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# 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 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 $$t$$.

# 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 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

# 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 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     

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