**Last updated:** 2019-03-31

**Checks:** 6 0

**Knit directory:** `fiveMinuteStats/analysis/`

This reproducible R Markdown analysis was created with workflowr (version 1.2.0). The *Report* tab describes the reproducibility checks that were applied when the results were created. The *Past versions* tab lists the development history.

`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! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated.

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: .Rhistory
Ignored: .Rproj.user/
Ignored: analysis/.Rhistory
Ignored: analysis/bernoulli_poisson_process_cache/
Untracked files:
Untracked: _workflowr.yml
Untracked: analysis/CI.Rmd
Untracked: analysis/gibbs_structure.Rmd
Untracked: analysis/libs/
Untracked: analysis/results.Rmd
Untracked: analysis/shiny/tester/
Untracked: docs/MH_intro_files/
Untracked: docs/citations.bib
Untracked: docs/figure/MH_intro.Rmd/
Untracked: docs/figure/hmm.Rmd/
Untracked: docs/hmm_files/
Untracked: docs/libs/
Untracked: docs/shiny/tester/
```

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 R Markdown and HTML files. If you’ve configured a remote Git repository (see `?wflow_git_remote`

), click on the hyperlinks in the table below to view them.

File | Version | Author | Date | Message |
---|---|---|---|---|

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 | fb0f6e3 | stephens999 | 2017-03-03 | Merge pull request #33 from mdavy86/f/review |

html | 0713277 | stephens999 | 2017-03-03 | Merge pull request #31 from mdavy86/f/review |

Rmd | d674141 | Marcus Davy | 2017-02-27 | 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 | bb814ef | jnovembre | 2016-01-31 | Initial commit |

An understanding of matrix multiplication and matrix powers.

Here we provide a quick introduction to discrete Markov Chains.

A Markov Chain is a discrete stochastic process with the *Markov property* : \(P(X_t|X_{t-1},\ldots,X_1)= P(X_t|X_{t-1})\). It is fully determined by a probability transition matrix \(P\) which defines the transition probabilities (\(P_ij=P(X_t=j|X_{t-1}=i)\) and an initial probability distribution specified by the vector \(x\) where \(x_i=P(X_0=i)\). The time-dependent random variable \(X_t\) is describing the state of our probabilistic system at time-step \(t\).

In Sheldon Ross’s Introduction to Probability Models, he has an example (4.3) of a Markov Chain for modeling Gary’s mood. Gary alternates between 3 state: Cheery (\(X=1\)), So-So (\(X=2\)), or Glum (\(X=3\)). Here we input the \(P\) matrix given by Ross and we input an arbitrary initial probability matrix.

```
# Define prob transition matrix
# (note matrix() takes vectors in column form so there is a transpose here to switch col's to row's)
P=t(matrix(c(c(0.5,0.4,0.1),c(0.3,0.4,0.3),c(0.2,0.3,0.5)),nrow=3))
# Check sum across = 1
apply(P,1,sum)
```

`[1] 1 1 1`

```
# Definte initial probability vector
x0=c(0.1,0.2,0.7)
# Check sums to 1
sum(x0)
```

`[1] 1`

If initial prob distribution \(x_0\) is \(3 \times 1\) column vector, then \(x_0^T P= x_1^T\). In R, the %*% operator automatically promotes a vector to the appropriate matrix to make the arguments conformable. In the case of multiplying a length 3 vector by a \(3\time 3\) matrix, it takes the vector to be a row-vector. This means our math can look simply:

```
# After one step
x0%*%P
```

```
[,1] [,2] [,3]
[1,] 0.25 0.33 0.42
```

And after two time-steps:

```
## The two-step prob trans matrix
P%*%P
```

```
[,1] [,2] [,3]
[1,] 0.39 0.39 0.22
[2,] 0.33 0.37 0.30
[3,] 0.29 0.35 0.36
```

```
## Multiplied by the initial state probability
x0%*%P%*%P
```

```
[,1] [,2] [,3]
[1,] 0.308 0.358 0.334
```

To generalize to an arbitrary number of time steps into the future, we can compute a the matrix power. In R, this can be done easily with the package expm. Let’s load the library and verify the second power is the same as we saw for P%*%P above.

```
# Load library
library(expm)
```

`Loading required package: Matrix`

```
Attaching package: 'expm'
```

```
The following object is masked from 'package:Matrix':
expm
```

```
# Verify the second power is P%*%P
P%^%2
```

```
[,1] [,2] [,3]
[1,] 0.39 0.39 0.22
[2,] 0.33 0.37 0.30
[3,] 0.29 0.35 0.36
```

And now let’s push this : Looking at the state of the chain after many steps, say 100. First let’s look at the probability transition matrix…

`P%^%100`

```
[,1] [,2] [,3]
[1,] 0.3387097 0.3709677 0.2903226
[2,] 0.3387097 0.3709677 0.2903226
[3,] 0.3387097 0.3709677 0.2903226
```

What do you notice about the rows? And let’s see what this does for various starting distributions:

`c(1,0,0) %*%(P%^%100)`

```
[,1] [,2] [,3]
[1,] 0.3387097 0.3709677 0.2903226
```

`c(0.2,0.5,0.3) %*%(P%^%100)`

```
[,1] [,2] [,3]
[1,] 0.3387097 0.3709677 0.2903226
```

Note that after a large number of steps the initial state does not matter any more, the probability of the chain being in any state \(j\) is independent of where we started. This is our first view of the *equilibrium distribuion* of a Markov Chain. These are also known as the *limiting probabilities of a Markov chain* or *stationary distribution*.

`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
other attached packages:
[1] expm_0.999-3 Matrix_1.2-15
loaded via a namespace (and not attached):
[1] workflowr_1.2.0 Rcpp_1.0.0 lattice_0.20-38 digest_0.6.18
[5] rprojroot_1.3-2 grid_3.5.2 backports_1.1.3 git2r_0.24.0
[9] magrittr_1.5 evaluate_0.12 stringi_1.2.4 fs_1.2.6
[13] whisker_0.3-2 rmarkdown_1.11 tools_3.5.2 stringr_1.3.1
[17] glue_1.3.0 xfun_0.4 yaml_2.2.0 compiler_3.5.2
[21] htmltools_0.3.6 knitr_1.21
```

This site was created with R Markdown