asymptotics

Determine Markov chain asymptotics

Syntax

xFix = asymptotics(mc)

[xFix,tMix] = asymptotics(mc)

Description

xFix = asymptotics(mc) returns the stationary distribution xFix of the discrete-time Markov chain mc.

[xFix,tMix] = asymptotics(mc) additionally returns an estimate of the mixing time tMix.

Examples

Determine Markov Chain Stationary Distribution

Consider this theoretical, right-stochastic transition matrix of a stochastic process.

$$P=\left[\begin{array}{ccccccc}0& 0& 1/2& 1/4& 1/4& 0& 0\\ 0& 0& 1/3& 0& 2/3& 0& 0\\ 0& 0& 0& 0& 0& 1/3& 2/3\\ 0& 0& 0& 0& 0& 1/2& 1/2\\ 0& 0& 0& 0& 0& 3/4& 1/4\\ 1/2& 1/2& 0& 0& 0& 0& 0\\ 1/4& 3/4& 0& 0& 0& 0& 0\end{array}\right].$$

Create the Markov chain that is characterized by the transition matrix P.

P = [ 0   0  1/2 1/4 1/4  0   0 ;
      0   0  1/3  0  2/3  0   0 ;
      0   0   0   0   0  1/3 2/3;
      0   0   0   0   0  1/2 1/2;
      0   0   0   0   0  3/4 1/4;
     1/2 1/2  0   0   0   0   0 ;
     1/4 3/4  0   0   0   0   0 ];
mc = dtmc(P);

Plot a directed graph of the Markov chain. Indicate the probability of transition by using edge colors.

figure;
graphplot(mc,'ColorEdges',true);

Determine the stationary distribution of the Markov chain.

xFix = asymptotics(mc)

xFix = 1×7

    0.1300    0.2034    0.1328    0.0325    0.1681    0.1866    0.1468

Because xFix is a row vector, it is the unique stationary distribution of mc.

Estimate Markov Chain Mixing Time

Create a five-state transition matrix of empirical counts by generating a block diagonal matrix composed of discrete uniform draws.

m = 100; % Maximal count
rng(1); % For reproducibility
P = blkdiag(randi(100,2) + 1,randi(100,3) + 1)

P = 5×5

    43     2     0     0     0
    74    32     0     0     0
     0     0    16    36    43
     0     0    11    41    70
     0     0    20    55    22

Create and plot a digraph of the Markov chain that is characterized by the transition matrix P.

mc = dtmc(P);

figure;
graphplot(mc)

Determine the stationary distribution and mixing time of the Markov chain.

[xFix,tMix] = asymptotics(mc)

xFix = 2×5

    0.9401    0.0599         0         0         0
         0         0    0.1497    0.4378    0.4125

tMix = 
0.8558

Rows of xFix correspond to the stationary distributions of the two independent recurrent classes of mc.

Create separate Markov chains representing the recurrent subchains of mc.

mc1 = subchain(mc,1);
mc2 = subchain(mc,3);

mc1 and mc2 are dtmc objects. mc1 is the recurrent class containing state 1, and mc2 is the recurrent class containing state 3.

Compare the mixing times of the subchains.

[x1,t1] = asymptotics(mc1)

x1 = 1×2

    0.9401    0.0599

t1 = 
0.7369

[x2,t2] = asymptotics(mc2)

x2 = 1×3

    0.1497    0.4378    0.4125

t2 = 
0.8558

mc1 approaches its stationary distribution more quickly than mc2.

Determine Dumbbell Chain Mixing Time

Create a "dumbbell" Markov chain containing 10 states in each "weight" and three states in the "bar."

• Specify random transition probabilities between states within each weight.

• If the Markov chain reaches the state in a weight that is closest to the bar, then specify a high probability of transitioning to the bar.

• Specify uniform transitions between states in the bar.

rng(1,"twister"); % For reproducibility
w = 10;                              % Dumbbell weights
DBar = [0 1 0; 1 0 1; 0 1 0];        % Dumbbell bar
DB = blkdiag(rand(w),DBar,rand(w));  % Transition matrix

% Connect dumbbell weights and bar
DB(w,w+1) = 1;                       
DB(w+1,w) = 1; 
DB(w+3,w+4) = 1; 
DB(w+4,w+3) = 1;

mc = dtmc(DB);

Visualize the transition matrix using a heatmap.

figure
imagesc(mc.P)
axis square
colorbar

Plot a directed graph of the Markov chain. Suppress node labels.

figure
h = graphplot(mc);
h.NodeLabel = {};

Plot the eigenvalues of the dumbbell chain.

figure
eigplot(mc)

The thin, red disc in the plot shows the spectral gap (the difference between the two largest eigenvalue moduli). The spectral gap determines the mixing time of the Markov chain. Large gaps indicate faster mixing, whereas thin gaps indicate slower mixing. In this case, the spectral gap is thin, indicating a long mixing time.

Estimate the mixing time of the dumbbell chain and determine whether the chain is ergodic.

[~,tMix] = asymptotics(mc)

tMix = 
85.3258

tf = isergodic(mc)

tf = logical
   1

On average, the time it takes for the total variation distance between any initial distribution and the stationary distribution to decay by a factor of $$e^1$$ is about 85 steps.

Input Arguments

mc — Discrete-time Markov chain

dtmc object

Discrete-time Markov chain with NumStates states and transition matrix P, specified as a dtmc object. P must be fully specified (no NaN entries).

Output Arguments

xFix — Stationary distribution

nonnegative numeric matrix

Stationary distribution, with xFix*P = xFix, returned as a nonnegative numeric matrix with NumStates columns. The number of rows of xFix is the number of independent recurrent classes in mc.

• For unichains, the distribution is unique, and xFix is a 1-by-NumStates vector.

• Otherwise, each row of xFix represents a distinct stationary distribution in mc.

tMix — Mixing time

positive numeric scalar

Mixing time, returned as a positive numeric scalar.

If μ, the second largest eigenvalue modulus (SLEM) of P, exists and is nonzero, then the estimated mixing time is $$-1/\log \left(\mu \right)$$.

Note

• If P is a nonnegative stochastic matrix, then the Markov chain mc it characterizes has a left eigenvector xFix with eigenvalue 1. The Perron-Frobenius Theorem [2] implies that if mc is a unichain (a chain with a single recurrent communicating class), then xFix is unique. For reducible chains with multiple recurrent classes, eigenvalue 1 has higher multiplicity, and xFix is nonunique. If a chain is periodic, xFix is stationary but not limiting because arbitrary initial distributions do not converge to it. xFix is both unique and limiting for ergodic chains only. See classify.

• For ergodic chains, tMix is a characteristic time for any initial distribution to converge to xFix. Specifically, it is the time for the total variation distance between an initial distribution and xFix to decay by a factor of e = exp(1). Mixing times are a measure of the relative connectivity of transition structures in different chains.

References

[1] Gallager, R.G. Stochastic Processes: Theory for Applications. Cambridge, UK: Cambridge University Press, 2013.

[2] Horn, R., and C. R. Johnson. Matrix Analysis. Cambridge, UK: Cambridge University Press, 1985.

[3] Seneta, E. Non-negative Matrices and Markov Chains. New York, NY: Springer-Verlag, 1981.

Version History

Introduced in R2017b