评估大型矩阵的平稳概率分布

Question

我需要计算与马尔可夫链相关的平稳分布。我尝试了三种不同的方法： 1）基于线性系统的解 2）基于固定分布的迭代 3）基于最大特征值（lambda = 1）的评估

我如何改进其中一种方法来处理大型稀疏矩阵（5000-7500个节点）。现在，对于完整矩阵，方法（2）是最快的。

import time
import numpy as np
from scipy.linalg import norm
def solveStationaryUsingSystem( A ):
    t0 = time.time()
    n = A.shape[0]
    a = np.eye( n ) - A
    a = np.vstack( (a.T, np.ones( n )) )
    b = np.matrix( [0] * n + [ 1 ] ).T
    print ("System time: ", time.time() - t0)
    return np.linalg.lstsq( a, b )[0]

def solveStationaryByIterating( A ):
    t0 = time.time()
    nStates=np.shape(A)[0]    
    p = np.random.rand(nStates,1)
    p /= p.sum()
    n=50
    k=0
    pOld=np.zeros([nStates,1])
    epsilon=1e-8
    theNorm=norm(p-pOld)
    pOld=p
    while k < n or theNorm>epsilon:
        p = A.T.dot(p)
        theNorm=norm(p-pOld)
        pOld=p
        k=k+1
    print ("Iterating time: ", time.time() - t0)
    return p

def solveStationaryByEigenvalue( A ):
    t0 = time.time()
    nStates=np.shape(A)[0]    
    p = np.random.rand(nStates,1)
    p /= p.sum()
    w, v = np.linalg.eig(A.T)
    j_stationary = np.argmin(abs(w - 1.0))
    p_stationary = v[:,j_stationary].real
    p_stationary /= p_stationary.sum()
    print ("Eigenvalue time: ", time.time() - t0)
    return(p_stationary)


if __name__=="__main__":
    nStates=5000
    transitionMatrix=np.random.rand(nStates, nStates)
    transitionMatrix /= transitionMatrix.sum(axis=1)[:,np.newaxis]
    print('system')
    pS=solveStationaryUsingSystem(transitionMatrix)
    print('iteratiion')
    pI=solveStationaryByIterating(transitionMatrix)
    print('eigenvalue')
    pE=solveStationaryByEigenvalue(transitionMatrix)