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This document assumes basic familiarity with Markov chains and linear algebra.


In this note, we illustrate one way of analytically obtaining the stationary distribution for a finite discrete Markov chain.

3x3 example

Assume our probability transition matrix is: \[P = \begin{bmatrix} 0.7 & 0.2 & 0.1 \\ 0.4 & 0.6 & 0 \\ 0 & 1 & 0 \end{bmatrix}\]

Since every state is accessible from every other state, this Markov chain is irreducible. Every irreducible finite state space Markov chain has a unique stationary distribution. Recall that the stationary distribution \(\pi\) is the row vector such that \[\pi = \pi P\].

Therefore, we can find our stationary distribution by solving the following linear system: \[\begin{align*} 0.7\pi_1 + 0.4\pi_2 &= \pi_1 \\ 0.2\pi_1 + 0.6\pi_2 + \pi_3 &= \pi_2 \\ 0.1\pi_1 &= \pi_3 \end{align*}\] subject to \(\pi_1 + \pi_2 + \pi_3 = 1\). Putting these four equations together and moving all of the variables to the left hand side, we get the following linear system: \[\begin{align*} -0.3\pi_1 + 0.4\pi_2 &= 0 \\ 0.2\pi_1 + -0.4\pi_2 + \pi_3 &= 0 \\ 0.1\pi_1 - \pi_3 &= 0 \\ \pi_1 + \pi_2 + \pi_3 &= 1 \end{align*}\]

We will define the linear system in matrix notation: \[\underbrace{\begin{bmatrix} -0.3 & 0.4 & 0 \\ 0.2 & -0.4 & 1 \\ 0.1 & 0 & -1 \\ 1 & 1 & 1 \end{bmatrix}}_A \begin{bmatrix} \pi_1 \\ \pi_2 \\ \pi_3 \end{bmatrix} = \underbrace{\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}}_b \\ A\pi^T = b\]

The stationary distribution, which is usually represented by a row vector, is transposed with \(\pi^T\).

Since this linear system has more equations than unknowns, it is an overdetermined system. Overdetermined systems can be solved using a QR decomposition, so we use that here. (In brief, qr.solve works by finding the QR decomposition of \(A\), \(A=QR\) with \(Q'Q=I\) and \(R\) an upper triangular matrix. Then if \(A\pi^T = b\) it must be the case that \(QR\pi^T=b\) which implies \(R\pi^T = Q'b\), and this can be solved easily because \(R\) is triangular.)

A        <- matrix(c(-0.3, 0.2, 0.1, 1, 0.4, -0.4, 0, 1, 0, 1, -1, 1 ), ncol=3,nrow=4)
b        <- c(0,0,0, 1)
pi        <- qr.solve(A,b)
names(pi) <- c('state.1', 'state.2', 'state.3')
   state.1    state.2    state.3 
0.54054054 0.40540541 0.05405405 

We find that: \[\begin{align*} \pi_1 \approx 0.54, \pi_2 \approx 0.41, \pi_3 \approx 0.05 \end{align*}\]

Therefore, under proper conditions, we expect the Markov chain to spend more time in states 1 and 2 as the chain evolves.

The General Approach

Recall that we are attempting to find a solution to \[\pi = \pi P\] such that \(\sum_i \pi_i =1\). First we rearrange the expression above to get: \[\begin{align} \pi - \pi P &= 0 \\ \pi (I - P) &= 0 \\ (I - P)^T\pi^T &= 0 \end{align}\]

One challenge though is that we need the constrained solution which respects that \(\pi\) describes a probability distribution (i.e. \(\sum \pi_i = 1\)). Luckily this is a linear constraint that is easily represented by adding another equation to the system. So as a small trick, we need to add a row of all 1’s to our \((I-P)^T\) (call this new matrix \(A\)) and a 1 to the last element of the zero vector on the right hand side (call this new vector \(b\)). Now we want to solve \(A\pi = b\) which is over-determined so we solve it as above using qr.solve.

The function stationary below implements the general approach, and we test it with the worked example above.

stationary <- function(transition) {
  stopifnot(is.matrix(transition) &&
        nrow(transition)==ncol(transition) &&
            all(transition>=0 & transition<=1))
  p <- diag(nrow(transition)) - transition
  A <- rbind(t(p),
         rep(1, ncol(transition)))
  b <- c(rep(0, nrow(transition)),
  res <- qr.solve(A, b)
  names(res) <- paste0("state.", 1:nrow(transition))
stationary(matrix(c(0.7, 0.2, 0.1, 0.4, 0.6, 0, 0, 1, 0),
                  nrow=3, byrow=TRUE))
   state.1    state.2    state.3 
0.54054054 0.40540541 0.05405405 

R version 3.6.2 (2019-12-12)
Platform: x86_64-apple-darwin18.7.0 (64-bit)
Running under: macOS Catalina 10.15.7

Matrix products: default
BLAS/LAPACK: /usr/local/Cellar/openblas/0.3.7/lib/libopenblasp-r0.3.7.dylib

[1] de_CH.UTF-8/de_CH.UTF-8/de_CH.UTF-8/C/de_CH.UTF-8/de_CH.UTF-8

attached base packages:
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 [1] workflowr_1.6.2   Rcpp_1.0.6        rprojroot_2.0.2   digest_0.6.27    
 [5] later_1.2.0       R6_2.5.0          git2r_0.28.0      magrittr_2.0.1   
 [9] evaluate_0.14     stringi_1.6.2     rlang_0.4.11      fs_1.5.0         
[13] promises_1.2.0.1  whisker_0.4       rmarkdown_2.8     tools_3.6.2      
[17] stringr_1.4.0     glue_1.4.2        httpuv_1.6.1      xfun_0.23        
[21] yaml_2.2.1        compiler_3.6.2    htmltools_0.5.1.1 knitr_1.33       

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