问题
I am trying to find the stationary distribution of a Markov chain. I have a transition probability matrix (TPM). Here is the code:
[V, D] = eigs(double(TPM'),1);
Py = abs(V)/sum(V);
My problem is that sum(V) < 0 so it gives me some negative values in the vector Py. I've tested the code using another probability matrix and it gives sum(V) > 0.
I don't know what is the problem, is it the TPM or the code I am using?
EDIT:
Here is a more "elaborated" version of the code (including the answer of "SecretAgentMan", see method 3):
function [Py1, Py2, Py3] = testStatDist(TPM)
% method 1
[V1, D1] = eigs(double(TPM'),1);
% method 2
[V2, D2] = iterator(TPM);
% method 3
V3 = null(eye(size(TPM))-TPM.');
if(abs(D1 - 1) > 1e-5)
warning(0, 'testStatDist.m: Check Transition Matrix');
exit;
end
if(abs(D2 - 1) > 1e-5)
warning(0, 'testStatDist.m: Check Transition Matrix');
exit;
end
% res 1
Py1 = abs(V1)/sum(V1);
% res 2
Py2 = abs(V2)/sum(V2);
% res 3
Py3 = abs(V3)/sum(V3);
end
Here is the function "iterator":
function [V, D] = iterator(TPM)
pi(1:length(TPM)) = 1/length(TPM);
error = 1;
budget = 1000;
while(error >= 1e-10)
pi2 = pi*TPM;
error = sum((pi - pi2).^2);
pi = pi2;
budget = budget - 1;
if(budget == 0)
break;
end
end
V = pi';
D = 1;
And here is one of the TPMs that I am using:
TPM = [0.987665198237886 0.000220264317180617 0.000440528634361233 0.000660792951541850 0.0110132158590308 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.000881057268722467 0.985682819383260 0.00110132158590308 0.00132158590308370 0 0.0110132158590308 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.00154185022026432 0.00176211453744493 0.983700440528634 0.00198237885462555 0 0 0.0110132158590308 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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Question 1:
Each of the 3 methods yields different result and I am not sure which one to use. For the iterator method, I know it's an approximation that can be used in the case of big TPMs.
Why there is such a difference in the values?
Question 2:
Why I am getting negative values in the stationary distribution? I used a bigger TPM of size 2600x2600 and I am not getting any negative values, but it's difficult to put 2600x2600 matrix here.
回答1:
No Toolbox Required:
We know the steady state probabilities are the unique solution to equations below for a Discrete Time Markov Chain (DTMC).
You can rewrite this to show that this distribution must lie in the nullspace of the transition probability matrix (see below). This permits using MATLAB's null().
Normalizing the sum gives the unique solution and satisfies the other equation.
P = [.2 .3 .3 0 .2;
.1 .2 .3 .4 0;
0 .5 .4 .1 0;
.2 .2 .2 .3 .1;
.5 .2 .1 .1 .1];
ssp0 = null(eye(size(P))-P.');
SteadyStateProb = ssp0./sum(ssp0) % [0.1263 0.3017 0.2969 0.2224 0.0528]
Notice the interim ssp0 has negative values yet the final answer, SteadyStateProb, is correct in that all values are valid probabilities and all probabilities sum to 1.
Update based on OP's edit: Conditioning the Matrix
One of the problems with your transition probability matrix, the 84×84 sized TPM in your edit, is that the row sum is greater than one. You can verify this with find(sum(TPM,2) > 1) to get the row indices where this occurs. Further, this "overage" is very small: max(sum(TPM,2))-1 shows the largest row sum exceeds 1 by 4.4409e-16 with for me is twice the eps (2.2204e-16).
Using the very low-tech code below, I normalized each row.
% Re-Normalize: Divide the kth row by the sum of that row
TPM2 = TPM;
for k = 1:size(TPM2,1)
TPM2(k,:) = TPM2(k,:)/sum(TPM2(k,:));
end
This results in no row sum exceeding one; you can verify with find(sum(TPM2,2) > 1) which returns a 0×1 empty double column vector.
Without further massaging the matrix we can obtain the distribution in steady state.
ssp0 = null(eye(size(TPM2))-TPM2.');
SteadyStateProb_TPM2 = ssp0./sum(ssp0);
Notice the min(SteadyStateProb_TPM2) is -9.6502e-16 (zero!) and the sum is essentially 1 (abs(1-sum(SteadyStateProb_TPM2))).
I would now just force non-negativity.
SteadyStateProb_TPM2(SteadyStateProb_TPM2<0) = 0;
Compare to an iterative approximation:
We know that TPM2^n converges to SteadyStateProb_TPM2 as n gets large enough. So let's check it.
We multiply the transition matrix by itself many times (here we get the 2500 step transition matrix). We can see it seems to be converging more or less on target.
Approx = TPM2^500;
Approx = Approx*Approx*Approx*Approx*Approx;
ind = find(SteadyStateProb_TPM2 > 0.05); % find states with steady state prob > 0.05
SteadyStateProb_TPM2(ind).'
Approx(1:6,ind)
Note the states are [4 8 12 16 20 24].
ans =
0.0821 0.2052 0.2565 0.2138 0.1336 0.0668ans =
0.0727 0.1818 0.2273 0.1894 0.1184 0.0592 0.0760 0.1900 0.2374 0.1979 0.1237 0.0618 0.0763 0.1908 0.2385 0.1988 0.1242 0.0621 0.0821 0.2052 0.2565 0.2138 0.1336 0.0668 0.0727 0.1818 0.2273 0.1894 0.1184 0.0592 0.0760 0.1900 0.2374 0.1979 0.1237 0.0618
Depending on your application, you may consider this iteration method good enough, but I wouldn't use it when the direct analytic methods are still viable. I imagine a 2600×2600 matrix is doable in MATLAB if you take time to ensure it is well-conditioned.
Update re: OP's comment: What about the eigenvalue method?
It works fine.
% Using the 5x5 P matrix from above
[V,D] = eigs(double(P.'),1);
Py = -abs(V)/sum(V)
Or alternatively,
Pyalt = V./sum(V)
You can verify this works on the 84x84 as well:
[V2,D2] = eigs(double(TPM2'),1);
Py_TPM2 = -abs(V2)/sum(V2);
Py_TPM2alt = V2./sum(V2);
% Evaluate to compare
SteadyStateProb_TPM2(ind).'
Py_TPM2(ind).'
Py_TPM2alt(ind).'
which gives the following. Note the states are [4 8 12 16 20 24].
SteadyStateProb_TPM2(ind).' = 0.0821 0.2052 0.2565 0.2138 0.1336 0.0668Py_TPM2(ind).' = 0.0821 0.2052 0.2565 0.2138 0.1336 0.0668Py_TPM2alt(ind).' = 0.0821 0.2052 0.2565 0.2138 0.1336 0.0668
来源：https://stackoverflow.com/questions/59950322/obtaining-the-stationary-distribution-for-a-markov-chain-using-eigenvectors-from