The normal markov chain as a multivariate normal

If you think a little you should be able to see that the above random walk simulation is actually simulating from a 1000-dimensional multivariate normal distribution!

Why?

Well, let’s write each of the \(N(0,1)\) variables we generate using rnorm() in that code as \(Z_1,Z_2,\dots\). Then: \[X_1 = Z_1\] \[X_2 = X_1 + Z_2 = Z_1 + Z_2\] \[X_3 = X_2 + Z_3 = Z_1 + Z_2 + Z_3\] etc.

So we can write \(X = AZ\) where \(A\) is the 1000 by 1000 matrix \[A = \begin{pmatrix} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 1 & 1 & 0 & \dots \\ \dots \end{pmatrix}.\]

Let’s take a look at what the covariance matrix Sigma looks like. (We get a good idea from just looking at the top left corner of the matrix what the pattern is)

A = matrix(0,nrow=1000,ncol=1000)
for(i in 1:1000){
    A[i,]=c(rep(1,i),rep(0,1000-i))
}
Sigma = A %*% t(A)
Sigma[1:10,1:10]

      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
 [1,]    1    1    1    1    1    1    1    1    1     1
 [2,]    1    2    2    2    2    2    2    2    2     2
 [3,]    1    2    3    3    3    3    3    3    3     3
 [4,]    1    2    3    4    4    4    4    4    4     4
 [5,]    1    2    3    4    5    5    5    5    5     5
 [6,]    1    2    3    4    5    6    6    6    6     6
 [7,]    1    2    3    4    5    6    7    7    7     7
 [8,]    1    2    3    4    5    6    7    8    8     8
 [9,]    1    2    3    4    5    6    7    8    9     9
[10,]    1    2    3    4    5    6    7    8    9    10

Now let us examine the precision matrix, \(\Omega\), which recall is the inverse of \(\Sigma\). Again we just show the top left corner of the precision matrix here.

Omega = chol2inv(chol(Sigma))
Omega[1:10,1:10]

      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
 [1,]    2   -1    0    0    0    0    0    0    0     0
 [2,]   -1    2   -1    0    0    0    0    0    0     0
 [3,]    0   -1    2   -1    0    0    0    0    0     0
 [4,]    0    0   -1    2   -1    0    0    0    0     0
 [5,]    0    0    0   -1    2   -1    0    0    0     0
 [6,]    0    0    0    0   -1    2   -1    0    0     0
 [7,]    0    0    0    0    0   -1    2   -1    0     0
 [8,]    0    0    0    0    0    0   -1    2   -1     0
 [9,]    0    0    0    0    0    0    0   -1    2    -1
[10,]    0    0    0    0    0    0    0    0   -1     2

Notice all the 0s in the precision matrix. This is because of the conditional independencies that occur in a Markov chain. In a Markov chain (any Markov chain) the conditional distribution of \(X_t\) given the other \(X_s\) (\(s \neq t\)) depends only on its neighbors \(X_{t-1}\) and \(X_{t+1}\). That is, \(X_{t}\) is conditionally independent of all other \(X_s\) given \(X_{t-1}\) and \(X_{t+1}\). This is exactly what we are seeing in the precision matrix above: the non-zero elements of the \(t\)th row are at coordinates \(t-1,t\) and \(t+1\).

Application

An application of this is imputation of missing values: suppose one of the \(X\) values is missing, say \(X_p\) is missing, but you know the covariance matrix and all the other \(X\) values. Then you could impute \(X_p\) by its conditional mean, which is a simple linear combination of the other values that can be read directly off the \(p\)th row of the precision matrix. This idea is the essence of Kriging.

Example

Consider the Markov chain above. The conditional distribution of \(X_1\) given all other \(X\) values is given by \[X_1 | X_2,X_3,\dots \sim N(X_2/2, 1/2).\]

And the conditional distribution of \(X_2\) given all other \(X\) values is \[X_2 | X_1,X_3, \dots \sim N((X_1+X_3)/2, 1/2).\] Similarly for \(X_p\) for \(p>2\). The intuition is that, if one wanted to guess what the value of \(X_p\) were given all other \(X\)s, the best guess would be the average of its neighbours.

Session information

sessionInfo()

R version 3.5.2 (2018-12-20)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.1

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] workflowr_1.2.0 Rcpp_1.0.0      digest_0.6.18   rprojroot_1.3-2
 [5] backports_1.1.3 git2r_0.24.0    magrittr_1.5    evaluate_0.12  
 [9] stringi_1.2.4   fs_1.2.6        whisker_0.3-2   rmarkdown_1.11 
[13] tools_3.5.2     stringr_1.3.1   glue_1.3.0      xfun_0.4       
[17] yaml_2.2.0      compiler_3.5.2  htmltools_0.3.6 knitr_1.21