Last updated: 2021-05-30

Checks: 6 1

Knit directory: fiveMinuteStats/analysis/

This reproducible R Markdown analysis was created with workflowr (version 1.6.2). The Checks tab describes the reproducibility checks that were applied when the results were created. The Past versions tab lists the development history.


The R Markdown file has unstaged changes. To know which version of the R Markdown file created these results, you’ll want to first commit it to the Git repo. If you’re still working on the analysis, you can ignore this warning. When you’re finished, you can run wflow_publish to commit the R Markdown file and build the HTML.

Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.

The command set.seed(12345) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.

Great job! Recording the operating system, R version, and package versions is critical for reproducibility.

Nice! There were no cached chunks for this analysis, so you can be confident that you successfully produced the results during this run.

Great job! Using relative paths to the files within your workflowr project makes it easier to run your code on other machines.

Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility.

The results in this page were generated with repository version b26f1bd. See the Past versions tab to see a history of the changes made to the R Markdown and HTML files.

Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:


Untracked files:
    Untracked:  analysis/.stationary_distribution.Rmd.swp

Unstaged changes:
    Modified:   analysis/stationary_distribution.Rmd

Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.


These are the previous versions of the repository in which changes were made to the R Markdown (analysis/stationary_distribution.Rmd) and HTML (docs/stationary_distribution.html) files. If you’ve configured a remote Git repository (see ?wflow_git_remote), click on the hyperlinks in the table below to view the files as they were in that past version.

File Version Author Date Message
html acd0a14 Matthew Stephens 2021-05-08 Build site.
Rmd fc5fe81 Matthew Stephens 2021-05-08 workflowr::wflow_publish(“analysis/stationary_distribution.Rmd”)
Rmd e4436fa GitHub 2021-04-20 fixes small error in typesetting
html 5f62ee6 Matthew Stephens 2019-03-31 Build site.
Rmd 0cd28bd Matthew Stephens 2019-03-31 workflowr::wflow_publish(all = TRUE)
html 34bcc51 John Blischak 2017-03-07 Build site.
Rmd 5fbc8b5 John Blischak 2017-03-07 Update workflowr project with wflow_update (version 0.4.0).
Rmd 391ba3c John Blischak 2017-03-07 Remove front and end matter of non-standard templates.
html 8e61683 Marcus Davy 2017-03-03 rendered html using wflow_build(all=TRUE)
html 5d0fa13 Marcus Davy 2017-03-02 wflow_build() rendered html files
Rmd d674141 Marcus Davy 2017-02-26 typos, refs
html c3b365a John Blischak 2017-01-03 Build site.
Rmd 67a8575 John Blischak 2017-01-03 Use external chunk to set knitr chunk options.
Rmd 5ec12c7 John Blischak 2017-01-03 Use session-info chunk.
Rmd 3bb3b73 mbonakda 2016-02-25 add two mixture model vignettes + merge redundant info in markov chain vignettes
Rmd f209a9a mbonakda 2016-01-30 fix up typos
Rmd 0f93e3c mbonakda 2016-01-30 split simulating discrete markov chains into three separate notes

Pre-requisites

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

Overview

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')
pi 
   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)),
         1)
  res <- qr.solve(A, b)
  names(res) <- paste0("state.", 1:nrow(transition))
  return(res)
}
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 

sessionInfo()
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

locale:
[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:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [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       

This site was created with R Markdown