在我的项目中，我需要多次计算 0-1 向量的熵。这是我的代码：
def entropy(labels):
""" Computes entropy of 0-1 vector. """
n_labels = len(labels)
if n_labels <= 1:
return 0
counts = np.bincount(labels)
probs = counts[np.nonzero(counts)] / n_labels
n_classes = len(probs)
if n_classes <= 1:
return 0
return - np.sum(probs * np.log(probs)) / np.log(n_classes)
有更快的方法吗？
@Sanjeet Gupta 的回答很好，但可以精简。这个问题专门询问“最快”的方式，但我只看到一个答案的时间，所以我将发布使用 scipy 和 numpy 与原始海报的 entropy2 答案的比较，稍作改动。
四种不同的方法： (1) scipy/numpy，(2) numpy/math，(3) pandas/numpy，(4) numpy
import numpy as np
from scipy.stats import entropy
from math import log, e
import pandas as pd
import timeit
def entropy1(labels, base=None):
value,counts = np.unique(labels, return_counts=True)
return entropy(counts, base=base)
def entropy2(labels, base=None):
""" Computes entropy of label distribution. """
n_labels = len(labels)
if n_labels <= 1:
return 0
value,counts = np.unique(labels, return_counts=True)
probs = counts / n_labels
n_classes = np.count_nonzero(probs)
if n_classes <= 1:
return 0
ent = 0.
# Compute entropy
base = e if base is None else base
for i in probs:
ent -= i * log(i, base)
return ent
def entropy3(labels, base=None):
vc = pd.Series(labels).value_counts(normalize=True, sort=False)
base = e if base is None else base
return -(vc * np.log(vc)/np.log(base)).sum()
def entropy4(labels, base=None):
value,counts = np.unique(labels, return_counts=True)
norm_counts = counts / counts.sum()
base = e if base is None else base
return -(norm_counts * np.log(norm_counts)/np.log(base)).sum()
时间操作：
repeat_number = 1000000
a = timeit.repeat(stmt='''entropy1(labels)''',
setup='''labels=[1,3,5,2,3,5,3,2,1,3,4,5];from __main__ import entropy1''',
repeat=3, number=repeat_number)
b = timeit.repeat(stmt='''entropy2(labels)''',
setup='''labels=[1,3,5,2,3,5,3,2,1,3,4,5];from __main__ import entropy2''',
repeat=3, number=repeat_number)
c = timeit.repeat(stmt='''entropy3(labels)''',
setup='''labels=[1,3,5,2,3,5,3,2,1,3,4,5];from __main__ import entropy3''',
repeat=3, number=repeat_number)
d = timeit.repeat(stmt='''entropy4(labels)''',
setup='''labels=[1,3,5,2,3,5,3,2,1,3,4,5];from __main__ import entropy4''',
repeat=3, number=repeat_number)
时间结果：
# for loop to print out results of timeit
for approach,timeit_results in zip(['scipy/numpy', 'numpy/math', 'pandas/numpy', 'numpy'], [a,b,c,d]):
print('Method: {}, Avg.: {:.6f}'.format(approach, np.array(timeit_results).mean()))
Method: scipy/numpy, Avg.: 63.315312
Method: numpy/math, Avg.: 49.256894
Method: pandas/numpy, Avg.: 884.644023
Method: numpy, Avg.: 60.026938
获胜者： numpy/math ( entropy2)
还值得注意的是，entropy2上面的函数可以处理数字和文本数据。例如：entropy2(list('abcdefabacdebcab'))。原始发帖人的答案来自 2013 年，并且有一个用于分箱整数的特定用例，但它不适用于文本。
将数据作为pd.Series和scipy.stats，计算给定数量的熵非常简单：
import pandas as pd
import scipy.stats
def ent(data):
"""Calculates entropy of the passed `pd.Series`
"""
p_data = data.value_counts() # counts occurrence of each value
entropy = scipy.stats.entropy(p_data) # get entropy from counts
return entropy
注意：scipy.stats将规范化提供的数据，所以这不需要明确地完成，即传递一个计数数组可以正常工作。
一个不依赖于 numpy 的答案，或者：
import math
from collections import Counter
def eta(data, unit='natural'):
base = {
'shannon' : 2.,
'natural' : math.exp(1),
'hartley' : 10.
}
if len(data) <= 1:
return 0
counts = Counter()
for d in data:
counts[d] += 1
ent = 0
probs = [float(c) / len(data) for c in counts.values()]
for p in probs:
if p > 0.:
ent -= p * math.log(p, base[unit])
return ent
这将接受您可以扔给它的任何数据类型：
>>> eta(['mary', 'had', 'a', 'little', 'lamb'])
1.6094379124341005
>>> eta([c for c in "mary had a little lamb"])
2.311097886212714
@Jarad 提供的答案也建议了时间。为此：
repeat_number = 1000000
e = timeit.repeat(
stmt='''eta(labels)''',
setup='''labels=[1,3,5,2,3,5,3,2,1,3,4,5];from __main__ import eta''',
repeat=3,
number=repeat_number)
Timeit 结果：（我相信这比最好的 numpy 方法快 4 倍）
print('Method: {}, Avg.: {:.6f}'.format("eta", np.array(e).mean()))
Method: eta, Avg.: 10.461799
根据 unutbu 的建议，我创建了一个纯 python 实现。
def entropy2(labels):
""" Computes entropy of label distribution. """
n_labels = len(labels)
if n_labels <= 1:
return 0
counts = np.bincount(labels)
probs = counts / n_labels
n_classes = np.count_nonzero(probs)
if n_classes <= 1:
return 0
ent = 0.
# Compute standard entropy.
for i in probs:
ent -= i * log(i, base=n_classes)
return ent
我缺少的一点是标签是一个大数组，但是概率是 3 或 4 个元素长。使用纯 python，我的应用程序现在速度是原来的两倍。
我最喜欢的熵函数如下：
def entropy(labels):
prob_dict = {x:labels.count(x)/len(labels) for x in labels}
probs = np.array(list(prob_dict.values()))
return - probs.dot(np.log2(probs))
我仍在寻找一种更好的方法来避免 dict -> values -> list -> np.array 转换。如果我找到它会再次评论。
均匀分布的数据（高熵）：
s=range(0,256)
香农熵逐步计算：
import collections
import math
# calculate probability for each byte as number of occurrences / array length
probabilities = [n_x/len(s) for x,n_x in collections.Counter(s).items()]
# [0.00390625, 0.00390625, 0.00390625, ...]
# calculate per-character entropy fractions
e_x = [-p_x*math.log(p_x,2) for p_x in probabilities]
# [0.03125, 0.03125, 0.03125, ...]
# sum fractions to obtain Shannon entropy
entropy = sum(e_x)
>>> entropy
8.0
单线（假设import collections）：
def H(s): return sum([-p_x*math.log(p_x,2) for p_x in [n_x/len(s) for x,n_x in collections.Counter(s).items()]])
适当的功能：
import collections
import math
def H(s):
probabilities = [n_x/len(s) for x,n_x in collections.Counter(s).items()]
e_x = [-p_x*math.log(p_x,2) for p_x in probabilities]
return sum(e_x)
测试用例 - 取自CyberChef entropy estimator的英文文本：
>>> H(range(0,256))
8.0
>>> H(range(0,64))
6.0
>>> H(range(0,128))
7.0
>>> H([0,1])
1.0
>>> H('Standard English text usually falls somewhere between 3.5 and 5')
4.228788210509104
这是我的方法：
labels = [0, 0, 1, 1]
from collections import Counter
from scipy import stats
stats.entropy(list(Counter(labels).values()), base=2)
此方法通过允许分箱扩展了其他解决方案。例如，bin=None（默认）不会分箱x并将计算 的每个元素的经验概率x，而在计算经验概率之前将bin=256块x分成 256 个箱。
import numpy as np
def entropy(x, bins=None):
N = x.shape[0]
if bins is None:
counts = np.bincount(x)
else:
counts = np.histogram(x, bins=bins)[0] # 0th idx is counts
p = counts[np.nonzero(counts)]/N # avoids log(0)
H = -np.dot( p, np.log2(p) )
return H
这是迄今为止我发现的最快的 Python 实现：
import numpy as np
def entropy(labels):
ps = np.bincount(labels) / len(labels)
return -np.sum([p * np.log2(p) for p in ps if p > 0])
BiEntropy 不会是计算熵的最快方法，但它是严格的，并且以明确定义的方式建立在 Shannon Entropy 之上。它已经在包括图像相关应用在内的各个领域进行了测试。它在 Github 上用 Python 实现。
派对迟到了，但我偶然发现了这一点，所有答案似乎都依赖于 Kullback-Leibler 散度，它没有上限，因此不符合我的需求。
在这里，我有一个TODO!从 [0,1] 开始的熵函数的近似值（可以改进）。
它计算单个列的偏差。
class Pandas_Dataframe_helper:
#some other methods here...
@staticmethod
def column_biass(df_column):
df_column_as_list = list(df_column)
N = len(df_column_as_list)
values,counts = np.unique(df_column_as_list, return_counts=True)
#generate synth list (TODO! what if not even number? Minimum Comun Multiple of(num_different_labels,[x for x in counts]))
num_different_labels = len(values)
num_items_per_label = N // num_different_labels
synthetic_list = []
for current_value in values:
synthetic_list.extend([current_value] * num_items_per_label)
#TODO! aproximacion
if(len(synthetic_list) != len(df_column_as_list)):
synthetic_list.extend([current_value] * (len(df_column_as_list) - len(synthetic_list)))
#now, extrapolate differences between sorted-input-list and synsthetic_list
df_column_as_list_sorted = sorted(df_column_as_list)
counter_unmatches = 0
for i in range(0,N):
if(df_column_as_list_sorted[i] != synthetic_list[i]):
counter_unmatches += 1
#upper_bound = g(N,num_different_labels)
#((K-1)M)-1 K==num_different_labels , M==num theorically perfect distribution's items per label
upper_bound = ((num_different_labels-1)*num_items_per_label)-1
return counter_unmatches/upper_bound
#---------------------------------------------------------------------------------------------------------------------
from collections import Counter
from scipy import stats
labels = [0.9, 0.09, 0.1]
stats.entropy(list(Counter(labels).keys()), base=2)
上面的答案很好，但是如果您需要一个可以沿不同轴操作的版本，这里有一个可行的实现。
def entropy(A, axis=None):
"""Computes the Shannon entropy of the elements of A. Assumes A is
an array-like of nonnegative ints whose max value is approximately
the number of unique values present.
>>> a = [0, 1]
>>> entropy(a)
1.0
>>> A = np.c_[a, a]
>>> entropy(A)
1.0
>>> A # doctest: +NORMALIZE_WHITESPACE
array([[0, 0], [1, 1]])
>>> entropy(A, axis=0) # doctest: +NORMALIZE_WHITESPACE
array([ 1., 1.])
>>> entropy(A, axis=1) # doctest: +NORMALIZE_WHITESPACE
array([[ 0.], [ 0.]])
>>> entropy([0, 0, 0])
0.0
>>> entropy([])
0.0
>>> entropy([5])
0.0
"""
if A is None or len(A) < 2:
return 0.
A = np.asarray(A)
if axis is None:
A = A.flatten()
counts = np.bincount(A) # needs small, non-negative ints
counts = counts[counts > 0]
if len(counts) == 1:
return 0. # avoid returning -0.0 to prevent weird doctests
probs = counts / float(A.size)
return -np.sum(probs * np.log2(probs))
elif axis == 0:
entropies = map(lambda col: entropy(col), A.T)
return np.array(entropies)
elif axis == 1:
entropies = map(lambda row: entropy(row), A)
return np.array(entropies).reshape((-1, 1))
else:
raise ValueError("unsupported axis: {}".format(axis))
def entropy(base, prob_a, prob_b ):
import math
base=2
x=prob_a
y=prob_b
expression =-((x*math.log(x,base)+(y*math.log(y,base))))
return [expression]