[英]How do I cope with negative steady state probabilities for a transition matrix of a markov chain?
[英]Steady State Probabilities (Markov Chain) Python Implementation
嗨,我正在尝试为转移概率矩阵生成稳态概率。 这是我正在使用的代码:
import numpy as np
one_step_transition = array([[0.125 , 0.42857143, 0.75 ],
[0.75 , 0.14285714, 0.25 ],
[0.125 , 0.42857143, 0. ]])
def steady_state_prop(p):
dim = p.shape[0]
q = (p-np.eye(dim))
ones = np.ones(dim)
q = np.c_[q,ones]
QTQ = np.dot(q, q.T)
bQT = np.ones(dim)
return np.linalg.solve(QTQ,bQT)
steady_state_matrix = steady_state_prop(one_step_transition.transpose())
print (steady_state_matrix)
#result is :
#array([0.38268793, 0.39863326, 0.21867882])
#Expected Result = (0.4,0.4,0.2)
我的问题是为什么结果与确切答案略有不同?
预期的结果是错误的。 对于稳态,过渡矩阵和稳态的乘积必须再次为稳态。
tobe = np.array(((0.4, 0.4, 0.2)))
print(tobe)
print(np.dot(one_step_transition.T, tobe))
print()
result = steady_state_prop(one_step_transition)
print(result)
print(np.dot(one_step_transition.T, result))
print()
输出是
[0.4 0.4 0.2]
[0.37142857 0.40714286 0.22142857]
[0.38268793 0.39863326 0.21867882]
[0.38268793 0.39863326 0.21867882]
因此,您的功能似乎是正确的,但预期结果却不正确。
我使用了不同的方法来解决这个问题:
def Markov_Steady_State_Prop(p):
p = p - np.eye(p.shape[0])
for ii in range(p.shape[0]):
p[0,ii] = 1
P0 = np.zeros((p.shape[0],1))
P0[0] = 1
return np.matmul(np.linalg.inv(p),P0)
结果与您的相同,我认为您的预期结果有些错误,或者它们是近似版本。
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