u3067\u4e88\u6e2c\u3059\u308b\u3002<\/p>\n\n\n\n
elnet.test.cv <-<\/span> predict<\/span>(<\/span>elnet.res.cv,<\/span> alpha=<\/span>1<\/span>,<\/span> s=<\/span>\"lambda.min\"<\/span>,<\/span> newx=<\/span>data.matrix<\/span>(<\/span>College[<\/span>-<\/span>sub,<\/span>-<\/span>4<\/span>]))<\/span>\nmean<\/span>((<\/span>elnet.test.cv-<\/span>College[<\/span>-<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
MSEP \u306f\u5b66\u7fd2\u30c7\u30fc\u30bf\u3088\u308a\u3082\u5c0f\u3055\u304f\u306a\u308a\u3001\u4e88\u6e2c\u6027\u80fd\u304c\u9ad8\u3044\u3068\u8a00\u3048\u308b\u3002<\/p>\n\n\n\n
><\/span> mean<\/span>((<\/span>elnet.test.cv-<\/span>College[<\/span>-<\/span>sub,]$<\/span>Enroll)<\/span>^<\/span>2<\/span>)<\/span>\n[<\/span>1<\/span>]<\/span> 45275.5<\/span>\n<\/code><\/pre>\n\n\n\n
\u307e\u3068\u3081<\/h2>\n\n\n\n
\u5b66\u7fd2\u30c7\u30fc\u30bf\u306b\u5f53\u3066\u306f\u307e\u308a\u3059\u304e\u306a\u3044\u3001\u4e88\u6e2c\u6027\u80fd\u306b\u512a\u308c\u305f\u30e2\u30c7\u30eb\u3092\u4f5c\u308b\u3001\u30ea\u30c3\u30b8\u3001\u30e9\u30c3\u30bd\u3001\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u3092\u7d39\u4ecb\u3057\u305f\u3002<\/p>\n\n\n\n
\u30ea\u30c3\u30b8\u304c\u3044\u3044\u304b\u3001\u30e9\u30c3\u30bd\u304c\u3044\u3044\u304b\u3001\u306f\u3001\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u3067 \u03b1 \u3092\u632f\u3063\u3066\u307f\u3066\u3001\u9078\u3076\u306e\u304c\u3088\u3044\u3002<\/p>\n\n\n\n
glmnetUtils \u30d1\u30c3\u30b1\u30fc\u30b8\u306e cva.glmnet() \u306f\u75d2\u3044\u3068\u3053\u308d\u306b\u624b\u304c\u5c4a\u304f\u95a2\u6570\u3067\u4f7f\u3044\u3084\u3059\u3044\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u30a8\u30e9\u30b9\u30c6\u30a3\u30c3\u30af\u30cd\u30c3\u30c8\u3092\u7c21\u5358\u306b\u89e3\u8aac R \u3067\u5b9f\u884c\u3059\u308b\u65b9\u6cd5\u3082\u89e3\u8aac<\/p>\n","protected":false},"author":2,"featured_media":2730,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"swell_btn_cv_data":"","_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[5,103],"tags":[],"class_list":["post-463","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-r","category-103"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"https:\/\/best-biostatistics.com\/toukei-er\/wp-content\/uploads\/2018\/09\/20180923152237.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts\/463","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/comments?post=463"}],"version-history":[{"count":4,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts\/463\/revisions"}],"predecessor-version":[{"id":2735,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts\/463\/revisions\/2735"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/media\/2730"}],"wp:attachment":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/media?parent=463"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/categories?post=463"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/tags?post=463"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}