Sparse PCA: reproduction of the synthetic example

The paper:

Hui Zou, Trevor Hastie, and Robert Tibshirani,

Sparse Principal Component Analysis,

Journal of computational and Graphical Statistics, 15(2): 265-286, 2006.

Reproduction of the Synthetic Example in Section 5.2 using R programming:

 library(elasticnet)

 ## sample version of SPCA
 n =
 v1 = rnorm(n,,sqrt())
 v2 = rnorm(n,,sqrt())
 v3 = -.*v1 + 0.925*v2 + rnorm(n)
 x1 = v1 + rnorm(n)
 x2 = v1 + rnorm(n)
 x3 = v1 + rnorm(n)
 x4 = v1 + rnorm(n)

 x5 = v2 + rnorm(n)
 x6 = v2 + rnorm(n)
 x7 = v2 + rnorm(n)
 x8 = v2 + rnorm(n)

 x9 = v3 + rnorm(n)
 x10 = v3 + rnorm(n)

 x = cbind(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10)
 x.cov = t(x) %*% x/n; head(x.cov)
 a = spca(x, , type='predictor', sparse='varnum', para=c(,), lambda=)
 a
 ## population version of SPCA
 g1 = matrix(, , )
 diag(g1) = 

 g2 = matrix(, , )
 diag(g2) = 

 g3 = matrix(283.7875, , )
 diag(g3) = diag(g3)+

 g1g3 = matrix(-, , )
 g2g3 = matrix(277.5, , )

 # construct the exact covariance matrix
 x.cov = matrix(, , )
 x.cov[:,:] = g1
 x.cov[:,:] = g2
 x.cov[:,:] = g3
 x.cov[:,:] = g1g3
 x.cov[:,:] = t(g1g3)
 x.cov[:,:] = g2g3
 x.cov[:,:] = t(g2g3)

 b = spca(x.cov, , type='Gram', sparse='varnum', para=c(,), lambda=)
 b

The results of the population version using exact covariance matrix are exactly as in the paper:

> b

Call:
spca(x = x.cov, K = , para = c(, ), type = "Gram", sparse = "varnum",
    lambda = )

 sparse PCs
Pct. of exp. var. : 40.9 39.5
Num. of non-zero loadings :
Sparse loadings
      PC1 PC2
 [,] 0.0 0.5
 [,] 0.0 0.5
 [,] 0.0 0.5
 [,] 0.0 0.5
 [,] 0.5 0.0
 [,] 0.5 0.0
 [,] 0.5 0.0
 [,] 0.5 0.0
 [,] 0.0 0.0
[,] 0.0 0.0

But the sample version may randomly vary a little.

> a

Call:
spca(x = x, K = , para = c(, ), type = "predictor", sparse = "varnum",
    lambda = )

 sparse PCs
Pct. of exp. var. : 37.9 37.6
Num. of non-zero loadings :
Sparse loadings
       PC1    PC2
x1   0.000 -0.303
x2   0.000 -0.533
x3   0.000 -0.576
x4   0.000 -0.540
x5  -0.492  0.000
x6  -0.287  0.000
x7  -0.481  0.000
x8  -0.666  0.000
x9   0.000  0.000
x10  0.000  0.000

Having fun learning sparse PCA!