r - Lasso 回归，生成系数矩阵

Question

我有这个套索回归代码，当我打印 beta 系数时，我得到许多组 beta 值，而不仅仅是一组。我没有为 lambda 指定任何值，但当我这样做时，我只得到一组 beta 值。我知道如何找到 lambda 的最佳值。我的问题是，当我没有指定 lambda 时，为什么会得到这么多组 beta？这些 beta 是连续变量吗？

Here is the code: 
library(MASS)
library(glmnet)
Boston=na.omit(Boston)
x=model.matrix(crim~.,Boston)[,-1]
y=as.matrix(Boston$crim)
lasso.mod =glmnet(x,y, alpha =1)
beta=coef(lasso.mod)

当我打印 beta 时，我得到了这些 beta(这里我只展示了一些 beta，因为它是维度为 14x77 的 beta 矩阵):

(Intercept) 3.613524 3.0893231 2.6116912 2.176491 1.7799525 1.4186414 1.0894283 0.7894616 0.5161430 0.10644553
zn          .        .         .         .        .         .         .         .         .         .         
indus       .        .         .         .        .         .         .         .         .         .         
chas        .        .         .         .        .         .         .         .         .         .         
nox         .        .         .         .        .         .         .         .         .         .         
rm          .        .         .         .        .         .         .         .         .         .         
age         .        .         .         .        .         .         .         .         .         .         
dis         .        .         .         .        .         .         .         .         .         .         
rad         .        0.0548935 0.1049104 0.150484 0.1920089 0.2298449 0.2643196 0.2957317 0.3243532 0.34314278
tax         .        .         .         .        .         .         .         .         .         .         
ptratio     .        .         .         .        .         .         .         .         .         .         
black       .        .         .         .        .         .         .         .         .         .         
lstat       .        .         .         .        .         .         .         .         .         0.01819859
medv        .        .         .         .        .         .         .         .         .         .         

(Intercept) -0.29224457 -0.65554971 -0.98654448 -1.2881346 -1.551777e+00 -1.3115723669 -1.023961164 -0.760703960
zn           .           .           .           .          .             .             .            .          
indus        .           .           .           .          .             .             .            .          
chas         .           .           .           .          .             .             .            .          
nox          .           .           .           .          .             .             .            .          
rm           .           .           .           .          .             .             .            .          
age          .           .           .           .          .             .             .            .          
dis          .           .           .           .          .             .             .            .          
rad          0.35910506  0.37366600  0.38691580  0.3989885  4.099887e-01  0.4167185339  0.423004227  0.428776109
tax          .           .           .           .          .             .             .            .          
ptratio      .           .           .           .          .             .             .            .          
black        .           .           .           .         -2.682197e-05 -0.0008416848 -0.001560914 -0.002216123
lstat        0.03766106  0.05538458  0.07154406  0.0862680  9.955836e-02  0.1059656295  0.109649285  0.112926619
medv         .           .           .           .          .            -0.0042117132 -0.010323462 -0.015921859

(Intercept) -0.520830886 -0.302267470 -0.105253730  0.076376939  0.241885979  0.392691730  0.627291211  0.864528799
zn           .            .            .            .            .            .            .            .          
indus        .            .            .            .            .            .            .            .          
chas         .            .            .            .            .            .            .            .          
nox          .            .            .            .            .            .            .            .          
rm           .            .            .            .            .            .            .            .          
age          .            .            .            .            .            .            .            .          
dis          .            .            .            .            .            .           -0.013081595 -0.027872125
rad          0.434035445  0.438827556  0.443126465  0.447110129  0.450740716  0.454048777  0.456008475  0.457602313
tax          .            .            .            .            .            .            .            .          
ptratio      .            .            .            .            .            .            .            .          
black       -0.002813124 -0.003357088 -0.003852845 -0.004304448 -0.004715927 -0.005090852 -0.005417568 -0.005712667
lstat        0.115912528  0.118633177  0.121243578  0.123491798  0.125539133  0.127404580  0.127149522  0.126384081
medv        -0.021022995 -0.025670960 -0.029854724 -0.033717719 -0.037237917 -0.040445393 -0.044169297 -0.047781258

(Intercept)  1.079254571  1.274889342  1.453144352  1.612076416  1.760570e+00  2.050760033  2.322171047
zn           .            .            .            .            1.285559e-05  0.004410478  0.008432185
indus        .            .            .            .            .             .            .          
chas         .            .            .            .            .             .           -0.037708532
nox          .            .            .            .            .             .            .          
rm           .            .            .            .            .             .            .          
age          .            .            .            .            .             .            .          
dis         -0.041210398 -0.053363055 -0.064436101 -0.074320056 -8.362340e-02 -0.125411018 -0.164769794
rad          0.459071435  0.460409659  0.461628996  0.462694814  4.637089e-01  0.463773651  0.463944084
tax          .            .            .            .            .             .            .          
ptratio      .            .            .            .            .             .            .          
black       -0.005981441 -0.006226339 -0.006449481 -0.006653008 -6.838219e-03 -0.006939683 -0.007027304
lstat        0.125716380  0.125108816  0.124555233  0.124200710  1.237351e-01  0.121952792  0.119996733
medv        -0.051057488 -0.054042369 -0.056762080 -0.059181727 -6.144805e-02 -0.066926875 -0.071842416

(Intercept)  2.549278186  2.762617045  2.952609432  3.1338778318  3.538132302  4.130162117  5.451162673  6.64884950
zn           0.012036563  0.015337882  0.018333706  0.0210755780  0.023426675  0.025313462  0.026747451  0.02813502
indus        .            .            .           -0.0001924235 -0.013776751 -0.025787090 -0.032420857 -0.03810011
chas        -0.124943128 -0.204109888 -0.276274870 -0.3416707320 -0.380135159 -0.421664111 -0.443464956 -0.46402519
nox          .            .            .            .             .           -0.003479190 -0.917150073 -1.74619114
rm           .            .            .            .             .            .            .            .         
age          .            .            .            .             .            .            .            .         
dis         -0.200411673 -0.233281676 -0.262884152 -0.2906434425 -0.335943512 -0.375943571 -0.430590855 -0.48105322
rad          0.463938072  0.464006464  0.464034620  0.4641320102  0.466554241  0.469838965  0.476096780  0.48167304
tax          .            .            .            .             .            .            .            .         
ptratio      .            .            .            .             .           -0.011535857 -0.035347459 -0.05667073
black       -0.007105454 -0.007175524 -0.007239875 -0.0072986549 -0.007367437 -0.007404653 -0.007449503 -0.00749179
lstat        0.118971687  0.117768685  0.116835186  0.1158446610  0.116189895  0.115858916  0.116384542  0.11661762
medv        -0.075653219 -0.079251279 -0.082455778 -0.0854666531 -0.090263034 -0.095771213 -0.102812853 -0.10924776

Answer 1

主要答案在?glmnet中给出:

lambda

A user supplied lambda sequence. Typical usage is to have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Supplying a value of lambda overrides this. WARNING: use with care. Avoid supplying a single value for lambda (for predictions after CV use predict() instead). Supply instead a decreasing sequence of lambda values. glmnet relies on its warms starts for speed, and its often faster to fit a whole path than compute a single fit.

现在，默认情况下，nlambda 是 100 而不是 77。最小的 lambda 由下式给出

lambda.min.ratio = ifelse(nobs<nvars,0.01,0.0001)

而最大的则所有系数都为零。最后，在

lasso.mod
# Call:  glmnet(x = x, y = y, alpha = 1) 
# 
# Df    %Dev   Lambda
#  [1,]  0 0.00000 5.375000
#  [2,]  1 0.06643 4.897000
#  [3,]  1 0.12160 4.462000
#  [4,]  1 0.16740 4.066000
# .....
# [73,] 13 0.45400 0.006627
# [74,] 13 0.45400 0.006038
# [75,] 13 0.45400 0.005501
# [76,] 13 0.45400 0.005013
# [77,] 13 0.45400 0.004567

我们看到百分比偏差似乎不再改变。因此，出于这个原因，lambda 序列会提前终止，而不会达到 100 个值。