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## ----setup, include=FALSE------------------------------------------------
knitr::opts_chunk$set(echo = TRUE)
## ---- message = FALSE, echo = TRUE, eval = FALSE-------------------------
#
# # oracle precision matrix
# Omega = matrix(0.9, ncol = 100, nrow = 100)
# diag(Omega = 1)
#
# # generate covariance matrix
# S = qr.solve(Omega)
#
# # generate data
# Z = matrix(rnorm(100*50), nrow = 50, ncol = 100)
# out = eigen(S, symmetric = TRUE)
# S.sqrt = out$vectors %*% diag(out$values^0.5) %*% t(out$vectors)
# X = Z %*% S.sqrt
#
## ---- message = FALSE, echo = TRUE, eval = FALSE-------------------------
#
# # oracle precision matrix
# Omega = matrix(0.9, ncol = 10, nrow = 10)
# diag(Omega = 1)
#
# # generate covariance matrix
# S = qr.solve(Omega)
#
# # generate data
# Z = matrix(rnorm(10*1000), nrow = 1000, ncol = 10)
# out = eigen(S, symmetric = TRUE)
# S.sqrt = out$vectors %*% diag(out$values^0.5) %*% t(out$vectors)
# X = Z %*% S.sqrt
#
## ---- message = FALSE, echo = TRUE, eval = FALSE, tidy = FALSE-----------
#
# # generate eigen values
# eigen = c(rep(1000, 5, rep(1, 100 - 5)))
#
# # randomly generate orthogonal basis (via QR)
# Q = matrix(rnorm(100*100), nrow = 100, ncol = 100) %>% qr %>% qr.Q
#
# # generate covariance matrix
# S = Q %*% diag(eigen) %*% t(Q)
#
# # generate data
# Z = matrix(rnorm(100*50), nrow = 50, ncol = 100)
# out = eigen(S, symmetric = TRUE)
# S.sqrt = out$vectors %*% diag(out$values^0.5) %*% t(out$vectors)
# X = Z %*% S.sqrt
#
## ---- message = FALSE, echo = TRUE, eval = FALSE, tidy = FALSE-----------
#
# # generate eigen values
# eigen = c(rep(1000, 5, rep(1, 10 - 5)))
#
# # randomly generate orthogonal basis (via QR)
# Q = matrix(rnorm(10*10), nrow = 10, ncol = 10) %>% qr %>% qr.Q
#
# # generate covariance matrix
# S = Q %*% diag(eigen) %*% t(Q)
#
# # generate data
# Z = matrix(rnorm(10*50), nrow = 50, ncol = 10)
# out = eigen(S, symmetric = TRUE)
# S.sqrt = out$vectors %*% diag(out$values^0.5) %*% t(out$vectors)
# X = Z %*% S.sqrt
#
## ---- message = FALSE, echo = TRUE, eval = FALSE, tidy = FALSE-----------
#
# # generate covariance matrix
# # (can confirm inverse is tri-diagonal)
# S = matrix(0, nrow = 100, ncol = 100)
# for (i in 1:100){
# for (j in 1:100){
# S[i, j] = 0.7^abs(i - j)
# }
# }
#
# # generate data
# Z = matrix(rnorm(10*50), nrow = 50, ncol = 10)
# out = eigen(S, symmetric = TRUE)
# S.sqrt = out$vectors %*% diag(out$values^0.5) %*% t(out$vectors)
# X = Z %*% S.sqrt
#
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