knitr::opts_chunk$set(echo = TRUE)
library(Ryacas)
Consider this model: $$ x_i = a x_0 + e_i, \quad i=1, \dots, 4 $$ and $x_0=e_0$. All terms $e_0, \dots, e_3$ are independent and $N(0,1)$ distributed. Let $e=(e_0, \dots, e_3)$ and $x=(x_0, \dots x_3)$. Isolating error terms gives that $$ e = L_1 x $$ where $L_1$ has the form
L1chr <- diag(4) L1chr[2:4, 1] <- "-a" L1 <- ysym(L1chr) L1
If error terms have variance $1$ then $\mathbf{Var}(e)=L \mathbf{Var}(x) L'$ so the covariance matrix is $V1=\mathbf{Var}(x) = L^- (L^-)'$ while the concentration matrix (the inverse covariances matrix) is $K=L' L$.
L1inv <- solve(L1) K1 <- t(L1) %*% L1 V1 <- L1inv %*% t(L1inv)
cat( "\\begin{align} K_1 &= ", tex(K1), " \\\\ V_1 &= ", tex(V1), " \\end{align}", sep = "")
Slightly more elaborate:
L2chr <- diag(4) L2chr[2:4, 1] <- c("-a1", "-a2", "-a3") L2 <- ysym(L2chr) L2 Vechr <- diag(4) Vechr[cbind(1:4, 1:4)] <- c("w1", "w2", "w2", "w2") Ve <- ysym(Vechr) Ve
L2inv <- solve(L2) K2 <- t(L2) %*% solve(Ve) %*% L2 V2 <- L2inv %*% Ve %*% t(L2inv)
cat( "\\begin{align} K_2 &= ", tex(K2), " \\\\ V_2 &= ", tex(V2), " \\end{align}", sep = "")
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