knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
library(caracas)
inline_code <- function(x) { x } if (!has_sympy()) { # SymPy not available, so the chunks shall not be evaluated knitr::opts_chunk$set(eval = FALSE) inline_code <- function(x) { deparse(substitute(x)) } }
N <- 3 L1 <- diag(4) L1[cbind(1 + (1:N), 1:N)] <- "-a" L1 <- as_sym(L1)
e <- as_sym(paste0("e", 0:3)) x <- as_sym(paste0("x", 0:3)) u <- as_sym(paste0("u", 1:3)) y <- as_sym(paste0("y", 1:3)) eu <- c(e, u) xy <- c(x, y)
Consider this model:
$$
x_i = a x_{i-1} + e_i, \quad i=1, \dots, 3
$$
and $x_0=e_0$. All terms $e_0, \dots, e_3$ are independent and $N(0,v^2)$ distributed.
Let $e=(e_0, \dots, e_3)$ and $x=(x_0, \dots x_3)$. Hence $e \sim N(0, v^2 I)$.
Isolating error terms gives
$$
e= r inline_code(tex(e))
= r inline_code(tex(L1))
r inline_code(tex(x))
= L_1 x
$$
<<ar1>>
Since $\mathbf{Var}(e)=v^2 I$ we have $\mathbf{Var}(e)=v^2 I=L \mathbf{Var}(x) L'$ so the covariance matrix of $x$ is $V_1=\mathbf{Var}(x) = v^2 L^- (L^-)'$ while the concentration matrix (the inverse covariances matrix) is $K=v^{-2}L' L$.
def_sym(v2) L1inv <- inv(L1) V1 <- v2 * L1inv %*% t(L1inv) K1 <- (t(L1) %*% L1) / v2
cat( "\\begin{align} K_1 &= ", tex(K1), " \\\\ V_1 &= ", tex(V1), " \\end{align}", sep = "")
N <- 3 L2 <- diag("1", 1 + 2*N) L2[cbind(1 + (1:N), 1:N)] <- "-a" L2[cbind(1 + N + (1:N), 1 + 1:N)] <- "-b" L2 <- as_sym(L2)
Augment the $AR(1)$ process above with $y_i=b x_i + u_i$ for
$i=1,2,3$. Suppose $u_i\sim N(0, w^2)$ and all $u_i$ are independent
and inpendent of $e$.
Then
$(e,u)$ can be expressed in terms of $(x,y)$ as
$$
(e,u) = r inline_code(tex(eu))
= r inline_code(tex(L2))
r inline_code(tex(xy))
= L_2 (x,y)
$$
where
<<L2>>
Veu <- diag(1, 7) diag(Veu)[1:4] <- "v2" diag(Veu)[5:7] <- "w2" Veu Veu <- as_sym(Veu) Veu L2inv <- inv(L2) V2 <- L2inv %*% Veu %*% t(L2inv) K2 <- t(L2) %*% inv(Veu) %*% L2
cat( "\\begin{align} K_2 &= ", tex(K2), " \\\\ V_2 &= ", tex(V2), " \\end{align}", sep = "")
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