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library(susieR)
library(Matrix)

data(N3finemapping)
attach(N3finemapping)
X0 = N3finemapping$X
## getting covariance matrix from the whole sample
## and examine the eigendecomposition to estimate numerical rank
R = cov(X0)
eig <- eigen(R)
plot(eig$values,
     main = "Eigenvalues of covariance matrix calculated using all samples",
     ylab = "Value",
     xlab = "Eigenvalue index")

Version Author Date
691542f dodat97 2025-12-01 
n0 = dim(X0)[1]
p0 = dim(X0)[2]
percent_explained = .95
eig_cumsum = cumsum(eig$values)
r_p = sum(eig_cumsum < percent_explained * eig_cumsum[p0]) ## percentage variance explained
sprintf("%d first principle components explain %.1f percent of variance", r_p, percent_explained*100)
[1] "82 first principle components explain 95.0 percent of variance"
snp_total = p0
sprintf("Total number of SNPs is %d", p0)
[1] "Total number of SNPs is 1001"
sprintf("Sample size %d", n0)
[1] "Sample size 574"

Now we proceed to split the data into half and look at the heatmap of the covariance matrices of two sub-samples.

Later we want to examine the Inverse-Wishart likelihood so I also sub-sample the SNPs here to make sure that the number of SNPs is less than the sample size.

#### randomly split the data into half
#### randomly select p consecutive SNPs where p < n so IW is proper
seed = 10
p = 50
# Start from a random point on the genome
indx_start = sample(1: (snp_total - p), 1)
X = X0[, indx_start:(indx_start + p -1)]
# View(cor(X)[1:10, 1:10])
## sub-sample into two
out_sample_size = n0 / 2
out_sample = sample(1:n0, out_sample_size)
X_out = X[out_sample, ]
X_in = X[setdiff(1:n0, out_sample), ]

rm_p = c(which(diag(cov(X_in))==0), which(diag(cov(X_out))==0))
indx_p = setdiff(1:p, rm_p)
X_in = X_in[, indx_p]
X_out = X_out[, indx_p]
## out-sample LD matrix
p = length(indx_p)
Rp = cov(X_out)
R0 = cov(X_in)
library(ggplot2)
library(reshape2)
df1 <- melt(R0)
df2 <- melt(Rp)
N_in = nrow(X_in)
N_out = nrow(X_out)
p1 <- ggplot(df1, aes(Var1, Var2, fill = value)) +
  geom_tile() +
  scale_fill_gradient2(low="blue", mid="white", high="red") +
  coord_fixed() +
  ggtitle(sprintf("In-sample Cov, %d samples", nrow(X_in)))
p2 <- ggplot(df2, aes(Var1, Var2, fill = value)) +
  geom_tile() +
  scale_fill_gradient2(low="blue", mid="white", high="red") +
  coord_fixed() +
  ggtitle(sprintf("Out-of-sample Cov, %d samples", nrow(X_out)))
library(gridExtra)
grid.arrange(p1, p2, ncol = 2)

Version Author Date
691542f dodat97 2025-12-01

They look pretty similar, let us see what is the MLE of \(\nu_0\) in the likelihood \(IW(R_0 | mean = R', df=\nu_0 + p + 1)\).

## IW and examine IW likelihood  
#### log IW(R0 | nu0 * Rp, nu0 + J + 1)
log_multigamma_vec <- function(a, p) {
  # vectorized multivariate gamma
  j <- 1:p
  # sum over j, but broadcasting a over j
  (p*(p-1)/4)*log(pi) +
    rowSums(matrix(lgamma(a), nrow=length(a), ncol=p, byrow=FALSE) +
              matrix((1 - j)/2, nrow=length(a), ncol=p, byrow=TRUE))
}

log_iw <- function(R0, Rp, nu_vec) {
  p <- nrow(R0)
  jitter = 1e-8
  R0 = R0 + jitter * diag(rep(1, p))
  Rp = Rp + jitter * diag(rep(1, p))
  # Precompute expensive shared quantities
  logdet_nu_Rp <- determinant(Rp, logarithm = TRUE)$modulus + p * log(nu_vec)
  logdetR0   <- determinant(R0,   logarithm = TRUE)$modulus
  tr_term   <- nu_vec * sum(t(Rp) * solve(R0))
  llhs = (.5 * (nu_vec + p + 1) * logdet_nu_Rp
          - .5 * (nu_vec + p + 1) * p * log(2)
          - log_multigamma_vec((nu_vec + p + 1) / 2, p)
          - .5 * (nu_vec + 2 * (p + 1)) * logdetR0
          - .5 * tr_term)
  as.numeric(llhs)
}

nu_vec = c(1:100) 
llhs = log_iw(R0, Rp, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood")

Version Author Date
691542f dodat97 2025-12-01

The likelihood just decreases linearly as \(\nu_0\) increases. It is because of the term \(\mathrm{trace}((R_0)^{-1} R')\) that dominates the log det term, and \(\nu_0\) is multiplied into this trace term in the log-likelihood. This numerical behavior is similar to when calculating \(v^\top R^{-1} v\) in the presentation in the group meeting we saw earlier (curse of dimensionality).

jitter = 1e-8
R0_ = R0 + jitter * diag(rep(1, p))
Rp_ = Rp + jitter * diag(rep(1, p))
trace_term = sum(t(Rp_) * solve(R0_))
logdet_nu_Rp <- determinant(Rp_, logarithm = TRUE)$modulus
logdetR0   <- determinant(R0_,   logarithm = TRUE)$modulus

print(trace_term)
[1] 299475.8
print(logdetR0)
[1] -227.7098
attr(,"logarithm")
[1] TRUE
print(logdet_nu_Rp)
[1] -217.3115
attr(,"logarithm")
[1] TRUE
## now let's try to turn it into the PCA-like form and learn nu again

eig <- eigen(Rp)
eig_cumsum = cumsum(eig$values)
percent_explained = .99
eig_cumsum = cumsum(eig$values)
q = sum(eig_cumsum < percent_explained * eig_cumsum[p]) 
sprintf("%d first principle components explain %.1f percent of variance", q, percent_explained*100)
[1] "26 first principle components explain 99.0 percent of variance"
lambda <- eig$values
U <- eig$vectors
sigma2_est <- mean(lambda[(q+1):p])
L_diag <- sqrt(lambda[1:q] - sigma2_est)
# L_diag <- sqrt(lambda[1:q])
W_ppca <- U[,1:q] %*% diag(L_diag)
Rp_PCA <- W_ppca %*% t(W_ppca) + sigma2_est * diag(p)

eig <- eigen(R0)
lambda <- eig$values
U <- eig$vectors
sigma2_est <- mean(lambda[(q+1):p])
L_diag <- sqrt(lambda[1:q] - sigma2_est)
# L_diag <- sqrt(lambda[1:q])
W_ppca <- U[,1:q] %*% diag(L_diag)
R0_PCA <- W_ppca %*% t(W_ppca) + sigma2_est * diag(p)

df3 <- melt(R0_PCA)
df4 <- melt(Rp_PCA)
p3 <- ggplot(df3, aes(Var1, Var2, fill = value)) +
  geom_tile() +
  scale_fill_gradient2(low="blue", mid="white", high="red") +
  coord_fixed() +
  ggtitle(paste0("In-sample PCA"))
p4 <- ggplot(df4, aes(Var1, Var2, fill = value)) +
  geom_tile() +
  scale_fill_gradient2(low="blue", mid="white", high="red") +
  coord_fixed() +
  ggtitle(paste0("Out-of-sample PCA"))

grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)

Version Author Date
691542f dodat97 2025-12-01 
nu_vec = c(1:200) 
llhs = log_iw(R0_PCA, Rp_PCA, nu_vec)
# llhs = log_iw(R0_PCA, Rp, nu_vec)
# llhs = log_iw(R0, Rp_PCA, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood")

Version Author Date
691542f dodat97 2025-12-01 
print(nu_vec[which.max(llhs)])
[1] 43

Seems like we get more reasonable estimate of \(\nu_0\) with 99% variance-explained PCA version of the covariance matrix.


sessionInfo()
R version 4.5.1 (2025-06-13)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.0

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: America/Chicago
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] gridExtra_2.3   reshape2_1.4.4  ggplot2_3.5.2   Matrix_1.7-3   
[5] susieR_0.14.2   workflowr_1.7.2

loaded via a namespace (and not attached):
 [1] sass_0.4.10        stringi_1.8.7      lattice_0.22-7     digest_0.6.37     
 [5] magrittr_2.0.3     evaluate_1.0.5     grid_4.5.1         RColorBrewer_1.1-3
 [9] fastmap_1.2.0      rprojroot_2.1.1    plyr_1.8.9         jsonlite_2.0.0    
[13] processx_3.8.6     whisker_0.4.1      reshape_0.8.10     ps_1.9.1          
[17] mixsqp_0.3-54      promises_1.5.0     httr_1.4.7         scales_1.4.0      
[21] jquerylib_0.1.4    cli_3.6.5          rlang_1.1.6        crayon_1.5.3      
[25] withr_3.0.2        cachem_1.1.0       yaml_2.3.10        otel_0.2.0        
[29] tools_4.5.1        httpuv_1.6.16      vctrs_0.6.5        R6_2.6.1          
[33] matrixStats_1.5.0  lifecycle_1.0.4    git2r_0.36.2       stringr_1.5.2     
[37] fs_1.6.6           irlba_2.3.5.1      pkgconfig_2.0.3    callr_3.7.6       
[41] pillar_1.11.0      bslib_0.9.0        later_1.4.4        gtable_0.3.6      
[45] glue_1.8.0         Rcpp_1.1.0         xfun_0.53          tibble_3.3.0      
[49] rstudioapi_0.17.1  knitr_1.50         farver_2.1.2       htmltools_0.5.8.1 
[53] labeling_0.4.3     rmarkdown_2.29     compiler_4.5.1     getPass_0.2-4