Investigate learn degree-of-freedom nu
Last updated: 2025-11-24
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Knit directory: Improved_LD_SuSiE/
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seed = 4 ## change this value to see other examples
library(susieR)
library(Matrix)
set.seed(seed)
setwd("~/Documents/Improved_LD_SuSiE")
gtex = readRDS("data/Thyroid_ENSG00000132855.rds")
maf = apply(gtex, 2, function(x) sum(x)/2/length(x))
X0 = gtex[, maf > 0.01]
dim(X0)[1] 574 7154X = na.omit(X0)
snp_total = ncol(X0)
n = nrow(X0)
p = 30
# 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 = 250
out_sample = sample(1:n, out_sample_size)
X_out = X[out_sample, ]
X_in = X[setdiff(1:n, 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(paste0("In-sample Cov, sample =", 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(paste0("Out-of-sample Cov, sample =", nrow(X_out)))
library(gridExtra)
grid.arrange(p1, p2, ncol = 2)
The in-sample R0 and out-of-sample Rp look pretty similar. Now let us plot the log density \(IW(R_0 | \nu R', \nu + p + 1)\) for various \(\nu\)
#### 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-6
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) / 10
llhs = log_iw(R0, Rp, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood")
The trace term is too large compared to other term:
R0 = R0 + 1e-6 * diag(rep(1, p))
Rp = Rp + 1e-6 * diag(rep(1, p))
print(sum(t(Rp) * solve(R0))) ## trace term[1] 11942.6determinant(R0, logarithm = TRUE)$modulus ## log det term[1] -202.9556
attr(,"logarithm")
[1] TRUEThis can be because both matrix R0 and Rp are almost low-rank (due to LD). Let us try two things (1) Plot the eigenvalues of R0 and Rp; (2) try this experiment again with full-rank population covariance matrix.
eig <- eigen(Rp)
plot(eig$values,
main = "Eigenvalues of Rp",
ylab = "Value",
xlab = "Eigenvalue index")
| Version | Author | Date |
|---|---|---|
| 701ab1e | dodat | 2025-11-24 |
print(eig$values) [1] 1.434889e+00 1.232171e+00 7.860183e-01 3.563626e-01 2.524908e-01
[6] 1.633243e-01 7.745190e-02 5.945554e-02 5.034201e-02 3.878367e-02
[11] 1.884320e-02 1.494342e-02 8.783745e-03 5.370828e-03 5.119441e-03
[16] 3.706699e-03 2.444832e-03 2.166237e-03 1.969695e-03 1.797069e-03
[21] 1.890547e-04 5.919595e-05 4.450706e-06 2.609377e-06 1.000000e-06
[26] 1.000000e-06 1.000000e-06 1.000000e-06 1.000000e-06 1.000000e-06- Let us try re-doing this experiment with the full-rank covariance matrix:
X_in = matrix(rnorm(N_in * p), nrow=N_in, ncol=p)
X_out = matrix(rnorm(N_out * p), nrow=N_out, ncol=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(paste0("In-sample Cov, sample =", 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(paste0("Out-of-sample Cov, sample =", nrow(X_out)))
library(gridExtra)
grid.arrange(p1, p2, ncol = 2)
| Version | Author | Date |
|---|---|---|
| 701ab1e | dodat | 2025-11-24 |
nu_vec = c(1:100)
llhs = log_iw(R0, Rp, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood")
print(paste0("the optimal nu for full-rank R is ", nu_vec[which.max(llhs)]))[1] "the optimal nu for full-rank R is 58"TO-DO: (1) Try to vary rank \(r\) in the low-rank-plus-diag (2) Try to directly model the low-rank part of \(R\), i.e., set \(R = V C V^\top\) for \(R \in \mathbb{R}^{p\times r}\) and \(C \in \mathbb{R}^{r\times r}\) and model \(C\) by Inverse-Wishart instead of \(R\). (Does this corresponds to the generalized Inverse-Wishart?)
sessionInfo()R version 4.5.1 (2025-06-13)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.6.1
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_1.7.1
loaded via a namespace (and not attached):
[1] sass_0.4.10 generics_0.1.4 stringi_1.8.7 lattice_0.22-7
[5] digest_0.6.37 magrittr_2.0.3 evaluate_1.0.4 grid_4.5.1
[9] RColorBrewer_1.1-3 fastmap_1.2.0 plyr_1.8.9 rprojroot_2.1.0
[13] jsonlite_2.0.0 processx_3.8.6 whisker_0.4.1 reshape_0.8.10
[17] ps_1.9.1 mixsqp_0.3-54 promises_1.3.3 httr_1.4.7
[21] scales_1.4.0 jquerylib_0.1.4 cli_3.6.5 rlang_1.1.6
[25] crayon_1.5.3 withr_3.0.2 cachem_1.1.0 yaml_2.3.10
[29] tools_4.5.1 dplyr_1.1.4 httpuv_1.6.16 vctrs_0.6.5
[33] R6_2.6.1 matrixStats_1.5.0 lifecycle_1.0.4 git2r_0.36.2
[37] stringr_1.5.1 fs_1.6.6 irlba_2.3.5.1 pkgconfig_2.0.3
[41] callr_3.7.6 pillar_1.11.0 bslib_0.9.0 later_1.4.2
[45] gtable_0.3.6 glue_1.8.0 Rcpp_1.1.0 xfun_0.52
[49] tibble_3.3.0 tidyselect_1.2.1 rstudioapi_0.17.1 knitr_1.50
[53] farver_2.1.2 htmltools_0.5.8.1 labeling_0.4.3 rmarkdown_2.29
[57] compiler_4.5.1 getPass_0.2-4