Pre-requisites

An understanding of matrix multiplication and matrix powers.

Overview

Here we provide a quick introduction to discrete Markov Chains.

Definition of a Markov Chain

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 specficied 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 \(t\).

Example: Gary’s mood

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 abitrary 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

What are the expected probability states after one or two steps?

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

What about an abitrary number of time steps?

To generalize to an abitrary 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 distrbutions:

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 3.3.2 (2016-10-31)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 14.04.5 LTS

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

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

other attached packages:
[1] expm_0.999-0   Matrix_1.2-7.1 rmarkdown_1.1 

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.7     lattice_0.20-34 gtools_3.5.0    digest_0.6.9   
 [5] assertthat_0.1  grid_3.3.2      formatR_1.4     magrittr_1.5   
 [9] evaluate_0.9    stringi_1.1.1   tools_3.3.2     stringr_1.0.0  
[13] yaml_2.1.13     htmltools_0.3.5 knitr_1.14      tibble_1.2