利用MATLAB中的大矩阵特征向量求马氏链的平稳分布
利用MATLAB中的大矩阵特征向量求马氏链的平稳分布
提问于 2020-01-28 13:51:46
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)。以下是代码：
[V, D] = eigs(double(TPM'),1);
Py = abs(V)/sum(V);我的问题是，sum(V) < 0，它给出了向量Py中的一些负值。我使用另一个概率矩阵测试了代码，它给出了sum(V) > 0。
我不知道问题是什么，是TPM还是我正在使用的代码？
编辑：
下面是代码的“详细说明”版本(包括"SecretAgentMan“的答案，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下面是“迭代器”函数：
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;以下是我所使用的其中一项城市规划措施：
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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这三种方法中的每一种都得到了不同的结果，我不确定使用哪种方法。对于迭代器方法，我知道这是一个近似，可以用于大TPM的情况。
为何价值观会有这么大的差异呢？
问题2:
为什么我在平稳分布中得到负值？我用了一个更大的尺寸为2600x2600的TPM，我没有得到任何负值，但是很难把2600x2600矩阵放在这里。
回答 1
Stack Overflow用户
回答已采纳
发布于 2020-01-28 14:30:04
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。
您可以重写它，以表明此分布必须位于转移概率矩阵的空空间中(见下文)。这允许使用MATLAB的null()。
归一化和给出唯一的解，并满足另一个方程。
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]注意，临时ssp0有负值，但最终答案SteadyStateProb是正确的，因为所有值都是有效概率，所有概率之和为1。
基于的更新:对矩阵的调理
转换概率矩阵(编辑中的84×84大小的TPM )的一个问题是，行和大于1。您可以使用find(sum(TPM,2) > 1)验证这一点，以获得发生这种情况的行索引。此外，这种“超限”非常小：max(sum(TPM,2))-1显示4.4409e-16的最大行和超过1，对我来说是eps (2.2204e-16)的两倍。
使用下面的非常低的技术代码，我标准化了每一行。
% 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这不会导致一行和超过1；您可以使用find(sum(TPM2,2) > 1)进行验证，后者返回0×1空的双列向量。
在不需要进一步按摩矩阵的情况下，可以得到稳态分布。
ssp0 = null(eye(size(TPM2))-TPM2.');
SteadyStateProb_TPM2 = ssp0./sum(ssp0);注意min(SteadyStateProb_TPM2)是-9.6502e-16 (0！)和本质上是1 (abs(1-sum(SteadyStateProb_TPM2)))。
我现在只想强迫你不要消极。
SteadyStateProb_TPM2(SteadyStateProb_TPM2<0) = 0; 与迭代逼近的比较：
我们知道，当TPM2^n变得足够大时，n就会收敛到SteadyStateProb_TPM2。所以让我们检查一下。
我们将转移矩阵本身乘以多次(这里我们得到了2500步转移矩阵)。我们可以看到，它似乎正或多或少地向目标靠拢。
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)注意，状态是[4 8 12 16 20 24]。
ans =
0.0821 0.2052 0.2565 0.2138 0.1336 0.0668
安=
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 0.1336 0.0668 0.0727 0.1818 0.2273 0.1894 0.0592 0.0760 0.1900 0.2374 0.1979 0.1237 0.0618
根据您的应用程序，您可能认为这个迭代方法足够好，但是当直接分析方法仍然可行时，我不会使用它。我想象一个2600×2600矩阵在MATLAB中是可行的，如果你花时间确保它是良好的条件。
更新re: ：特征值方法怎么办？
效果很好。
% Using the 5x5 P matrix from above
[V,D] = eigs(double(P.'),1);
Py = -abs(V)/sum(V)或者说，
Pyalt = V./sum(V)您也可以在84x84上验证这一点：
[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).'这就说明了以下几点。注意，状态是[4 8 12 16 20 24]。
SteadyStateProb_TPM2(ind).' = 0.0821 0.2052 0.2565 0.2138 0.1336 0.0668
Py_TPM2(ind).' = 0.0821 0.2052 0.2565 0.2138 0.1336 0.0668
Py_TPM2alt(ind).' = 0.0821 0.2052 0.2565 0.2138 0.1336 0.0668
页面原文内容由Stack Overflow提供。腾讯云小微IT领域专用引擎提供翻译支持
原文链接：
https://stackoverflow.com/questions/59950322
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