distplot

Plot Markov chain redistributions

Syntax

distplot(mc,X)

distplot(mc,X,Name,Value)

distplot(ax,___)

h = distplot(___)

Description

distplot(mc,X) creates a heatmap from the data X showing the evolution of a distribution of states in the discrete-time Markov chain mc.

distplot(mc,X,Name,Value) uses additional options specified by one or more name-value pair arguments. For example, specify the type of plot or the frame rate for animated plots.

distplot(ax,___) plots on the axes specified by ax instead of the current axes (gca) using any of the input arguments in the previous syntaxes. The option ax can precede any of the input argument combinations in the previous syntaxes.

h = distplot(___) returns a handle to the distribution plot. Use h to modify properties of the plot after you create it.

Examples

Visualize Evolution of State Distribution

Create a four-state Markov chain from a randomly generated transition matrix containing eight infeasible transitions.

rng('default'); % For reproducibility 
mc = mcmix(4,'Zeros',8);

mc is a dtmc object.

Plot a digraph of the Markov chain.

figure;
graphplot(mc);

State 4 is an absorbing state.

Compute the state redistributions at each step for 10 discrete time steps. Assume an initial uniform distribution over the states.

X = redistribute(mc,10)

X = 11×4

    0.2500    0.2500    0.2500    0.2500
    0.0869    0.2577    0.3088    0.3467
    0.1073    0.2990    0.1536    0.4402
    0.0533    0.2133    0.1844    0.5489
    0.0641    0.2010    0.1092    0.6257
    0.0379    0.1473    0.1162    0.6985
    0.0404    0.1316    0.0765    0.7515
    0.0266    0.0997    0.0746    0.7991
    0.0259    0.0864    0.0526    0.8351
    0.0183    0.0670    0.0484    0.8663
      ⋮

X is an 11-by-4 matrix. Rows correspond to time steps, and columns correspond to states.

Visualize the state redistribution.

figure;
distplot(mc,X)

After 10 transitions, the distribution appears to settle with a majority of the probability mass in state 4.

Animate Evolution of Markov Chain

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);

Compute the state redistributions at each step for 20 discrete time steps.

X = redistribute(mc,20);

Animate the redistributions in a histogram. Specify a half-second frame rate.

figure;
distplot(mc,X,'Type','histogram','FrameRate',0.5);

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).

X — Evolution of state probabilities

nonnegative numeric matrix

Evolution of state probabilities, specified as a (1 + numSteps)-by-NumStates nonnegative numeric matrix returned by redistribute. The first row is the initial state distribution. Subsequent rows are the redistributions at each step. distplot normalizes the rows by their respective sums before plotting.

Data Types: double

ax — Axes on which to plot

Axes object

Axes on which to plot, specified as an Axes object.

By default, distplot plots to the current axes (gca).

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Type','graph','FrameRate',3 creates an animated plot of the redistributions using a frame rate of 3 seconds.

'Type' — Plot type

'evolution' (default) | 'histogram' | 'graph'

Plot type, specified as the comma-separated pair consisting of 'Type' and a value in this table.

Value	Description
'evolution'	Evolution of the initial distribution. The plot is a (1 + NumSteps)-by-NumStates heatmap. Row i displays the redistribution at step i.
'histogram'	Animated histogram of the redistributions. The vertical axis displays probability mass, and the horizontal axis displays states. The 'FrameRate' name-value pair argument controls the animation progress.
'graph'	Animated graph of the redistributions. distplot colors the nodes by their probability mass at each step. The 'FrameRate' name-value pair argument controls the animation progress.

Example: 'Type','graph'

Data Types: string | char

'FrameRate' — Length of discrete time steps

positive scalar

Length of discrete time steps, in seconds, for animated plots, specified as the comma-separated pair consisting of 'FrameRate' and a positive scalar.

The default is a pause at each time step. The animation proceeds when you press the space bar.

Example: 'FrameRate',3

Data Types: double

Output Arguments

h — Handle to distribution plot

graphics object

Handle to the distribution plot, returned as a graphics object. h contains a unique plot identifier, which you can use to query or modify properties of the plot.