利用MATLAB中的大矩阵特征向量求马尔可夫链的平稳分布

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）。 代码如下：

[V, D] = eigs(double(TPM'),1); 
Py = abs(V)/sum(V);

我的问题是

sum（V）<0

，所以它在向量

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，它给出了

sum（V）>0

我不知道是什么问题，是TPM还是我正在使用的代码

编辑：

下面是一个更“详细”的代码版本（包括“秘书长”的答案，见方法3）：

下面是函数“迭代器”：

以下是我正在使用的胎压监测系统之一：

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
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
0   0   0   0.988986784140969   0   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
0.00440528634361233 0   0   0   0.983259911894273   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   0.00440528634361233 0   0   0.000881057268722467    0.981277533039648   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   0.00440528634361233 0   0.00154185022026432 0.00176211453744493 0.979295154185022   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   0.00440528634361233 0   0   0   0.984581497797357   0   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   0.00881057268722467 0   0   0   0.978854625550661   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   0.00881057268722467 0   0   0.000881057268722467    0.976872246696035   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   0.00881057268722467 0   0.00154185022026432 0.00176211453744493 0.974889867841410   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   0.00881057268722467 0   0   0   0.980176211453745   0   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   0.0132158590308370  0   0   0   0.974449339207049   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   0.0132158590308370  0   0   0.000881057268722467    0.972466960352423   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   0.0132158590308370  0   0.00154185022026432 0.00176211453744493 0.970484581497797   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   0.0132158590308370  0   0   0   0.975770925110132   0   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   0.0176211453744493  0   0   0   0.970044052863436   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   0.0176211453744493  0   0   0.000881057268722467    0.968061674008811   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   0.0176211453744493  0   0.00154185022026432 0.00176211453744493 0.966079295154185   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   0.0176211453744493  0   0   0   0.971365638766520   0   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   0.0220264317180617  0   0   0   0.965638766519824   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   0.0220264317180617  0   0   0.000881057268722467    0.963656387665198   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   0.0220264317180617  0   0.00154185022026432 0.00176211453744493 0.961674008810573   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   0.0220264317180617  0   0   0   0.966960352422908   0   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   0.0264317180616740  0   0   0   0.961233480176212   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   0.0264317180616740  0   0   0.000881057268722467    0.959251101321586   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   0.0264317180616740  0   0.00154185022026432 0.00176211453744493 0.957268722466960   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   0.0264317180616740  0   0   0   0.962555066079295   0   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   0.0308370044052863  0   0   0   0.956828193832599   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   0.0308370044052863  0   0   0.000881057268722467    0.954845814977974   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   0.0308370044052863  0   0.00154185022026432 0.00176211453744493 0.952863436123348   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   0.0308370044052863  0   0   0   0.958149779735683   0   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   0.0352422907488987  0   0   0   0.952422907488987   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   0.0352422907488987  0   0   0.000881057268722467    0.950440528634361   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   0.0352422907488987  0   0.00154185022026432 0.00176211453744493 0.948458149779736   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   0.0352422907488987  0   0   0   0.953744493392071   0   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   0.0396475770925110  0   0   0   0.948017621145375   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   0.0396475770925110  0   0   0.000881057268722467    0.946035242290749   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   0.0396475770925110  0   0.00154185022026432 0.00176211453744493 0.944052863436123   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   0.0396475770925110  0   0   0   0.949339207048458   0   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   0.0440528634361234  0   0   0   0.943612334801762   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   0.0440528634361234  0   0   0.000881057268722467    0.941629955947137   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   0.0440528634361234  0   0.00154185022026432 0.00176211453744493 0.939647577092511   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   0.0440528634361234  0   0   0   0.944933920704846   0   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   0.0484581497797357  0   0   0   0.939207048458150   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   0.0484581497797357  0   0   0.000881057268722467    0.937224669603524   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   0.0484581497797357  0   0.00154185022026432 0.00176211453744493 0.935242290748899   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   0.0484581497797357  0   0   0   0.940528634361234   0   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   0.0528634361233480  0   0   0   0.934801762114538   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   0.0528634361233480  0   0   0.000881057268722467    0.932819383259912   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   0.0528634361233480  0   0.00154185022026432 0.00176211453744493 0.930837004405286   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   0.0528634361233480  0   0   0   0.936123348017621   0   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   0.0572687224669604  0   0   0   0.930396475770925   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   0.0572687224669604  0   0   0.000881057268722467    0.928414096916300   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   0.0572687224669604  0   0.00154185022026432 0.00176211453744493 0.926431718061674   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   0.0572687224669604  0   0   0   0.931718061674009   0   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   0.0616740088105727  0   0   0   0.925991189427313   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   0.0616740088105727  0   0   0.000881057268722467    0.924008810572687   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   0.0616740088105727  0   0.00154185022026432 0.00176211453744493 0.922026431718062   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   0.0616740088105727  0   0   0   0.927312775330397   0   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   0.0660792951541850  0   0   0   0.921585903083701   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   0.0660792951541850  0   0   0.000881057268722467    0.919603524229075   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   0.0660792951541850  0   0.00154185022026432 0.00176211453744493 0.917621145374449   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   0.0660792951541850  0   0   0   0.922907488986784   0   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   0.0704845814977974  0   0   0   0.917180616740088   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   0.0704845814977974  0   0   0.000881057268722467    0.915198237885463   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   0.0704845814977974  0   0.00154185022026432 0.00176211453744493 0.913215859030837   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   0.0704845814977974  0   0   0   0.918502202643172   0   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   0.0748898678414097  0   0   0   0.912775330396476   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   0.0748898678414097  0   0   0.000881057268722467    0.910792951541850   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   0.0748898678414097  0   0.00154185022026432 0.00176211453744493 0.908810572687225   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   0.0748898678414097  0   0   0   0.914096916299560   0   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   0.0792951541850220  0   0   0   0.908370044052864   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   0.0792951541850220  0   0   0.000881057268722467    0.906387665198238   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   0.0792951541850220  0   0.00154185022026432 0.00176211453744493 0.904405286343612   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   0.0792951541850220  0   0   0   0.909691629955947   0   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   0.0837004405286344  0   0   0   0.903964757709251   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   0.0837004405286344  0   0   0.000881057268722467    0.901982378854626   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   0.0837004405286344  0   0.00154185022026432 0.00176211453744493 0.900000000000000   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   0.0837004405286344  0   0   0   0.905286343612335   0   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   0.0881057268722467  0   0   0   0.910572687224670   0.000220264317180617    0.000440528634361233    0.000660792951541850
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0.0881057268722467  0   0   0.000881057268722467    0.908590308370044   0.00110132158590308 0.00132158590308370
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0.0881057268722467  0   0.00154185022026432 0.00176211453744493 0.906607929515419   0.00198237885462555
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0.0881057268722467  0   0   0   0.911894273127753];

问题1:

三种方法中的每一种都会产生不同的结果，我不确定使用哪一种。 对于迭代器方法，我知道它是一种近似方法，可用于大型TPM

为什么价值观会有如此大的差异

问题2:

为什么我在平稳分布中得到负值？

---

我使用了尺寸为2600x2600的更大TPM，没有得到任何负值，但很难将2600x2600矩阵放在这里。

无需工具箱：
我们知道，对于离散时间马尔可夫链（DTMC），稳态概率是下面方程的唯一解

您可以重写它，以表明此分布必须位于转移概率矩阵的零空间中（请参见下文）。这允许使用MATLAB的

归一化求和得到唯一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]

请注意，中间的

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，所有概率总和为1

---

更新依据：调整矩阵

转换概率矩阵（编辑中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）

来验证这一点，以获取发生这种情况的行索引。此外，这个“超额”非常小：

max（sum（TPM，2））-1

显示最大的行和超过1的

4.4409e-16

，对于我来说是

eps

（

2.2204e-16

）的两倍

使用下面的低技术代码，我规范化了每一行

% 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

这不会导致行和超过1；您可以使用

find（sum（TPM2,2）>1）

进行验证，它返回一个0×1的空双列向量

不需要进一步修改矩阵，我们就可以得到稳态下的分布

ssp0 = null(eye(size(TPM2))-TPM2.');
SteadyStateProb_TPM2 = ssp0./sum(ssp0);

注意

min（SteadyStateProb\u TPM2）

是

-9.6502e-16

（零！），总和基本上是1（

abs（1-sum（SteadyStateProb\u TPM2））

）

我现在要强制非消极性

SteadyStateProb_TPM2(SteadyStateProb_TPM2<0) = 0; 

注：状态为

[4 8 12 16 20 24]

ans=

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
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

ans=

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
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

根据您的应用程序，您可以考虑这种迭代方法足够好，但当直接分析方法仍然可行时，我不会使用它。我想，如果您花时间确保2600×2600矩阵条件良好，那么它在MATLAB中是可行的

更新re：：特征值法如何？

它很好用

% Using the 5x5 P matrix from above
[V,D] = eigs(double(P.'),1);
Py = -abs(V)/sum(V)

或者

Pyalt = V./sum(V)

您也可以验证这在84x84上是否有效：

[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).'

这就给出了以下结论。注：状态为

[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

---

（1） 您是否已首先确认转移概率矩阵中的所有条目均为正值且在区间

[0,1]

？（2） 您确认所有行的行和等于1了吗<代码>总和（TPM，2）@SecretAgentMan是的，我不确定我是否理解“ssp0有负值，但最终答案是正确的”您的意思是统计距离可以包含负值？另外，我想比较您的代码和我的代码的结果，并使用我确信是正确的TPM（即，验证您询问的条件），结果存在差异。你能帮我吗？我已经编辑了我的答案。这是否更清楚？另外，您能否将您的问题包括在TPM中？（如果你的TPM很大，考虑构造一个较小的例子来代表这个问题）谢谢。我编辑了我的问题，我隐藏了一个指向我正在谈论的TPM的链接。它在驱动器上，希望它会有用。出于安全目的，如果您编辑问题，将TPM包含在单个代码块中（可以右滚动）会更好。然后我们可以复制粘贴。否则，您是否发现代码与此方法之间存在差异？我发布的例子怎么样？@Afo我已经更新了答案。这回答了你的问题吗？

[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).'