redistribute
Compute Markov chain redistributions
Description
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
.
Specify Initial Distribution for State Redistribution
Consider this theoretical, right-stochastic transition matrix of a stochastic process.
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);
Compute a 20-step redistribution of the Markov chain using random initial values.
rng(1); % For reproducibility x0 = rand(mc.NumStates,1); rd = redistribute(mc,20,'X0',x0);
Plot the redistribution.
figure; distplot(mc,rd);
The redistribution suggests that the chain is periodic with a period of three.
Remove periodicity by creating a lazy version of the Markov chain.
lc = lazy(mc);
Compute a 20-step redistribution of the lazy chain using random initial values. Plot the redistribution.
x0 = rand(mc.NumStates,1);
lrd1 = redistribute(lc,20,'X0',x0);
figure;
distplot(lc,lrd1);
The redistribution appears to settle after several 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).
numSteps
— Number of discrete time steps
positive integer
Number of discrete time steps to compute, specified as a positive integer.
Data Types: double
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'X0',[0.5 0.25 0.25]
specifies an initial state distribution of [0.5 0.25 0.25]
.
X0
— Initial distribution
nonnegative numeric vector
Initial distribution, specified as the comma-separated pair consisting of 'X0'
and a nonnegative numeric vector of NumStates
length. redistribute
normalizes X0
so that it sums to 1
.
The default is a uniform distribution of states.
Example: 'X0',[0.5 0.25 0.25]
Data Types: double
Output Arguments
X
— Evolution of state probabilities
nonnegative numeric matrix
Evolution of state probabilities, returned as a (1 + numSteps)
-by-NumStates
nonnegative numeric matrix. The first row is X0
. Subsequent rows are the redistributions at each step, which redistribute
determines by the transition matrix P
.
Note
If mc
is ergodic, and numSteps
is sufficiently large, X(end,:)
approximates x = asymptotics(mc)
. See asymptotics
.
Tips
To visualize the data created by redistribute
, use distplot
.
Version History
Introduced in R2017b
See Also
Objects
Functions
MATLAB 명령
다음 MATLAB 명령에 해당하는 링크를 클릭했습니다.
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