Last updated: 2021-05-08
Checks: 7 0
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.
Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.
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 fc5fe81. 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:
Ignored files:
Ignored: .Rproj.user/
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 |
|---|---|---|---|---|
| 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 | 2017-03-06 | Build site. |
| Rmd | 5fbc8b5 | John Blischak | 2017-03-06 | Update workflowr project with wflow_update (version 0.4.0). |
| Rmd | 391ba3c | John Blischak | 2017-03-06 | 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 | 2017-01-02 | Build site. |
| Rmd | 67a8575 | John Blischak | 2017-01-02 | Use external chunk to set knitr chunk options. |
| Rmd | 5ec12c7 | John Blischak | 2017-01-02 | Use session-info chunk. |
| Rmd | 3bb3b73 | mbonakda | 2016-02-24 | 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 |
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.
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 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: \[\underbrace{\begin{bmatrix} -0.3 & 0.4 & 0 \\ 0.2 & -0.4 & 1 \\ 0.1 & 0 & -1 \\ 1 & 1 & 1 \end{bmatrix}}_A \begin{bmatrix} \pi_1 \\ \pi_2 \\ \pi_3 \end{bmatrix} = \underbrace{\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}}_b \\ A\pi = b\]
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 \(A\pi = b\) it must be the case that \(QR\pi=b\) which implies \(R\pi = Q'b\), and this can be solved easily because \(R\) is triangular.)
library(Matrix)
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.
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: \[\begin{align} \pi - \pi P &= 0 \\ (I - P)\pi &= 0 \end{align}\]
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 \((I-P)\) (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 overdetermine so we solve it as above using qr.solve.
sessionInfo()R version 4.0.5 (2021-03-31)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/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
other attached packages:
[1] Matrix_1.3-2
loaded via a namespace (and not attached):
[1] Rcpp_1.0.6 whisker_0.4 knitr_1.32 magrittr_2.0.1
[5] workflowr_1.6.2 lattice_0.20-41 R6_2.5.0 rlang_0.4.10
[9] fansi_0.4.2 stringr_1.4.0 tools_4.0.5 grid_4.0.5
[13] xfun_0.22 utf8_1.2.1 git2r_0.28.0 htmltools_0.5.1.1
[17] ellipsis_0.3.2 rprojroot_2.0.2 yaml_2.2.1 digest_0.6.27
[21] tibble_3.1.1 lifecycle_1.0.0 crayon_1.4.1 later_1.2.0
[25] vctrs_0.3.8 fs_1.5.0 promises_1.2.0.1 glue_1.4.2
[29] evaluate_0.14 rmarkdown_2.7 stringi_1.5.3 compiler_4.0.5
[33] pillar_1.6.0 httpuv_1.5.5 pkgconfig_2.0.3 This site was created with R Markdown