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See here for a PDF version of this vignette.
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] 1If 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 \(3 \times
3\) matrix, it takes the vector to be a row-vector. This means
our math can be written simply as:
# After one step
x0%*%P
# [,1] [,2] [,3]
# [1,] 0.25 0.33 0.42And 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 0.334To 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 0.36And now let’s push this by 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 0.2903226What do you notice about the rows? And let’s see what this does for various different 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 0.2903226Note 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 4.3.3 (2024-02-29)
# Platform: aarch64-apple-darwin20 (64-bit)
# Running under: macOS 15.7.1
#
# Matrix products: default
# BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
#
# locale:
# [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#
# time zone: America/Chicago
# tzcode source: internal
#
# attached base packages:
# [1] stats graphics grDevices utils datasets methods base
#
# other attached packages:
# [1] expm_1.0-0 Matrix_1.6-5
#
# loaded via a namespace (and not attached):
# [1] vctrs_0.6.5 cli_3.6.5 knitr_1.50 rlang_1.1.6
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# [13] htmltools_0.5.8.1 httpuv_1.6.14 sass_0.4.10 rmarkdown_2.29
# [17] grid_4.3.3 evaluate_1.0.4 jquerylib_0.1.4 tibble_3.3.0
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# [33] R6_2.6.1 pillar_1.11.0 magrittr_2.0.3 bslib_0.9.0
# [37] tools_4.3.3 cachem_1.1.0