Last updated: 2025-12-01

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Rmd b9f08b6 dodat 2025-12-01 wflow_publish(c("analysis/df-nu-low-rank.Rmd")) 
seed = 4  ## change this value to see other examples
library(susieR)
Warning: package 'susieR' was built under R version 4.3.3
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 7154
X = 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)
Warning: package 'ggplot2' was built under R version 4.3.3
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)

eig <- eigen(Rp)
plot(eig$values,
        main = "Eigenvalues of Rp",
        ylab = "Value",
        xlab = "Eigenvalue index")

print(eig$values)
 [1]  1.434888e+00  1.232170e+00  7.860173e-01  3.563616e-01  2.524898e-01
 [6]  1.633233e-01  7.745090e-02  5.945454e-02  5.034101e-02  3.878267e-02
[11]  1.884220e-02  1.494242e-02  8.782745e-03  5.369828e-03  5.118441e-03
[16]  3.705699e-03  2.443832e-03  2.165237e-03  1.968695e-03  1.796069e-03
[21]  1.880547e-04  5.819595e-05  3.450706e-06  1.609377e-06  3.925945e-16
[26]  3.028957e-16  1.453242e-17  2.324990e-18 -3.267752e-17 -4.824910e-17
### Select rank
eig <- eigen(Rp)
eig_cumsum = cumsum(eig$values)
eig_cumsum[p]
[1] 4.516665
r_p = sum(eig_cumsum < 0.95 * eig_cumsum[p]) ## percentage variance explained
print(paste0("Chosen rank is ", r_p))
[1] "Chosen rank is 6"
Vp = eig$vectors[, c(1:r_p)]
Dp = diag(eig$values[c(1:r_p)])
print(sum((Rp - Vp %*% Dp %*% t(Vp))**2))
[1] 0.0143138
Rp_FA = Vp %*% Dp %*% t(Vp) + diag(rep(1, p)) * sum(eig$values[c(r_p + 1, p)])

eig <- eigen(R0)
V0 = eig$vectors[, c(1:r_p)]
D0 = diag(eig$values[c(1:r_p)])
print(sum((R0 - V0 %*% D0 %*% t(V0))**2))
[1] 0.005421943
R0_FA = V0 %*% D0 %*% t(V0) + diag(rep(1, p)) * sum(eig$values[c(r_p + 1, p)])

df3 <- melt(R0_FA)
df4 <- melt(Rp_FA)
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 FA"))
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 FA"))

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

#### 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-10
  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_FA, Rp_FA, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood")

print(nu_vec[which.max(llhs)])
[1] 67

We can look at nu around the optimal point more closely.

nu_opt = nu_vec[which.max(llhs)]
nu_vec1 = c((nu_opt - 10): (nu_opt + 10)) 
llhs1 = log_iw(R0_FA, Rp_FA, nu_vec1)
plot(nu_vec1, llhs1, xlab = "nu value", ylab = "log-likelihood")

snp_total = ncol(X0)
n = nrow(X0)
p = 1000
# Start from a random point on the genome
indx_start = sample(1: (snp_total - p), 1)
X = X0[, indx_start:(indx_start + p -1)]
X_subsample = X[c(1:100), ]
# X_subsample = X
R_sample = cor(t(X_subsample))
df = melt(R_sample)
ggplot(df, aes(Var1, Var2, fill = value)) +
  geom_tile() +
  scale_fill_gradient2(low="blue", mid="white", high="red") +
  coord_fixed() +
  ggtitle(paste0("Correlation matrix of 100 individuals"))


sessionInfo()
R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS 15.6.1

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0

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_1.6-1.1 
[5] susieR_0.14.2   workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] sass_0.4.7         utf8_1.2.4         generics_0.1.3     stringi_1.7.12    
 [5] lattice_0.22-5     digest_0.6.33      magrittr_2.0.3     evaluate_0.23     
 [9] grid_4.3.2         RColorBrewer_1.1-3 fastmap_1.1.1      plyr_1.8.9        
[13] rprojroot_2.0.3    jsonlite_1.8.7     processx_3.8.2     whisker_0.4.1     
[17] reshape_0.8.10     mixsqp_0.3-54      ps_1.7.5           promises_1.3.3    
[21] httr_1.4.7         fansi_1.0.5        scales_1.4.0       jquerylib_0.1.4   
[25] cli_3.6.1          rlang_1.1.1        crayon_1.5.2       withr_2.5.2       
[29] cachem_1.0.8       yaml_2.3.7         tools_4.3.2        dplyr_1.1.3       
[33] httpuv_1.6.16      vctrs_0.6.4        R6_2.5.1           matrixStats_1.0.0 
[37] lifecycle_1.0.3    git2r_0.36.2       stringr_1.5.0      fs_1.6.3          
[41] irlba_2.3.5.1      pkgconfig_2.0.3    callr_3.7.3        pillar_1.9.0      
[45] bslib_0.5.1        later_1.4.2        gtable_0.3.6       glue_1.6.2        
[49] Rcpp_1.1.0         xfun_0.52          tibble_3.2.1       tidyselect_1.2.0  
[53] highr_0.10         rstudioapi_0.15.0  knitr_1.45         farver_2.1.1      
[57] htmltools_0.5.7    labeling_0.4.3     rmarkdown_2.25     compiler_4.3.2    
[61] getPass_0.2-4