北京大学R语言教程(李东风)第42章：正则化与惩罚回归

42.1 介绍

这一章用Hitters数据集演示线性回归、回归自变量选择，岭回归、lasso回归，以及如何进行超参数调优。

考虑ISLR包的Hitters数据集。此数据集有322个运动员的20个变量的数据，其中的变量Salary（工资）是我们关心的。变量包括:

library(tidyverse)
library(ISLR) # 参考书对应的包
data(Hitters)
names(Hitters)

##  [1] "AtBat"     "Hits"      "HmRun"     "Runs"      "RBI"       "Walks"     "Years"     "CAtBat"    "CHits"     "CHmRun"    "CRuns"     "CRBI"      "CWalks"    "League"    "Division"  "PutOuts"   "Assists"   "Errors"    "Salary"    "NewLeague"

数据集的详细变量信息如下:

glimpse(Hitters)

## Rows: 322
## Columns: 20
## $ AtBat     <int> 293, 315, 479, 496, 321, 594, 185, 298, 323, 401, 574, 202, 418, 239, 196, 183, 568, 190, 407, 127, 413, 426, 22, 472, 629, 587, 324, 474, 550, 513, 313, 419, 517, 583, 204, 379, 161, 268, 346, 241, 181, 216, 200, 217, 194, 254, 416, 205, 542, 526, 457, 214, 19, 591, 403, 405, 244, 235, 313, 627, 416, 155, 236, 216, 24, 585, 191, 199, 521, 419, 311, 138, 512, 507, 529, 424, 351, 195, 388, 339, 561, 255, 677, 227, 614, 329, 637, 280, 155, 458, 314, 475, 317, 511, 278, 382, 565…
## $ Hits      <int> 66, 81, 130, 141, 87, 169, 37, 73, 81, 92, 159, 53, 113, 60, 43, 39, 158, 46, 104, 32, 92, 109, 10, 116, 168, 163, 73, 129, 152, 137, 84, 108, 141, 168, 49, 106, 36, 60, 98, 61, 41, 54, 57, 46, 40, 68, 132, 57, 140, 146, 101, 53, 7, 168, 101, 102, 58, 61, 78, 177, 113, 44, 56, 53, 3, 139, 37, 53, 142, 113, 81, 31, 131, 122, 137, 119, 97, 55, 103, 96, 118, 70, 238, 46, 163, 83, 174, 82, 41, 114, 83, 123, 78, 138, 69, 119, 148, 71, 115, 110, 151, 132, 49, 106, 114, 37, 95, 154,…
## $ HmRun     <int> 1, 7, 18, 20, 10, 4, 1, 0, 6, 17, 21, 4, 13, 0, 7, 3, 20, 2, 6, 8, 16, 3, 1, 16, 18, 4, 4, 10, 6, 20, 9, 6, 27, 17, 6, 10, 0, 5, 5, 1, 1, 0, 6, 7, 7, 2, 7, 8, 12, 13, 14, 2, 0, 19, 12, 18, 9, 3, 6, 25, 24, 6, 0, 1, 0, 31, 4, 5, 20, 1, 3, 8, 26, 29, 26, 6, 4, 5, 15, 4, 35, 7, 31, 7, 29, 9, 31, 16, 12, 13, 13, 27, 7, 25, 3, 13, 24, 2, 27, 15, 17, 9, 2, 16, 23, 8, 23, 22, 31, 4, 16, 16, 24, 31, 14, 34, 12, 14, 4, 3, 21, 16, 5, 11, 2, 16, 13, 5, 15, 21, 14, 10, 7, 1, 5, 4, 40, 6,…
## $ Runs      <int> 30, 24, 66, 65, 39, 74, 23, 24, 26, 49, 107, 31, 48, 30, 29, 20, 89, 24, 57, 16, 72, 55, 4, 60, 73, 92, 32, 50, 92, 90, 42, 55, 70, 83, 23, 38, 19, 24, 31, 34, 15, 21, 23, 32, 19, 28, 57, 34, 46, 71, 42, 30, 1, 80, 45, 49, 28, 24, 32, 98, 58, 21, 27, 31, 1, 93, 12, 29, 67, 44, 42, 18, 69, 78, 86, 57, 55, 24, 59, 37, 70, 49, 117, 23, 89, 50, 89, 44, 21, 67, 39, 76, 35, 76, 24, 54, 90, 27, 97, 70, 61, 69, 41, 48, 67, 15, 55, 76, 101, 19, 70, 33, 81, 91, 30, 91, 63, 45, 42, 30, …
## $ RBI       <int> 29, 38, 72, 78, 42, 51, 8, 24, 32, 66, 75, 26, 61, 11, 27, 15, 75, 8, 43, 22, 48, 43, 2, 62, 102, 51, 18, 56, 37, 95, 30, 36, 87, 80, 25, 60, 10, 25, 53, 12, 21, 18, 14, 19, 29, 26, 49, 32, 75, 70, 63, 29, 2, 72, 53, 85, 25, 39, 41, 81, 69, 23, 15, 15, 0, 94, 17, 22, 86, 27, 30, 21, 96, 85, 97, 46, 29, 33, 47, 29, 94, 35, 113, 20, 83, 39, 116, 45, 29, 57, 46, 93, 35, 96, 21, 58, 104, 29, 71, 47, 84, 47, 23, 56, 67, 19, 58, 84, 108, 18, 73, 52, 105, 101, 42, 108, 54, 47, 36, 4…
## $ Walks     <int> 14, 39, 76, 37, 30, 35, 21, 7, 8, 65, 59, 27, 47, 22, 30, 11, 73, 15, 65, 14, 65, 62, 1, 74, 40, 70, 22, 40, 81, 90, 39, 22, 52, 56, 12, 30, 17, 15, 30, 14, 33, 15, 14, 9, 30, 22, 33, 9, 41, 84, 22, 23, 1, 39, 39, 20, 35, 21, 12, 70, 16, 15, 11, 22, 2, 62, 14, 21, 45, 44, 26, 38, 52, 91, 97, 13, 39, 30, 39, 23, 33, 43, 53, 12, 75, 56, 56, 47, 22, 48, 16, 72, 32, 61, 29, 36, 77, 14, 68, 36, 78, 54, 18, 35, 53, 15, 37, 43, 41, 11, 80, 37, 62, 64, 24, 52, 30, 26, 66, 20, 60, 41,…
## $ Years     <int> 1, 14, 3, 11, 2, 11, 2, 3, 2, 13, 10, 9, 4, 6, 13, 3, 15, 5, 12, 8, 1, 1, 6, 6, 18, 6, 7, 10, 5, 14, 17, 3, 9, 5, 7, 14, 4, 2, 16, 1, 2, 18, 9, 4, 11, 6, 3, 5, 16, 6, 17, 2, 4, 9, 12, 6, 4, 14, 12, 6, 1, 16, 4, 4, 3, 17, 4, 3, 4, 12, 17, 3, 14, 18, 15, 9, 4, 8, 6, 4, 16, 15, 5, 5, 11, 9, 14, 2, 16, 4, 5, 4, 1, 3, 8, 12, 14, 15, 3, 7, 10, 2, 8, 10, 13, 6, 3, 14, 5, 1, 14, 5, 13, 3, 18, 6, 4, 16, 9, 8, 15, 20, 5, 5, 11, 13, 5, 8, 5, 7, 7, 5, 18, 4, 9, 3, 6, 15, 5, 2, 2, 4, 12, …
## $ CAtBat    <int> 293, 3449, 1624, 5628, 396, 4408, 214, 509, 341, 5206, 4631, 1876, 1512, 1941, 3231, 201, 8068, 479, 5233, 727, 413, 426, 84, 1924, 8424, 2695, 1931, 2331, 2308, 5201, 6890, 591, 3571, 1646, 1309, 6207, 1053, 350, 5913, 241, 232, 7318, 2516, 694, 4183, 999, 932, 756, 7099, 2648, 6521, 226, 41, 4478, 5150, 950, 1335, 3926, 3742, 3210, 416, 6631, 1115, 926, 159, 7546, 773, 514, 815, 4484, 8247, 244, 5347, 7761, 6661, 3651, 1258, 1313, 2174, 1064, 6677, 6311, 2223, 1325, 5017, 3…
## $ CHits     <int> 66, 835, 457, 1575, 101, 1133, 42, 108, 86, 1332, 1300, 467, 392, 510, 825, 42, 2273, 102, 1478, 180, 92, 109, 26, 489, 2464, 747, 491, 604, 633, 1382, 1833, 149, 994, 452, 308, 1906, 244, 78, 1615, 61, 50, 1926, 684, 160, 1069, 236, 273, 192, 2130, 715, 1767, 59, 13, 1307, 1429, 231, 333, 1029, 968, 927, 113, 1634, 270, 210, 28, 1982, 163, 120, 205, 1231, 2198, 53, 1397, 1947, 1785, 1046, 353, 338, 555, 290, 1575, 1661, 737, 324, 1388, 948, 2024, 113, 1338, 298, 405, 471, 78…
## $ CHmRun    <int> 1, 69, 63, 225, 12, 19, 1, 0, 6, 253, 90, 15, 41, 4, 36, 3, 177, 5, 100, 24, 16, 3, 2, 67, 164, 17, 13, 61, 32, 166, 224, 8, 215, 44, 27, 146, 3, 5, 235, 1, 4, 46, 46, 32, 64, 21, 24, 32, 235, 77, 281, 2, 1, 113, 166, 29, 49, 35, 35, 133, 24, 98, 1, 9, 0, 315, 16, 8, 22, 32, 100, 12, 221, 347, 291, 32, 16, 25, 80, 11, 442, 154, 93, 44, 266, 145, 247, 25, 181, 28, 28, 108, 7, 28, 32, 41, 305, 60, 45, 38, 275, 14, 7, 86, 241, 36, 31, 131, 92, 4, 209, 71, 271, 53, 348, 107, 14, …
## $ CRuns     <int> 30, 321, 224, 828, 48, 501, 30, 41, 32, 784, 702, 192, 205, 309, 376, 20, 1045, 65, 643, 67, 72, 55, 9, 242, 1008, 442, 291, 246, 349, 763, 1033, 80, 545, 219, 126, 859, 156, 34, 784, 34, 20, 796, 371, 86, 486, 108, 113, 117, 987, 352, 1003, 32, 3, 634, 747, 99, 164, 441, 409, 529, 58, 698, 116, 118, 20, 1141, 61, 57, 99, 612, 950, 33, 712, 1175, 1082, 461, 196, 144, 285, 123, 901, 1019, 349, 156, 813, 575, 978, 61, 746, 160, 156, 292, 35, 87, 258, 287, 1135, 753, 156, 335, 8…
## $ CRBI      <int> 29, 414, 266, 838, 46, 336, 9, 37, 34, 890, 504, 186, 204, 103, 290, 16, 993, 23, 658, 82, 48, 43, 9, 251, 1072, 198, 108, 327, 182, 734, 864, 46, 652, 208, 132, 803, 86, 29, 901, 12, 29, 627, 230, 76, 493, 117, 121, 107, 1089, 342, 977, 32, 4, 563, 666, 138, 179, 401, 321, 472, 69, 661, 64, 69, 12, 1179, 74, 40, 103, 344, 909, 32, 815, 1152, 949, 301, 110, 149, 274, 108, 1210, 608, 401, 158, 822, 528, 1093, 70, 805, 123, 159, 343, 35, 110, 192, 294, 1234, 596, 119, 174, 1015…
## $ CWalks    <int> 14, 375, 263, 354, 33, 194, 24, 12, 8, 866, 488, 161, 203, 207, 238, 11, 732, 39, 653, 56, 65, 62, 3, 240, 402, 317, 180, 166, 308, 784, 1087, 31, 337, 136, 66, 571, 107, 18, 560, 14, 45, 483, 195, 32, 608, 118, 80, 51, 431, 289, 619, 27, 4, 319, 526, 64, 194, 333, 170, 313, 16, 777, 57, 114, 9, 727, 52, 39, 78, 422, 690, 55, 548, 1380, 989, 112, 117, 153, 186, 55, 608, 820, 171, 67, 617, 635, 495, 63, 875, 122, 76, 267, 32, 71, 162, 227, 791, 259, 99, 258, 709, 90, 106, 248,…
## $ League    <fct> A, N, A, N, N, A, N, A, N, A, A, N, N, A, N, A, N, A, A, N, N, A, A, N, A, A, N, N, N, A, A, N, N, A, A, N, A, N, A, N, A, N, N, A, A, A, N, A, A, N, A, N, A, A, A, N, N, A, N, A, A, N, A, N, A, A, N, A, A, A, N, N, A, A, A, A, N, N, A, A, A, N, A, A, N, A, N, A, A, A, A, N, A, A, N, N, A, N, N, N, A, A, A, A, A, A, N, A, A, A, A, N, N, N, N, A, A, A, N, A, N, N, A, N, N, A, A, A, N, A, N, N, A, A, N, N, A, A, A, A, A, A, N, N, A, N, A, A, A, A, A, A, N, N, N, A, N, N, A, A, …
## $ Division  <fct> E, W, W, E, E, W, E, W, W, E, E, W, E, E, E, W, W, W, W, W, E, W, W, W, E, E, E, W, W, W, W, W, W, E, W, W, E, W, E, W, E, W, W, E, E, E, W, E, E, W, W, E, E, W, E, W, W, E, W, E, E, E, W, W, W, E, E, W, E, E, W, E, W, E, E, E, W, E, W, W, W, E, E, W, W, W, W, E, W, W, W, E, E, W, W, W, E, W, W, W, E, E, E, E, E, E, W, W, E, E, W, W, E, W, E, W, W, W, W, E, E, W, W, E, W, W, W, W, E, W, E, E, W, W, W, E, E, E, E, W, W, E, E, W, W, E, E, E, E, E, W, W, W, W, E, W, E, E, W, W, …
## $ PutOuts   <int> 446, 632, 880, 200, 805, 282, 76, 121, 143, 0, 238, 304, 211, 121, 80, 118, 105, 102, 912, 202, 280, 361, 812, 518, 1067, 434, 222, 732, 262, 267, 127, 226, 1378, 109, 419, 72, 70, 442, 0, 166, 326, 103, 69, 307, 325, 359, 73, 58, 697, 303, 389, 109, 0, 67, 316, 161, 142, 425, 106, 240, 203, 53, 125, 73, 80, 0, 391, 152, 107, 211, 153, 244, 119, 808, 280, 224, 226, 83, 182, 104, 463, 51, 1377, 92, 303, 276, 278, 148, 165, 246, 533, 226, 45, 157, 142, 59, 292, 360, 274, 292, 1…
## $ Assists   <int> 33, 43, 82, 11, 40, 421, 127, 283, 290, 0, 445, 45, 11, 151, 45, 0, 290, 177, 88, 22, 9, 22, 84, 55, 157, 9, 3, 83, 329, 5, 221, 7, 102, 292, 46, 170, 149, 59, 0, 172, 29, 84, 1, 25, 22, 30, 177, 4, 61, 9, 39, 7, 0, 147, 6, 10, 14, 43, 206, 482, 70, 88, 199, 152, 4, 0, 38, 3, 242, 2, 223, 21, 216, 108, 10, 286, 7, 2, 9, 213, 32, 54, 100, 2, 6, 6, 9, 4, 9, 389, 40, 10, 122, 7, 210, 156, 9, 32, 2, 6, 88, 327, 132, 41, 2, 115, 10, 439, 17, 5, 218, 87, 62, 111, 4, 334, 377, 6, 48…
## $ Errors    <int> 20, 10, 14, 3, 4, 25, 7, 9, 19, 0, 22, 11, 7, 6, 8, 0, 10, 16, 9, 2, 5, 2, 11, 3, 14, 3, 3, 13, 16, 3, 7, 4, 8, 25, 5, 24, 12, 6, 0, 10, 5, 5, 1, 1, 2, 4, 18, 4, 9, 9, 4, 3, 0, 4, 5, 3, 2, 4, 7, 13, 10, 3, 13, 11, 0, 0, 8, 5, 23, 1, 10, 4, 12, 2, 5, 8, 3, 1, 4, 9, 8, 8, 6, 2, 6, 2, 9, 2, 1, 18, 4, 6, 26, 8, 10, 9, 5, 5, 7, 3, 13, 20, 10, 7, 4, 15, 7, 10, 10, 12, 16, 3, 8, 11, 4, 21, 26, 5, 19, 12, 9, 16, 7, 4, 20, 0, 5, 1, 4, 5, 15, 20, 0, 16, 1, 2, 3, 13, 5, 9, 14, 6, 3, 4, …
## $ Salary    <dbl> NA, 475.000, 480.000, 500.000, 91.500, 750.000, 70.000, 100.000, 75.000, 1100.000, 517.143, 512.500, 550.000, 700.000, 240.000, NA, 775.000, 175.000, NA, 135.000, 100.000, 115.000, NA, 600.000, 776.667, 765.000, 708.333, 750.000, 625.000, 900.000, NA, 110.000, NA, 612.500, 300.000, 850.000, NA, 90.000, NA, NA, 67.500, NA, NA, 180.000, NA, 305.000, 215.000, 247.500, NA, 815.000, 875.000, 70.000, NA, 1200.000, 675.000, 415.000, 340.000, NA, 416.667, 1350.000, 90.000, 275.000, 2…
## $ NewLeague <fct> A, N, A, N, N, A, A, A, N, A, A, N, N, A, N, A, N, A, A, N, N, N, A, N, A, A, N, N, N, A, A, N, N, A, A, N, A, N, A, N, A, N, N, A, A, A, N, A, A, N, A, N, A, A, A, N, N, A, N, A, A, N, A, N, A, A, N, A, A, A, N, N, A, A, A, N, A, N, A, A, A, N, A, A, N, A, N, A, A, A, A, N, A, A, N, N, A, N, N, N, A, A, A, A, A, A, N, A, A, A, A, A, N, N, N, A, A, A, N, A, N, N, A, A, N, A, A, A, N, A, N, N, A, A, N, N, A, A, N, N, A, A, N, N, A, N, A, A, A, A, A, A, N, N, N, A, N, N, A, A, …

希望以Salary为因变量，查看其缺失值个数:

sum( is.na(Hitters$Salary) )

## [1] 59

为简单起见，去掉有缺失值的观测：

da_hit <- na.omit(Hitters); dim(da_hit)

## [1] 263  20

42.2 划分训练集和测试集

rsample包的initial_split可以将一个数据集随机拆分为两个数据集，称为训练集和测试集，用prop指定比例，用strata指定分层抽样基于的变量。基于因变量使用分层抽样法划分训练集、测试集可以更具有代表性。

library(rsample)
set.seed(101)
hit_split <- initial_split(
  da_hit, prop = 0.80, strata = Salary)
hit_train <- training(hit_split)
hit_test <- testing(hit_split)

42.3 回归自变量选择

42.3.1 最优子集选择

用leaps包的regsubsets()函数计算最优子集回归，办法是对某个试验性的子集自变量个数p̂ 值，都找到p̂ 固定情况下残差平方和最小的变量子集，这样只要在这些不同p̂ 的最优子集中挑选就可以了。挑选可以用AIC、BIC等方法。

可以先进行一个包含所有自变量的全集回归：

regfit.full <- regsubsets(
  Salary ~ ., data=hit_train, nvmax=19)
reg.summary <- summary(regfit.full)
reg.summary

## Subset selection object
## Call: regsubsets.formula(Salary ~ ., data = hit_train, nvmax = 19)
## 19 Variables  (and intercept)
##            Forced in Forced out
## AtBat          FALSE      FALSE
## Hits           FALSE      FALSE
## HmRun          FALSE      FALSE
## Runs           FALSE      FALSE
## RBI            FALSE      FALSE
## Walks          FALSE      FALSE
## Years          FALSE      FALSE
## CAtBat         FALSE      FALSE
## CHits          FALSE      FALSE
## CHmRun         FALSE      FALSE
## CRuns          FALSE      FALSE
## CRBI           FALSE      FALSE
## CWalks         FALSE      FALSE
## LeagueN        FALSE      FALSE
## DivisionW      FALSE      FALSE
## PutOuts        FALSE      FALSE
## Assists        FALSE      FALSE
## Errors         FALSE      FALSE
## NewLeagueN     FALSE      FALSE
## 1 subsets of each size up to 19
## Selection Algorithm: exhaustive
##           AtBat Hits HmRun Runs RBI Walks Years CAtBat CHits CHmRun CRuns CRBI CWalks LeagueN DivisionW PutOuts Assists Errors NewLeagueN
## 1  ( 1 )  " "   " "  " "   " "  " " " "   " "   " "    " "   " "    " "   "*"  " "    " "     " "       " "     " "     " "    " "       
## 2  ( 1 )  " "   "*"  " "   " "  " " " "   " "   " "    " "   " "    " "   "*"  " "    " "     " "       " "     " "     " "    " "       
## 3  ( 1 )  " "   "*"  " "   " "  " " " "   " "   " "    " "   " "    " "   "*"  " "    " "     "*"       " "     " "     " "    " "       
## 4  ( 1 )  " "   "*"  " "   " "  " " " "   " "   " "    " "   " "    " "   "*"  " "    " "     "*"       "*"     " "     " "    " "       
## 5  ( 1 )  "*"   "*"  " "   " "  " " " "   " "   " "    " "   " "    " "   "*"  " "    " "     "*"       "*"     " "     " "    " "       
## 6  ( 1 )  "*"   "*"  " "   " "  " " "*"   " "   " "    " "   " "    " "   "*"  " "    " "     "*"       "*"     " "     " "    " "       
## 7  ( 1 )  "*"   "*"  " "   " "  " " "*"   " "   " "    " "   " "    " "   "*"  "*"    " "     "*"       "*"     " "     " "    " "       
## 8  ( 1 )  "*"   "*"  " "   " "  " " "*"   " "   " "    " "   " "    "*"   "*"  "*"    " "     "*"       "*"     " "     " "    " "       
## 9  ( 1 )  "*"   "*"  " "   " "  " " "*"   " "   " "    " "   "*"    "*"   " "  "*"    " "     "*"       "*"     "*"     " "    " "       
## 10  ( 1 ) "*"   "*"  " "   " "  " " "*"   " "   "*"    " "   " "    "*"   "*"  "*"    " "     "*"       "*"     "*"     " "    " "       
## 11  ( 1 ) "*"   "*"  " "   " "  " " "*"   " "   "*"    " "   " "    "*"   "*"  "*"    "*"     "*"       "*"     "*"     " "    " "       
## 12  ( 1 ) "*"   "*"  " "   "*"  " " "*"   " "   "*"    " "   " "    "*"   "*"  "*"    "*"     "*"       "*"     "*"     " "    " "       
## 13  ( 1 ) "*"   "*"  " "   "*"  " " "*"   "*"   "*"    " "   " "    "*"   "*"  "*"    "*"     "*"       "*"     "*"     " "    " "       
## 14  ( 1 ) "*"   "*"  " "   "*"  "*" "*"   "*"   "*"    " "   " "    "*"   "*"  "*"    "*"     "*"       "*"     "*"     " "    " "       
## 15  ( 1 ) "*"   "*"  " "   "*"  "*" "*"   "*"   "*"    " "   " "    "*"   "*"  "*"    "*"     "*"       "*"     "*"     "*"    " "       
## 16  ( 1 ) "*"   "*"  " "   "*"  "*" "*"   "*"   "*"    "*"   "*"    "*"   "*"  "*"    "*"     "*"       "*"     "*"     " "    " "       
## 17  ( 1 ) "*"   "*"  " "   "*"  "*" "*"   "*"   "*"    "*"   "*"    "*"   "*"  "*"    "*"     "*"       "*"     "*"     "*"    " "       
## 18  ( 1 ) "*"   "*"  " "   "*"  "*" "*"   "*"   "*"    "*"   "*"    "*"   "*"  "*"    "*"     "*"       "*"     "*"     "*"    "*"       
## 19  ( 1 ) "*"   "*"  "*"   "*"  "*" "*"   "*"   "*"    "*"   "*"    "*"   "*"  "*"    "*"     "*"       "*"     "*"     "*"    "*"

这里用nvmax=指定了允许所有的自变量都参加，缺省行为是限制最多个数的。上述结果表格中每一行给出了固定p̂ 条件下的最优子集。

试比较这些最优模型的BIC值:

reg.summary$bic

##  [1] -63.90242 -86.59469 -90.68877 -93.51559 -96.29865 -96.35699 -95.24328 -94.33547 -91.79438 -89.31463 -85.07463 -80.40798 -75.33025 -70.12122 -64.82873 -59.53306 -54.25553 -48.92352 -43.58870

plot(reg.summary$bic)

图42.1: Hitters数据最优子集回归BIC

其中p̂ =5,6的值相近，都很低，取p̂ =6。用coef()加id=6指定第六种子集：

coef(regfit.full, id=6)

##  (Intercept)        AtBat         Hits        Walks         CRBI    DivisionW      PutOuts 
##  149.0951521   -2.1064928    8.2070703    3.2517011    0.6351933 -136.2935330    0.2646021

这种方法实现了选取BIC最小的自变量子集，有6个自变量。

42.3.2 逐步回归方法

在用lm()做了全集回归后，把全集回归结果输入到stats::step()函数中可以执行逐步回归。如:

lm.full <- lm(Salary ~ ., data = hit_train)
print(summary(lm.full))

## 
## Call:
## lm(formula = Salary ~ ., data = hit_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -918.96 -183.16  -35.62  138.30 1799.45 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  241.67291  109.57064   2.206  0.02862 * 
## AtBat         -2.48494    0.76899  -3.231  0.00145 **
## Hits           8.15485    2.84403   2.867  0.00461 **
## HmRun         -0.37929    7.64779  -0.050  0.96050   
## Runs          -2.12109    3.59273  -0.590  0.55564   
## RBI            0.76668    3.11770   0.246  0.80602   
## Walks          6.27568    2.18144   2.877  0.00448 **
## Years         -7.18987   15.10209  -0.476  0.63457   
## CAtBat        -0.14891    0.16372  -0.909  0.36425   
## CHits          0.23486    0.78151   0.301  0.76411   
## CHmRun         0.50158    1.97716   0.254  0.80002   
## CRuns          1.11476    0.92330   1.207  0.22881   
## CRBI           0.70183    0.84282   0.833  0.40606   
## CWalks        -0.83644    0.37968  -2.203  0.02881 * 
## LeagueN       47.02170   94.26262   0.499  0.61848   
## DivisionW   -120.60207   48.51038  -2.486  0.01379 * 
## PutOuts        0.26292    0.09121   2.883  0.00440 **
## Assists        0.38272    0.26915   1.422  0.15670   
## Errors        -1.28251    5.36074  -0.239  0.81118   
## NewLeagueN    -7.16809   94.61668  -0.076  0.93969   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 336.8 on 188 degrees of freedom
## Multiple R-squared:  0.5146, Adjusted R-squared:  0.4655 
## F-statistic: 10.49 on 19 and 188 DF,  p-value: < 2.2e-16

stats::step(lm.full)

## Start:  AIC=2439.89
## Salary ~ AtBat + Hits + HmRun + Runs + RBI + Walks + Years + 
##     CAtBat + CHits + CHmRun + CRuns + CRBI + CWalks + League + 
##     Division + PutOuts + Assists + Errors + NewLeague
## 
##             Df Sum of Sq      RSS    AIC
## - HmRun      1       279 21327132 2437.9
## - NewLeague  1       651 21327504 2437.9
## - Errors     1      6493 21333346 2437.9
## - RBI        1      6860 21333713 2438.0
## - CHmRun     1      7301 21334153 2438.0
## - CHits      1     10245 21337098 2438.0
## - Years      1     25712 21352565 2438.1
## - League     1     28228 21355081 2438.2
## - Runs       1     39540 21366393 2438.3
## - CRBI       1     78662 21405515 2438.7
## - CAtBat     1     93836 21420689 2438.8
## - CRuns      1    165367 21492220 2439.5
## <none>                   21326853 2439.9
## - Assists    1    229372 21556225 2440.1
## - CWalks     1    550572 21877425 2443.2
## - Division   1    701147 22028000 2444.6
## - Hits       1    932679 22259532 2446.8
## - Walks      1    938864 22265716 2446.8
## - PutOuts    1    942588 22269441 2446.9
## - AtBat      1   1184571 22511424 2449.1
## 
## Step:  AIC=2437.89
## Salary ~ AtBat + Hits + Runs + RBI + Walks + Years + CAtBat + 
##     CHits + CHmRun + CRuns + CRBI + CWalks + League + Division + 
##     PutOuts + Assists + Errors + NewLeague
## 
##             Df Sum of Sq      RSS    AIC
## - NewLeague  1       566 21327698 2435.9
## - Errors     1      6443 21333575 2436.0
## - CHmRun     1      7539 21334671 2436.0
## - CHits      1      9986 21337118 2436.0
## - RBI        1     12495 21339627 2436.0
## - Years      1     25478 21352610 2436.1
## - League     1     27950 21355082 2436.2
## - Runs       1     53429 21380561 2436.4
## - CAtBat     1     94340 21421471 2436.8
## - CRBI       1     96689 21423821 2436.8
## - CRuns      1    185367 21512499 2437.7
## <none>                   21327132 2437.9
## - Assists    1    235593 21562725 2438.2
## - CWalks     1    575407 21902539 2441.4
## - Division   1    720408 22047540 2442.8
## - PutOuts    1    947076 22274208 2444.9
## - Walks      1   1002501 22329633 2445.4
## - Hits       1   1073306 22400438 2446.1
## - AtBat      1   1185325 22512457 2447.1
## 
## Step:  AIC=2435.9
## Salary ~ AtBat + Hits + Runs + RBI + Walks + Years + CAtBat + 
##     CHits + CHmRun + CRuns + CRBI + CWalks + League + Division + 
##     PutOuts + Assists + Errors
## 
##            Df Sum of Sq      RSS    AIC
## - Errors    1      6155 21333853 2434.0
## - CHmRun    1      7339 21335037 2434.0
## - CHits     1      9541 21337239 2434.0
## - RBI       1     12817 21340515 2434.0
## - Years     1     25398 21353097 2434.2
## - Runs      1     53335 21381033 2434.4
## - League    1     75071 21402769 2434.6
## - CAtBat    1     93812 21421510 2434.8
## - CRBI      1     98282 21425981 2434.9
## - CRuns     1    190610 21518308 2435.8
## <none>                  21327698 2435.9
## - Assists   1    236010 21563708 2436.2
## - CWalks    1    577288 21904986 2439.4
## - Division  1    720061 22047759 2440.8
## - PutOuts   1    948064 22275762 2442.9
## - Walks     1   1003786 22331484 2443.5
## - Hits      1   1091940 22419639 2444.3
## - AtBat     1   1223590 22551289 2445.5
## 
## Step:  AIC=2433.96
## Salary ~ AtBat + Hits + Runs + RBI + Walks + Years + CAtBat + 
##     CHits + CHmRun + CRuns + CRBI + CWalks + League + Division + 
##     PutOuts + Assists
## 
##            Df Sum of Sq      RSS    AIC
## - CHmRun    1      6724 21340577 2432.0
## - CHits     1      7824 21341677 2432.0
## - RBI       1     11220 21345072 2432.1
## - Years     1     24104 21357956 2432.2
## - Runs      1     57526 21391379 2432.5
## - League    1     70922 21404775 2432.7
## - CAtBat    1     90644 21424497 2432.8
## - CRBI      1    100984 21434837 2432.9
## - CRuns     1    201382 21535235 2433.9
## <none>                  21333853 2434.0
## - Assists   1    313674 21647527 2435.0
## - CWalks    1    593539 21927392 2437.7
## - Division  1    722945 22056798 2438.9
## - PutOuts   1    942739 22276592 2440.9
## - Walks     1   1040700 22374553 2441.9
## - Hits      1   1161864 22495717 2443.0
## - AtBat     1   1281359 22615212 2444.1
## 
## Step:  AIC=2432.03
## Salary ~ AtBat + Hits + Runs + RBI + Walks + Years + CAtBat + 
##     CHits + CRuns + CRBI + CWalks + League + Division + PutOuts + 
##     Assists
## 
##            Df Sum of Sq      RSS    AIC
## - CHits     1      2192 21342770 2430.1
## - RBI       1     12586 21353163 2430.2
## - Years     1     24971 21365548 2430.3
## - Runs      1     63054 21403631 2430.6
## - League    1     71042 21411619 2430.7
## - CAtBat    1     86281 21426858 2430.9
## <none>                  21340577 2432.0
## - Assists   1    306971 21647548 2433.0
## - CRuns     1    433335 21773912 2434.2
## - CWalks    1    631568 21972145 2436.1
## - Division  1    716579 22057157 2436.9
## - PutOuts   1    954537 22295114 2439.1
## - CRBI      1   1001899 22342476 2439.6
## - Walks     1   1036407 22376984 2439.9
## - Hits      1   1187105 22527683 2441.3
## - AtBat     1   1283747 22624325 2442.2
## 
## Step:  AIC=2430.05
## Salary ~ AtBat + Hits + Runs + RBI + Walks + Years + CAtBat + 
##     CRuns + CRBI + CWalks + League + Division + PutOuts + Assists
## 
##            Df Sum of Sq      RSS    AIC
## - RBI       1     13190 21355960 2428.2
## - Years     1     29638 21372407 2428.3
## - League    1     72742 21415512 2428.8
## - Runs      1     81521 21424290 2428.8
## <none>                  21342770 2430.1
## - CAtBat    1    230265 21573034 2430.3
## - Assists   1    307170 21649939 2431.0
## - CRuns     1    713710 22056479 2434.9
## - Division  1    715586 22058356 2434.9
## - CWalks    1    929774 22272544 2436.9
## - PutOuts   1    978714 22321484 2437.4
## - CRBI      1   1002770 22345540 2437.6
## - Walks     1   1086910 22429680 2438.4
## - AtBat     1   1599684 22942453 2443.1
## - Hits      1   1779918 23122687 2444.7
## 
## Step:  AIC=2428.18
## Salary ~ AtBat + Hits + Runs + Walks + Years + CAtBat + CRuns + 
##     CRBI + CWalks + League + Division + PutOuts + Assists
## 
##            Df Sum of Sq      RSS    AIC
## - Years     1     26692 21382651 2426.4
## - League    1     70307 21426266 2426.9
## - Runs      1     73753 21429713 2426.9
## <none>                  21355960 2428.2
## - CAtBat    1    249406 21605365 2428.6
## - Assists   1    295538 21651497 2429.0
## - CRuns     1    702284 22058244 2432.9
## - Division  1    734085 22090044 2433.2
## - CWalks    1    937348 22293308 2435.1
## - PutOuts   1   1002301 22358261 2435.7
## - Walks     1   1086003 22441962 2436.5
## - CRBI      1   1439193 22795152 2439.7
## - AtBat     1   1640165 22996124 2441.6
## - Hits      1   1787801 23143761 2442.9
## 
## Step:  AIC=2426.43
## Salary ~ AtBat + Hits + Runs + Walks + CAtBat + CRuns + CRBI + 
##     CWalks + League + Division + PutOuts + Assists
## 
##            Df Sum of Sq      RSS    AIC
## - Runs      1     69079 21451730 2425.1
## - League    1     87548 21470199 2425.3
## <none>                  21382651 2426.4
## - Assists   1    314039 21696690 2427.5
## - CAtBat    1    492567 21875218 2429.2
## - Division  1    725175 22107827 2431.4
## - CRuns     1    880113 22262764 2432.8
## - CWalks    1    988001 22370652 2433.8
## - PutOuts   1   1049648 22432299 2434.4
## - Walks     1   1079896 22462547 2434.7
## - CRBI      1   1420036 22802687 2437.8
## - AtBat     1   1614330 22996981 2439.6
## - Hits      1   1772982 23155633 2441.0
## 
## Step:  AIC=2425.11
## Salary ~ AtBat + Hits + Walks + CAtBat + CRuns + CRBI + CWalks + 
##     League + Division + PutOuts + Assists
## 
##            Df Sum of Sq      RSS    AIC
## - League    1    113492 21565223 2424.2
## <none>                  21451730 2425.1
## - Assists   1    399827 21851557 2426.9
## - CAtBat    1    428452 21880182 2427.2
## - Division  1    727359 22179089 2430.0
## - CRuns     1    811308 22263038 2430.8
## - CWalks    1    947776 22399506 2432.1
## - Walks     1   1029714 22481444 2432.9
## - PutOuts   1   1153252 22604982 2434.0
## - CRBI      1   1434607 22886337 2436.6
## - AtBat     1   1793723 23245454 2439.8
## - Hits      1   1825947 23277677 2440.1
## 
## Step:  AIC=2424.2
## Salary ~ AtBat + Hits + Walks + CAtBat + CRuns + CRBI + CWalks + 
##     Division + PutOuts + Assists
## 
##            Df Sum of Sq      RSS    AIC
## <none>                  21565223 2424.2
## - CAtBat    1    366456 21931678 2425.7
## - Assists   1    423017 21988240 2426.2
## - CRuns     1    756041 22321264 2429.4
## - Division  1    762166 22327389 2429.4
## - CWalks    1    998625 22563847 2431.6
## - Walks     1   1124976 22690198 2432.8
## - PutOuts   1   1245275 22810497 2433.9
## - CRBI      1   1393594 22958817 2435.2
## - Hits      1   1785448 23350671 2438.8
## - AtBat     1   1830070 23395292 2439.2

## 
## Call:
## lm(formula = Salary ~ AtBat + Hits + Walks + CAtBat + CRuns + 
##     CRBI + CWalks + Division + PutOuts + Assists, data = hit_train)
## 
## Coefficients:
## (Intercept)        AtBat         Hits        Walks       CAtBat        CRuns         CRBI       CWalks    DivisionW      PutOuts      Assists  
##    235.9278      -2.5863       7.7364       5.9827      -0.1210       1.2468       0.9302      -0.9100    -123.4092       0.2893       0.3770

最后保留了10个自变量。

42.3.3 预测根均方误差计算

仅用训练集估计模型。为了在测试集和交叉验证集上用模型进行预报并估计预测均方误差，需要自己写一个预测函数：

predict.regsubsets <- function(object, newdata, id, ...){
  form <- as.formula(object$call[[2]])
  mat <- model.matrix(form, newdata)
  coefi <- coef(object, id=id)
  xvars <- names(coefi)
  mat[, xvars] %*% coefi
}

42.3.4 用10折交叉验证方法选择最优子集

用交叉验证方法比较不同的模型，使用tidymodels扩展包有标准的做法，参见47.3。这里为了对方法进行更直接的演示，直接调用交叉验证函数进行超参数调优并在测试集上计算预测精度指标。

下列程序对数据中每一行分配一个折号：

set.seed(102)
hit_fold <- vfold_cv(hit_train, v = 10)

下面，对10折中每一折都分别当作测试集一次，得到不同子集大小的根均方误差：

cv.errors <- matrix( as.numeric(NA), 10, 19, dimnames=list(NULL, paste(1:19)) )
for(j in 1:10){ # 折
  d_ana <- analysis(hit_fold$splits[[j]])
  d_ass <- assessment((hit_fold$splits[[j]]))
  best.fit <- regsubsets(
    Salary ~ ., 
    data = d_ana, nvmax=19)
  for(i in 1:19){
    pred <- predict( 
      best.fit, d_ass, id=i)
    cv.errors[j, i] <- 
      mean( (d_ass[["Salary"]] - pred)^2 ) |> sqrt()
  }
}
cv.errors[1:3, 1:5]

##             1        2        3        4        5
## [1,] 527.1116 448.7947 541.2805 486.2844 500.2707
## [2,] 380.8413 417.8030 339.0055 320.3588 294.5357
## [3,] 425.7064 407.8210 401.5712 381.7351 365.8554

cv.errors是一个10×19矩阵，每行对应一折作为测试集（或称评估集）的情形，每列是一个子集大小，元素值是预测的根均方误差。

对每列的10个元素求平均，可以得到每个子集大小的平均根均方误差:

mean.cv.errors <- rowMeans(cv.errors)
mean.cv.errors

##  [1] 446.5360 348.7238 370.0678 519.2774 403.6128 254.5081 298.6319 302.1066 387.7379 353.0201

best.id <- which.min(mean.cv.errors)
plot(mean.cv.errors, type='b', 
  main = "RMSE",
  xlab = "p")

图42.2: Hitters数据CV均方误差

这样找到的最优子集大小是6， RMSE=254.5。注意，一般不需要用户自己进行这种交叉验证调参，机器学习的函数一般都集成了这个功能。

用这种方法找到最优子集大小后，可以对全数据集重新建模但是选择最优子集大小为6:

reg.best <- regsubsets(Salary ~ ., data = da_hit, nvmax=19)
coef(reg.best, id=best.id)

##  (Intercept)        AtBat         Hits        Walks         CRBI    DivisionW      PutOuts 
##   91.5117981   -1.8685892    7.6043976    3.6976468    0.6430169 -122.9515338    0.2643076

这样的模型可以用于同一问题的新增数据的预测。

42.4 岭回归

当自变量个数太多时，模型复杂度高，可能有过度拟合，模型不稳定。自变量子集选择是降低复杂度的一种方法。

另一种方法是对较大的模型系数施加二次惩罚，把最小二乘问题变成带有二次惩罚项的惩罚最小二乘问题：

min∑i=1n(yi−β0−β1xi1−⋯−βpxip)2+λ∑j=1pβ2j.

这比通常最小二乘得到的回归系数绝对值变小，但是求解的稳定性增加了，避免了共线问题。这种方法称为“正则化”(regularization)，其中的∑pj=1β2j称为正则项或者L2惩罚项。

实际上, 与线性模型Y=Xβ+ε 的普通最小二乘解 β̂ =(XTX)−1XTY 相比，岭回归问题的解为

β̃ =(XTX+sI)−1XTY

其中I为单位阵，s>0与λ有关。

λ称为调节参数，λ越大，相当于模型复杂度越低。适当选择λ可以在方差与偏差之间找到适当的折衷，从而减小预测误差。这样的参数不能从数据中直接估计，称为“超参数”，需要用模型比较的方法获得最优值。

由于量纲问题，在不同自变量不可比时，数据集应该进行标准化。

用R的glmnet包计算岭回归。用glmnet()函数，指定参数alpha=0时执行的是岭回归。用参数lambda=指定一个调节参数网格，岭回归的算法可以进行一轮计算就获得所有这些调节参数上对应的参数估计。用coef()从回归结果中取得不同调节参数对应的回归系数估计，结果是一个矩阵，每列对应于一个调节参数。

仍采用上面去掉了缺失值的Hitters数据集结果da_hit。

glmnet包不支持R的公式界面，所以用如下程序把回归的设计阵与因变量提取出来：

x <- model.matrix(Salary ~ ., hit_train)[,-1]
y <- hit_train$Salary

岭回归涉及到调节参数λ的选择，为了绘图，先选择λ的一个网格：

grid <- 10^seq(10, -2, length=100)

用所有数据针对这样的调节参数网格计算岭回归结果，注意glmnet()函数允许调节参数λ输入多个值：

ridge.mod <- glmnet(x, y, alpha=0, lambda=grid)
dim(coef(ridge.mod))

## [1]  20 100

glmnet()函数默认对数据进行标准化。
coef()的结果是一个矩阵，每列对应一个调节参数值，其中的数值是回归系数估计值。

42.4.1 用10折交叉验证选取调节参数

如何进行超参数调优并在测试集上计算性能， tidymodels有系统的方法，参见47.3。这里为了对方法进行更直接的演示，直接调用交叉验证函数进行超参数调优并在测试集上计算预测精度指标。

在训练集用交叉验证选择调节参数，称为参数调优或者超参数调优。 cv.glmnet()函数本身可以执行交叉验证，不需要自己划分折：

set.seed(1)
cv.out <- cv.glmnet(x, y, alpha=0)
plot(cv.out)

图42.3: Hitters数据岭回归参数选择

bestlam <- cv.out$lambda.min
bestlam

## [1] 25.22831

这样获得了最优调节参数λ= 25.2283126。用最优调节参数对测试集作预测，得到预测根均方误差：

ridge.pred <- predict(
  ridge.mod, s = bestlam, 
  newx = model.matrix(Salary ~ ., hit_test)[,-1])
mean( (ridge.pred - hit_test$Salary)^2 ) |> sqrt()

## [1] 240.7377

根均方误差240.7，比最优自变量子集方法的254.5要好。

最后，用选取的最优调节系数对全数据集建模，得到相应的岭回归系数估计：

x <- model.matrix(Salary ~ ., da_hit)[,-1]
y <- da_hit$Salary
out <- glmnet(x, y, alpha=0)
predict(out, type='coefficients', s=bestlam)[1:20,]

##   (Intercept)         AtBat          Hits         HmRun          Runs           RBI         Walks         Years        CAtBat         CHits        CHmRun         CRuns          CRBI        CWalks       LeagueN     DivisionW       PutOuts       Assists        Errors    NewLeagueN 
##  8.112693e+01 -6.815959e-01  2.772312e+00 -1.365680e+00  1.014826e+00  7.130224e-01  3.378558e+00 -9.066800e+00 -1.199478e-03  1.361029e-01  6.979958e-01  2.958896e-01  2.570711e-01 -2.789666e-01  5.321272e+01 -1.228345e+02  2.638876e-01  1.698796e-01 -3.685645e+00 -1.810510e+01

这样的模型可以用在同一问题的新数据预测上。

42.5 Lasso回归

另一种对回归系数的惩罚是L1惩罚：

min∑i=1n(yi−β0−β1xi1−⋯−βpxip)2+λ∑j=1p|βj|.(42.1)

奇妙地是，当调节参数λ较大时，可以使得部分回归系数变成零，达到了即减小回归系数的绝对值又挑选重要变量子集的效果。

事实上，(42.1)等价于约束最小值问题

min∑i=1n(yi−β0−β1xi1−⋯−βpxip)2s.t.∑j=1p|βj|≤s.

其中s与λ一一对应。这样的约束区域是带有顶点的凸集，而目标函数是二次函数，最小值点经常在约束区域顶点达到，这些顶点是某些坐标等于零的点。见图42.4。图中阴影部分是约束区域，注意4个顶点处都有一个回归系数等于0；同心的椭圆线是目标函数的等值线，椭圆中心处是目标函数的无约束最小值点，即普通最小二乘的解，而约束区域与目标函数值最小的等值线的交点出现在顶点处，该处的β1=0。

knitr::include_graphics("figs/lasso-min.png")

图42.4: Lasso约束优化问题图示

对于每个调节参数λ，都应该解出(42.1)的相应解，记为β̂ (λ)。幸运的是，不需要对每个λ去解最小值问题(42.1)，存在巧妙的算法使得问题的计算量与求解一次最小二乘相仿。

通常选取λ的格子点，计算相应的惩罚回归系数。用交叉验证方法估计预测的均方误差。选取使得交叉验证均方误差最小的调节参数（一般R函数中已经作为选项）。

用R的glmnet包计算lasso。用glmnet()函数，指定参数alpha=1时执行的是lasso。用参数lambda=指定一个调节参数网格， lasso将输出这些调节参数对应的结果。对回归结果使用plot()函数可以画出调节参数变化时系数估计的变化情况。

仍使用gmlnet包的glmnet()函数计算Lasso回归，指定一个调节参数网格（沿用前面的网格）：

x <- model.matrix(Salary ~ ., hit_train)[,-1]
y <- hit_train$Salary
lasso.mod <- glmnet(x, y, alpha=1, lambda=grid)
plot(lasso.mod)

## Warning in regularize.values(x, y, ties, missing(ties), na.rm = na.rm): collapsing to unique 'x' values

图42.5: Hitters数据lasso轨迹

对lasso结果使用plot()函数可以绘制延调节参数网格变化的各回归系数估计，横坐标不是调节参数而是调节参数对应的系数绝对值和，可以看出随着系数绝对值和增大，实际是调节参数变小，更多地自变量进入模型。

42.5.1 用交叉验证估计调节参数

按照前面划分的训练集与测试集，仅使用训练集数据做交叉验证估计最优调节参数：

set.seed(1)
cv.out <- cv.glmnet(x, y, alpha=1)
plot(cv.out)

bestlam <- cv.out$lambda.min; bestlam

## [1] 2.19423

得到调节参数估计后，对测试集计算预测均方误差：

lasso.pred <- predict(
  lasso.mod, s = bestlam, 
  newx = model.matrix(Salary ~ ., hit_test)[,-1])
mean( (lasso.pred - hit_test$Salary)^2 ) |> sqrt()

## [1] 242.0375

RMSE=242.0，这个效果比岭回归(RMSE=240.7)效果略差，比最优子集方法(RMSE=254.5)好。

为了充分利用数据，使用前面获得的最优调节参数，对全数据集建模：

x <- model.matrix(Salary ~ ., da_hit)[,-1]
y <- da_hit$Salary
out <- glmnet(x, y, alpha=1, lambda=grid)
lasso.coef <- predict(
  out, type='coefficients', s=bestlam)[1:20,]
lasso.coef[lasso.coef != 0]

##   (Intercept)         AtBat          Hits         HmRun         Walks         Years        CAtBat        CHmRun         CRuns          CRBI        CWalks       LeagueN     DivisionW       PutOuts       Assists        Errors 
##  1.348925e+02 -1.689582e+00  5.971182e+00  9.734402e-02  4.978211e+00 -1.019167e+01 -9.794493e-05  5.650266e-01  7.036826e-01  3.867695e-01 -5.851131e-01  3.305686e+01 -1.193420e+02  2.760478e-01  2.008473e-01 -2.277618e+00

选择的自变量子集有15个自变量。

42.6 附录

42.6.1 Hitters数据

knitr::kable(Hitters)

	AtBat	Hits	HmRun	Runs	RBI	Walks	Years	CAtBat	CHits	CHmRun	CRuns	CRBI	CWalks	League	Division	PutOuts	Assists	Errors	Salary	NewLeague
-Andy Allanson	293	66	1	30	29	14	1	293	66	1	30	29	14	A	E	446	33	20	NA	A
-Alan Ashby	315	81	7	24	38	39	14	3449	835	69	321	414	375	N	W	632	43	10	475.000	N
-Alvin Davis	479	130	18	66	72	76	3	1624	457	63	224	266	263	A	W	880	82	14	480.000	A
-Andre Dawson	496	141	20	65	78	37	11	5628	1575	225	828	838	354	N	E	200	11	3	500.000	N
-Andres Galarraga	321	87	10	39	42	30	2	396	101	12	48	46	33	N	E	805	40	4	91.500	N
-Alfredo Griffin	594	169	4	74	51	35	11	4408	1133	19	501	336	194	A	W	282	421	25	750.000	A
-Al Newman	185	37	1	23	8	21	2	214	42	1	30	9	24	N	E	76	127	7	70.000	A
-Argenis Salazar	298	73	0	24	24	7	3	509	108	0	41	37	12	A	W	121	283	9	100.000	A
-Andres Thomas	323	81	6	26	32	8	2	341	86	6	32	34	8	N	W	143	290	19	75.000	N
-Andre Thornton	401	92	17	49	66	65	13	5206	1332	253	784	890	866	A	E	0	0	0	1100.000	A
-Alan Trammell	574	159	21	107	75	59	10	4631	1300	90	702	504	488	A	E	238	445	22	517.143	A
-Alex Trevino	202	53	4	31	26	27	9	1876	467	15	192	186	161	N	W	304	45	11	512.500	N
-Andy VanSlyke	418	113	13	48	61	47	4	1512	392	41	205	204	203	N	E	211	11	7	550.000	N
-Alan Wiggins	239	60	0	30	11	22	6	1941	510	4	309	103	207	A	E	121	151	6	700.000	A
-Bill Almon	196	43	7	29	27	30	13	3231	825	36	376	290	238	N	E	80	45	8	240.000	N
-Billy Beane	183	39	3	20	15	11	3	201	42	3	20	16	11	A	W	118	0	0	NA	A
-Buddy Bell	568	158	20	89	75	73	15	8068	2273	177	1045	993	732	N	W	105	290	10	775.000	N
-Buddy Biancalana	190	46	2	24	8	15	5	479	102	5	65	23	39	A	W	102	177	16	175.000	A
-Bruce Bochte	407	104	6	57	43	65	12	5233	1478	100	643	658	653	A	W	912	88	9	NA	A
-Bruce Bochy	127	32	8	16	22	14	8	727	180	24	67	82	56	N	W	202	22	2	135.000	N
-Barry Bonds	413	92	16	72	48	65	1	413	92	16	72	48	65	N	E	280	9	5	100.000	N
-Bobby Bonilla	426	109	3	55	43	62	1	426	109	3	55	43	62	A	W	361	22	2	115.000	N
-Bob Boone	22	10	1	4	2	1	6	84	26	2	9	9	3	A	W	812	84	11	NA	A
-Bob Brenly	472	116	16	60	62	74	6	1924	489	67	242	251	240	N	W	518	55	3	600.000	N
-Bill Buckner	629	168	18	73	102	40	18	8424	2464	164	1008	1072	402	A	E	1067	157	14	776.667	A
-Brett Butler	587	163	4	92	51	70	6	2695	747	17	442	198	317	A	E	434	9	3	765.000	A
-Bob Dernier	324	73	4	32	18	22	7	1931	491	13	291	108	180	N	E	222	3	3	708.333	N
-Bo Diaz	474	129	10	50	56	40	10	2331	604	61	246	327	166	N	W	732	83	13	750.000	N
-Bill Doran	550	152	6	92	37	81	5	2308	633	32	349	182	308	N	W	262	329	16	625.000	N
-Brian Downing	513	137	20	90	95	90	14	5201	1382	166	763	734	784	A	W	267	5	3	900.000	A
-Bobby Grich	313	84	9	42	30	39	17	6890	1833	224	1033	864	1087	A	W	127	221	7	NA	A
-Billy Hatcher	419	108	6	55	36	22	3	591	149	8	80	46	31	N	W	226	7	4	110.000	N
-Bob Horner	517	141	27	70	87	52	9	3571	994	215	545	652	337	N	W	1378	102	8	NA	N
-Brook Jacoby	583	168	17	83	80	56	5	1646	452	44	219	208	136	A	E	109	292	25	612.500	A
-Bob Kearney	204	49	6	23	25	12	7	1309	308	27	126	132	66	A	W	419	46	5	300.000	A
-Bill Madlock	379	106	10	38	60	30	14	6207	1906	146	859	803	571	N	W	72	170	24	850.000	N
-Bobby Meacham	161	36	0	19	10	17	4	1053	244	3	156	86	107	A	E	70	149	12	NA	A
-Bob Melvin	268	60	5	24	25	15	2	350	78	5	34	29	18	N	W	442	59	6	90.000	N
-Ben Oglivie	346	98	5	31	53	30	16	5913	1615	235	784	901	560	A	E	0	0	0	NA	A
-Bip Roberts	241	61	1	34	12	14	1	241	61	1	34	12	14	N	W	166	172	10	NA	N
-BillyJo Robidoux	181	41	1	15	21	33	2	232	50	4	20	29	45	A	E	326	29	5	67.500	A
-Bill Russell	216	54	0	21	18	15	18	7318	1926	46	796	627	483	N	W	103	84	5	NA	N
-Billy Sample	200	57	6	23	14	14	9	2516	684	46	371	230	195	N	W	69	1	1	NA	N
-Bill Schroeder	217	46	7	32	19	9	4	694	160	32	86	76	32	A	E	307	25	1	180.000	A
-Butch Wynegar	194	40	7	19	29	30	11	4183	1069	64	486	493	608	A	E	325	22	2	NA	A
-Chris Bando	254	68	2	28	26	22	6	999	236	21	108	117	118	A	E	359	30	4	305.000	A
-Chris Brown	416	132	7	57	49	33	3	932	273	24	113	121	80	N	W	73	177	18	215.000	N
-Carmen Castillo	205	57	8	34	32	9	5	756	192	32	117	107	51	A	E	58	4	4	247.500	A
-Cecil Cooper	542	140	12	46	75	41	16	7099	2130	235	987	1089	431	A	E	697	61	9	NA	A
-Chili Davis	526	146	13	71	70	84	6	2648	715	77	352	342	289	N	W	303	9	9	815.000	N
-Carlton Fisk	457	101	14	42	63	22	17	6521	1767	281	1003	977	619	A	W	389	39	4	875.000	A
-Curt Ford	214	53	2	30	29	23	2	226	59	2	32	32	27	N	E	109	7	3	70.000	N
-Cliff Johnson	19	7	0	1	2	1	4	41	13	1	3	4	4	A	E	0	0	0	NA	A
-Carney Lansford	591	168	19	80	72	39	9	4478	1307	113	634	563	319	A	W	67	147	4	1200.000	A
-Chet Lemon	403	101	12	45	53	39	12	5150	1429	166	747	666	526	A	E	316	6	5	675.000	A
-Candy Maldonado	405	102	18	49	85	20	6	950	231	29	99	138	64	N	W	161	10	3	415.000	N
-Carmelo Martinez	244	58	9	28	25	35	4	1335	333	49	164	179	194	N	W	142	14	2	340.000	N
-Charlie Moore	235	61	3	24	39	21	14	3926	1029	35	441	401	333	A	E	425	43	4	NA	A
-Craig Reynolds	313	78	6	32	41	12	12	3742	968	35	409	321	170	N	W	106	206	7	416.667	N
-Cal Ripken	627	177	25	98	81	70	6	3210	927	133	529	472	313	A	E	240	482	13	1350.000	A
-Cory Snyder	416	113	24	58	69	16	1	416	113	24	58	69	16	A	E	203	70	10	90.000	A
-Chris Speier	155	44	6	21	23	15	16	6631	1634	98	698	661	777	N	E	53	88	3	275.000	N
-Curt Wilkerson	236	56	0	27	15	11	4	1115	270	1	116	64	57	A	W	125	199	13	230.000	A
-Dave Anderson	216	53	1	31	15	22	4	926	210	9	118	69	114	N	W	73	152	11	225.000	N
-Doug Baker	24	3	0	1	0	2	3	159	28	0	20	12	9	A	W	80	4	0	NA	A
-Don Baylor	585	139	31	93	94	62	17	7546	1982	315	1141	1179	727	A	E	0	0	0	950.000	A
-Dann Bilardello	191	37	4	12	17	14	4	773	163	16	61	74	52	N	E	391	38	8	NA	N
-Daryl Boston	199	53	5	29	22	21	3	514	120	8	57	40	39	A	W	152	3	5	75.000	A
-Darnell Coles	521	142	20	67	86	45	4	815	205	22	99	103	78	A	E	107	242	23	105.000	A
-Dave Collins	419	113	1	44	27	44	12	4484	1231	32	612	344	422	A	E	211	2	1	NA	A
-Dave Concepcion	311	81	3	42	30	26	17	8247	2198	100	950	909	690	N	W	153	223	10	320.000	N
-Darren Daulton	138	31	8	18	21	38	3	244	53	12	33	32	55	N	E	244	21	4	NA	N
-Doug DeCinces	512	131	26	69	96	52	14	5347	1397	221	712	815	548	A	W	119	216	12	850.000	A
-Darrell Evans	507	122	29	78	85	91	18	7761	1947	347	1175	1152	1380	A	E	808	108	2	535.000	A
-Dwight Evans	529	137	26	86	97	97	15	6661	1785	291	1082	949	989	A	E	280	10	5	933.333	A
-Damaso Garcia	424	119	6	57	46	13	9	3651	1046	32	461	301	112	A	E	224	286	8	850.000	N
-Dan Gladden	351	97	4	55	29	39	4	1258	353	16	196	110	117	N	W	226	7	3	210.000	A
-Danny Heep	195	55	5	24	33	30	8	1313	338	25	144	149	153	N	E	83	2	1	NA	N
-Dave Henderson	388	103	15	59	47	39	6	2174	555	80	285	274	186	A	W	182	9	4	325.000	A
-Donnie Hill	339	96	4	37	29	23	4	1064	290	11	123	108	55	A	W	104	213	9	275.000	A
-Dave Kingman	561	118	35	70	94	33	16	6677	1575	442	901	1210	608	A	W	463	32	8	NA	A
-Davey Lopes	255	70	7	49	35	43	15	6311	1661	154	1019	608	820	N	E	51	54	8	450.000	N
-Don Mattingly	677	238	31	117	113	53	5	2223	737	93	349	401	171	A	E	1377	100	6	1975.000	A
-Darryl Motley	227	46	7	23	20	12	5	1325	324	44	156	158	67	A	W	92	2	2	NA	A
-Dale Murphy	614	163	29	89	83	75	11	5017	1388	266	813	822	617	N	W	303	6	6	1900.000	N
-Dwayne Murphy	329	83	9	50	39	56	9	3828	948	145	575	528	635	A	W	276	6	2	600.000	A
-Dave Parker	637	174	31	89	116	56	14	6727	2024	247	978	1093	495	N	W	278	9	9	1041.667	N
-Dan Pasqua	280	82	16	44	45	47	2	428	113	25	61	70	63	A	E	148	4	2	110.000	A
-Darrell Porter	155	41	12	21	29	22	16	5409	1338	181	746	805	875	A	W	165	9	1	260.000	A
-Dick Schofield	458	114	13	67	57	48	4	1350	298	28	160	123	122	A	W	246	389	18	475.000	A
-Don Slaught	314	83	13	39	46	16	5	1457	405	28	156	159	76	A	W	533	40	4	431.500	A
-Darryl Strawberry	475	123	27	76	93	72	4	1810	471	108	292	343	267	N	E	226	10	6	1220.000	N
-Dale Sveum	317	78	7	35	35	32	1	317	78	7	35	35	32	A	E	45	122	26	70.000	A
-Danny Tartabull	511	138	25	76	96	61	3	592	164	28	87	110	71	A	W	157	7	8	145.000	A
-Dickie Thon	278	69	3	24	21	29	8	2079	565	32	258	192	162	N	W	142	210	10	NA	N
-Denny Walling	382	119	13	54	58	36	12	2133	594	41	287	294	227	N	W	59	156	9	595.000	N
-Dave Winfield	565	148	24	90	104	77	14	7287	2083	305	1135	1234	791	A	E	292	9	5	1861.460	A
-Enos Cabell	277	71	2	27	29	14	15	5952	1647	60	753	596	259	N	W	360	32	5	NA	N
-Eric Davis	415	115	27	97	71	68	3	711	184	45	156	119	99	N	W	274	2	7	300.000	N
-Eddie Milner	424	110	15	70	47	36	7	2130	544	38	335	174	258	N	W	292	6	3	490.000	N
-Eddie Murray	495	151	17	61	84	78	10	5624	1679	275	884	1015	709	A	E	1045	88	13	2460.000	A
-Ernest Riles	524	132	9	69	47	54	2	972	260	14	123	92	90	A	E	212	327	20	NA	A
-Ed Romero	233	49	2	41	23	18	8	1350	336	7	166	122	106	A	E	102	132	10	375.000	A
-Ernie Whitt	395	106	16	48	56	35	10	2303	571	86	266	323	248	A	E	709	41	7	NA	A
-Fred Lynn	397	114	23	67	67	53	13	5589	1632	241	906	926	716	A	E	244	2	4	NA	A
-Floyd Rayford	210	37	8	15	19	15	6	994	244	36	107	114	53	A	E	40	115	15	NA	A
-Franklin Stubbs	420	95	23	55	58	37	3	646	139	31	77	77	61	N	W	206	10	7	NA	N
-Frank White	566	154	22	76	84	43	14	6100	1583	131	743	693	300	A	W	316	439	10	750.000	A
-George Bell	641	198	31	101	108	41	5	2129	610	92	297	319	117	A	E	269	17	10	1175.000	A
-Glenn Braggs	215	51	4	19	18	11	1	215	51	4	19	18	11	A	E	116	5	12	70.000	A
-George Brett	441	128	16	70	73	80	14	6675	2095	209	1072	1050	695	A	W	97	218	16	1500.000	A
-Greg Brock	325	76	16	33	52	37	5	1506	351	71	195	219	214	N	W	726	87	3	385.000	A
-Gary Carter	490	125	24	81	105	62	13	6063	1646	271	847	999	680	N	E	869	62	8	1925.571	N
-Glenn Davis	574	152	31	91	101	64	3	985	260	53	148	173	95	N	W	1253	111	11	215.000	N
-George Foster	284	64	14	30	42	24	18	7023	1925	348	986	1239	666	N	E	96	4	4	NA	N
-Gary Gaetti	596	171	34	91	108	52	6	2862	728	107	361	401	224	A	W	118	334	21	900.000	A
-Greg Gagne	472	118	12	63	54	30	4	793	187	14	102	80	50	A	W	228	377	26	155.000	A
-George Hendrick	283	77	14	45	47	26	16	6840	1910	259	915	1067	546	A	W	144	6	5	700.000	A
-Glenn Hubbard	408	94	4	42	36	66	9	3573	866	59	429	365	410	N	W	282	487	19	535.000	N
-Garth Iorg	327	85	3	30	44	20	8	2140	568	16	216	208	93	A	E	91	185	12	362.500	A
-Gary Matthews	370	96	21	49	46	60	15	6986	1972	231	1070	955	921	N	E	137	5	9	733.333	N
-Graig Nettles	354	77	16	36	55	41	20	8716	2172	384	1172	1267	1057	N	W	83	174	16	200.000	N
-Gary Pettis	539	139	5	93	58	69	5	1469	369	12	247	126	198	A	W	462	9	7	400.000	A
-Gary Redus	340	84	11	62	33	47	5	1516	376	42	284	141	219	N	E	185	8	4	400.000	A
-Garry Templeton	510	126	2	42	44	35	11	5562	1578	44	703	519	256	N	W	207	358	20	737.500	N
-Gorman Thomas	315	59	16	45	36	58	13	4677	1051	268	681	782	697	A	W	0	0	0	NA	A
-Greg Walker	282	78	13	37	51	29	5	1649	453	73	211	280	138	A	W	670	57	5	500.000	A
-Gary Ward	380	120	5	54	51	31	8	3118	900	92	444	419	240	A	W	237	8	1	600.000	A
-Glenn Wilson	584	158	15	70	84	42	5	2358	636	58	265	316	134	N	E	331	20	4	662.500	N
-Harold Baines	570	169	21	72	88	38	7	3754	1077	140	492	589	263	A	W	295	15	5	950.000	A
-Hubie Brooks	306	104	14	50	58	25	7	2954	822	55	313	377	187	N	E	116	222	15	750.000	N
-Howard Johnson	220	54	10	30	39	31	5	1185	299	40	145	154	128	N	E	50	136	20	297.500	N
-Hal McRae	278	70	7	22	37	18	18	7186	2081	190	935	1088	643	A	W	0	0	0	325.000	A
-Harold Reynolds	445	99	1	46	24	29	4	618	129	1	72	31	48	A	W	278	415	16	87.500	A
-Harry Spilman	143	39	5	18	30	15	9	639	151	16	80	97	61	N	W	138	15	1	175.000	N
-Herm Winningham	185	40	4	23	11	18	3	524	125	7	58	37	47	N	E	97	2	2	90.000	N
-Jesse Barfield	589	170	40	107	108	69	6	2325	634	128	371	376	238	A	E	368	20	3	1237.500	A
-Juan Beniquez	343	103	6	48	36	40	15	4338	1193	70	581	421	325	A	E	211	56	13	430.000	A
-Juan Bonilla	284	69	1	33	18	25	5	1407	361	6	139	98	111	A	E	122	140	5	NA	N
-John Cangelosi	438	103	2	65	32	71	2	440	103	2	67	32	71	A	W	276	7	9	100.000	N
-Jose Canseco	600	144	33	85	117	65	2	696	173	38	101	130	69	A	W	319	4	14	165.000	A
-Joe Carter	663	200	29	108	121	32	4	1447	404	57	210	222	68	A	E	241	8	6	250.000	A
-Jack Clark	232	55	9	34	23	45	12	4405	1213	194	702	705	625	N	E	623	35	3	1300.000	N
-Jose Cruz	479	133	10	48	72	55	17	7472	2147	153	980	1032	854	N	W	237	5	4	773.333	N
-Julio Cruz	209	45	0	38	19	42	10	3859	916	23	557	279	478	A	W	132	205	5	NA	A
-Jody Davis	528	132	21	61	74	41	6	2641	671	97	273	383	226	N	E	885	105	8	1008.333	N
-Jim Dwyer	160	39	8	18	31	22	14	2128	543	56	304	268	298	A	E	33	3	0	275.000	A
-Julio Franco	599	183	10	80	74	32	5	2482	715	27	330	326	158	A	E	231	374	18	775.000	A
-Jim Gantner	497	136	7	58	38	26	11	3871	1066	40	450	367	241	A	E	304	347	10	850.000	A
-Johnny Grubb	210	70	13	32	51	28	15	4040	1130	97	544	462	551	A	E	0	0	0	365.000	A
-Jerry Hairston	225	61	5	32	26	26	11	1568	408	25	202	185	257	A	W	132	9	0	NA	A
-Jack Howell	151	41	4	26	21	19	2	288	68	9	45	39	35	A	W	28	56	2	95.000	A
-John Kruk	278	86	4	33	38	45	1	278	86	4	33	38	45	N	W	102	4	2	110.000	N
-Jeffrey Leonard	341	95	6	48	42	20	10	2964	808	81	379	428	221	N	W	158	4	5	100.000	N
-Jim Morrison	537	147	23	58	88	47	10	2744	730	97	302	351	174	N	E	92	257	20	277.500	N
-John Moses	399	102	3	56	34	34	5	670	167	4	89	48	54	A	W	211	9	3	80.000	A
-Jerry Mumphrey	309	94	5	37	32	26	13	4618	1330	57	616	522	436	N	E	161	3	3	600.000	N
-Joe Orsulak	401	100	2	60	19	28	4	876	238	2	126	44	55	N	E	193	11	4	NA	N
-Jorge Orta	336	93	9	35	46	23	15	5779	1610	128	730	741	497	A	W	0	0	0	NA	A
-Jim Presley	616	163	27	83	107	32	3	1437	377	65	181	227	82	A	W	110	308	15	200.000	A
-Jamie Quirk	219	47	8	24	26	17	12	1188	286	23	100	125	63	A	W	260	58	4	NA	A
-Johnny Ray	579	174	7	67	78	58	6	3053	880	32	366	337	218	N	E	280	479	5	657.000	N
-Jeff Reed	165	39	2	13	9	16	3	196	44	2	18	10	18	A	W	332	19	2	75.000	N
-Jim Rice	618	200	20	98	110	62	13	7127	2163	351	1104	1289	564	A	E	330	16	8	2412.500	A
-Jerry Royster	257	66	5	31	26	32	14	3910	979	33	518	324	382	N	W	87	166	14	250.000	A
-John Russell	315	76	13	35	60	25	3	630	151	24	68	94	55	N	E	498	39	13	155.000	N
-Juan Samuel	591	157	16	90	78	26	4	2020	541	52	310	226	91	N	E	290	440	25	640.000	N
-John Shelby	404	92	11	54	49	18	6	1354	325	30	188	135	63	A	E	222	5	5	300.000	A
-Joel Skinner	315	73	5	23	37	16	4	450	108	6	38	46	28	A	W	227	15	3	110.000	A
-Jeff Stone	249	69	6	32	19	20	4	702	209	10	97	48	44	N	E	103	8	2	NA	N
-Jim Sundberg	429	91	12	41	42	57	13	5590	1397	83	578	579	644	A	W	686	46	4	825.000	N
-Jim Traber	212	54	13	28	44	18	2	233	59	13	31	46	20	A	E	243	23	5	NA	A
-Jose Uribe	453	101	3	46	43	61	3	948	218	6	96	72	91	N	W	249	444	16	195.000	N
-Jerry Willard	161	43	4	17	26	22	3	707	179	21	77	99	76	A	W	300	12	2	NA	A
-Joel Youngblood	184	47	5	20	28	18	11	3327	890	74	419	382	304	N	W	49	2	0	450.000	N
-Kevin Bass	591	184	20	83	79	38	5	1689	462	40	219	195	82	N	W	303	12	5	630.000	N
-Kal Daniels	181	58	6	34	23	22	1	181	58	6	34	23	22	N	W	88	0	3	86.500	N
-Kirk Gibson	441	118	28	84	86	68	8	2723	750	126	433	420	309	A	E	190	2	2	1300.000	A
-Ken Griffey	490	150	21	69	58	35	14	6126	1839	121	983	707	600	A	E	96	5	3	1000.000	N
-Keith Hernandez	551	171	13	94	83	94	13	6090	1840	128	969	900	917	N	E	1199	149	5	1800.000	N
-Kent Hrbek	550	147	29	85	91	71	6	2816	815	117	405	474	319	A	W	1218	104	10	1310.000	A
-Ken Landreaux	283	74	4	34	29	22	10	3919	1062	85	505	456	283	N	W	145	5	7	737.500	N
-Kevin McReynolds	560	161	26	89	96	66	4	1789	470	65	233	260	155	N	W	332	9	8	625.000	N
-Kevin Mitchell	328	91	12	51	43	33	2	342	94	12	51	44	33	N	E	145	59	8	125.000	N
-Keith Moreland	586	159	12	72	79	53	9	3082	880	83	363	477	295	N	E	181	13	4	1043.333	N
-Ken Oberkfell	503	136	5	62	48	83	10	3423	970	20	408	303	414	N	W	65	258	8	725.000	N
-Ken Phelps	344	85	24	69	64	88	7	911	214	64	150	156	187	A	W	0	0	0	300.000	A
-Kirby Puckett	680	223	31	119	96	34	3	1928	587	35	262	201	91	A	W	429	8	6	365.000	A
-Kurt Stillwell	279	64	0	31	26	30	1	279	64	0	31	26	30	N	W	107	205	16	75.000	N
-Leon Durham	484	127	20	66	65	67	7	3006	844	116	436	458	377	N	E	1231	80	7	1183.333	N
-Len Dykstra	431	127	8	77	45	58	2	667	187	9	117	64	88	N	E	283	8	3	202.500	N
-Larry Herndon	283	70	8	33	37	27	12	4479	1222	94	557	483	307	A	E	156	2	2	225.000	A
-Lee Lacy	491	141	11	77	47	37	15	4291	1240	84	615	430	340	A	E	239	8	2	525.000	A
-Len Matuszek	199	52	9	26	28	21	6	805	191	30	113	119	87	N	W	235	22	5	265.000	N
-Lloyd Moseby	589	149	21	89	86	64	7	3558	928	102	513	471	351	A	E	371	6	6	787.500	A
-Lance Parrish	327	84	22	53	62	38	10	4273	1123	212	577	700	334	A	E	483	48	6	800.000	N
-Larry Parrish	464	128	28	67	94	52	13	5829	1552	210	740	840	452	A	W	0	0	0	587.500	A
-Luis Rivera	166	34	0	20	13	17	1	166	34	0	20	13	17	N	E	64	119	9	NA	N
-Larry Sheets	338	92	18	42	60	21	3	682	185	36	88	112	50	A	E	0	0	0	145.000	A
-Lonnie Smith	508	146	8	80	44	46	9	3148	915	41	571	289	326	A	W	245	5	9	NA	A
-Lou Whitaker	584	157	20	95	73	63	10	4704	1320	93	724	522	576	A	E	276	421	11	420.000	A
-Mike Aldrete	216	54	2	27	25	33	1	216	54	2	27	25	33	N	W	317	36	1	75.000	N
-Marty Barrett	625	179	4	94	60	65	5	1696	476	12	216	163	166	A	E	303	450	14	575.000	A
-Mike Brown	243	53	4	18	26	27	4	853	228	23	101	110	76	N	E	107	3	3	NA	N
-Mike Davis	489	131	19	77	55	34	7	2051	549	62	300	263	153	A	W	310	9	9	780.000	A
-Mike Diaz	209	56	12	22	36	19	2	216	58	12	24	37	19	N	E	201	6	3	90.000	N
-Mariano Duncan	407	93	8	47	30	30	2	969	230	14	121	69	68	N	W	172	317	25	150.000	N
-Mike Easler	490	148	14	64	78	49	13	3400	1000	113	445	491	301	A	E	0	0	0	700.000	N
-Mike Fitzgerald	209	59	6	20	37	27	4	884	209	14	66	106	92	N	E	415	35	3	NA	N
-Mel Hall	442	131	18	68	77	33	6	1416	398	47	210	203	136	A	E	233	7	7	550.000	A
-Mickey Hatcher	317	88	3	40	32	19	8	2543	715	28	269	270	118	A	W	220	16	4	NA	A
-Mike Heath	288	65	8	30	36	27	9	2815	698	55	315	325	189	N	E	259	30	10	650.000	A
-Mike Kingery	209	54	3	25	14	12	1	209	54	3	25	14	12	A	W	102	6	3	68.000	A
-Mike LaValliere	303	71	3	18	30	36	3	344	76	3	20	36	45	N	E	468	47	6	100.000	N
-Mike Marshall	330	77	19	47	53	27	6	1928	516	90	247	288	161	N	W	149	8	6	670.000	N
-Mike Pagliarulo	504	120	28	71	71	54	3	1085	259	54	150	167	114	A	E	103	283	19	175.000	A
-Mark Salas	258	60	8	28	33	18	3	638	170	17	80	75	36	A	W	358	32	8	137.000	A
-Mike Schmidt	20	1	0	0	0	0	2	41	9	2	6	7	4	N	E	78	220	6	2127.333	N
-Mike Scioscia	374	94	5	36	26	62	7	1968	519	26	181	199	288	N	W	756	64	15	875.000	N
-Mickey Tettleton	211	43	10	26	35	39	3	498	116	14	59	55	78	A	W	463	32	8	120.000	A
-Milt Thompson	299	75	6	38	23	26	3	580	160	8	71	33	44	N	E	212	1	2	140.000	N
-Mitch Webster	576	167	8	89	49	57	4	822	232	19	132	83	79	N	E	325	12	8	210.000	N
-Mookie Wilson	381	110	9	61	45	32	7	3015	834	40	451	249	168	N	E	228	7	5	800.000	N
-Marvell Wynne	288	76	7	34	37	15	4	1644	408	16	198	120	113	N	W	203	3	3	240.000	N
-Mike Young	369	93	9	43	42	49	5	1258	323	54	181	177	157	A	E	149	1	6	350.000	A
-Nick Esasky	330	76	12	35	41	47	4	1367	326	55	167	198	167	N	W	512	30	5	NA	N
-Ozzie Guillen	547	137	2	58	47	12	2	1038	271	3	129	80	24	A	W	261	459	22	175.000	A
-Oddibe McDowell	572	152	18	105	49	65	2	978	249	36	168	91	101	A	W	325	13	3	200.000	A
-Omar Moreno	359	84	4	46	27	21	12	4992	1257	37	699	386	387	N	W	151	8	5	NA	N
-Ozzie Smith	514	144	0	67	54	79	9	4739	1169	13	583	374	528	N	E	229	453	15	1940.000	N
-Ozzie Virgil	359	80	15	45	48	63	7	1493	359	61	176	202	175	N	W	682	93	13	700.000	N
-Phil Bradley	526	163	12	88	50	77	4	1556	470	38	245	167	174	A	W	250	11	1	750.000	A
-Phil Garner	313	83	9	43	41	30	14	5885	1543	104	751	714	535	N	W	58	141	23	450.000	N
-Pete Incaviglia	540	135	30	82	88	55	1	540	135	30	82	88	55	A	W	157	6	14	172.000	A
-Paul Molitor	437	123	9	62	55	40	9	4139	1203	79	676	390	364	A	E	82	170	15	1260.000	A
-Pete O’Brien	551	160	23	86	90	87	5	2235	602	75	278	328	273	A	W	1224	115	11	NA	A
-Pete Rose	237	52	0	15	25	30	24	14053	4256	160	2165	1314	1566	N	W	523	43	6	750.000	N
-Pat Sheridan	236	56	6	41	19	21	5	1257	329	24	166	125	105	A	E	172	1	4	190.000	A
-Pat Tabler	473	154	6	61	48	29	6	1966	566	29	250	252	178	A	E	846	84	9	580.000	A
-Rafael Belliard	309	72	0	33	31	26	5	354	82	0	41	32	26	N	E	117	269	12	130.000	N
-Rick Burleson	271	77	5	35	29	33	12	4933	1358	48	630	435	403	A	W	62	90	3	450.000	A
-Randy Bush	357	96	7	50	45	39	5	1394	344	43	178	192	136	A	W	167	2	4	300.000	A
-Rick Cerone	216	56	4	22	18	15	12	2796	665	43	266	304	198	A	E	391	44	4	250.000	A
-Ron Cey	256	70	13	42	36	44	16	7058	1845	312	965	1128	990	N	E	41	118	8	1050.000	A
-Rob Deer	466	108	33	75	86	72	3	652	142	44	102	109	102	A	E	286	8	8	215.000	A
-Rick Dempsey	327	68	13	42	29	45	18	3949	939	78	438	380	466	A	E	659	53	7	400.000	A
-Rich Gedman	462	119	16	49	65	37	7	2131	583	69	244	288	150	A	E	866	65	6	NA	A
-Ron Hassey	341	110	9	45	49	46	9	2331	658	50	249	322	274	A	E	251	9	4	560.000	A
-Rickey Henderson	608	160	28	130	74	89	8	4071	1182	103	862	417	708	A	E	426	4	6	1670.000	A
-Reggie Jackson	419	101	18	65	58	92	20	9528	2510	548	1509	1659	1342	A	W	0	0	0	487.500	A
-Ricky Jones	33	6	0	2	4	7	1	33	6	0	2	4	7	A	W	205	5	4	NA	A
-Ron Kittle	376	82	21	42	60	35	5	1770	408	115	238	299	157	A	W	0	0	0	425.000	A
-Ray Knight	486	145	11	51	76	40	11	3967	1102	67	410	497	284	N	E	88	204	16	500.000	A
-Randy Kutcher	186	44	7	28	16	11	1	186	44	7	28	16	11	N	W	99	3	1	NA	N
-Rudy Law	307	80	1	42	36	29	7	2421	656	18	379	198	184	A	W	145	2	2	NA	A
-Rick Leach	246	76	5	35	39	13	6	912	234	12	102	96	80	A	E	44	0	1	250.000	A
-Rick Manning	205	52	8	31	27	17	12	5134	1323	56	643	445	459	A	E	155	3	2	400.000	A
-Rance Mulliniks	348	90	11	50	45	43	10	2288	614	43	295	273	269	A	E	60	176	6	450.000	A
-Ron Oester	523	135	8	52	44	52	9	3368	895	39	377	284	296	N	W	367	475	19	750.000	N
-Rey Quinones	312	68	2	32	22	24	1	312	68	2	32	22	24	A	E	86	150	15	70.000	A
-Rafael Ramirez	496	119	8	57	33	21	7	3358	882	36	365	280	165	N	W	155	371	29	875.000	N
-Ronn Reynolds	126	27	3	8	10	5	4	239	49	3	16	13	14	N	E	190	2	9	190.000	N
-Ron Roenicke	275	68	5	42	42	61	6	961	238	16	128	104	172	N	E	181	3	2	191.000	N
-Ryne Sandberg	627	178	14	68	76	46	6	3146	902	74	494	345	242	N	E	309	492	5	740.000	N
-Rafael Santana	394	86	1	38	28	36	4	1089	267	3	94	71	76	N	E	203	369	16	250.000	N
-Rick Schu	208	57	8	32	25	18	3	653	170	17	98	54	62	N	E	42	94	13	140.000	N
-Ruben Sierra	382	101	16	50	55	22	1	382	101	16	50	55	22	A	W	200	7	6	97.500	A
-Roy Smalley	459	113	20	59	57	68	12	5348	1369	155	713	660	735	A	W	0	0	0	740.000	A
-Robby Thompson	549	149	7	73	47	42	1	549	149	7	73	47	42	N	W	255	450	17	140.000	N
-Rob Wilfong	288	63	3	25	33	16	10	2682	667	38	315	259	204	A	W	135	257	7	341.667	A
-Reggie Williams	303	84	4	35	32	23	2	312	87	4	39	32	23	N	W	179	5	3	NA	N
-Robin Yount	522	163	9	82	46	62	13	7037	2019	153	1043	827	535	A	E	352	9	1	1000.000	A
-Steve Balboni	512	117	29	54	88	43	6	1750	412	100	204	276	155	A	W	1236	98	18	100.000	A
-Scott Bradley	220	66	5	20	28	13	3	290	80	5	27	31	15	A	W	281	21	3	90.000	A
-Sid Bream	522	140	16	73	77	60	4	730	185	22	93	106	86	N	E	1320	166	17	200.000	N
-Steve Buechele	461	112	18	54	54	35	2	680	160	24	76	75	49	A	W	111	226	11	135.000	A
-Shawon Dunston	581	145	17	66	68	21	2	831	210	21	106	86	40	N	E	320	465	32	155.000	N
-Scott Fletcher	530	159	3	82	50	47	6	1619	426	11	218	149	163	A	W	196	354	15	475.000	A
-Steve Garvey	557	142	21	58	81	23	18	8759	2583	271	1138	1299	478	N	W	1160	53	7	1450.000	N
-Steve Jeltz	439	96	0	44	36	65	4	711	148	1	68	56	99	N	E	229	406	22	150.000	N
-Steve Lombardozzi	453	103	8	53	33	52	2	507	123	8	63	39	58	A	W	289	407	6	105.000	A
-Spike Owen	528	122	1	67	45	51	4	1716	403	12	211	146	155	A	W	209	372	17	350.000	A
-Steve Sax	633	210	6	91	56	59	6	3070	872	19	420	230	274	N	W	367	432	16	90.000	N
-Tony Armas	16	2	0	1	0	0	2	28	4	0	1	0	0	A	E	247	4	8	NA	A
-Tony Bernazard	562	169	17	88	73	53	8	3181	841	61	450	342	373	A	E	351	442	17	530.000	A
-Tom Brookens	281	76	3	42	25	20	8	2658	657	48	324	300	179	A	E	106	144	7	341.667	A
-Tom Brunansky	593	152	23	69	75	53	6	2765	686	133	369	384	321	A	W	315	10	6	940.000	A
-Tony Fernandez	687	213	10	91	65	27	4	1518	448	15	196	137	89	A	E	294	445	13	350.000	A
-Tim Flannery	368	103	3	48	28	54	8	1897	493	9	207	162	198	N	W	209	246	3	326.667	N
-Tom Foley	263	70	1	26	23	30	4	888	220	9	83	82	86	N	E	81	147	4	250.000	N
-Tony Gwynn	642	211	14	107	59	52	5	2364	770	27	352	230	193	N	W	337	19	4	740.000	N
-Terry Harper	265	68	8	26	30	29	7	1337	339	32	135	163	128	N	W	92	5	3	425.000	A
-Toby Harrah	289	63	7	36	41	44	17	7402	1954	195	1115	919	1153	A	W	166	211	7	NA	A
-Tommy Herr	559	141	2	48	61	73	8	3162	874	16	421	349	359	N	E	352	414	9	925.000	N
-Tim Hulett	520	120	17	53	44	21	4	927	227	22	106	80	52	A	W	70	144	11	185.000	A
-Terry Kennedy	19	4	1	2	3	1	1	19	4	1	2	3	1	N	W	692	70	8	920.000	A
-Tito Landrum	205	43	2	24	17	20	7	854	219	12	105	99	71	N	E	131	6	1	286.667	N
-Tim Laudner	193	47	10	21	29	24	6	1136	256	42	129	139	106	A	W	299	13	5	245.000	A
-Tom O’Malley	181	46	1	19	18	17	5	937	238	9	88	95	104	A	E	37	98	9	NA	A
-Tom Paciorek	213	61	4	17	22	3	17	4061	1145	83	488	491	244	A	W	178	45	4	235.000	A
-Tony Pena	510	147	10	56	52	53	7	2872	821	63	307	340	174	N	E	810	99	18	1150.000	N
-Terry Pendleton	578	138	1	56	59	34	3	1399	357	7	149	161	87	N	E	133	371	20	160.000	N
-Tony Perez	200	51	2	14	29	25	23	9778	2732	379	1272	1652	925	N	W	398	29	7	NA	N
-Tony Phillips	441	113	5	76	52	76	5	1546	397	17	226	149	191	A	W	160	290	11	425.000	A
-Terry Puhl	172	42	3	17	14	15	10	4086	1150	57	579	363	406	N	W	65	0	0	900.000	N
-Tim Raines	580	194	9	91	62	78	8	3372	1028	48	604	314	469	N	E	270	13	6	NA	N
-Ted Simmons	127	32	4	14	25	12	19	8396	2402	242	1048	1348	819	N	W	167	18	6	500.000	N
-Tim Teufel	279	69	4	35	31	32	4	1359	355	31	180	148	158	N	E	133	173	9	277.500	N
-Tim Wallach	480	112	18	50	71	44	7	3031	771	110	338	406	239	N	E	94	270	16	750.000	N
-Vince Coleman	600	139	0	94	29	60	2	1236	309	1	201	69	110	N	E	300	12	9	160.000	N
-Von Hayes	610	186	19	107	98	74	6	2728	753	69	399	366	286	N	E	1182	96	13	1300.000	N
-Vance Law	360	81	5	37	44	37	7	2268	566	41	279	257	246	N	E	170	284	3	525.000	N
-Wally Backman	387	124	1	67	27	36	7	1775	506	6	272	125	194	N	E	186	290	17	550.000	N
-Wade Boggs	580	207	8	107	71	105	5	2778	978	32	474	322	417	A	E	121	267	19	1600.000	A
-Will Clark	408	117	11	66	41	34	1	408	117	11	66	41	34	N	W	942	72	11	120.000	N
-Wally Joyner	593	172	22	82	100	57	1	593	172	22	82	100	57	A	W	1222	139	15	165.000	A
-Wayne Krenchicki	221	53	2	21	23	22	8	1063	283	15	107	124	106	N	E	325	58	6	NA	N
-Willie McGee	497	127	7	65	48	37	5	2703	806	32	379	311	138	N	E	325	9	3	700.000	N
-Willie Randolph	492	136	5	76	50	94	12	5511	1511	39	897	451	875	A	E	313	381	20	875.000	A
-Wayne Tolleson	475	126	3	61	43	52	6	1700	433	7	217	93	146	A	W	37	113	7	385.000	A
-Willie Upshaw	573	144	9	85	60	78	8	3198	857	97	470	420	332	A	E	1314	131	12	960.000	A
-Willie Wilson	631	170	9	77	44	31	11	4908	1457

北京大学R语言教程(李东风)第42章： 正则化与惩罚回归

42.1 介绍

42.2 划分训练集和测试集

42.3 回归自变量选择

42.3.1 最优子集选择

42.3.2 逐步回归方法

42.3.3 预测根均方误差计算

42.3.4 用10折交叉验证方法选择最优子集

42.4 岭回归

42.4.1 用10折交叉验证选取调节参数

42.5 Lasso回归

42.5.1 用交叉验证估计调节参数

42.6 附录

42.6.1 Hitters数据

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北京大学R语言教程(李东风)第42章：正则化与惩罚回归