在给定转移矩阵的情况下有效地将转移应用于状态矩阵

Question

我希望将状态更改应用于具有 k 个类别的大型分类矩阵 (M)，其中我知道每个类别到 k (T) 中每个其他类别的转移概率

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。

我已经尝试了一些解决方案：

蛮力嵌套索引循环（太长了）
numba 辅助嵌套 for 循环（~500 毫秒，这对我来说太长了）
每个类别和替换的预先计算抽奖（~400 毫秒）

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)

此方法有效，但速度很慢，如有任何关于更有效的实施方法的建议，我将不胜感激

Answer 1

这里有一个方法可以让我在这个例子上加速 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')

样本运行：

[[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

Answer 2

始终提供您迄今为止尝试过的所有实现

我尝试了一个简单的实现，例如 here. 它应该比您的解决方案快大约 20-80 倍，具体取决于您有多少个可用内核。

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)

时间

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)

在给定转移矩阵的情况下有效地将转移应用于状态矩阵

Efficiently applying transition to state matrix given transition matrix

python

transition

numpy

numba