Strange results with asymptotics function

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xiaohan wang
xiaohan wang 2024-5-29
回答： Aneela 2024-7-26
Ran in:
I have a Markov chain with K communication classes, each communication class forms an ergodic subchain. Then I use the asymptotics function to get the K limit distributions of this Markov chain. However, results from the asymptotics function seems very sensitive to the transition matrix. Then I impliment the asymptotics function by my self following the Spedicato's algorithm (ref), i.e.:
And I found that Spedicato's algorithm is not that sensitive to the transition matrix.
Here is the code:
%% load data
clear;clc;
load('data.mat'); % load the example transition matrix A and indicator matrix C
% note: A is a left stochastic matrix, i.e., all cols sum to 1
% A2, C2 is slightly changed from A1, C1
% disp(sum(sum(abs(A1-A2)))); %7.1385e-05
if true
A=A1;
C=C1;
else
A=A2;
C=C2;
end
%% matlab asymptotics function
K = 3;
S=size(A,1);
mc = dtmc(A');
[pi, tmix] = asymptotics(mc);
pi=pi';
prss = sum(C*pi);
fprintf('prss=%s\n', mat2str(prss));
prss=[0.802175510871664 0.802802946112983 0.801362094610701]
fprintf('prss diff = %.3f%%\n', (max(prss)-min(prss))/min(prss)*100);
prss diff = 0.180%
%% Spedicato's algorithm
[bins,ClassStates,ClassRecurrence,ClassPeriod] = classify(mc);
num_classes = size(ClassStates, 2);
prss = zeros(1, num_classes);
for c =1 : num_classes
sc = subchain(mc,str2num(ClassStates{c}(1)));
Cs = zeros(size(C,1), sc.NumStates);
for i=1:sc.NumStates
idx = str2num(sc.StateNames(i));
Cs(:,i) = C(:, idx);
end
As = sc.P';
% get stationary distribution
% the condition number is not very large, so I think using inv() is ok
pi = inv(ones(sc.NumStates)+eye(sc.NumStates)-As)*ones(sc.NumStates,1);
prss(c) = sum(Cs * pi);
end
fprintf('prss=%s\n', mat2str(prss));
prss=[0.802972576205566 0.800424333889229 0.802175510871663]
fprintf('prss diff = %.3f%%\n', (max(prss)-min(prss))/min(prss)*100);
prss diff = 0.318%

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回答（1 个）

Aneela
Aneela 2024-7-26
Hi @xiaohan wang
Asymptotics function:
  • MATLAB's “asymptotics” function is part of the “dtmc” class for discrete-time Markov chains. It calculates the stationary distribution and mixing time of the Markov chain.
  • The asymptotics function directly works with the entire transition matrix and can handle a wider range of transition matrices more robustly.
Spedicato's Algorithm:
  • It breaks down the problem into smaller subproblems (subchains), which can lead to differences in the final aggregated result due to accumulated numerical errors.
  • It can sometimes help to mitigate the numerical instability but that doesn’t inherently make Spedicato’s algorithm less sensitive to the transition matrix.

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