These methods are also called Shrinkage models. Regularization works well on high-dimensional data.
Ridge Regression keeps all variables as it shrinks parameters close to zero. Imposes penalty on the size of the coefficients to reduce the variance of the estimates.
Lasso Regression performs implicit variable selection as it shrinks some parameters exactly to zero.
#ridge regression
library(glmnet)## Warning: package 'glmnet' was built under R version 3.5.2## Loading required package: Matrix## Loading required package: foreach## Warning: package 'foreach' was built under R version 3.5.2## Loaded glmnet 2.0-16library(ISLR)## Warning: package 'ISLR' was built under R version 3.5.2#removing missing values from Hitters dataset of ISLR package
Hit <- na.omit(Hitters)
str(Hit)## 'data.frame': 263 obs. of 20 variables:
## $ AtBat : int 315 479 496 321 594 185 298 323 401 574 ...
## $ Hits : int 81 130 141 87 169 37 73 81 92 159 ...
## $ HmRun : int 7 18 20 10 4 1 0 6 17 21 ...
## $ Runs : int 24 66 65 39 74 23 24 26 49 107 ...
## $ RBI : int 38 72 78 42 51 8 24 32 66 75 ...
## $ Walks : int 39 76 37 30 35 21 7 8 65 59 ...
## $ Years : int 14 3 11 2 11 2 3 2 13 10 ...
## $ CAtBat : int 3449 1624 5628 396 4408 214 509 341 5206 4631 ...
## $ CHits : int 835 457 1575 101 1133 42 108 86 1332 1300 ...
## $ CHmRun : int 69 63 225 12 19 1 0 6 253 90 ...
## $ CRuns : int 321 224 828 48 501 30 41 32 784 702 ...
## $ CRBI : int 414 266 838 46 336 9 37 34 890 504 ...
## $ CWalks : int 375 263 354 33 194 24 12 8 866 488 ...
## $ League : Factor w/ 2 levels "A","N": 2 1 2 2 1 2 1 2 1 1 ...
## $ Division : Factor w/ 2 levels "E","W": 2 2 1 1 2 1 2 2 1 1 ...
## $ PutOuts : int 632 880 200 805 282 76 121 143 0 238 ...
## $ Assists : int 43 82 11 40 421 127 283 290 0 445 ...
## $ Errors : int 10 14 3 4 25 7 9 19 0 22 ...
## $ Salary : num 475 480 500 91.5 750 ...
## $ NewLeague: Factor w/ 2 levels "A","N": 2 1 2 2 1 1 1 2 1 1 ...
## - attr(*, "na.action")= 'omit' Named int 1 16 19 23 31 33 37 39 40 42 ...
## ..- attr(*, "names")= chr "-Andy Allanson" "-Billy Beane" "-Bruce Bochte" "-Bob Boone" ...#convert factor variables League & Division into numeric variables
Hit.data <- model.matrix( Salary ~ ., data = Hit)The command, model.matrix, converts factor variable into numeric variable but it by default creates intercept variable, hence exclude the intercept variable.
#creating data with only independent variables by excluding intercept variable
x <- Hit.data[,-1]
#creating data with only target variable.
y <- Hit$Salary#Ridge regression, alpha = 0
ridge.reg <- glmnet(x,y, alpha=0)
ridge.reg##
## Call: glmnet(x = x, y = y, alpha = 0)
##
## Df %Dev Lambda
## [1,] 19 6.185e-36 255300.00
## [2,] 19 1.161e-02 232600.00
## [3,] 19 1.272e-02 211900.00
## [4,] 19 1.393e-02 193100.00
## [5,] 19 1.525e-02 176000.00
## [6,] 19 1.670e-02 160300.00
## [7,] 19 1.827e-02 146100.00
## [8,] 19 1.999e-02 133100.00
## [9,] 19 2.186e-02 121300.00
## [10,] 19 2.391e-02 110500.00
## [11,] 19 2.613e-02 100700.00
## [12,] 19 2.855e-02 91740.00
## [13,] 19 3.118e-02 83590.00
## [14,] 19 3.404e-02 76170.00
## [15,] 19 3.714e-02 69400.00
## [16,] 19 4.050e-02 63240.00
## [17,] 19 4.414e-02 57620.00
## [18,] 19 4.808e-02 52500.00
## [19,] 19 5.233e-02 47840.00
## [20,] 19 5.692e-02 43590.00
## [21,] 19 6.186e-02 39710.00
## [22,] 19 6.717e-02 36190.00
## [23,] 19 7.287e-02 32970.00
## [24,] 19 7.897e-02 30040.00
## [25,] 19 8.549e-02 27370.00
## [26,] 19 9.244e-02 24940.00
## [27,] 19 9.983e-02 22730.00
## [28,] 19 1.077e-01 20710.00
## [29,] 19 1.160e-01 18870.00
## [30,] 19 1.247e-01 17190.00
## [31,] 19 1.339e-01 15660.00
## [32,] 19 1.435e-01 14270.00
## [33,] 19 1.536e-01 13000.00
## [34,] 19 1.640e-01 11850.00
## [35,] 19 1.748e-01 10800.00
## [36,] 19 1.860e-01 9837.00
## [37,] 19 1.974e-01 8963.00
## [38,] 19 2.091e-01 8167.00
## [39,] 19 2.210e-01 7442.00
## [40,] 19 2.330e-01 6781.00
## [41,] 19 2.451e-01 6178.00
## [42,] 19 2.572e-01 5629.00
## [43,] 19 2.692e-01 5129.00
## [44,] 19 2.812e-01 4674.00
## [45,] 19 2.929e-01 4258.00
## [46,] 19 3.045e-01 3880.00
## [47,] 19 3.157e-01 3535.00
## [48,] 19 3.266e-01 3221.00
## [49,] 19 3.371e-01 2935.00
## [50,] 19 3.471e-01 2674.00
## [51,] 19 3.568e-01 2437.00
## [52,] 19 3.659e-01 2220.00
## [53,] 19 3.746e-01 2023.00
## [54,] 19 3.827e-01 1843.00
## [55,] 19 3.904e-01 1680.00
## [56,] 19 3.976e-01 1530.00
## [57,] 19 4.043e-01 1394.00
## [58,] 19 4.106e-01 1271.00
## [59,] 19 4.164e-01 1158.00
## [60,] 19 4.218e-01 1055.00
## [61,] 19 4.269e-01 961.10
## [62,] 19 4.316e-01 875.70
## [63,] 19 4.359e-01 797.90
## [64,] 19 4.400e-01 727.10
## [65,] 19 4.437e-01 662.50
## [66,] 19 4.473e-01 603.60
## [67,] 19 4.506e-01 550.00
## [68,] 19 4.537e-01 501.10
## [69,] 19 4.566e-01 456.60
## [70,] 19 4.593e-01 416.00
## [71,] 19 4.619e-01 379.10
## [72,] 19 4.643e-01 345.40
## [73,] 19 4.667e-01 314.70
## [74,] 19 4.689e-01 286.80
## [75,] 19 4.710e-01 261.30
## [76,] 19 4.731e-01 238.10
## [77,] 19 4.751e-01 216.90
## [78,] 19 4.771e-01 197.70
## [79,] 19 4.789e-01 180.10
## [80,] 19 4.808e-01 164.10
## [81,] 19 4.826e-01 149.50
## [82,] 19 4.845e-01 136.20
## [83,] 19 4.863e-01 124.10
## [84,] 19 4.880e-01 113.10
## [85,] 19 4.898e-01 103.10
## [86,] 19 4.916e-01 93.90
## [87,] 19 4.934e-01 85.56
## [88,] 19 4.952e-01 77.96
## [89,] 19 4.970e-01 71.03
## [90,] 19 4.988e-01 64.72
## [91,] 19 5.006e-01 58.97
## [92,] 19 5.024e-01 53.73
## [93,] 19 5.042e-01 48.96
## [94,] 19 5.060e-01 44.61
## [95,] 19 5.077e-01 40.65
## [96,] 19 5.095e-01 37.04
## [97,] 19 5.113e-01 33.75
## [98,] 19 5.130e-01 30.75
## [99,] 19 5.148e-01 28.02
## [100,] 19 5.164e-01 25.53#Print lambda values
ridge.reg$lambda## [1] 255282.09651 232603.53866 211939.68139 193111.54424 175956.04690
## [6] 160324.59666 146081.80138 133104.29678 121279.67791 110505.52560
## [11] 100688.51928 91743.62874 83593.37763 76167.17236 69400.69070
## [16] 63235.32462 57617.67267 52499.07743 47835.20409 43585.65640
## [21] 39713.62682 36185.57767 32970.95069 30041.90230 27373.06250
## [26] 24941.31507 22725.59739 20706.71795 18867.19020 17191.08102
## [31] 15663.87277 14272.33748 13004.42236 11849.14532 10796.49991
## [36] 9837.36861 8963.44390 8167.15625 7441.60860 6780.51660
## [41] 6178.15419 5629.30400 5129.21215 4673.54708 4258.36204
## [46] 3880.06089 3535.36698 3221.29472 2935.12377 2674.37547
## [51] 2436.79132 2220.31350 2023.06697 1843.34327 1679.58574
## [56] 1530.37597 1394.42159 1270.54502 1157.67330 1054.82879
## [61] 961.12071 875.73740 797.93930 727.05257 662.46322
## [66] 603.61182 549.98861 501.12914 456.61020 416.04621
## [71] 379.08581 345.40887 314.72370 286.76452 261.28915
## [76] 238.07694 216.92684 197.65566 180.09647 164.09720
## [81] 149.51926 136.23638 124.13351 113.10583 103.05782
## [86] 93.90245 85.56042 77.95946 71.03376 64.72332
## [91] 58.97348 53.73443 48.96082 44.61127 40.64813
## [96] 37.03706 33.74679 30.74882 28.01718 25.52821#print all ridge coefficients
coef(ridge.reg)## 20 x 100 sparse Matrix of class "dgCMatrix"## [[ suppressing 100 column names 's0', 's1', 's2' ... ]]##
## (Intercept) 5.359259e+02 5.279266e+02 5.271582e+02 5.263173e+02
## AtBat 1.221172e-36 2.308443e-03 2.530141e-03 2.772795e-03
## Hits 4.429736e-36 8.386408e-03 9.193196e-03 1.007650e-02
## HmRun 1.784944e-35 3.369009e-02 3.692020e-02 4.045446e-02
## Runs 7.491019e-36 1.417283e-02 1.553533e-02 1.702684e-02
## RBI 7.912870e-36 1.495834e-02 1.639502e-02 1.796744e-02
## Walks 9.312961e-36 1.762818e-02 1.932377e-02 2.118006e-02
## Years 3.808598e-35 7.183685e-02 7.871918e-02 8.624857e-02
## CAtBat 1.048494e-37 1.980310e-04 2.170325e-04 2.378257e-04
## CHits 3.858759e-37 7.292122e-04 7.992257e-04 8.758489e-04
## CHmRun 2.910036e-36 5.498215e-03 6.026006e-03 6.603596e-03
## CRuns 7.741531e-37 1.462967e-03 1.603433e-03 1.757158e-03
## CRBI 7.989430e-37 1.509842e-03 1.654812e-03 1.813467e-03
## CWalks 8.452752e-37 1.595772e-03 1.748819e-03 1.916278e-03
## LeagueN -1.301217e-35 -2.302321e-02 -2.506692e-02 -2.726959e-02
## DivisionW -1.751460e-34 -3.339842e-01 -3.663713e-01 -4.018806e-01
## PutOuts 4.891197e-37 9.299203e-04 1.019803e-03 1.118291e-03
## Assists 7.989093e-38 1.517222e-04 1.663687e-04 1.824136e-04
## Errors -3.725027e-37 -7.293365e-04 -8.021006e-04 -8.822976e-04
## NewLeagueN -2.585026e-36 -3.616980e-03 -3.829358e-03 -4.034365e-03
##
## (Intercept) 5.253971e+02 5.243905e+02 5.232897e+02 5.220865e+02
## AtBat 3.038296e-03 3.328712e-03 3.646278e-03 3.993410e-03
## Hits 1.104332e-02 1.210125e-02 1.325858e-02 1.452424e-02
## HmRun 4.432017e-02 4.854702e-02 5.316710e-02 5.821503e-02
## Runs 1.865911e-02 2.044493e-02 2.239818e-02 2.453385e-02
## RBI 1.968794e-02 2.156991e-02 2.362784e-02 2.587740e-02
## Walks 2.321176e-02 2.543486e-02 2.786670e-02 3.052604e-02
## Years 9.448282e-02 1.034849e-01 1.133226e-01 1.240693e-01
## CAtBat 2.605725e-04 2.854488e-04 3.126446e-04 3.423652e-04
## CHits 9.596818e-04 1.051375e-03 1.151634e-03 1.261220e-03
## CHmRun 7.235506e-03 7.926636e-03 8.682286e-03 9.508181e-03
## CRuns 1.925348e-03 2.109308e-03 2.310454e-03 2.530310e-03
## CRBI 1.987051e-03 2.176912e-03 2.384511e-03 2.611422e-03
## CWalks 2.099451e-03 2.299751e-03 2.518700e-03 2.757945e-03
## LeagueN -2.963873e-02 -3.218103e-02 -3.490201e-02 -3.780550e-02
## DivisionW -4.408093e-01 -4.834822e-01 -5.302543e-01 -5.815128e-01
## PutOuts 1.226192e-03 1.344385e-03 1.473830e-03 1.615569e-03
## Assists 1.999873e-04 2.192317e-04 2.403011e-04 2.633633e-04
## Errors -9.707209e-04 -1.068256e-03 -1.175890e-03 -1.294729e-03
## NewLeagueN -4.225737e-03 -4.395524e-03 -4.533726e-03 -4.627857e-03
##
## (Intercept) 5.207716e+02 5.193354e+02 5.177674e+02 5.160563e+02
## AtBat 4.372716e-03 4.787004e-03 5.239292e-03 5.732818e-03
## Hits 1.590790e-02 1.742001e-02 1.907183e-02 2.087547e-02
## HmRun 6.372805e-02 6.974621e-02 7.631240e-02 8.347254e-02
## Runs 2.686815e-02 2.941856e-02 3.220389e-02 3.524437e-02
## RBI 2.833550e-02 3.102036e-02 3.395155e-02 3.715009e-02
## Walks 3.343318e-02 3.661001e-02 4.008014e-02 4.386895e-02
## Years 1.358036e-01 1.486101e-01 1.625792e-01 1.778076e-01
## CAtBat 3.748319e-04 4.102827e-04 4.489731e-04 4.911767e-04
## CHits 1.380952e-03 1.511717e-03 1.654462e-03 1.810209e-03
## CHmRun 1.041050e-02 1.139588e-02 1.247147e-02 1.364492e-02
## CRuns 2.770526e-03 3.032875e-03 3.319264e-03 3.631737e-03
## CRBI 2.859348e-03 3.130119e-03 3.425705e-03 3.748215e-03
## CWalks 3.019257e-03 3.304540e-03 3.615839e-03 3.955341e-03
## LeagueN -4.089313e-02 -4.416363e-02 -4.761196e-02 -5.122831e-02
## DivisionW -6.376806e-01 -6.992187e-01 -7.666298e-01 -8.404614e-01
## PutOuts 1.770736e-03 1.940563e-03 2.126387e-03 2.329658e-03
## Assists 2.886002e-04 3.162091e-04 3.464037e-04 3.794152e-04
## Errors -1.426011e-03 -1.571121e-03 -1.731619e-03 -1.909256e-03
## NewLeagueN -4.662425e-03 -4.618317e-03 -4.472067e-03 -4.194993e-03
##
## (Intercept) 5.141900e+02 5.121557e+02 5.099397e+02 5.075264e+02
## AtBat 6.271048e-03 6.857679e-03 7.496648e-03 8.192805e-03
## Hits 2.284394e-02 2.499119e-02 2.733212e-02 2.988423e-02
## HmRun 9.127555e-02 9.977347e-02 1.090214e-01 1.190848e-01
## Runs 3.856170e-02 4.217909e-02 4.612129e-02 5.041627e-02
## RBI 4.063847e-02 4.444070e-02 4.858233e-02 5.309210e-02
## Walks 4.800372e-02 5.251365e-02 5.742993e-02 6.278724e-02
## Years 1.943982e-01 2.124603e-01 2.321095e-01 2.534840e-01
## CAtBat 5.371862e-04 5.873132e-04 6.418886e-04 7.012864e-04
## CHits 1.980047e-03 2.165140e-03 2.366725e-03 2.586160e-03
## CHmRun 1.492443e-02 1.631872e-02 1.783708e-02 1.948950e-02
## CRuns 3.972483e-03 4.343837e-03 4.748283e-03 5.188464e-03
## CRBI 4.099909e-03 4.483199e-03 4.900651e-03 5.354977e-03
## CWalks 4.325379e-03 4.728439e-03 5.167155e-03 5.644308e-03
## LeagueN -5.499691e-02 -5.889454e-02 -6.288881e-02 -6.693503e-02
## DivisionW -9.213097e-01 -1.009823e+00 -1.106707e+00 -1.212726e+00
## PutOuts 2.551947e-03 2.794952e-03 3.060508e-03 3.350592e-03
## Assists 4.154929e-04 4.549059e-04 4.979437e-04 5.449142e-04
## Errors -2.106007e-03 -2.324099e-03 -2.566050e-03 -2.834762e-03
## NewLeagueN -3.752182e-03 -3.101295e-03 -2.191175e-03 -9.595900e-04
##
## (Intercept) 5.049021e+02 5.020499e+02 4.989526e+02 4.955925e+02
## AtBat 8.949413e-03 9.771646e-03 1.066438e-02 1.163271e-02
## Hits 3.266166e-02 3.568358e-02 3.896891e-02 4.253756e-02
## HmRun 1.300125e-01 1.418739e-01 1.547358e-01 1.686669e-01
## Runs 5.508922e-02 6.017095e-02 6.569256e-02 7.168666e-02
## RBI 5.799597e-02 6.332539e-02 6.911206e-02 7.538906e-02
## Walks 6.861852e-02 7.496229e-02 8.185807e-02 8.934739e-02
## Years 2.766833e-01 3.018534e-01 3.291327e-01 3.586635e-01
## CAtBat 7.658359e-04 8.359434e-04 9.120145e-04 9.944705e-04
## CHits 2.824757e-03 3.084012e-03 3.365454e-03 3.670681e-03
## CHmRun 2.128613e-02 2.323800e-02 2.535657e-02 2.765375e-02
## CRuns 5.667151e-03 6.187283e-03 6.751930e-03 7.364295e-03
## CRBI 5.849067e-03 6.385943e-03 6.968777e-03 7.600879e-03
## CWalks 6.162839e-03 6.725813e-03 7.336432e-03 7.998010e-03
## LeagueN -7.097770e-02 -7.494234e-02 -7.873553e-02 -8.224030e-02
## DivisionW -1.328711e+00 -1.455563e+00 -1.594257e+00 -1.745848e+00
## PutOuts 3.667340e-03 4.013041e-03 4.390155e-03 4.801315e-03
## Assists 5.961548e-04 6.520195e-04 7.128874e-04 7.791613e-04
## Errors -3.133391e-03 -3.465650e-03 -3.835727e-03 -4.248398e-03
## NewLeagueN 6.663878e-04 2.775656e-03 5.473674e-03 8.885759e-03
##
## (Intercept) 4.919511e+02 4.880093e+02 483.747661592 479.146314293
## AtBat 1.268189e-02 1.381736e-02 0.015044666 0.016369408
## Hits 4.641037e-02 5.060905e-02 0.055156096 0.060074668
## HmRun 1.837376e-01 2.000197e-01 0.217585175 0.236505646
## Runs 7.818726e-02 8.522959e-02 0.092849976 0.101085540
## RBI 8.219067e-02 8.955224e-02 0.097509906 0.106100310
## Walks 9.747368e-02 1.062822e-01 0.115819593 0.126134099
## Years 3.905913e-01 4.250635e-01 0.462227267 0.502228207
## CAtBat 1.083746e-03 1.180288e-03 0.001284549 0.001396984
## CHits 4.001345e-03 4.359152e-03 0.004745841 0.005163178
## CHmRun 3.014187e-02 3.283362e-02 0.035741917 0.038879848
## CRuns 8.027696e-03 8.745552e-03 0.009521354 0.010358643
## CRBI 8.285675e-03 9.026700e-03 0.009827563 0.010691924
## CWalks 8.713957e-03 9.487753e-03 0.010322913 0.011222956
## LeagueN -8.531179e-02 -8.777229e-02 -0.089405568 -0.089950537
## DivisionW -1.911474e+00 -2.092364e+00 -2.289842195 -2.505332119
## PutOuts 5.249336e-03 5.737218e-03 0.006268152 0.006845522
## Assists 8.512677e-04 9.296566e-04 0.001014800 0.001107192
## Errors -4.709121e-03 -5.224140e-03 -0.005800612 -0.006446752
## NewLeagueN 1.316032e-02 1.847250e-02 0.025028251 0.033068851
##
## (Intercept) 474.185313045 468.844820657 463.105377067 456.948207629
## AtBat 0.017797210 0.019333626 0.020984065 0.022753690
## Hits 0.065388435 0.071121398 0.077297668 0.083941203
## HmRun 0.256851123 0.278688786 0.302081508 0.327086165
## Runs 0.109973935 0.119552961 0.129860139 0.140932201
## RBI 0.115360215 0.125326067 0.136033474 0.147516599
## Walks 0.137274755 0.149291235 0.162233326 0.176150370
## Years 0.545207628 0.591299931 0.640629455 0.693306925
## CAtBat 0.001518047 0.001648183 0.001787822 0.001937367
## CHits 0.005612933 0.006096860 0.006616675 0.007174028
## CHmRun 0.042260488 0.045896759 0.049801233 0.053985923
## CRuns 0.011260970 0.012231856 0.013274742 0.014392938
## CRBI 0.011623456 0.012625803 0.013702528 0.014857059
## CWalks 0.012191351 0.013231466 0.014346505 0.015539437
## LeagueN -0.089094216 -0.086463942 -0.081618904 -0.074041051
## DivisionW -2.740362651 -2.996572699 -3.275715907 -3.579665051
## PutOuts 0.007472906 0.008154075 0.008892993 0.009693807
## Assists 0.001207344 0.001315787 0.001433062 0.001559722
## Errors -0.007172000 -0.007987217 -0.008904911 -0.009939494
## NewLeagueN 0.042875822 0.054776282 0.069148642 0.086428568
##
## (Intercept) 450.355572640 443.311156774 435.800495283 427.811431715
## AtBat 0.024647307 0.026669246 0.028823214 0.031112155
## Hits 0.091075510 0.098723311 0.106906168 0.115644080
## HmRun 0.353751746 0.382117244 0.412209365 0.444040072
## Runs 0.152804511 0.165510412 0.179080505 0.193541867
## RBI 0.159807478 0.172935255 0.186925343 0.201798518
## Walks 0.191090618 0.207100510 0.224223877 0.242501069
## Years 0.749425497 0.809056419 0.872244375 0.939002552
## CAtBat 0.002097190 0.002267620 0.002448929 0.002641325
## CHits 0.007770471 0.008407423 0.009086134 0.009807647
## CHmRun 0.058462034 0.063239703 0.068327710 0.073733174
## CRuns 0.015589554 0.016867437 0.018229095 0.019676620
## CRBI 0.016092623 0.017412178 0.018818334 0.020313277
## CWalks 0.016812912 0.018169177 0.019609978 0.021136460
## LeagueN -0.063125467 -0.048170386 -0.028367058 -0.002789760
## DivisionW -3.910416019 -4.270091300 -4.660942851 -5.085354236
## PutOuts 0.010560844 0.011498598 0.012511718 0.013604991
## Assists 0.001696321 0.001843413 0.002001540 0.002171225
## Errors -0.011107583 -0.012428341 -0.013923860 -0.015619598
## NewLeagueN 0.107115096 0.131776734 0.161057318 0.195681347
##
## (Intercept) 419.334599510 410.363917329 400.897085328 390.936067133
## AtBat 0.033538086 0.036101933 0.038803355 0.041640582
## Hits 0.124955049 0.134854634 0.145355502 0.156466974
## HmRun 0.477603991 0.512875750 0.549807288 0.588325228
## Runs 0.208917221 0.225224063 0.242473788 0.260670813
## RBI 0.217569965 0.234248292 0.251834532 0.270321167
## Walks 0.261968041 0.282655394 0.304587407 0.327781089
## Years 1.009307551 1.083094248 1.160250757 1.240613665
## CAtBat 0.002844938 0.003059805 0.003285864 0.003522938
## CHits 0.010572757 0.011381972 0.012235468 0.013133062
## CHmRun 0.079461241 0.085514771 0.091894035 0.098596432
## CRuns 0.021211604 0.022835064 0.024547359 0.026348120
## CRBI 0.021898684 0.023575640 0.025344562 0.027205123
## CWalks 0.022749060 0.024447405 0.026230202 0.028095145
## LeagueN 0.029613681 0.070030403 0.119789528 0.180367095
## DivisionW -5.545841909 -6.045055496 -6.585776918 -7.170918186
## PutOuts 0.014783328 0.016051734 0.017415295 0.018879147
## Assists 0.002352967 0.002547223 0.002754403 0.002974856
## Errors -0.017544869 -0.019733383 -0.022223838 -0.025060567
## NewLeagueN 0.236458460 0.284286613 0.340153513 0.405135721
##
## (Intercept) 380.492001784 369.568362783 358.186065513 346.367852244
## AtBat 0.044601862 0.047696486 0.050910899 0.054236024
## Hits 0.168180638 0.180522659 0.193478909 0.207041822
## HmRun 0.628287833 0.669638545 0.712181235 0.755720834
## Runs 0.279806688 0.299879722 0.320864302 0.342729951
## RBI 0.289690984 0.309918827 0.330965687 0.352782287
## Walks 0.352247366 0.377984110 0.404982592 0.433223258
## Years 1.323965458 1.410041670 1.498497785 1.588934652
## CAtBat 0.003770853 0.004028993 0.004296876 0.004573803
## CHits 0.014074659 0.015058492 0.016083473 0.017147736
## CHmRun 0.105619085 0.112948331 0.120573793 0.128479555
## CRuns 0.028236573 0.030210051 0.032265967 0.034400551
## CRBI 0.029156255 0.031195810 0.033320966 0.035527920
## CWalks 0.030038719 0.032056449 0.034142435 0.036289544
## LeagueN 0.253341924 0.340565448 0.443918334 0.565448488
## DivisionW -7.803531736 -8.486752347 -9.223862979 -10.018238760
## PutOuts 0.020448554 0.022128498 0.023924185 0.025840685
## Assists 0.003209162 0.003457036 0.003718803 0.003994483
## Errors -0.028289576 -0.031976068 -0.036182095 -0.040981539
## NewLeagueN 0.480374030 0.567144196 0.666736238 0.780506780
##
## (Intercept) 334.142203618 321.543327245 308.611009893 295.390327867
## AtBat 0.057660699 0.061171635 0.064753402 0.068388463
## Hits 0.221199117 0.235933811 0.251224367 0.267044985
## HmRun 0.800028666 0.844842352 0.889866431 0.934773736
## Runs 0.365436270 0.388932818 0.413159235 0.438045590
## RBI 0.375307668 0.398469148 0.422182538 0.446352642
## Walks 0.462676142 0.493300787 0.525046401 0.557852273
## Years 1.680890574 1.773842516 1.867208708 1.960352606
## CAtBat 0.004858935 0.005151298 0.005449792 0.005753196
## CHits 0.018248980 0.019384479 0.020551121 0.021745444
## CHmRun 0.136646242 0.145051152 0.153668504 0.162469784
## CRuns 0.036609149 0.038886258 0.041225592 0.043620173
## CRBI 0.037811979 0.040167593 0.042588428 0.045067462
## CWalks 0.038489350 0.040732151 0.043007010 0.045301835
## LeagueN 0.707320749 0.871795427 1.061200615 1.277898182
## DivisionW -10.873344870 -11.792720308 -12.779957814 -13.838679352
## PutOuts 0.027882981 0.030055940 0.032364288 0.034812580
## Assists 0.004283996 0.004587163 0.004903716 0.005233310
## Errors -0.046457523 -0.052703088 -0.059821784 -0.067928129
## NewLeagueN 0.909841612 1.056128713 1.220725354 1.404919249
##
## (Intercept) 281.931216326 268.287904623 254.518230732 240.682852868
## AtBat 0.072057246 0.075738253 0.079408204 0.083042199
## Hits 0.283366032 0.300154605 0.317375220 0.334990594
## HmRun 0.979207509 1.022784254 1.065097242 1.105720592
## Runs 0.463512982 0.489474364 0.515835606 0.542496737
## RBI 0.470874048 0.495632198 0.520504722 0.545363021
## Walks 0.591648458 0.626356704 0.661891620 0.698162047
## Years 2.052588075 2.143185625 2.231379423 2.316374816
## CAtBat 0.006060191 0.006369369 0.006679258 0.006988341
## CHits 0.022963700 0.024201921 0.025455999 0.026721778
## CHmRun 0.171424207 0.180499284 0.189661477 0.198876945
## CRuns 0.046062448 0.048544437 0.051057906 0.053594554
## CRBI 0.047597117 0.050169414 0.052776152 0.055409116
## CWalks 0.047603487 0.049897906 0.052170266 0.054405147
## LeagueN 1.524243647 1.802540410 2.114989214 2.463634043
## DivisionW -14.972506497 -16.185025086 -17.479743461 -18.860043740
## PutOuts 0.037405174 0.040146198 0.043039515 0.046088692
## Assists 0.005575545 0.005930000 0.006296277 0.006674057
## Errors -0.077147908 -0.087618226 -0.099487299 -0.112913894
## NewLeagueN 1.609883955 1.836629069 2.085946053 2.358350945
##
## (Intercept) 226.844379891 213.066444060 199.418113624 185.946732481
## AtBat 0.086613902 0.090095728 0.093426871 0.096634022
## Hits 0.352962515 0.371252755 0.389767263 0.408580477
## HmRun 1.144213851 1.180126954 1.212875007 1.242303538
## Runs 0.569353372 0.596298285 0.623229048 0.650047294
## RBI 0.570074066 0.594502389 0.618547529 0.642033634
## Walks 0.735072618 0.772525465 0.810467707 0.848737420
## Years 2.397356090 2.473494235 2.544170910 2.608433223
## CAtBat 0.007295083 0.007597952 0.007897059 0.008188531
## CHits 0.027995153 0.029272172 0.030554662 0.031829975
## CHmRun 0.208112349 0.217335715 0.226545984 0.235663247
## CRuns 0.056146219 0.058705097 0.061265846 0.063816873
## CRBI 0.058060281 0.060722036 0.063384832 0.066045116
## CWalks 0.056586702 0.058698830 0.060720300 0.062642350
## LeagueN 2.850306094 3.276567808 3.743295031 4.252099472
## DivisionW -20.329125634 -21.889942546 -23.545192292 -25.296959245
## PutOuts 0.049296951 0.052667119 0.056202373 0.059902888
## Assists 0.007063169 0.007463678 0.007879196 0.008305300
## Errors -0.128066380 -0.145121335 -0.164203267 -0.185603401
## NewLeagueN 2.654025549 2.972759111 3.313773161 3.676189320
##
## (Intercept) 172.720339542 159.796625704 147.229940279 135.070668968
## AtBat 0.099662970 0.102483884 0.105066772 0.107381506
## Hits 0.427613302 0.446840518 0.466242140 0.485804000
## HmRun 1.267838795 1.289060569 1.305566982 1.316978600
## Runs 0.676642658 0.702915317 0.728770397 0.754119185
## RBI 0.664847505 0.686866068 0.707973643 0.728063747
## Walks 0.887265878 0.925962427 0.964741607 1.003524587
## Years 2.665510662 2.714623467 2.755005485 2.785904099
## CAtBat 0.008472029 0.008746278 0.009010094 0.009262397
## CHits 0.033099124 0.034359576 0.035609341 0.036847056
## CHmRun 0.244686353 0.253594870 0.262373079 0.271010647
## CRuns 0.066354566 0.068874010 0.071371505 0.073844717
## CRBI 0.068696462 0.071334608 0.073956619 0.076561000
## CWalks 0.064445823 0.066114944 0.067634293 0.068988790
## LeagueN 4.803143578 5.396487429 6.031739769 6.708052714
## DivisionW -27.147059491 -29.096663728 -31.146274915 -33.295635026
## PutOuts 0.063770572 0.067805862 0.072008376 0.076376801
## Assists 0.008745578 0.009201998 0.009677269 0.010174962
## Errors -0.209468234 -0.235989097 -0.265353793 -0.297742859
## NewLeagueN 4.058198317 4.457548059 4.871395868 5.296302663
##
## (Intercept) 123.364687896 112.152893255 101.470804106 91.34824416
## AtBat 0.109397777 0.111084965 0.112411969 0.11334700
## Hits 0.505518240 0.525383722 0.545406468 0.56560020
## HmRun 1.322942515 1.323136611 1.317274357 1.30511042
## Runs 0.778880335 0.802981121 0.826358772 0.84896187
## RBI 0.747040927 0.764822642 0.781341162 0.79654539
## Walks 1.042240653 1.080828728 1.119238917 1.15743392
## Years 2.806580422 2.816310761 2.814390631 2.80014228
## CAtBat 0.009502226 0.009728753 0.009941306 0.01013939
## CHits 0.038072060 0.039284480 0.040485305 0.04167644
## CHmRun 0.279503233 0.287852974 0.296068770 0.30416623
## CRuns 0.076292787 0.078716386 0.081117693 0.08350026
## CRBI 0.079147743 0.081718296 0.084275441 0.08682308
## CWalks 0.070163610 0.071144012 0.071915095 0.07246150
## LeagueN 7.424132485 8.178267752 8.968376868 9.79207432
## DivisionW -35.543634366 -37.888226969 -40.326355895 -42.85389265
## PutOuts 0.080908779 0.085600796 0.090448054 0.09544436
## Assists 0.010699624 0.011256892 0.011853584 0.01249776
## Errors -0.333325577 -0.372255918 -0.414668745 -0.46067665
## NewLeagueN 5.728243831 6.162638103 6.594395070 7.01798093
##
## (Intercept) 81.80912119 72.87133790 64.54687932 56.84211884
## AtBat 0.11385741 0.11390958 0.11346895 0.11250016
## Hits 0.58598709 0.60659877 0.62747760 0.64867794
## HmRun 1.28644724 1.26114229 1.22911527 1.19035371
## Runs 0.87075166 0.89170313 0.91180536 0.93106082
## RBI 0.81040237 0.82289826 0.83403818 0.84384486
## Walks 1.19539015 1.23309825 1.27056270 1.30780054
## Years 2.77292594 2.73215348 2.67730126 2.60791691
## CAtBat 0.01032270 0.01049116 0.01064486 0.01078408
## CHits 0.04286072 0.04404185 0.04522429 0.04641299
## CHmRun 0.31216718 0.32009858 0.32799094 0.33587641
## CRuns 0.08586877 0.08822869 0.09058589 0.09294633
## CRBI 0.08936592 0.09190920 0.09445842 0.09701925
## CWalks 0.07276715 0.07281501 0.07258712 0.07206472
## LeagueN 10.64675536 11.52969556 12.43815958 13.36951042
## DivisionW -45.46559519 -48.15508912 -50.91487654 -53.73637654
## PutOuts 0.10058200 0.10585167 0.11124236 0.11674136
## Assists 0.01319872 0.01396697 0.01481403 0.01575224
## Errors -0.51036787 -0.56380551 -0.62102829 -0.68205215
## NewLeagueN 7.42750076 7.81679432 8.17954077 8.50936701
##
## (Intercept) 49.66071253 43.19811550 37.35435452 32.11979366
## AtBat 0.11100896 0.10892860 0.10621244 0.10282090
## Hits 0.67065961 0.69282433 0.71552124 0.73884964
## HmRun 1.14705569 1.09527613 1.03694724 0.97229778
## Runs 0.94979681 0.96731179 0.98399499 0.99988070
## RBI 0.85258323 0.85970919 0.86561767 0.87037858
## Walks 1.34484630 1.38155943 1.41814249 1.45465328
## Years 2.53231022 2.43283529 2.31716859 2.18495419
## CAtBat 0.01091417 0.01102056 0.01111261 0.01119052
## CHits 0.04758007 0.04877693 0.04999657 0.05124466
## CHmRun 0.34338316 0.35123985 0.35920440 0.36730879
## CRuns 0.09521714 0.09760090 0.10001535 0.10246862
## CRBI 0.09956753 0.10219578 0.10486285 0.10757714
## CWalks 0.07127246 0.07014231 0.06866213 0.06681191
## LeagueN 14.32669056 15.29747611 16.28421405 17.28537184
## DivisionW -56.60831617 -59.52349944 -62.46922340 -65.43365772
## PutOuts 0.12232144 0.12799058 0.13372132 0.13949521
## Assists 0.01678133 0.01793012 0.01920903 0.02063139
## Errors -0.74749993 -0.81627639 -0.88872621 -0.96476647
## NewLeagueN 8.80224343 9.04758049 9.24139657 9.37812039
##
## (Intercept) 27.48122385 23.35433066 19.88127229 16.95362750
## AtBat 0.09871046 0.09395554 0.08832442 0.08182542
## Hits 0.76291617 0.78834250 0.81428191 0.84131229
## HmRun 0.90161601 0.82742595 0.74548748 0.65860519
## Runs 1.01501167 1.02948109 1.04302309 1.05593994
## RBI 0.87407730 0.87674606 0.87837874 0.87925295
## Walks 1.49116764 1.52742252 1.56402050 1.60093596
## Years 2.03595690 1.87852192 1.69351221 1.49123197
## CAtBat 0.01125484 0.01129989 0.01132964 0.01134725
## CHits 0.05252781 0.05377450 0.05512854 0.05653905
## CHmRun 0.37558468 0.38342242 0.39208576 0.40100461
## CRuns 0.10496767 0.10742946 0.11007712 0.11279509
## CRBI 0.11034553 0.11322294 0.11617340 0.11920092
## CWalks 0.06457003 0.06205832 0.05900961 0.05550229
## LeagueN 18.29984048 19.33207105 20.37082664 21.42171930
## DivisionW -68.40455396 -71.36780965 -74.31450740 -77.23051972
## PutOuts 0.14529280 0.15108124 0.15686653 0.16261472
## Assists 0.02221041 0.02392623 0.02584713 0.02796086
## Errors -1.04428666 -1.12800772 -1.21408593 -1.30312467
## NewLeagueN 9.45260305 9.46236555 9.39876389 9.26063415
##
## (Intercept) 14.55643468 12.63381060 11.26303439 10.35569021
## AtBat 0.07441067 0.06620425 0.05681886 0.04633830
## Hits 0.86958000 0.89973720 0.93088191 0.96376522
## HmRun 0.56719135 0.47343820 0.37390139 0.27163150
## Runs 1.06824527 1.07970774 1.09069163 1.10118079
## RBI 0.87942459 0.87868071 0.87757438 0.87606196
## Walks 1.63826451 1.67551222 1.71399148 1.75331031
## Years 1.27103489 1.03942211 0.78041117 0.50454902
## CAtBat 0.01135160 0.01132298 0.01129281 0.01124891
## CHits 0.05801169 0.05944063 0.06105328 0.06274116
## CHmRun 0.41020878 0.41896545 0.42884413 0.43896753
## CRuns 0.11560090 0.11847102 0.12153596 0.12471202
## CRBI 0.12231947 0.12568482 0.12905186 0.13253839
## CWalks 0.05151658 0.04725138 0.04225513 0.03672947
## LeagueN 22.48506168 23.56451409 24.65329547 25.75710221
## DivisionW -80.10405700 -82.92243236 -85.67844190 -88.36043501
## PutOuts 0.16830574 0.17391052 0.17943174 0.18483877
## Assists 0.03027682 0.03276148 0.03550485 0.03847012
## Errors -1.39490474 -1.48995450 -1.58633587 -1.68470903
## NewLeagueN 9.04508554 8.75201941 8.37544584 7.91725605
##
## (Intercept) 9.88559249 9.88291317 10.275516171 11.089425330
## AtBat 0.03490091 0.02203798 0.008088528 -0.007441506
## Hits 0.99895354 1.03565510 1.075046172 1.116358483
## HmRun 0.16809480 0.06098960 -0.046053680 -0.155150648
## Runs 1.11069309 1.12003913 1.128360907 1.136603508
## RBI 0.87378443 0.87157484 0.868722584 0.866089829
## Walks 1.79291526 1.83449226 1.876642472 1.921230858
## Years 0.21529405 -0.09893987 -0.425145088 -0.774801522
## CAtBat 0.01116099 0.01108217 0.010952415 0.010834527
## CHits 0.06439189 0.06627447 0.068113575 0.070207840
## CHmRun 0.44865817 0.45953755 0.469845084 0.481333855
## CRuns 0.12805488 0.13156927 0.135302716 0.139206329
## CRBI 0.13636588 0.14012418 0.144283368 0.148340356
## CWalks 0.03088644 0.02419398 0.017175670 0.009253631
## LeagueN 26.87720380 28.01101288 29.161665128 30.327721487
## DivisionW -90.95870286 -93.46888824 -95.881062917 -98.193701089
## PutOuts 0.19011040 0.19524711 0.200221069 0.205034904
## Assists 0.04162273 0.04505173 0.048671621 0.052568534
## Errors -1.78525046 -1.88631880 -1.988817461 -2.091238621
## NewLeagueN 7.37938543 6.75760342 6.058926704 5.279536552
##
## (Intercept) 1.226775e+01 1.381737e+01 15.70484847 17.90970899
## AtBat -2.419845e-02 -4.269835e-02 -0.06256388 -0.08413294
## Hits 1.160708e+00 1.207444e+00 1.25757731 1.31086031
## HmRun -2.631285e-01 -3.713857e-01 -0.47748576 -0.58154451
## Runs 1.143697e+00 1.150774e+00 1.15651180 1.16164000
## RBI 8.628801e-01 8.599947e-01 0.85653655 0.85311884
## Walks 1.966769e+00 2.015222e+00 2.06502273 2.11744797
## Years -1.136139e+00 -1.517933e+00 -1.91097346 -2.31727741
## CAtBat 1.065732e-02 1.049592e-02 0.01026407 0.01001251
## CHits 7.226562e-02 7.459755e-02 0.07690091 0.07936051
## CHmRun 4.922058e-01 5.041878e-01 0.51551046 0.52708730
## CRuns 1.434083e-01 1.477624e-01 0.15251315 0.15751258
## CRBI 1.528435e-01 1.572011e-01 0.16204498 0.16699038
## CWalks 9.709713e-04 -8.252262e-03 -0.01787305 -0.02821953
## LeagueN 3.150933e+01 3.270677e+01 33.91642380 35.13804585
## DivisionW -1.003989e+02 -1.024976e+02 -104.48392249 -106.35913149
## PutOuts 2.096675e-01 2.141219e-01 0.21838467 0.22245636
## Assists 5.665892e-02 6.101918e-02 0.06557221 0.07035289
## Errors -2.194190e+00 -2.296448e+00 -2.39828780 -2.49896136
## NewLeagueN 4.428640e+00 3.503041e+00 2.51422817 1.46288664
##
## (Intercept) 2.042569e+01 2.322775e+01 2.628708e+01 2.959266e+01
## AtBat -1.076199e-01 -1.326867e-01 -1.596436e-01 -1.886732e-01
## Hits 1.367370e+00 1.427859e+00 1.492213e+00 1.560599e+00
## HmRun -6.831310e-01 -7.806457e-01 -8.736810e-01 -9.615323e-01
## Runs 1.166491e+00 1.169713e+00 1.172052e+00 1.173737e+00
## RBI 8.498208e-01 8.459378e-01 8.419675e-01 8.378370e-01
## Walks 2.173219e+00 2.231001e+00 2.291941e+00 2.356685e+00
## Years -2.739338e+00 -3.169167e+00 -3.606777e+00 -4.053798e+00
## CAtBat 9.765985e-03 9.437049e-03 9.081117e-03 8.726137e-03
## CHits 8.208094e-02 8.479218e-02 8.766490e-02 9.080053e-02
## CHmRun 5.394969e-01 5.511666e-01 5.629019e-01 5.752954e-01
## CRuns 1.627131e-01 1.684242e-01 1.744269e-01 1.806427e-01
## CRBI 1.718068e-01 1.771098e-01 1.824907e-01 1.876852e-01
## CWalks -3.948045e-02 -5.119556e-02 -6.363472e-02 -7.696108e-02
## LeagueN 3.637129e+01 3.760851e+01 3.884910e+01 4.009234e+01
## DivisionW -1.081248e+02 -1.097775e+02 -1.113213e+02 -1.127601e+02
## PutOuts 2.263377e-01 2.300240e-01 2.335185e-01 2.368253e-01
## Assists 7.537864e-02 8.058875e-02 8.600338e-02 9.163105e-02
## Errors -2.597976e+00 -2.695216e+00 -2.790180e+00 -2.882509e+00
## NewLeagueN 3.517716e-01 -8.053887e-01 -2.006618e+00 -3.248330e+00
##
## (Intercept) 3.313791e+01 3.687610e+01 4.080513e+01 4.489952e+01
## AtBat -2.194352e-01 -2.521934e-01 -2.869253e-01 -3.236275e-01
## Hits 1.633540e+00 1.711071e+00 1.793412e+00 1.880772e+00
## HmRun -1.043506e+00 -1.118487e+00 -1.186104e+00 -1.245525e+00
## Runs 1.173320e+00 1.171549e+00 1.167987e+00 1.162431e+00
## RBI 8.329465e-01 8.276589e-01 8.216738e-01 8.148652e-01
## Walks 2.423953e+00 2.494675e+00 2.568828e+00 2.646408e+00
## Years -4.504274e+00 -4.955202e+00 -5.406734e+00 -5.855617e+00
## CAtBat 8.270074e-03 7.776529e-03 7.226223e-03 6.615397e-03
## CHits 9.394021e-02 9.722801e-02 1.006546e-01 1.042079e-01
## CHmRun 5.868901e-01 5.983159e-01 6.096034e-01 6.206532e-01
## CRuns 1.875131e-01 1.947169e-01 2.023873e-01 2.105312e-01
## CRBI 1.933475e-01 1.990595e-01 2.048436e-01 2.106881e-01
## CWalks -9.077116e-02 -1.052462e-01 -1.203776e-01 -1.361228e-01
## LeagueN 4.132647e+01 4.255111e+01 4.376094e+01 4.495100e+01
## DivisionW -1.140917e+02 -1.153218e+02 -1.164536e+02 -1.174905e+02
## PutOuts 2.399476e-01 2.428889e-01 2.456566e-01 2.482565e-01
## Assists 9.742921e-02 1.033987e-01 1.095343e-01 1.158216e-01
## Errors -2.971818e+00 -3.057862e+00 -3.140289e+00 -3.218841e+00
## NewLeagueN -4.514032e+00 -5.801048e+00 -7.100963e+00 -8.405519e+00
##
## (Intercept) 4.913958e+01 5.350440e+01 5.797287e+01 6.252374e+01
## AtBat -3.622792e-01 -4.028403e-01 -4.452517e-01 -4.894343e-01
## Hits 1.973313e+00 2.071175e+00 2.174458e+00 2.283221e+00
## HmRun -1.296148e+00 -1.337393e+00 -1.368755e+00 -1.389807e+00
## Runs 1.154631e+00 1.144325e+00 1.131239e+00 1.115101e+00
## RBI 8.070830e-01 7.981701e-01 7.879725e-01 7.763423e-01
## Walks 2.727395e+00 2.811719e+00 2.899267e+00 2.989885e+00
## Years -6.299563e+00 -6.736089e+00 -7.162738e+00 -7.577105e+00
## CAtBat 5.938497e-03 5.189262e-03 4.361308e-03 3.447937e-03
## CHits 1.078833e-01 1.116714e-01 1.155620e-01 1.195433e-01
## CHmRun 6.314485e-01 6.419584e-01 6.521595e-01 6.620336e-01
## CRuns 2.191741e-01 2.283429e-01 2.380637e-01 2.483633e-01
## CRBI 2.165690e-01 2.224670e-01 2.283610e-01 2.342300e-01
## CWalks -1.524506e-01 -1.693195e-01 -1.866829e-01 -2.044893e-01
## LeagueN 4.611629e+01 4.725194e+01 4.835330e+01 4.941609e+01
## DivisionW -1.184363e+02 -1.192949e+02 -1.200702e+02 -1.207661e+02
## PutOuts 2.506956e-01 2.529811e-01 2.551205e-01 2.571211e-01
## Assists 1.222480e-01 1.287998e-01 1.354622e-01 1.422202e-01
## Errors -3.293277e+00 -3.363384e+00 -3.428986e+00 -3.489938e+00
## NewLeagueN -9.706721e+00 -1.099665e+01 -1.226767e+01 -1.351256e+01
##
## (Intercept) 6.713573e+01 71.78758387 7.645824e+01 8.112693e+01
## AtBat -5.352890e-01 -0.58269657 -6.315180e-01 -6.815959e-01
## Hits 2.397469e+00 2.51715271 2.642160e+00 2.772312e+00
## HmRun -1.400213e+00 -1.39973429 -1.388233e+00 -1.365680e+00
## Runs 1.095639e+00 1.07259572 1.045729e+00 1.014826e+00
## RBI 7.631426e-01 0.74825248 7.315713e-01 7.130225e-01
## Walks 3.083376e+00 3.17950552 3.278001e+00 3.378558e+00
## Years -7.976859e+00 -8.35976896 -8.723734e+00 -9.066800e+00
## CAtBat 2.442244e-03 0.00133718 1.256355e-04 -1.199478e-03
## CHits 1.236025e-01 0.12772556 1.318975e-01 1.361029e-01
## CHmRun 6.715658e-01 0.68074413 6.895578e-01 6.979958e-01
## CRuns 2.592688e-01 0.27080732 2.830055e-01 2.958896e-01
## CRBI 2.400538e-01 0.24581306 2.514905e-01 2.570711e-01
## CWalks -2.226824e-01 -0.24120197 -2.599851e-01 -2.789666e-01
## LeagueN 5.043646e+01 51.41107137 5.233720e+01 5.321272e+01
## DivisionW -1.213866e+02 -121.93563374 -1.224170e+02 -1.228345e+02
## PutOuts 2.589908e-01 0.26073685 2.623667e-01 2.638876e-01
## Assists 1.490576e-01 0.15595798 1.629044e-01 1.698796e-01
## Errors -3.546133e+00 -3.59749877 -3.644002e+00 -3.685645e+00
## NewLeagueN -1.472457e+01 -15.89754176 -1.702598e+01 -1.810510e+01#print 50th ridge coefficient
coef(ridge.reg)[,50]## (Intercept) AtBat Hits HmRun Runs
## 213.066444060 0.090095728 0.371252755 1.180126954 0.596298285
## RBI Walks Years CAtBat CHits
## 0.594502389 0.772525465 2.473494235 0.007597952 0.029272172
## CHmRun CRuns CRBI CWalks LeagueN
## 0.217335715 0.058705097 0.060722036 0.058698830 3.276567808
## DivisionW PutOuts Assists Errors NewLeagueN
## -21.889942546 0.052667119 0.007463678 -0.145121335 2.972759111summary(ridge.reg)## Length Class Mode
## a0 100 -none- numeric
## beta 1900 dgCMatrix S4
## df 100 -none- numeric
## dim 2 -none- numeric
## lambda 100 -none- numeric
## dev.ratio 100 -none- numeric
## nulldev 1 -none- numeric
## npasses 1 -none- numeric
## jerr 1 -none- numeric
## offset 1 -none- logical
## call 4 -none- call
## nobs 1 -none- numeric*Use cross-validation method to choose minimum lambda value. cv.glmnet by default selects minimum lambda value which is the best fit model.
#choose minimum lambda value with cross-validation
cv.ridge.reg <- cv.glmnet(x,y,alpha=0)
cv.ridge.reg## $lambda
## [1] 255282.09651 232603.53866 211939.68139 193111.54424 175956.04690
## [6] 160324.59666 146081.80138 133104.29678 121279.67791 110505.52560
## [11] 100688.51928 91743.62874 83593.37763 76167.17236 69400.69070
## [16] 63235.32462 57617.67267 52499.07743 47835.20409 43585.65640
## [21] 39713.62682 36185.57767 32970.95069 30041.90230 27373.06250
## [26] 24941.31507 22725.59739 20706.71795 18867.19020 17191.08102
## [31] 15663.87277 14272.33748 13004.42236 11849.14532 10796.49991
## [36] 9837.36861 8963.44390 8167.15625 7441.60860 6780.51660
## [41] 6178.15419 5629.30400 5129.21215 4673.54708 4258.36204
## [46] 3880.06089 3535.36698 3221.29472 2935.12377 2674.37547
## [51] 2436.79132 2220.31350 2023.06697 1843.34327 1679.58574
## [56] 1530.37597 1394.42159 1270.54502 1157.67330 1054.82879
## [61] 961.12071 875.73740 797.93930 727.05257 662.46322
## [66] 603.61182 549.98861 501.12914 456.61020 416.04621
## [71] 379.08581 345.40887 314.72370 286.76452 261.28915
## [76] 238.07694 216.92684 197.65566 180.09647 164.09720
## [81] 149.51926 136.23638 124.13351 113.10583 103.05782
## [86] 93.90245 85.56042 77.95946 71.03376 64.72332
## [91] 58.97348 53.73443 48.96082 44.61127 40.64813
## [96] 37.03706 33.74679 30.74882 28.01718
##
## $cvm
## [1] 204417.8 202915.5 202334.9 202022.1 201758.3 201470.1 201155.7
## [8] 200812.6 200438.6 200031.2 199587.5 199104.8 198580.0 198010.1
## [15] 197391.7 196721.2 195995.5 195210.7 194363.2 193449.5 192465.8
## [22] 191408.6 190274.7 189060.8 187764.1 186382.4 184913.6 183356.5
## [29] 181710.7 179976.4 178155.0 176248.7 174261.1 172196.8 170061.8
## [36] 167863.5 165610.4 163311.9 160979.0 158623.4 156257.7 153894.9
## [43] 151548.4 149231.5 146957.1 144737.8 142585.2 140509.6 138520.2
## [50] 136624.6 134828.9 133137.1 131552.6 130076.8 128709.4 127449.0
## [57] 126293.0 125238.0 124279.7 123413.3 122633.5 121935.0 121312.2
## [64] 120758.9 120269.9 119844.6 119474.6 119153.2 118875.9 118640.3
## [71] 118444.4 118281.5 118146.5 118042.3 117957.0 117892.3 117846.3
## [78] 117810.9 117788.2 117773.3 117763.6 117757.2 117754.7 117746.8
## [85] 117742.4 117726.2 117714.4 117684.2 117662.6 117614.2 117577.1
## [92] 117508.4 117455.9 117368.2 117301.1 117194.1 117113.4 116992.2
## [99] 116898.9
##
## $cvsd
## [1] 24426.33 24542.70 24330.92 24276.44 24265.46 24253.44 24240.31
## [8] 24225.95 24210.27 24193.15 24174.46 24154.07 24131.85 24107.64
## [15] 24081.29 24052.64 24021.47 23987.63 23950.94 23911.19 23868.18
## [22] 23821.70 23771.55 23717.54 23659.46 23597.15 23530.43 23459.17
## [29] 23383.24 23302.58 23217.16 23127.00 23032.18 22932.87 22829.30
## [36] 22721.80 22610.76 22496.76 22380.36 22262.28 22143.33 22024.38
## [43] 21906.37 21790.29 21677.14 21567.91 21463.56 21364.97 21272.95
## [50] 21188.15 21110.98 21041.80 20980.94 20928.34 20883.81 20847.01
## [57] 20817.49 20794.65 20777.84 20766.33 20759.38 20756.25 20756.25
## [64] 20758.65 20763.18 20769.97 20777.22 20784.76 20792.44 20800.28
## [71] 20808.89 20816.85 20824.58 20832.12 20839.16 20846.16 20853.17
## [78] 20859.90 20866.30 20872.99 20879.29 20885.62 20892.47 20898.35
## [85] 20905.38 20911.34 20918.08 20924.05 20931.61 20937.64 20945.20
## [92] 20951.52 20959.85 20966.92 20976.79 20984.68 20996.35 21006.71
## [99] 21020.09
##
## $cvup
## [1] 228844.2 227458.2 226665.8 226298.6 226023.7 225723.6 225396.0
## [8] 225038.6 224648.9 224224.3 223761.9 223258.9 222711.9 222117.8
## [15] 221473.0 220773.9 220016.9 219198.3 218314.2 217360.7 216334.0
## [22] 215230.3 214046.2 212778.3 211423.6 209979.5 208444.0 206815.7
## [29] 205094.0 203279.0 201372.1 199375.7 197293.2 195129.6 192891.1
## [36] 190585.3 188221.1 185808.6 183359.3 180885.7 178401.0 175919.3
## [43] 173454.8 171021.8 168634.3 166305.8 164048.7 161874.5 159793.1
## [50] 157812.8 155939.9 154178.9 152533.6 151005.2 149593.2 148296.0
## [57] 147110.5 146032.7 145057.6 144179.6 143392.9 142691.2 142068.4
## [64] 141517.5 141033.0 140614.6 140251.8 139938.0 139668.3 139440.6
## [71] 139253.3 139098.4 138971.1 138874.5 138796.1 138738.4 138699.5
## [78] 138670.8 138654.5 138646.3 138642.9 138642.8 138647.2 138645.1
## [85] 138647.8 138637.5 138632.5 138608.3 138594.2 138551.8 138522.3
## [92] 138459.9 138415.7 138335.1 138277.9 138178.8 138109.8 137998.9
## [99] 137919.0
##
## $cvlo
## [1] 179991.50 178372.77 178003.93 177745.68 177492.81 177216.70 176915.37
## [8] 176586.68 176228.37 175838.01 175413.03 174950.71 174448.19 173902.47
## [15] 173310.38 172668.61 171974.00 171223.04 170412.29 169538.29 168597.61
## [22] 167586.94 166503.12 165343.23 164104.67 162785.21 161383.15 159897.38
## [29] 158327.48 156673.86 154937.83 153121.70 151228.87 149263.89 147232.49
## [36] 145141.67 142999.60 140815.12 138598.61 136361.12 134114.38 131870.56
## [43] 129642.04 127441.19 125280.00 123169.93 121121.60 119144.60 117247.23
## [50] 115436.50 113717.96 112095.26 110571.69 109148.48 107825.61 106602.00
## [57] 105475.56 104443.39 103501.89 102646.96 101874.13 101178.73 100555.94
## [64] 100000.21 99506.69 99074.67 98697.37 98368.45 98083.43 97840.07
## [71] 97635.53 97464.65 97321.92 97210.22 97117.81 97046.10 96993.18
## [78] 96950.97 96921.86 96900.28 96884.30 96871.58 96862.28 96848.42
## [85] 96837.01 96814.87 96796.29 96760.17 96731.03 96676.55 96631.85
## [92] 96556.88 96496.03 96401.26 96324.29 96209.45 96117.07 95985.45
## [99] 95878.80
##
## $nzero
## s0 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15 s16 s17
## 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19
## s18 s19 s20 s21 s22 s23 s24 s25 s26 s27 s28 s29 s30 s31 s32 s33 s34 s35
## 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19
## s36 s37 s38 s39 s40 s41 s42 s43 s44 s45 s46 s47 s48 s49 s50 s51 s52 s53
## 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19
## s54 s55 s56 s57 s58 s59 s60 s61 s62 s63 s64 s65 s66 s67 s68 s69 s70 s71
## 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19
## s72 s73 s74 s75 s76 s77 s78 s79 s80 s81 s82 s83 s84 s85 s86 s87 s88 s89
## 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19
## s90 s91 s92 s93 s94 s95 s96 s97 s98
## 19 19 19 19 19 19 19 19 19
##
## $name
## mse
## "Mean-Squared Error"
##
## $glmnet.fit
##
## Call: glmnet(x = x, y = y, alpha = 0)
##
## Df %Dev Lambda
## [1,] 19 6.185e-36 255300.00
## [2,] 19 1.161e-02 232600.00
## [3,] 19 1.272e-02 211900.00
## [4,] 19 1.393e-02 193100.00
## [5,] 19 1.525e-02 176000.00
## [6,] 19 1.670e-02 160300.00
## [7,] 19 1.827e-02 146100.00
## [8,] 19 1.999e-02 133100.00
## [9,] 19 2.186e-02 121300.00
## [10,] 19 2.391e-02 110500.00
## [11,] 19 2.613e-02 100700.00
## [12,] 19 2.855e-02 91740.00
## [13,] 19 3.118e-02 83590.00
## [14,] 19 3.404e-02 76170.00
## [15,] 19 3.714e-02 69400.00
## [16,] 19 4.050e-02 63240.00
## [17,] 19 4.414e-02 57620.00
## [18,] 19 4.808e-02 52500.00
## [19,] 19 5.233e-02 47840.00
## [20,] 19 5.692e-02 43590.00
## [21,] 19 6.186e-02 39710.00
## [22,] 19 6.717e-02 36190.00
## [23,] 19 7.287e-02 32970.00
## [24,] 19 7.897e-02 30040.00
## [25,] 19 8.549e-02 27370.00
## [26,] 19 9.244e-02 24940.00
## [27,] 19 9.983e-02 22730.00
## [28,] 19 1.077e-01 20710.00
## [29,] 19 1.160e-01 18870.00
## [30,] 19 1.247e-01 17190.00
## [31,] 19 1.339e-01 15660.00
## [32,] 19 1.435e-01 14270.00
## [33,] 19 1.536e-01 13000.00
## [34,] 19 1.640e-01 11850.00
## [35,] 19 1.748e-01 10800.00
## [36,] 19 1.860e-01 9837.00
## [37,] 19 1.974e-01 8963.00
## [38,] 19 2.091e-01 8167.00
## [39,] 19 2.210e-01 7442.00
## [40,] 19 2.330e-01 6781.00
## [41,] 19 2.451e-01 6178.00
## [42,] 19 2.572e-01 5629.00
## [43,] 19 2.692e-01 5129.00
## [44,] 19 2.812e-01 4674.00
## [45,] 19 2.929e-01 4258.00
## [46,] 19 3.045e-01 3880.00
## [47,] 19 3.157e-01 3535.00
## [48,] 19 3.266e-01 3221.00
## [49,] 19 3.371e-01 2935.00
## [50,] 19 3.471e-01 2674.00
## [51,] 19 3.568e-01 2437.00
## [52,] 19 3.659e-01 2220.00
## [53,] 19 3.746e-01 2023.00
## [54,] 19 3.827e-01 1843.00
## [55,] 19 3.904e-01 1680.00
## [56,] 19 3.976e-01 1530.00
## [57,] 19 4.043e-01 1394.00
## [58,] 19 4.106e-01 1271.00
## [59,] 19 4.164e-01 1158.00
## [60,] 19 4.218e-01 1055.00
## [61,] 19 4.269e-01 961.10
## [62,] 19 4.316e-01 875.70
## [63,] 19 4.359e-01 797.90
## [64,] 19 4.400e-01 727.10
## [65,] 19 4.437e-01 662.50
## [66,] 19 4.473e-01 603.60
## [67,] 19 4.506e-01 550.00
## [68,] 19 4.537e-01 501.10
## [69,] 19 4.566e-01 456.60
## [70,] 19 4.593e-01 416.00
## [71,] 19 4.619e-01 379.10
## [72,] 19 4.643e-01 345.40
## [73,] 19 4.667e-01 314.70
## [74,] 19 4.689e-01 286.80
## [75,] 19 4.710e-01 261.30
## [76,] 19 4.731e-01 238.10
## [77,] 19 4.751e-01 216.90
## [78,] 19 4.771e-01 197.70
## [79,] 19 4.789e-01 180.10
## [80,] 19 4.808e-01 164.10
## [81,] 19 4.826e-01 149.50
## [82,] 19 4.845e-01 136.20
## [83,] 19 4.863e-01 124.10
## [84,] 19 4.880e-01 113.10
## [85,] 19 4.898e-01 103.10
## [86,] 19 4.916e-01 93.90
## [87,] 19 4.934e-01 85.56
## [88,] 19 4.952e-01 77.96
## [89,] 19 4.970e-01 71.03
## [90,] 19 4.988e-01 64.72
## [91,] 19 5.006e-01 58.97
## [92,] 19 5.024e-01 53.73
## [93,] 19 5.042e-01 48.96
## [94,] 19 5.060e-01 44.61
## [95,] 19 5.077e-01 40.65
## [96,] 19 5.095e-01 37.04
## [97,] 19 5.113e-01 33.75
## [98,] 19 5.130e-01 30.75
## [99,] 19 5.148e-01 28.02
## [100,] 19 5.164e-01 25.53
##
## $lambda.min
## [1] 28.01718
##
## $lambda.1se
## [1] 2674.375
##
## attr(,"class")
## [1] "cv.glmnet"By default it specifies the minimum lambda value, that is 28.01718, 99th observation. This lamda would give minimum error when model is built.
#print minimum lambda coefficients
coef(cv.ridge.reg, s="lambda.min")## 20 x 1 sparse Matrix of class "dgCMatrix"
## 1
## (Intercept) 7.645824e+01
## AtBat -6.315180e-01
## Hits 2.642160e+00
## HmRun -1.388233e+00
## Runs 1.045729e+00
## RBI 7.315713e-01
## Walks 3.278001e+00
## Years -8.723734e+00
## CAtBat 1.256355e-04
## CHits 1.318975e-01
## CHmRun 6.895578e-01
## CRuns 2.830055e-01
## CRBI 2.514905e-01
## CWalks -2.599851e-01
## LeagueN 5.233720e+01
## DivisionW -1.224170e+02
## PutOuts 2.623667e-01
## Assists 1.629044e-01
## Errors -3.644002e+00
## NewLeagueN -1.702598e+01#Prediction with minimum lambda values / coefficeints
#x is like training dataset
pred.ridge <- predict(cv.ridge.reg, newx = x, s="lambda.min")
head(pred.ridge)## 1
## -Alan Ashby 445.3238
## -Alvin Davis 673.7466
## -Andre Dawson 1053.6486
## -Andres Galarraga 518.8045
## -Alfredo Griffin 543.7695
## -Al Newman 215.5220#model evaluation
library(DMwR)## Warning: package 'DMwR' was built under R version 3.5.3## Loading required package: lattice## Loading required package: gridround(regr.eval(y, pred.ridge), 3)## mae mse rmse mape
## 217.535 98376.082 313.650 0.700Least Absolute Shrinkage and Selection Operator. Lasso usually results into sparse models that are easier to interpret
#For LASSO regression use library(glmnet)
#lasso regression, alpha=1
library(glmnet)
lasso.reg <- glmnet(x,y, alpha=1)
#print
lasso.reg##
## Call: glmnet(x = x, y = y, alpha = 1)
##
## Df %Dev Lambda
## [1,] 0 0.00000 255.3000
## [2,] 1 0.05458 232.6000
## [3,] 2 0.10050 211.9000
## [4,] 2 0.13910 193.1000
## [5,] 3 0.17610 176.0000
## [6,] 4 0.21910 160.3000
## [7,] 4 0.25680 146.1000
## [8,] 4 0.28820 133.1000
## [9,] 4 0.31420 121.3000
## [10,] 4 0.33580 110.5000
## [11,] 5 0.35410 100.7000
## [12,] 5 0.37300 91.7400
## [13,] 5 0.38870 83.5900
## [14,] 5 0.40170 76.1700
## [15,] 6 0.41510 69.4000
## [16,] 6 0.42730 63.2400
## [17,] 6 0.43750 57.6200
## [18,] 6 0.44600 52.5000
## [19,] 6 0.45300 47.8400
## [20,] 6 0.45880 43.5900
## [21,] 6 0.46360 39.7100
## [22,] 6 0.46760 36.1900
## [23,] 6 0.47100 32.9700
## [24,] 6 0.47370 30.0400
## [25,] 6 0.47600 27.3700
## [26,] 6 0.47790 24.9400
## [27,] 6 0.47950 22.7300
## [28,] 6 0.48080 20.7100
## [29,] 6 0.48190 18.8700
## [30,] 7 0.48290 17.1900
## [31,] 7 0.48400 15.6600
## [32,] 7 0.48480 14.2700
## [33,] 9 0.48570 13.0000
## [34,] 9 0.48660 11.8500
## [35,] 9 0.48730 10.8000
## [36,] 9 0.48790 9.8370
## [37,] 9 0.48840 8.9630
## [38,] 11 0.49120 8.1670
## [39,] 11 0.49600 7.4420
## [40,] 12 0.50270 6.7810
## [41,] 12 0.50820 6.1780
## [42,] 13 0.51320 5.6290
## [43,] 13 0.51770 5.1290
## [44,] 13 0.52140 4.6740
## [45,] 13 0.52450 4.2580
## [46,] 13 0.52700 3.8800
## [47,] 13 0.52910 3.5350
## [48,] 13 0.53090 3.2210
## [49,] 13 0.53240 2.9350
## [50,] 13 0.53360 2.6740
## [51,] 13 0.53460 2.4370
## [52,] 14 0.53550 2.2200
## [53,] 15 0.53620 2.0230
## [54,] 15 0.53720 1.8430
## [55,] 17 0.53840 1.6800
## [56,] 17 0.53960 1.5300
## [57,] 17 0.54060 1.3940
## [58,] 17 0.54150 1.2710
## [59,] 17 0.54220 1.1580
## [60,] 17 0.54280 1.0550
## [61,] 17 0.54320 0.9611
## [62,] 17 0.54360 0.8757
## [63,] 17 0.54400 0.7979
## [64,] 17 0.54430 0.7271
## [65,] 18 0.54450 0.6625
## [66,] 18 0.54480 0.6036
## [67,] 18 0.54490 0.5500
## [68,] 17 0.54510 0.5011
## [69,] 18 0.54520 0.4566
## [70,] 18 0.54530 0.4160
## [71,] 18 0.54540 0.3791
## [72,] 18 0.54550 0.3454
## [73,] 18 0.54560 0.3147
## [74,] 18 0.54570 0.2868
## [75,] 18 0.54570 0.2613
## [76,] 18 0.54580 0.2381
## [77,] 18 0.54580 0.2169
## [78,] 18 0.54590 0.1977
## [79,] 18 0.54590 0.1801
## [80,] 19 0.54590 0.1641Lasso regression model by default builds 80 models.
#lasso regression with cross-validation method
cv.lasso.reg <- cv.glmnet(x,y,alpha=1)
cv.lasso.reg## $lambda
## [1] 255.2820965 232.6035387 211.9396814 193.1115442 175.9560469
## [6] 160.3245967 146.0818014 133.1042968 121.2796779 110.5055256
## [11] 100.6885193 91.7436287 83.5933776 76.1671724 69.4006907
## [16] 63.2353246 57.6176727 52.4990774 47.8352041 43.5856564
## [21] 39.7136268 36.1855777 32.9709507 30.0419023 27.3730625
## [26] 24.9413151 22.7255974 20.7067180 18.8671902 17.1910810
## [31] 15.6638728 14.2723375 13.0044224 11.8491453 10.7964999
## [36] 9.8373686 8.9634439 8.1671562 7.4416086 6.7805166
## [41] 6.1781542 5.6293040 5.1292122 4.6735471 4.2583620
## [46] 3.8800609 3.5353670 3.2212947 2.9351238 2.6743755
## [51] 2.4367913 2.2203135 2.0230670 1.8433433 1.6795857
## [56] 1.5303760 1.3944216 1.2705450 1.1576733 1.0548288
## [61] 0.9611207 0.8757374 0.7979393 0.7270526 0.6624632
## [66] 0.6036118 0.5499886 0.5011291 0.4566102 0.4160462
## [71] 0.3790858 0.3454089 0.3147237 0.2867645
##
## $cvm
## [1] 204141.5 198391.4 189258.1 181621.5 174584.1 167236.1 160514.7
## [8] 154680.5 149857.8 145828.7 142469.6 139537.4 136854.0 134207.8
## [15] 131731.4 129607.7 127743.8 126005.2 124564.1 123377.2 122403.9
## [22] 121610.5 120959.0 120436.2 120008.0 119690.0 119510.6 119398.8
## [29] 119360.9 119365.6 119398.9 119429.5 119633.4 119889.5 120208.7
## [36] 120569.0 120979.1 121312.0 121243.6 120694.6 120115.5 119434.9
## [43] 118800.1 118300.1 117909.7 117631.5 117406.3 117203.8 117068.3
## [50] 117108.7 117409.3 117813.9 118220.8 118592.5 118961.0 119334.0
## [57] 119594.2 119820.8 120068.2 120260.8 120434.0 120572.3 120690.0
## [64] 120798.6 120912.4 121043.6 121140.1 121252.1 121348.8 121440.6
## [71] 121536.6 121613.3 121698.7 121789.3
##
## $cvsd
## [1] 26018.21 26007.95 24689.32 23663.50 22970.88 22483.70 22095.20
## [8] 21980.08 22030.28 22195.52 22458.58 22794.74 23167.27 23477.22
## [15] 23804.11 24119.66 24351.51 24436.09 24514.27 24590.28 24664.09
## [22] 24735.34 24804.35 24870.76 24933.49 24994.25 25057.37 25113.59
## [29] 25156.59 25193.58 25228.53 25258.30 25314.33 25373.93 25420.79
## [36] 25447.27 25457.54 25473.97 25498.59 25462.87 25352.94 25135.18
## [43] 24921.34 24720.94 24538.77 24368.19 24206.42 24039.73 23890.84
## [50] 23763.09 23668.20 23590.63 23507.62 23417.28 23343.95 23283.38
## [57] 23233.19 23170.80 23091.91 23018.23 22955.07 22910.30 22875.33
## [64] 22840.86 22808.12 22778.80 22756.62 22736.00 22716.05 22700.87
## [71] 22687.66 22685.90 22682.26 22675.34
##
## $cvup
## [1] 230159.7 224399.3 213947.4 205285.0 197555.0 189719.8 182609.9
## [8] 176660.6 171888.1 168024.2 164928.1 162332.1 160021.2 157685.0
## [15] 155535.5 153727.3 152095.3 150441.3 149078.4 147967.5 147067.9
## [22] 146345.9 145763.4 145306.9 144941.5 144684.2 144568.0 144512.4
## [29] 144517.5 144559.2 144627.4 144687.8 144947.7 145263.4 145629.5
## [36] 146016.3 146436.6 146785.9 146742.2 146157.4 145468.4 144570.1
## [43] 143721.4 143021.1 142448.4 141999.7 141612.7 141243.5 140959.1
## [50] 140871.8 141077.5 141404.6 141728.4 142009.8 142305.0 142617.4
## [57] 142827.4 142991.6 143160.1 143279.0 143389.1 143482.6 143565.4
## [64] 143639.5 143720.6 143822.4 143896.8 143988.1 144064.9 144141.5
## [71] 144224.3 144299.2 144381.0 144464.6
##
## $cvlo
## [1] 178123.26 172383.40 164568.73 157957.96 151613.22 144752.40 138419.53
## [8] 132700.45 127827.55 123633.14 120010.97 116742.62 113686.68 110730.59
## [15] 107927.33 105488.03 103392.31 101569.11 100049.85 98786.94 97739.76
## [22] 96875.18 96154.65 95565.40 95074.54 94695.71 94453.25 94285.20
## [29] 94204.32 94172.04 94170.32 94171.22 94319.07 94515.58 94787.94
## [36] 95121.71 95521.54 95838.01 95745.03 95231.71 94762.54 94299.74
## [43] 93878.77 93579.18 93370.88 93263.31 93199.89 93164.08 93177.43
## [50] 93345.66 93741.12 94223.32 94713.20 95175.26 95617.08 96050.60
## [57] 96361.01 96649.97 96976.26 97242.57 97478.95 97662.04 97814.70
## [64] 97957.75 98104.32 98264.82 98383.51 98516.12 98632.77 98739.73
## [71] 98848.97 98927.41 99016.45 99113.94
##
## $nzero
## s0 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15 s16 s17
## 0 1 2 2 3 4 4 4 4 4 5 5 5 5 6 6 6 6
## s18 s19 s20 s21 s22 s23 s24 s25 s26 s27 s28 s29 s30 s31 s32 s33 s34 s35
## 6 6 6 6 6 6 6 6 6 6 6 7 7 7 9 9 9 9
## s36 s37 s38 s39 s40 s41 s42 s43 s44 s45 s46 s47 s48 s49 s50 s51 s52 s53
## 9 11 11 12 12 13 13 13 13 13 13 13 13 13 13 14 15 15
## s54 s55 s56 s57 s58 s59 s60 s61 s62 s63 s64 s65 s66 s67 s68 s69 s70 s71
## 17 17 17 17 17 17 17 17 17 17 18 18 18 17 18 18 18 18
## s72 s73
## 18 18
##
## $name
## mse
## "Mean-Squared Error"
##
## $glmnet.fit
##
## Call: glmnet(x = x, y = y, alpha = 1)
##
## Df %Dev Lambda
## [1,] 0 0.00000 255.3000
## [2,] 1 0.05458 232.6000
## [3,] 2 0.10050 211.9000
## [4,] 2 0.13910 193.1000
## [5,] 3 0.17610 176.0000
## [6,] 4 0.21910 160.3000
## [7,] 4 0.25680 146.1000
## [8,] 4 0.28820 133.1000
## [9,] 4 0.31420 121.3000
## [10,] 4 0.33580 110.5000
## [11,] 5 0.35410 100.7000
## [12,] 5 0.37300 91.7400
## [13,] 5 0.38870 83.5900
## [14,] 5 0.40170 76.1700
## [15,] 6 0.41510 69.4000
## [16,] 6 0.42730 63.2400
## [17,] 6 0.43750 57.6200
## [18,] 6 0.44600 52.5000
## [19,] 6 0.45300 47.8400
## [20,] 6 0.45880 43.5900
## [21,] 6 0.46360 39.7100
## [22,] 6 0.46760 36.1900
## [23,] 6 0.47100 32.9700
## [24,] 6 0.47370 30.0400
## [25,] 6 0.47600 27.3700
## [26,] 6 0.47790 24.9400
## [27,] 6 0.47950 22.7300
## [28,] 6 0.48080 20.7100
## [29,] 6 0.48190 18.8700
## [30,] 7 0.48290 17.1900
## [31,] 7 0.48400 15.6600
## [32,] 7 0.48480 14.2700
## [33,] 9 0.48570 13.0000
## [34,] 9 0.48660 11.8500
## [35,] 9 0.48730 10.8000
## [36,] 9 0.48790 9.8370
## [37,] 9 0.48840 8.9630
## [38,] 11 0.49120 8.1670
## [39,] 11 0.49600 7.4420
## [40,] 12 0.50270 6.7810
## [41,] 12 0.50820 6.1780
## [42,] 13 0.51320 5.6290
## [43,] 13 0.51770 5.1290
## [44,] 13 0.52140 4.6740
## [45,] 13 0.52450 4.2580
## [46,] 13 0.52700 3.8800
## [47,] 13 0.52910 3.5350
## [48,] 13 0.53090 3.2210
## [49,] 13 0.53240 2.9350
## [50,] 13 0.53360 2.6740
## [51,] 13 0.53460 2.4370
## [52,] 14 0.53550 2.2200
## [53,] 15 0.53620 2.0230
## [54,] 15 0.53720 1.8430
## [55,] 17 0.53840 1.6800
## [56,] 17 0.53960 1.5300
## [57,] 17 0.54060 1.3940
## [58,] 17 0.54150 1.2710
## [59,] 17 0.54220 1.1580
## [60,] 17 0.54280 1.0550
## [61,] 17 0.54320 0.9611
## [62,] 17 0.54360 0.8757
## [63,] 17 0.54400 0.7979
## [64,] 17 0.54430 0.7271
## [65,] 18 0.54450 0.6625
## [66,] 18 0.54480 0.6036
## [67,] 18 0.54490 0.5500
## [68,] 17 0.54510 0.5011
## [69,] 18 0.54520 0.4566
## [70,] 18 0.54530 0.4160
## [71,] 18 0.54540 0.3791
## [72,] 18 0.54550 0.3454
## [73,] 18 0.54560 0.3147
## [74,] 18 0.54570 0.2868
## [75,] 18 0.54570 0.2613
## [76,] 18 0.54580 0.2381
## [77,] 18 0.54580 0.2169
## [78,] 18 0.54590 0.1977
## [79,] 18 0.54590 0.1801
## [80,] 19 0.54590 0.1641
##
## $lambda.min
## [1] 2.935124
##
## $lambda.1se
## [1] 91.74363
##
## attr(,"class")
## [1] "cv.glmnet"#print coefficient of each variable
coef(cv.lasso.reg, s="lambda.min")## 20 x 1 sparse Matrix of class "dgCMatrix"
## 1
## (Intercept) 117.5258436
## AtBat -1.4742901
## Hits 5.4994256
## HmRun .
## Runs .
## RBI .
## Walks 4.5991651
## Years -9.1918308
## CAtBat .
## CHits .
## CHmRun 0.4806743
## CRuns 0.6354799
## CRBI 0.3956153
## CWalks -0.4993240
## LeagueN 31.6238173
## DivisionW -119.2516409
## PutOuts 0.2704287
## Assists 0.1594997
## Errors -1.9426357
## NewLeagueN .In the above output, the dotted ones are dropped variables which are exactly to zero. So Lasso regression penalizes coefficients to shrink towards exactly zero. By this, the model is performing variable selection as well. So it drops variable which are not important to the model.
#prediction
lasso.pred <- predict(cv.lasso.reg, newx=x, s="lambda.min")
#print top 5 predicted values
head(lasso.pred)## 1
## -Alan Ashby 433.6688
## -Alvin Davis 699.3764
## -Andre Dawson 1101.4861
## -Andres Galarraga 528.2402
## -Alfredo Griffin 570.2200
## -Al Newman 196.4160#model evaluation
library(DMwR)
round(regr.eval(y,lasso.pred), 3)## mae mse rmse mape
## 215.307 94801.378 307.898 0.695It generalizes both Ridge and LASSO regressions.
#Elasticnet regression use alpha=0.5
Elastic.reg <- glmnet(x,y,alpha=0.5)
Elastic.reg##
## Call: glmnet(x = x, y = y, alpha = 0.5)
##
## Df %Dev Lambda
## [1,] 0 0.00000 510.6000
## [2,] 2 0.04280 465.2000
## [3,] 3 0.08286 423.9000
## [4,] 3 0.11870 386.2000
## [5,] 5 0.15570 351.9000
## [6,] 7 0.19950 320.6000
## [7,] 7 0.23670 292.2000
## [8,] 7 0.26850 266.2000
## [9,] 8 0.29550 242.6000
## [10,] 8 0.31850 221.0000
## [11,] 9 0.33970 201.4000
## [12,] 9 0.35940 183.5000
## [13,] 9 0.37600 167.2000
## [14,] 9 0.39010 152.3000
## [15,] 10 0.40430 138.8000
## [16,] 10 0.41730 126.5000
## [17,] 10 0.42830 115.2000
## [18,] 10 0.43760 105.0000
## [19,] 10 0.44550 95.6700
## [20,] 10 0.45210 87.1700
## [21,] 10 0.45770 79.4300
## [22,] 10 0.46250 72.3700
## [23,] 10 0.46650 65.9400
## [24,] 10 0.46990 60.0800
## [25,] 10 0.47280 54.7500
## [26,] 8 0.47510 49.8800
## [27,] 8 0.47710 45.4500
## [28,] 8 0.47870 41.4100
## [29,] 8 0.48010 37.7300
## [30,] 9 0.48140 34.3800
## [31,] 9 0.48260 31.3300
## [32,] 9 0.48370 28.5400
## [33,] 10 0.48460 26.0100
## [34,] 10 0.48560 23.7000
## [35,] 10 0.48640 21.5900
## [36,] 10 0.48710 19.6700
## [37,] 10 0.48770 17.9300
## [38,] 10 0.48820 16.3300
## [39,] 12 0.49060 14.8800
## [40,] 13 0.49490 13.5600
## [41,] 13 0.50020 12.3600
## [42,] 14 0.50470 11.2600
## [43,] 14 0.50930 10.2600
## [44,] 14 0.51330 9.3470
## [45,] 14 0.51680 8.5170
## [46,] 14 0.52000 7.7600
## [47,] 14 0.52270 7.0710
## [48,] 14 0.52510 6.4430
## [49,] 14 0.52720 5.8700
## [50,] 14 0.52900 5.3490
## [51,] 14 0.53050 4.8740
## [52,] 14 0.53190 4.4410
## [53,] 15 0.53310 4.0460
## [54,] 16 0.53410 3.6870
## [55,] 18 0.53510 3.3590
## [56,] 18 0.53620 3.0610
## [57,] 18 0.53720 2.7890
## [58,] 18 0.53830 2.5410
## [59,] 18 0.53930 2.3150
## [60,] 18 0.54020 2.1100
## [61,] 18 0.54090 1.9220
## [62,] 18 0.54160 1.7510
## [63,] 18 0.54220 1.5960
## [64,] 18 0.54270 1.4540
## [65,] 18 0.54310 1.3250
## [66,] 18 0.54350 1.2070
## [67,] 19 0.54380 1.1000
## [68,] 19 0.54420 1.0020
## [69,] 19 0.54440 0.9132
## [70,] 19 0.54470 0.8321
## [71,] 19 0.54490 0.7582
## [72,] 19 0.54510 0.6908
## [73,] 19 0.54520 0.6294
## [74,] 19 0.54530 0.5735
## [75,] 19 0.54540 0.5226
## [76,] 19 0.54560 0.4762
## [77,] 19 0.54560 0.4339
## [78,] 19 0.54570 0.3953
## [79,] 19 0.54570 0.3602summary(Elastic.reg)## Length Class Mode
## a0 79 -none- numeric
## beta 1501 dgCMatrix S4
## df 79 -none- numeric
## dim 2 -none- numeric
## lambda 79 -none- numeric
## dev.ratio 79 -none- numeric
## nulldev 1 -none- numeric
## npasses 1 -none- numeric
## jerr 1 -none- numeric
## offset 1 -none- logical
## call 4 -none- call
## nobs 1 -none- numeric#print coefficeints of 3rd model
coef(Elastic.reg, 3)## 20 x 1 sparse Matrix of class "dgCMatrix"
## 1
## (Intercept) 1.378568e+02
## AtBat -1.668179e+00
## Hits 5.844264e+00
## HmRun 1.708095e-01
## Runs .
## RBI 2.970095e-02
## Walks 5.008051e+00
## Years -1.037298e+01
## CAtBat -6.221306e-03
## CHits 4.304375e-03
## CHmRun 5.520112e-01
## CRuns 7.357503e-01
## CRBI 4.056508e-01
## CWalks -6.007500e-01
## LeagueN 3.910736e+01
## DivisionW -1.201353e+02
## PutOuts 2.783122e-01
## Assists 2.255262e-01
## Errors -2.710374e+00
## NewLeagueN -4.759615e+00#elasticnet regression using cross-validation method
cv.elastic.reg <- cv.glmnet(x,y,alpha=0.5)
cv.elastic.reg## $lambda
## [1] 510.5641930 465.2070773 423.8793628 386.2230885 351.9120938
## [6] 320.6491933 292.1636028 266.2085936 242.5593558 221.0110512
## [11] 201.3770386 183.4872575 167.1867553 152.3343447 138.8013814
## [16] 126.4706492 115.2353453 104.9981549 95.6704082 87.1713128
## [21] 79.4272536 72.3711553 65.9419014 60.0838046 54.7461250
## [26] 49.8826301 45.4511948 41.4134359 37.7343804 34.3821620
## [31] 31.3277455 28.5446750 26.0088447 23.6982906 21.5929998
## [36] 19.6747372 17.9268878 16.3343125 14.8832172 13.5610332
## [41] 12.3563084 11.2586080 10.2584243 9.3470942 8.5167241
## [46] 7.7601218 7.0707340 6.4425894 5.8702475 5.3487509
## [51] 4.8735826 4.4406270 4.0461339 3.6866865 3.3591715
## [56] 3.0607519 2.7888432 2.5410900 2.3153466 2.1096576
## [61] 1.9222414 1.7514748 1.5958786 1.4541051 1.3249264
## [66] 1.2072236 1.0999772 1.0022583 0.9132204 0.8320924
## [71] 0.7581716 0.6908177 0.6294474 0.5735290
##
## $cvm
## [1] 203130.0 196802.9 188834.2 181751.7 175135.2 168654.2 162151.4
## [8] 156195.1 151192.4 147096.1 143498.6 140105.5 137163.0 134683.3
## [15] 132483.2 130500.4 128629.5 126843.5 125295.2 123994.5 122903.8
## [22] 121991.0 121230.6 120598.8 120079.7 119661.3 119355.5 119127.8
## [29] 118977.7 118884.3 118829.2 118791.0 118780.5 118792.7 118820.0
## [36] 118867.3 119036.5 119271.0 119500.2 119594.2 119476.6 119027.2
## [43] 118547.8 118136.7 117782.1 117400.8 116865.1 116454.6 116143.5
## [50] 115878.4 115658.5 115485.9 115388.4 115363.5 115405.7 115424.3
## [57] 115413.9 115438.3 115417.8 115423.7 115427.0 115436.5 115449.0
## [64] 115452.9 115462.2 115477.0 115517.3 115556.7 115633.9 115708.5
## [71] 115802.6 115852.5 115935.1 115987.5
##
## $cvsd
## [1] 33846.65 34002.93 33124.40 32317.41 31612.03 30963.89 30182.79
## [8] 29453.77 28841.61 28349.62 27940.52 27619.21 27361.94 27166.76
## [15] 27049.40 26939.66 26732.04 26420.42 26121.01 25860.10 25632.92
## [22] 25434.50 25260.34 25106.86 24972.59 24854.82 24743.71 24642.26
## [29] 24554.57 24483.88 24421.00 24367.34 24318.53 24274.80 24239.63
## [36] 24214.54 24189.22 24149.36 24095.52 24048.28 23991.90 23906.83
## [43] 23803.53 23711.50 23631.26 23528.82 23385.02 23288.92 23232.61
## [50] 23191.77 23169.50 23156.42 23149.45 23154.39 23167.56 23182.94
## [57] 23205.79 23224.87 23238.41 23255.89 23275.71 23297.72 23318.72
## [64] 23338.58 23355.87 23372.88 23387.71 23404.56 23423.67 23445.78
## [71] 23469.20 23483.88 23505.38 23519.64
##
## $cvup
## [1] 236976.6 230805.8 221958.6 214069.2 206747.3 199618.1 192334.2
## [8] 185648.9 180034.0 175445.7 171439.1 167724.8 164524.9 161850.0
## [15] 159532.6 157440.0 155361.5 153264.0 151416.2 149854.6 148536.7
## [22] 147425.5 146490.9 145705.7 145052.3 144516.1 144099.3 143770.1
## [29] 143532.2 143368.1 143250.2 143158.3 143099.0 143067.5 143059.6
## [36] 143081.8 143225.7 143420.4 143595.7 143642.4 143468.5 142934.0
## [43] 142351.3 141848.2 141413.4 140929.6 140250.1 139743.5 139376.1
## [50] 139070.2 138828.0 138642.4 138537.8 138517.9 138573.3 138607.2
## [57] 138619.7 138663.2 138656.2 138679.6 138702.8 138734.2 138767.7
## [64] 138791.5 138818.1 138849.9 138905.1 138961.2 139057.6 139154.3
## [71] 139271.9 139336.4 139440.5 139507.1
##
## $cvlo
## [1] 169283.34 162799.98 155709.77 149434.34 143523.21 137690.33 131968.64
## [8] 126741.35 122350.81 118746.49 115558.09 112486.33 109801.06 107516.50
## [15] 105433.75 103560.69 101897.41 100423.13 99174.16 98134.43 97270.85
## [22] 96556.52 95970.24 95491.96 95107.11 94806.48 94611.83 94485.56
## [29] 94423.08 94400.38 94408.22 94423.61 94461.99 94517.86 94580.35
## [36] 94652.76 94847.30 95121.65 95404.64 95545.89 95484.67 95120.33
## [43] 94744.26 94425.15 94150.85 93871.99 93480.04 93165.68 92910.90
## [50] 92686.67 92488.97 92329.52 92238.92 92209.13 92238.15 92241.35
## [57] 92208.08 92213.44 92179.40 92167.85 92151.33 92138.75 92130.28
## [64] 92114.31 92106.35 92104.10 92129.64 92152.11 92210.24 92262.71
## [71] 92333.45 92368.65 92429.69 92467.82
##
## $nzero
## s0 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15 s16 s17
## 0 2 3 3 5 7 7 7 8 8 9 9 9 9 10 10 10 10
## s18 s19 s20 s21 s22 s23 s24 s25 s26 s27 s28 s29 s30 s31 s32 s33 s34 s35
## 10 10 10 10 10 10 10 8 8 8 8 9 9 9 10 10 10 10
## s36 s37 s38 s39 s40 s41 s42 s43 s44 s45 s46 s47 s48 s49 s50 s51 s52 s53
## 10 10 12 13 13 14 14 14 14 14 14 14 14 14 14 14 15 16
## s54 s55 s56 s57 s58 s59 s60 s61 s62 s63 s64 s65 s66 s67 s68 s69 s70 s71
## 18 18 18 18 18 18 18 18 18 18 18 18 19 19 19 19 19 19
## s72 s73
## 19 19
##
## $name
## mse
## "Mean-Squared Error"
##
## $glmnet.fit
##
## Call: glmnet(x = x, y = y, alpha = 0.5)
##
## Df %Dev Lambda
## [1,] 0 0.00000 510.6000
## [2,] 2 0.04280 465.2000
## [3,] 3 0.08286 423.9000
## [4,] 3 0.11870 386.2000
## [5,] 5 0.15570 351.9000
## [6,] 7 0.19950 320.6000
## [7,] 7 0.23670 292.2000
## [8,] 7 0.26850 266.2000
## [9,] 8 0.29550 242.6000
## [10,] 8 0.31850 221.0000
## [11,] 9 0.33970 201.4000
## [12,] 9 0.35940 183.5000
## [13,] 9 0.37600 167.2000
## [14,] 9 0.39010 152.3000
## [15,] 10 0.40430 138.8000
## [16,] 10 0.41730 126.5000
## [17,] 10 0.42830 115.2000
## [18,] 10 0.43760 105.0000
## [19,] 10 0.44550 95.6700
## [20,] 10 0.45210 87.1700
## [21,] 10 0.45770 79.4300
## [22,] 10 0.46250 72.3700
## [23,] 10 0.46650 65.9400
## [24,] 10 0.46990 60.0800
## [25,] 10 0.47280 54.7500
## [26,] 8 0.47510 49.8800
## [27,] 8 0.47710 45.4500
## [28,] 8 0.47870 41.4100
## [29,] 8 0.48010 37.7300
## [30,] 9 0.48140 34.3800
## [31,] 9 0.48260 31.3300
## [32,] 9 0.48370 28.5400
## [33,] 10 0.48460 26.0100
## [34,] 10 0.48560 23.7000
## [35,] 10 0.48640 21.5900
## [36,] 10 0.48710 19.6700
## [37,] 10 0.48770 17.9300
## [38,] 10 0.48820 16.3300
## [39,] 12 0.49060 14.8800
## [40,] 13 0.49490 13.5600
## [41,] 13 0.50020 12.3600
## [42,] 14 0.50470 11.2600
## [43,] 14 0.50930 10.2600
## [44,] 14 0.51330 9.3470
## [45,] 14 0.51680 8.5170
## [46,] 14 0.52000 7.7600
## [47,] 14 0.52270 7.0710
## [48,] 14 0.52510 6.4430
## [49,] 14 0.52720 5.8700
## [50,] 14 0.52900 5.3490
## [51,] 14 0.53050 4.8740
## [52,] 14 0.53190 4.4410
## [53,] 15 0.53310 4.0460
## [54,] 16 0.53410 3.6870
## [55,] 18 0.53510 3.3590
## [56,] 18 0.53620 3.0610
## [57,] 18 0.53720 2.7890
## [58,] 18 0.53830 2.5410
## [59,] 18 0.53930 2.3150
## [60,] 18 0.54020 2.1100
## [61,] 18 0.54090 1.9220
## [62,] 18 0.54160 1.7510
## [63,] 18 0.54220 1.5960
## [64,] 18 0.54270 1.4540
## [65,] 18 0.54310 1.3250
## [66,] 18 0.54350 1.2070
## [67,] 19 0.54380 1.1000
## [68,] 19 0.54420 1.0020
## [69,] 19 0.54440 0.9132
## [70,] 19 0.54470 0.8321
## [71,] 19 0.54490 0.7582
## [72,] 19 0.54510 0.6908
## [73,] 19 0.54520 0.6294
## [74,] 19 0.54530 0.5735
## [75,] 19 0.54540 0.5226
## [76,] 19 0.54560 0.4762
## [77,] 19 0.54560 0.4339
## [78,] 19 0.54570 0.3953
## [79,] 19 0.54570 0.3602
##
## $lambda.min
## [1] 3.686687
##
## $lambda.1se
## [1] 167.1868
##
## attr(,"class")
## [1] "cv.glmnet"So minimum lambda is 4.046134, which 53rd model
#coefficeints of 53rd model
coef(cv.elastic.reg, 53)## 20 x 1 sparse Matrix of class "dgCMatrix"
## 1
## (Intercept) 50.11488143
## AtBat .
## Hits 1.68428289
## HmRun .
## Runs 0.02153104
## RBI 0.01071734
## Walks 2.13116793
## Years .
## CAtBat .
## CHits 0.05545304
## CHmRun 0.34027355
## CRuns 0.17186673
## CRBI 0.22799415
## CWalks .
## LeagueN .
## DivisionW -81.65977827
## PutOuts 0.19066430
## Assists .
## Errors .
## NewLeagueN .#prediction
elastic.pred <- predict(cv.elastic.reg, newx = x, s="lambda.min")
round(regr.eval(y,elastic.pred), 3)## mae mse rmse mape
## 215.380 94443.976 307.317 0.697