Create random Markov chain with specified mixing structure
Generate a six-state Markov chain from a random transition matrix.
rng(1); % For reproducibility mc = mcmix(6);
mc is a
Display the transition matrix.
ans = 6×6 0.2732 0.1116 0.1145 0.1957 0.0407 0.2642 0.3050 0.2885 0.0475 0.0195 0.1513 0.1882 0.0078 0.0439 0.0082 0.2439 0.2950 0.4013 0.2480 0.1481 0.2245 0.0485 0.1369 0.1939 0.2708 0.2488 0.0580 0.1614 0.0137 0.2474 0.2791 0.1095 0.0991 0.2611 0.1999 0.0513
Plot a digraph of the Markov chain. Specify coloring the edges according to the probability of transition.
Generate random transition matrices containing a specified number of zeros in random locations. A zero in location (i, j) indicates that state i does not transition to state j.
Generate two 10-state Markov chains from random transition matrices. Specify the random placement of 10 zeros within one chain and 30 zeros within the other chain.
rng(1); % For reproducibility numStates = 10; mc1 = mcmix(numStates,'Zeros',10); mc2 = mcmix(numStates,'Zeros',30);
Estimate the mixing times for each Markov chain.
[~,tMix1] = asymptotics(mc1)
tMix1 = 0.7567
[~,tMix2] = asymptotics(mc2)
tMix2 = 0.8137
mc1, the Markov chain with higher connectivity, mixes more quickly than
Generate a Markov chain characterized by a partially random transition matrix. Also, decrease the number of feasible transitions.
Generate a 4-by-4 matrix of missing (
NaN) values, which represents the transition matrix.
P = NaN(4);
Specify that state 1 transitions to state 2 with probability 0.5, and that state 2 transitions to state 1 with the same probability.
P(1,2) = 0.5; P(2,1) = 0.5;
Create a Markov chain characterized by the partially known transition matrix. For the remaining unknown transition probabilities, specify that five transitions are infeasible for 5 random transitions. An infeasible transition is a transition whose probability of occurring is zero.
rng(1); % For reproducibility mc = mcmix(4,'Fix',P,'Zeros',5);
mc is a
dtmc object. With the exception of the fixed elements (1,2) and (2,1) of the transition matrix,
mcmix places five zeros in random locations and generates random probabilities for the remaining nine locations. The probabilities in a particular row sum to 1.
Display the transition matrix and plot a digraph of the Markov chain. In the plot, indicate transition probabilities by specifying edge colors.
P = mc.P
P = 4×4 0 0.5000 0.1713 0.3287 0.5000 0 0.1829 0.3171 0.1632 0 0.8368 0 0 0.5672 0.1676 0.2652
numStates— Number of states
Number of states, specified as a positive integer.
If you do not specify any name-value pair arguments,
mcmix constructs a Markov chain with random transition probabilities.
comma-separated pairs of
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
0at 10 random locations in the transition matrix.
'Fix'— Locations and values of fixed transition probabilities
NaN(numStates)(default) | numeric matrix
Locations and values of fixed transition probabilities, specified as the comma-separated pair consisting of
'Fix' and a
numStates numeric matrix.
Probabilities in any row must have a sum less than or equal to
1. Rows that sum to
1 also fix
0 values in the rest of the row.
mcmix assigns random probabilities to locations containing
'Fix',[0.5 NaN NaN; NaN 0.5 NaN; NaN NaN 0.5]
'Zeros'— Number of zero-valued transition probabilities
0(default) | positive integer
Number of zero-valued transition probabilities to assign to random locations in the transition matrix, specified as the comma-separated pair consisting of
'Zeros' and a positive integer less than
mcmix function assigns
Zeros zeros to the locations containing a
'StateNames'— State labels
string(1:numStates)(default) | string vector | cell vector of character vectors | numeric vector
State labels, specified as the comma-separated pair consisting of a string vector, cell vector of character vectors, or numeric vector of
numStates length. Elements correspond to rows and columns of the transition matrix.
'StateNames',["Depression" "Recession" "Stagnant" "Boom"]
 Gallager, R.G. Stochastic Processes: Theory for Applications. Cambridge, UK: Cambridge University Press, 2013.
 Horn, R., and C. R. Johnson. Matrix Analysis. Cambridge, UK: Cambridge University Press, 1985.