Last updated: 2021-01-23

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File Version Author Date Message
Rmd 32ba051 Matthew Stephens 2021-01-23 workflowr::wflow_publish(“prec_vs_cov.Rmd”)

Introduction

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

Low dimension case (n>p)

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

High dimension case (n<p)

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

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