在给定转移矩阵的状态矩阵中有效地应用转移

Question

我想把一个状态变化应用到一个大的分类矩阵（m）中，其中有k个类别，我知道k（T）中每个类别到另一个类别的转移概率

本质上，我希望能够有效地获取M中的每个元素，模拟给定T中概率的状态变化，并用计算出的变化替换元素。你知道吗

我尝试了一些解决方案：

• 带索引的循环的强制嵌套（太长）
• numba辅助嵌套for循环（~500ms对于我来说太长了）
• 每种类型和更换的预计算牵引力（~400ms）

import numpy as np


def categorical_transition(mat, t_mat, k=4):

    transformed_mat = mat.copy()
    cat_counts = np.bincount(mat.reshape(-1,))

    for i in range(k):
        rand_vec = np.random.multinomial(1, t_mat[i], cat_counts[i])

        choice = np.where(rand_vec)[1]

        transformed_mat[mat == i] = choice

    return transformed_mat


# load data
mat = np.random.choice(4, (16000, 256))
t_mat = np.random.random((4, 4))

# normalize transition matrix
for i in range(t_mat.shape[0]):
    t_mat[i] = t_mat[i] / t_mat[i].sum()

transformed_mat = categorical_transition(mat, t_mat)

这个方法是可行的，但它是缓慢的，我将感谢任何建议，更有效的方法来实现它

Answer 1

始终提供您迄今为止尝试的所有实现

例如，我尝试了一个简单的实现，如decribedhere.，它应该比您的解决方案快20-80倍，这取决于您有多少可用的内核。你知道吗

实施

@nb.njit(parallel=True)  
def categorical_transition_nb(mat_in, t_mat):
    mat=np.reshape(mat_in,-1)
    transformed_mat = np.empty_like(mat)
    for i in nb.prange(mat.shape[0]):
        rand_number=np.random.rand()
        probabilities=t_mat[mat[i],:]
        if rand_number<probabilities[0]:
            transformed_mat[i]=0
        else:
            for j in range(1,probabilities.shape[0]):
                if rand_number>=probabilities[j-1] and rand_number<probabilities[j]:
                    transformed_mat[i]=j

    return transformed_mat.reshape(mat_in.shape)

计时

import numpy as np
import numba as nb

# load data
mat = np.random.choice(4, (16_000,256))
t_mat = np.random.random((4, 4))

# normalize transition matrix
for i in range(t_mat.shape[0]):
    t_mat[i] = t_mat[i] / t_mat[i].sum()

t_mat_2=np.cumsum(t_mat,axis=1)
%timeit transformed_mat_2 = categorical_transition_nb(mat, t_mat_2)
21.7 ms ± 1.85 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)

Answer 2

这里有一个方法，使我的例子加速了5倍

import numpy as np

def categorical_transition(mat, t_mat, k=4):

    transformed_mat = mat.copy()
    cat_counts = np.bincount(mat.reshape(-1,))

    for i in range(k):
        rand_vec = np.random.multinomial(1, t_mat[i], cat_counts[i])

        choice = np.where(rand_vec)[1]

        transformed_mat[mat == i] = choice

    return transformed_mat

def pp(mat,t_mat):
    ps = t_mat.cumsum(1)
    ps /= ps[:,-1:]
    return (np.random.random(mat.shape+(1,))<ps[mat]).argmax(-1)


# load data
mat = np.random.choice(4, (16000, 256))
t_mat = np.random.random((4, 4))

# normalize transition matrix
for i in range(t_mat.shape[0]):
    t_mat[i] = t_mat[i] / t_mat[i].sum()

transformed_mat = categorical_transition(mat, t_mat)
transformed_mat_pp = pp(mat, t_mat)

# check correctness
from pprint import pprint
np.set_printoptions(3)

cnts = np.bincount(mat.ravel())
pprint([[np.bincount(tm[mat==i])/cnts[i] for tm in (transformed_mat,transformed_mat_pp)] + [t_mat[i]] for i in range(4)])

from timeit import timeit

print('OP',timeit(lambda:categorical_transition(mat, t_mat),number=10)*100,'ms')
print('pp',timeit(lambda:pp(mat, t_mat),number=10)*100,'ms')

运行示例：

[[array([0.186, 0.1  , 0.078, 0.637]),
  array([0.186, 0.099, 0.078, 0.637]),
  array([0.186, 0.099, 0.078, 0.637])],
 [array([0.303, 0.517, 0.088, 0.092]),
  array([0.303, 0.517, 0.089, 0.092]),
  array([0.303, 0.517, 0.088, 0.092])],
 [array([0.319, 0.27 , 0.329, 0.082]),
  array([0.319, 0.271, 0.328, 0.083]),
  array([0.318, 0.27 , 0.329, 0.082])],
 [array([0.408, 0.106, 0.264, 0.222]),
  array([0.409, 0.107, 0.263, 0.221]),
  array([0.408, 0.107, 0.264, 0.221])]]
OP 872.7993675973266 ms
pp 170.54899749346077 ms