Associated with each value of \(\lambda\) is a vector of ridge regression coefficients, stored in a matrix that can be accessed by coef(). In this case, it is a 20×100 matrix, with 20 rows (one for each predictor, plus an intercept) and 100 columns (one for each value of \(\lambda\)).

> dim(coef(ridge.mod))
[1]  20 100

We expect the coefficient estimates to be much smaller, in terms of \(\ell_2\) norm, when a large value of \(\lambda\) is used, as compared to when a small value of \(\lambda\) is used. These are the coefficients when \(\lambda = 11,498\), along with their \(\ell_2\) norm:

> ridge.mod$lambda[50]
[1] 11497.57
> coef(ridge.mod)[, 50]
  (Intercept)         AtBat          Hits         HmRun 
407.356050200   0.036957182   0.138180344   0.524629976 
         Runs           RBI         Walks         Years 
  0.230701523   0.239841459   0.289618741   1.107702929 
       CAtBat         CHits        CHmRun         CRuns 
  0.003131815   0.011653637   0.087545670   0.023379882 
         CRBI        CWalks       LeagueN     DivisionW 
  0.024138320   0.025015421   0.085028114  -6.215440973 
      PutOuts       Assists        Errors    NewLeagueN 
  0.016482577   0.002612988  -0.020502690   0.301433531 
> sqrt(sum(coef(ridge.mod)[-1, 50]^2))
[1] 6.360612

In contrast, here are the coefficients when \(\lambda\) = 705, along with their \(\ell_2\) norm. Note the much larger \(\ell_2\) norm of the coefficients associated with this smaller value of \(\lambda\).

> ridge.mod$lambda[60]
[1] 705.4802
> coef(ridge.mod)[, 60]
 (Intercept)        AtBat         Hits        HmRun 
 54.32519950   0.11211115   0.65622409   1.17980910 
        Runs          RBI        Walks        Years 
  0.93769713   0.84718546   1.31987948   2.59640425 
      CAtBat        CHits       CHmRun        CRuns 
  0.01083413   0.04674557   0.33777318   0.09355528 
        CRBI       CWalks      LeagueN    DivisionW 
  0.09780402   0.07189612  13.68370191 -54.65877750 
     PutOuts      Assists       Errors   NewLeagueN 
  0.11852289   0.01606037  -0.70358655   8.61181213 
> sqrt(sum(coef(ridge.mod)[-1, 60]^2))

At index 40, \(\lambda = 43\). Try calculating the corresponding \(\ell_2\) and store it in ell2:

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Assume that:

• The ISLR2 and glmnet libraries have been loaded
• The Hitters dataset has been loaded and attached