Last updated: 2021-01-23
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Here I look at the difference between the precision and covariance for ridge regression.
With suitable scaling, the precision matrix (Omega) is \((X'X+I)\) and covariance (Sigma) is inverse of this. I’m going to look at this when the elements of \(X\) are iid N(0,1).
First I am going to check I have got the right formulae to compute the diagonal elements of Omega and Sigma from an SVD of X, by comparing with direct inversion:
n = 10
p = 20
X = matrix(rnorm(n*p),nrow=n)
Omega = t(X) %*% X + diag(p)
Sigma = chol2inv(chol(Omega))
compute_diags = function(X){
udv = svd(X)
d2 = udv$d^2
v = udv$v
Sigma_diag = 1 - colSums((d2/(1+d2))*t(v^2))
Omega_diag = 1 + colSums(d2*t(v^2))
return(list(Sigma_diag=Sigma_diag,Omega_diag = Omega_diag, inv_Omega_diag = 1/Omega_diag))
}
res = compute_diags(X)
res$Sigma_diag - diag(Sigma)
[1] -3.330669e-16 -4.440892e-16 -5.551115e-16 -1.110223e-16 4.440892e-16
[6] 0.000000e+00 1.110223e-16 0.000000e+00 3.330669e-16 1.110223e-16
[11] 2.220446e-16 -2.220446e-16 8.881784e-16 7.771561e-16 4.996004e-16
[16] -1.665335e-16 2.220446e-16 -6.106227e-16 4.440892e-16 2.220446e-16
res$Omega_diag - diag(Omega)
[1] 8.881784e-16 8.881784e-15 1.953993e-14 6.217249e-15 -5.329071e-15
[6] -3.552714e-15 1.421085e-14 1.776357e-15 7.993606e-15 3.552714e-15
[11] -5.329071e-15 4.440892e-15 1.776357e-15 3.552714e-15 1.776357e-14
[16] 3.552714e-15 -1.776357e-15 8.881784e-15 1.065814e-14 1.776357e-15
If n>>p then diagonal of Sigma and inverse of diagonal of Omega are similar, and look like 1/n, independent of p.
compare_means = function(n,p){
X = matrix(rnorm(n*p),nrow=n)
res = compute_diags(X)
lapply(res,mean)
}
compare_means(1000,100)
$Sigma_diag
[1] 0.00110587
$Omega_diag
[1] 1008.187
$inv_Omega_diag
[1] 0.0009940994
compare_means(10000,100)
$Sigma_diag
[1] 0.0001010482
$Omega_diag
[1] 9997.386
$inv_Omega_diag
[1] 0.0001000428
compare_means(100,20)
$Sigma_diag
[1] 0.0127932
$Omega_diag
[1] 99.12597
$inv_Omega_diag
[1] 0.01021614
compare_means(1000,20)
$Sigma_diag
[1] 0.001006495
$Omega_diag
[1] 1013.262
$inv_Omega_diag
[1] 0.0009898052
compare_means(10000,20)
$Sigma_diag
[1] 9.944003e-05
$Omega_diag
[1] 10080.04
$inv_Omega_diag
[1] 9.922732e-05
If p>n then the inverse of Omega_diag continues to look like 1/n but Sigma looks like 1-(n/p).
compare_means(100,100)
$Sigma_diag
[1] 0.09546693
$Omega_diag
[1] 100.3718
$inv_Omega_diag
[1] 0.01013801
compare_means(100,1000)
$Sigma_diag
[1] 0.9001111
$Omega_diag
[1] 101.0955
$inv_Omega_diag
[1] 0.01010052
compare_means(100,10000)
$Sigma_diag
[1] 0.990001
$Omega_diag
[1] 101.1118
$inv_Omega_diag
[1] 0.01008871
compare_means(100,100000)
$Sigma_diag
[1] 0.999
$Omega_diag
[1] 101.0076
$inv_Omega_diag
[1] 0.01009906
For n=p inv_Omega looks like 1/n, Sigma looks like 1/sqrt(p) [= 1/sqrt(n)].
compare_means(10,10)
$Sigma_diag
[1] 0.296318
$Omega_diag
[1] 9.731093
$inv_Omega_diag
[1] 0.1176032
compare_means(20,20)
$Sigma_diag
[1] 0.2015544
$Omega_diag
[1] 20.64975
$inv_Omega_diag
[1] 0.04976618
compare_means(50,50)
$Sigma_diag
[1] 0.1344954
$Omega_diag
[1] 53.87387
$inv_Omega_diag
[1] 0.01932045
compare_means(100,100)
$Sigma_diag
[1] 0.1020224
$Omega_diag
[1] 100.9052
$inv_Omega_diag
[1] 0.01011604
compare_means(200,200)
$Sigma_diag
[1] 0.07067105
$Omega_diag
[1] 201.0988
$inv_Omega_diag
[1] 0.005013635
compare_means(500,500)
$Sigma_diag
[1] 0.04502965
$Omega_diag
[1] 501.131
$inv_Omega_diag
[1] 0.002003956
compare_means(1000,1000)
$Sigma_diag
[1] 0.03162624
$Omega_diag
[1] 1002.299
$inv_Omega_diag
[1] 0.0009998656
pvec = c(10,20,50,100,200,500,1000)
res = rep(0,length(pvec))
for(i in 1:length(pvec)){
res[i] = compare_means(pvec[i],pvec[i])$Sigma_diag
}
plot(pvec,res)
plot(log(pvec),log(res))
lm(log(res)~log(pvec))
Call:
lm(formula = log(res) ~ log(pvec))
Coefficients:
(Intercept) log(pvec)
0.02073 -0.50734
plot(pvec^(-0.5),res)
abline(a=0,b=1)
sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] Rcpp_1.0.6 rstudioapi_0.11 whisker_0.4 knitr_1.29
[5] magrittr_1.5 workflowr_1.6.2 R6_2.4.1 rlang_0.4.8
[9] stringr_1.4.0 tools_3.6.0 xfun_0.16 git2r_0.27.1
[13] htmltools_0.5.0 ellipsis_0.3.1 yaml_2.2.1 digest_0.6.27
[17] rprojroot_1.3-2 tibble_3.0.4 lifecycle_0.2.0 crayon_1.3.4
[21] later_1.1.0.1 vctrs_0.3.4 fs_1.5.0 promises_1.1.1
[25] glue_1.4.2 evaluate_0.14 rmarkdown_2.3 stringi_1.4.6
[29] compiler_3.6.0 pillar_1.4.6 backports_1.1.10 httpuv_1.5.4
[33] pkgconfig_2.0.3