[使用MATLAB中大型矩阵的特征向量获得马尔可夫链的平稳分布
问题描述 投票：0回答：1
我试图找到马尔可夫链的平稳分布。我有一个转移概率矩阵（TPM）。这是代码：
[V, D] = eigs(double(TPM'),1);
Py = abs(V)/sum(V);
我的问题是sum(V) < 0，所以它在矢量Py中给了我一些负值。我已经使用另一个概率矩阵测试了代码，结果为sum(V) > 0。
我不知道这是什么问题，是TPM还是我使用的代码？
编辑：
这里是代码的更“精巧”版本（包括“ SecretAgentMan”的答案，请参见方法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
这里是函数“迭代器”：
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;
这是我正在使用的TPM之一：
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
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问题1：
这3种方法中的每一种都会产生不同的结果，我不确定该使用哪种方法。对于迭代器方法，我知道在大TPM的情况下可以使用近似值。
为什么值会有这样的差异？
问题2：
为什么我在平稳分布中得到负值？我使用了较大的TPM，大小为2600x2600，但没有得到任何负值，但是很难在此处放置2600x2600矩阵。
1个回答
0
投票
投票
不需要工具箱：我们知道稳态概率是离散时间马尔可夫链（DTMC）下方程的唯一解。
您可以重写此内容以表明此分布必须位于转换概率矩阵的零空间中（请参见下文）。这允许使用MATLAB的null()。
null()
对总和进行归一化给出唯一的解，并且满足另一个方程式。
注意，临时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]
具有负值，但最终答案ssp0是正确的，因为所有值均为有效概率，所有概率之和为1。
基于SteadyStateProb的更新：条件矩阵
您的转换概率矩阵的问题之一，即您编辑中的84×84大小OP's edit，是行总和大于一。您可以使用TPM进行验证，以获取发生这种情况的行索引。此外，此“超量”非常小：find(sum(TPM,2) > 1)显示最大行总和超过1乘以max(sum(TPM,2))-1，对于我来说，是4.4409e-16（eps）的两倍。
使用下面技术含量很低的代码，我对每一行进行了规范化。
2.2204e-16这将导致行总数不超过一；您可以使用% 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
进行验证，该方法将返回一个0×1的空双列向量。
无需进一步按摩矩阵，我们就可以获得稳态下的分布。
find(sum(TPM2,2) > 1)注意ssp0 = null(eye(size(TPM2))-TPM2.');
SteadyStateProb_TPM2 = ssp0./sum(ssp0);
为min(SteadyStateProb_TPM2)（零！），且总和基本上为1（-9.6502e-16）。
我现在要强制非负性。
abs(1-sum(SteadyStateProb_TPM2))比较迭代近似值：我们知道，随着SteadyStateProb_TPM2(SteadyStateProb_TPM2<0) = 0;
变得足够大，TPM2^n会收敛到SteadyStateProb_TPM2。因此，让我们检查一下。
我们将转移矩阵本身乘以很多次（这里我们得到2500步的转移矩阵）。我们可以看到它似乎或多或少地集中在目标上。
nans =
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)ans =
0.0821 0.2052 0.2565 0.2138 0.1336 0.0668
取决于您的应用程序，您可能会认为此迭代方法足够好，但是当直接分析方法仍然可行时，我将不使用它。我想如果您花点时间确保条件良好，那么在MATLAB中2600×2600矩阵是可行的。
更新版本：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
：特征值方法呢？
效果很好。
OP's comment
或替代地，
% Using the 5x5 P matrix from above
[V,D] = eigs(double(P.'),1);
Py = -abs(V)/sum(V)
[您也可以在84x84上验证此功能：
Pyalt = V./sum(V)
给出以下内容。
[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).'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 0.0668
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