如何使用NumPy广播以加快此相关性计算? [英] How can I use broadcasting with NumPy to speed up this correlation calculation?
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问题描述
我正在尝试利用 NumPy广播和后端数组计算以显着加快此功能.不幸的是，它不能很好地扩展，因此我希望大大提高其性能.目前，该代码未正确利用广播进行计算.
I'm trying to take advantage of NumPy broadcasting and backend array computations to significantly speed up this function. Unfortunately, it doesn't scale so well so I'm hoping to greatly improve the performance of this. Right now the code isn't properly utilizing broadcasting for the computations.
我正在使用 WGCNA的bicor函数以此作为黄金标准是目前我所知道的最快的实现. Python版本输出与R函数相同的结果.
I'm using WGCNA's bicor function as a gold standard as this is the fastest implementation I know of at the moment. The Python version outputs the same results as the R function.
# ==============================================================================
# Imports
# ==============================================================================
# Built-ins
import os, sys, time, multiprocessing
# 3rd party
import numpy as np
import pandas as pd
# ==============================================================================
# R Imports
# ==============================================================================
from rpy2 import robjects, rinterface
from rpy2.robjects.packages import importr
from rpy2.robjects import pandas2ri
pandas2ri.activate()
R = robjects.r
NULL = robjects.rinterface.NULL
rinterface.set_writeconsole_regular(None)
WGCNA = importr("WGCNA")
# Python
def _biweight_midcorrelation(a, b):
a_median = np.median(a)
b_median = np.median(b)
# Median absolute deviation
a_mad = np.median(np.abs(a - a_median))
b_mad = np.median(np.abs(b - b_median))
u = (a - a_median) / (9 * a_mad)
v = (b - b_median) / (9 * b_mad)
w_a = np.square(1 - np.square(u)) * ((1 - np.abs(u)) > 0)
w_b = np.square(1 - np.square(v)) * ((1 - np.abs(v)) > 0)
a_item = (a - a_median) * w_a
b_item = (b - b_median) * w_b
return (a_item * b_item).sum() / (
np.sqrt(np.square(a_item).sum()) *
np.sqrt(np.square(b_item).sum()))
def biweight_midcorrelation(X):
return X.corr(method=_biweight_midcorrelation)
# # OLD IMPLEMENTATION
# def biweight_midcorrelation(X):
# median = X.median()
# mad = (X - median).abs().median()
# U = (X - median) / (9 * mad)
# adjacency = np.square(1 - np.square(U)) * ((1 - U.abs()) > 0)
# estimator = (X - median) * adjacency
# bicor_matrix = np.empty((X.shape[1], X.shape[1]), dtype=float)
# for i, ac in enumerate(estimator):
# for j, bc in enumerate(estimator):
# a = estimator[ac]
# b = estimator[bc]
# c = (a * b).sum() / (
# np.sqrt(np.square(a).sum()) * np.sqrt(np.square(b).sum()))
# bicor_matrix[i, j] = c
# bicor_matrix[j, i] = c
# return pd.DataFrame(bicor_matrix, index=X.columns, columns=X.columns)
# R
def biweight_midcorrelation_r_wrapper(X, n_jobs=-1, r_package=None):
"""
WGCNA: bicor
function (x, y = NULL, robustX = TRUE, robustY = TRUE, use = "all.obs",
maxPOutliers = 1, qu <...> dian absolute deviation, or zero variance."))
"""
if r_package is None:
r_package = importr("WGCNA")
if n_jobs == -1:
n_jobs = multiprocessing.cpu_count()
labels = X.columns
r_df_sim = r_package.bicor(pandas2ri.py2ri(X), nThreads=n_jobs)
df_bicor = pd.DataFrame(pandas2ri.ri2py(r_df_sim), index=labels, columns=labels)
return df_bicor
# X.shape = (150,4)
X = pd.DataFrame({'sepal_length': {'iris_0': 5.1, 'iris_1': 4.9, 'iris_2': 4.7, 'iris_3': 4.6, 'iris_4': 5.0, 'iris_5': 5.4, 'iris_6': 4.6, 'iris_7': 5.0, 'iris_8': 4.4, 'iris_9': 4.9, 'iris_10': 5.4, 'iris_11': 4.8, 'iris_12': 4.8, 'iris_13': 4.3, 'iris_14': 5.8, 'iris_15': 5.7, 'iris_16': 5.4, 'iris_17': 5.1, 'iris_18': 5.7, 'iris_19': 5.1, 'iris_20': 5.4, 'iris_21': 5.1, 'iris_22': 4.6, 'iris_23': 5.1, 'iris_24': 4.8, 'iris_25': 5.0, 'iris_26': 5.0, 'iris_27': 5.2, 'iris_28': 5.2, 'iris_29': 4.7, 'iris_30': 4.8, 'iris_31': 5.4, 'iris_32': 5.2, 'iris_33': 5.5, 'iris_34': 4.9, 'iris_35': 5.0, 'iris_36': 5.5, 'iris_37': 4.9, 'iris_38': 4.4, 'iris_39': 5.1, 'iris_40': 5.0, 'iris_41': 4.5, 'iris_42': 4.4, 'iris_43': 5.0, 'iris_44': 5.1, 'iris_45': 4.8, 'iris_46': 5.1, 'iris_47': 4.6, 'iris_48': 5.3, 'iris_49': 5.0, 'iris_50': 7.0, 'iris_51': 6.4, 'iris_52': 6.9, 'iris_53': 5.5, 'iris_54': 6.5, 'iris_55': 5.7, 'iris_56': 6.3, 'iris_57': 4.9, 'iris_58': 6.6, 'iris_59': 5.2, 'iris_60': 5.0, 'iris_61': 5.9, 'iris_62': 6.0, 'iris_63': 6.1, 'iris_64': 5.6, 'iris_65': 6.7, 'iris_66': 5.6, 'iris_67': 5.8, 'iris_68': 6.2, 'iris_69': 5.6, 'iris_70': 5.9, 'iris_71': 6.1, 'iris_72': 6.3, 'iris_73': 6.1, 'iris_74': 6.4, 'iris_75': 6.6, 'iris_76': 6.8, 'iris_77': 6.7, 'iris_78': 6.0, 'iris_79': 5.7, 'iris_80': 5.5, 'iris_81': 5.5, 'iris_82': 5.8, 'iris_83': 6.0, 'iris_84': 5.4, 'iris_85': 6.0, 'iris_86': 6.7, 'iris_87': 6.3, 'iris_88': 5.6, 'iris_89': 5.5, 'iris_90': 5.5, 'iris_91': 6.1, 'iris_92': 5.8, 'iris_93': 5.0, 'iris_94': 5.6, 'iris_95': 5.7, 'iris_96': 5.7, 'iris_97': 6.2, 'iris_98': 5.1, 'iris_99': 5.7, 'iris_100': 6.3, 'iris_101': 5.8, 'iris_102': 7.1, 'iris_103': 6.3, 'iris_104': 6.5, 'iris_105': 7.6, 'iris_106': 4.9, 'iris_107': 7.3, 'iris_108': 6.7, 'iris_109': 7.2, 'iris_110': 6.5, 'iris_111': 6.4, 'iris_112': 6.8, 'iris_113': 5.7, 'iris_114': 5.8, 'iris_115': 6.4, 'iris_116': 6.5, 'iris_117': 7.7, 'iris_118': 7.7, 'iris_119': 6.0, 'iris_120': 6.9, 'iris_121': 5.6, 'iris_122': 7.7, 'iris_123': 6.3, 'iris_124': 6.7, 'iris_125': 7.2, 'iris_126': 6.2, 'iris_127': 6.1, 'iris_128': 6.4, 'iris_129': 7.2, 'iris_130': 7.4, 'iris_131': 7.9, 'iris_132': 6.4, 'iris_133': 6.3, 'iris_134': 6.1, 'iris_135': 7.7, 'iris_136': 6.3, 'iris_137': 6.4, 'iris_138': 6.0, 'iris_139': 6.9, 'iris_140': 6.7, 'iris_141': 6.9, 'iris_142': 5.8, 'iris_143': 6.8, 'iris_144': 6.7, 'iris_145': 6.7, 'iris_146': 6.3, 'iris_147': 6.5, 'iris_148': 6.2, 'iris_149': 5.9}, 'sepal_width': {'iris_0': 3.5, 'iris_1': 3.0, 'iris_2': 3.2, 'iris_3': 3.1, 'iris_4': 3.6, 'iris_5': 3.9, 'iris_6': 3.4, 'iris_7': 3.4, 'iris_8': 2.9, 'iris_9': 3.1, 'iris_10': 3.7, 'iris_11': 3.4, 'iris_12': 3.0, 'iris_13': 3.0, 'iris_14': 4.0, 'iris_15': 4.4, 'iris_16': 3.9, 'iris_17': 3.5, 'iris_18': 3.8, 'iris_19': 3.8, 'iris_20': 3.4, 'iris_21': 3.7, 'iris_22': 3.6, 'iris_23': 3.3, 'iris_24': 3.4, 'iris_25': 3.0, 'iris_26': 3.4, 'iris_27': 3.5, 'iris_28': 3.4, 'iris_29': 3.2, 'iris_30': 3.1, 'iris_31': 3.4, 'iris_32': 4.1, 'iris_33': 4.2, 'iris_34': 3.1, 'iris_35': 3.2, 'iris_36': 3.5, 'iris_37': 3.6, 'iris_38': 3.0, 'iris_39': 3.4, 'iris_40': 3.5, 'iris_41': 2.3, 'iris_42': 3.2, 'iris_43': 3.5, 'iris_44': 3.8, 'iris_45': 3.0, 'iris_46': 3.8, 'iris_47': 3.2, 'iris_48': 3.7, 'iris_49': 3.3, 'iris_50': 3.2, 'iris_51': 3.2, 'iris_52': 3.1, 'iris_53': 2.3, 'iris_54': 2.8, 'iris_55': 2.8, 'iris_56': 3.3, 'iris_57': 2.4, 'iris_58': 2.9, 'iris_59': 2.7, 'iris_60': 2.0, 'iris_61': 3.0, 'iris_62': 2.2, 'iris_63': 2.9, 'iris_64': 2.9, 'iris_65': 3.1, 'iris_66': 3.0, 'iris_67': 2.7, 'iris_68': 2.2, 'iris_69': 2.5, 'iris_70': 3.2, 'iris_71': 2.8, 'iris_72': 2.5, 'iris_73': 2.8, 'iris_74': 2.9, 'iris_75': 3.0, 'iris_76': 2.8, 'iris_77': 3.0, 'iris_78': 2.9, 'iris_79': 2.6, 'iris_80': 2.4, 'iris_81': 2.4, 'iris_82': 2.7, 'iris_83': 2.7, 'iris_84': 3.0, 'iris_85': 3.4, 'iris_86': 3.1, 'iris_87': 2.3, 'iris_88': 3.0, 'iris_89': 2.5, 'iris_90': 2.6, 'iris_91': 3.0, 'iris_92': 2.6, 'iris_93': 2.3, 'iris_94': 2.7, 'iris_95': 3.0, 'iris_96': 2.9, 'iris_97': 2.9, 'iris_98': 2.5, 'iris_99': 2.8, 'iris_100': 3.3, 'iris_101': 2.7, 'iris_102': 3.0, 'iris_103': 2.9, 'iris_104': 3.0, 'iris_105': 3.0, 'iris_106': 2.5, 'iris_107': 2.9, 'iris_108': 2.5, 'iris_109': 3.6, 'iris_110': 3.2, 'iris_111': 2.7, 'iris_112': 3.0, 'iris_113': 2.5, 'iris_114': 2.8, 'iris_115': 3.2, 'iris_116': 3.0, 'iris_117': 3.8, 'iris_118': 2.6, 'iris_119': 2.2, 'iris_120': 3.2, 'iris_121': 2.8, 'iris_122': 2.8, 'iris_123': 2.7, 'iris_124': 3.3, 'iris_125': 3.2, 'iris_126': 2.8, 'iris_127': 3.0, 'iris_128': 2.8, 'iris_129': 3.0, 'iris_130': 2.8, 'iris_131': 3.8, 'iris_132': 2.8, 'iris_133': 2.8, 'iris_134': 2.6, 'iris_135': 3.0, 'iris_136': 3.4, 'iris_137': 3.1, 'iris_138': 3.0, 'iris_139': 3.1, 'iris_140': 3.1, 'iris_141': 3.1, 'iris_142': 2.7, 'iris_143': 3.2, 'iris_144': 3.3, 'iris_145': 3.0, 'iris_146': 2.5, 'iris_147': 3.0, 'iris_148': 3.4, 'iris_149': 3.0}, 'petal_length': {'iris_0': 1.4, 'iris_1': 1.4, 'iris_2': 1.3, 'iris_3': 1.5, 'iris_4': 1.4, 'iris_5': 1.7, 'iris_6': 1.4, 'iris_7': 1.5, 'iris_8': 1.4, 'iris_9': 1.5, 'iris_10': 1.5, 'iris_11': 1.6, 'iris_12': 1.4, 'iris_13': 1.1, 'iris_14': 1.2, 'iris_15': 1.5, 'iris_16': 1.3, 'iris_17': 1.4, 'iris_18': 1.7, 'iris_19': 1.5, 'iris_20': 1.7, 'iris_21': 1.5, 'iris_22': 1.0, 'iris_23': 1.7, 'iris_24': 1.9, 'iris_25': 1.6, 'iris_26': 1.6, 'iris_27': 1.5, 'iris_28': 1.4, 'iris_29': 1.6, 'iris_30': 1.6, 'iris_31': 1.5, 'iris_32': 1.5, 'iris_33': 1.4, 'iris_34': 1.5, 'iris_35': 1.2, 'iris_36': 1.3, 'iris_37': 1.4, 'iris_38': 1.3, 'iris_39': 1.5, 'iris_40': 1.3, 'iris_41': 1.3, 'iris_42': 1.3, 'iris_43': 1.6, 'iris_44': 1.9, 'iris_45': 1.4, 'iris_46': 1.6, 'iris_47': 1.4, 'iris_48': 1.5, 'iris_49': 1.4, 'iris_50': 4.7, 'iris_51': 4.5, 'iris_52': 4.9, 'iris_53': 4.0, 'iris_54': 4.6, 'iris_55': 4.5, 'iris_56': 4.7, 'iris_57': 3.3, 'iris_58': 4.6, 'iris_59': 3.9, 'iris_60': 3.5, 'iris_61': 4.2, 'iris_62': 4.0, 'iris_63': 4.7, 'iris_64': 3.6, 'iris_65': 4.4, 'iris_66': 4.5, 'iris_67': 4.1, 'iris_68': 4.5, 'iris_69': 3.9, 'iris_70': 4.8, 'iris_71': 4.0, 'iris_72': 4.9, 'iris_73': 4.7, 'iris_74': 4.3, 'iris_75': 4.4, 'iris_76': 4.8, 'iris_77': 5.0, 'iris_78': 4.5, 'iris_79': 3.5, 'iris_80': 3.8, 'iris_81': 3.7, 'iris_82': 3.9, 'iris_83': 5.1, 'iris_84': 4.5, 'iris_85': 4.5, 'iris_86': 4.7, 'iris_87': 4.4, 'iris_88': 4.1, 'iris_89': 4.0, 'iris_90': 4.4, 'iris_91': 4.6, 'iris_92': 4.0, 'iris_93': 3.3, 'iris_94': 4.2, 'iris_95': 4.2, 'iris_96': 4.2, 'iris_97': 4.3, 'iris_98': 3.0, 'iris_99': 4.1, 'iris_100': 6.0, 'iris_101': 5.1, 'iris_102': 5.9, 'iris_103': 5.6, 'iris_104': 5.8, 'iris_105': 6.6, 'iris_106': 4.5, 'iris_107': 6.3, 'iris_108': 5.8, 'iris_109': 6.1, 'iris_110': 5.1, 'iris_111': 5.3, 'iris_112': 5.5, 'iris_113': 5.0, 'iris_114': 5.1, 'iris_115': 5.3, 'iris_116': 5.5, 'iris_117': 6.7, 'iris_118': 6.9, 'iris_119': 5.0, 'iris_120': 5.7, 'iris_121': 4.9, 'iris_122': 6.7, 'iris_123': 4.9, 'iris_124': 5.7, 'iris_125': 6.0, 'iris_126': 4.8, 'iris_127': 4.9, 'iris_128': 5.6, 'iris_129': 5.8, 'iris_130': 6.1, 'iris_131': 6.4, 'iris_132': 5.6, 'iris_133': 5.1, 'iris_134': 5.6, 'iris_135': 6.1, 'iris_136': 5.6, 'iris_137': 5.5, 'iris_138': 4.8, 'iris_139': 5.4, 'iris_140': 5.6, 'iris_141': 5.1, 'iris_142': 5.1, 'iris_143': 5.9, 'iris_144': 5.7, 'iris_145': 5.2, 'iris_146': 5.0, 'iris_147': 5.2, 'iris_148': 5.4, 'iris_149': 5.1}, 'petal_width': {'iris_0': 0.2, 'iris_1': 0.2, 'iris_2': 0.2, 'iris_3': 0.2, 'iris_4': 0.2, 'iris_5': 0.4, 'iris_6': 0.3, 'iris_7': 0.2, 'iris_8': 0.2, 'iris_9': 0.1, 'iris_10': 0.2, 'iris_11': 0.2, 'iris_12': 0.1, 'iris_13': 0.1, 'iris_14': 0.2, 'iris_15': 0.4, 'iris_16': 0.4, 'iris_17': 0.3, 'iris_18': 0.3, 'iris_19': 0.3, 'iris_20': 0.2, 'iris_21': 0.4, 'iris_22': 0.2, 'iris_23': 0.5, 'iris_24': 0.2, 'iris_25': 0.2, 'iris_26': 0.4, 'iris_27': 0.2, 'iris_28': 0.2, 'iris_29': 0.2, 'iris_30': 0.2, 'iris_31': 0.4, 'iris_32': 0.1, 'iris_33': 0.2, 'iris_34': 0.2, 'iris_35': 0.2, 'iris_36': 0.2, 'iris_37': 0.1, 'iris_38': 0.2, 'iris_39': 0.2, 'iris_40': 0.3, 'iris_41': 0.3, 'iris_42': 0.2, 'iris_43': 0.6, 'iris_44': 0.4, 'iris_45': 0.3, 'iris_46': 0.2, 'iris_47': 0.2, 'iris_48': 0.2, 'iris_49': 0.2, 'iris_50': 1.4, 'iris_51': 1.5, 'iris_52': 1.5, 'iris_53': 1.3, 'iris_54': 1.5, 'iris_55': 1.3, 'iris_56': 1.6, 'iris_57': 1.0, 'iris_58': 1.3, 'iris_59': 1.4, 'iris_60': 1.0, 'iris_61': 1.5, 'iris_62': 1.0, 'iris_63': 1.4, 'iris_64': 1.3, 'iris_65': 1.4, 'iris_66': 1.5, 'iris_67': 1.0, 'iris_68': 1.5, 'iris_69': 1.1, 'iris_70': 1.8, 'iris_71': 1.3, 'iris_72': 1.5, 'iris_73': 1.2, 'iris_74': 1.3, 'iris_75': 1.4, 'iris_76': 1.4, 'iris_77': 1.7, 'iris_78': 1.5, 'iris_79': 1.0, 'iris_80': 1.1, 'iris_81': 1.0, 'iris_82': 1.2, 'iris_83': 1.6, 'iris_84': 1.5, 'iris_85': 1.6, 'iris_86': 1.5, 'iris_87': 1.3, 'iris_88': 1.3, 'iris_89': 1.3, 'iris_90': 1.2, 'iris_91': 1.4, 'iris_92': 1.2, 'iris_93': 1.0, 'iris_94': 1.3, 'iris_95': 1.2, 'iris_96': 1.3, 'iris_97': 1.3, 'iris_98': 1.1, 'iris_99': 1.3, 'iris_100': 2.5, 'iris_101': 1.9, 'iris_102': 2.1, 'iris_103': 1.8, 'iris_104': 2.2, 'iris_105': 2.1, 'iris_106': 1.7, 'iris_107': 1.8, 'iris_108': 1.8, 'iris_109': 2.5, 'iris_110': 2.0, 'iris_111': 1.9, 'iris_112': 2.1, 'iris_113': 2.0, 'iris_114': 2.4, 'iris_115': 2.3, 'iris_116': 1.8, 'iris_117': 2.2, 'iris_118': 2.3, 'iris_119': 1.5, 'iris_120': 2.3, 'iris_121': 2.0, 'iris_122': 2.0, 'iris_123': 1.8, 'iris_124': 2.1, 'iris_125': 1.8, 'iris_126': 1.8, 'iris_127': 1.8, 'iris_128': 2.1, 'iris_129': 1.6, 'iris_130': 1.9, 'iris_131': 2.0, 'iris_132': 2.2, 'iris_133': 1.5, 'iris_134': 1.4, 'iris_135': 2.3, 'iris_136': 2.4, 'iris_137': 1.8, 'iris_138': 1.8, 'iris_139': 2.1, 'iris_140': 2.4, 'iris_141': 2.3, 'iris_142': 1.9, 'iris_143': 2.3, 'iris_144': 2.5, 'iris_145': 2.3, 'iris_146': 1.9, 'iris_147': 2.0, 'iris_148': 2.3, 'iris_149': 1.8}})
# Python computation
start_time = time.time()
df_bicor__python = biweight_midcorrelation(X)
# R computation
df_bicor__r = biweight_midcorrelation_r_wrapper(X)
np.allclose(df_bicor__python, df_bicor__r)
推荐答案
摘要
一个人大约可以写出这种计算. (对于您指定的输入)快一个数量级:
Summary
One could write this computation approx. one order of magnitude faster (for the input you specified) with:
import numpy as np
def biweight_midcorrelation(arr):
n, m = arr.shape
arr = arr - np.median(arr, axis=0, keepdims=True)
v = 1 - (arr / (9 * np.median(np.abs(arr), axis=0, keepdims=True))) ** 2
arr = arr * v ** 2 * (v > 0)
norms = np.sqrt(np.sum(arr ** 2, axis=0))
return np.einsum('mi,mj->ij', arr, arr) / norms[:, None] / norms[None, :]
通过以下方式桥接到Pandas数据框:
to be bridged to a Pandas dataframe by:
import pandas as pd
def corr_np2pd(df, func):
return pd.DataFrame(func(np.array(df)), index=df.columns, columns=df.columns)
其用法是:
corr_df = corr_np2pd(df, biweight_midcorrelation)
通过使用Numba进行最后的计算，可以使速度更快.
This could be made even faster by implementing the last computation with Numba.
我不太确定为什么您希望广播在当前代码中有所帮助. 您可能是指向量化吗? 无论如何，我相信可以编写更快的代码，并且您的旧"方法的矢量化版本将胜过您当前的方法. 使用Numba可以使速度更快.
I am not quite sure why you expect broadcasting to be of help in the current code. Did you perhaps mean vectorizing? Anyway, I believe that it is possible to write faster code, and a vectorized version of your "old" approach would outperform your current approach. This could be made even faster using Numba.
有两种实用的方法可以解决您的问题:
There are two practical approaches to your problem:
- 手动计算相关矩阵
- 生成相关函数以传递给
pd.DataFrame.corr()
进行(1)时，如果不计算相关矩阵的不必要部分，可能无法避免显式循环.
When doing (1), an explicit looping may not be avoidable without computing unnecessary parts of the correlation matrix.
When doing (2), it will be necessary to compute the auxiliary value of the computation for each (symmetric) pair of the 1D inputs (2 * comb(n, 2) times), as opposed to computing the auxiliary values only once for each of the 1D inputs (n times). For example, for the input specified in the question, one would need to perform n == 4 pre-computations, but, if done in pairwise fashion, this number becomes 2 * comb(4, 2) == 12.
让我们看看如何在两种情况下都能提高演出效果.
Let us see how can we push the performances in both cases.
让我们首先定义一个函数以充当Pandas到NumPy的桥梁:
Let us first define a function to serve as a Pandas-to-NumPy bridge:
import numpy as np
import pandas as pd
def corr_np2pd(df, func):
return pd.DataFrame(func(np.array(df)), index=df.columns, columns=df.columns)
注释中现在带有显式循环的函数属于此类别，下面将其报告为biweight_midcorrelation_pd_OP():
def biweight_midcorrelation_pd_OP(X):
median = X.median()
mad = (X - median).abs().median()
U = (X - median) / (9 * mad)
adjacency = np.square(1 - np.square(U)) * ((1 - U.abs()) > 0)
estimator = (X - median) * adjacency
bicor_matrix = np.empty((X.shape[1], X.shape[1]), dtype=float)
for i, ac in enumerate(estimator):
for j, bc in enumerate(estimator):
a = estimator[ac]
b = estimator[bc]
c = (a * b).sum() / (
np.sqrt(np.square(a).sum()) * np.sqrt(np.square(b).sum()))
bicor_matrix[i, j] = c
bicor_matrix[j, i] = c
return pd.DataFrame(bicor_matrix, index=X.columns, columns=X.columns)
对此稍作修改的版本，其中的计算完全在NumPy中完成，应与corr_np2pd()一起使用，内容为:
A slightly modified version of that, where the computation is done entirely in NumPy and which should be used with corr_np2pd(), reads:
def biweight_midcorrelation_OP(arr):
n, m = arr.shape
med = np.median(arr, axis=0, keepdims=True)
mad = np.median(np.abs(arr - med), axis=0, keepdims=True)
u = (arr - med) / (9 * mad)
adj = ((1 - u ** 2) ** 2) * ((1 - np.abs(u)) > 0)
est = (arr - med) * adj
result = np.empty((m, m))
for i in range(m):
for j in range(m):
a = est[:, i]
b = est[:, j]
c = (a * b).sum() / (
np.sqrt(np.sum(a ** 2)) * np.sqrt(np.sum(b ** 2)))
result[i, j] = result[j, i] = c
return result
现在，这有一些改进点:
Now, this has some points of improvement:
- 中间计算可以减少
- 可以使最终的嵌套循环更高效
可以通过以下两种方法来改善最后一点:
This last point could be improved with two ways:
- 通过只计算一次对称索引，得出
biweight_midcorrelation_np() - 通过矢量化形式将其写入，从而生成
biweight_midcorrelation_npv()
- by only computing the symmetric indices once, resulting in
biweight_midcorrelation_np() - by writing it in vectorized form, resulting in
biweight_midcorrelation_npv()
def biweight_midcorrelation_np(arr):
n, m = arr.shape
arr = arr - np.median(arr, axis=0, keepdims=True)
v = 1 - (arr / (9 * np.median(np.abs(arr), axis=0, keepdims=True))) ** 2
arr = arr * v ** 2 * (v > 0)
norms = np.sqrt(np.sum(arr ** 2, axis=0))
result = np.empty((m, m))
np.fill_diagonal(result, 1.0)
for i, j in zip(*np.triu_indices(m, 1)):
result[i, j] = result[j, i] = \
np.sum(arr[:, i] * arr[:, j]) / norms[i] / norms[j]
return result
def biweight_midcorrelation_npv(arr):
n, m = arr.shape
arr = arr - np.median(arr, axis=0, keepdims=True)
v = 1 - (arr / (9 * np.median(np.abs(arr), axis=0, keepdims=True))) ** 2
arr = arr * v ** 2 * (v > 0)
norms = np.sqrt(np.sum(arr ** 2, axis=0))
return np.einsum('mi,mj->ij', arr, arr) / norms[:, None] / norms[None, :]
由于显式循环，只要m小，第一个就会很快.
第二个通常会很快，但是两次计算矩阵的某些项似乎效率不高.
为了克服这两个问题，可以使用Numba重写最后一个循环:
The first one will be fast as long as m is small, because of the explicit looping.
The second one will generally be fast, but it seems inefficient to compute some of the entries of the matrix twice.
To overcome both issues, one could rewrite the final loop with Numba:
import numba as nb
@nb.jit
def _biweight_midcorrelation_triu_nb(n, m, est, norms, result):
for i in range(m):
for j in range(i + 1, m):
x = 0
for k in range(n):
x += est[k, i] * est[k, j]
result[i, j] = result[j, i] = x / norms[i] / norms[j]
def biweight_midcorrelation_nb(arr):
n, m = arr.shape
arr = arr - np.median(arr, axis=0, keepdims=True)
v = 1 - (arr / (9 * np.median(np.abs(arr), axis=0, keepdims=True))) ** 2
arr = arr * v ** 2 * (v > 0)
norms = np.sqrt(np.sum(arr ** 2, axis=0))
result = np.empty((m, m))
np.fill_diagonal(result, 1.0)
_biweight_midcorrelation_triu_nb(n, m, arr, norms, result)
return result
成对相关函数
您现在提议的方法的稍作修改的版本属于此类别:
Pairwise Correlation Function
A slightly modified version of your now proposed approach belongs to this category:
def pairwise_biweight_midcorrelation_OP(a, b):
a_median = np.median(a)
b_median = np.median(b)
a_mad = np.median(np.abs(a - a_median))
b_mad = np.median(np.abs(b - b_median))
u_a = (a - a_median) / (9 * a_mad)
u_b = (b - b_median) / (9 * b_mad)
adj_a = (1 - u_a ** 2) ** 2 * ((1 - np.abs(u_a)) > 0)
adj_b = (1 - u_b ** 2) ** 2 * ((1 - np.abs(u_b)) > 0)
a = (a - a_median) * adj_a
b = (b - b_median) * adj_b
return np.sum(a * b) / (np.sqrt(np.sum(a ** 2)) * np.sqrt(np.sum(b ** 2)))
使用与上述类似的简化方法，可以使代码写得更简洁一些:
This may be written a bit more concisely, using similar simplifications as above, resuling in:
def pairwise_biweight_midcorrelation_opt(a, b):
a = a - np.median(a)
b = b - np.median(b)
v_a = 1 - (a / (9 * np.median(np.abs(a)))) ** 2
v_b = 1 - (b / (9 * np.median(np.abs(b)))) ** 2
a = a * v_a ** 2 * (v_a > 0)
b = b * v_b ** 2 * (v_b > 0)
return np.sum(a * b) / (np.sqrt(np.sum(a ** 2)) * np.sqrt(np.sum(b ** 2)))
最后一个操作是在a和b上执行三次求和，但实际上可以在一个循环中完成，这可以通过Numba再次提高:
The last operation is performing summation over a and b three times, but it could actually be done in a single loop, which could be again made fast with Numba:
@nb.jit
def pairwise_biweight_midcorrelation_nb(a, b):
n = a.size
a = a - np.median(a)
b = b - np.median(b)
v_a = 1 - (a / (9 * np.median(np.abs(a)))) ** 2
v_b = 1 - (b / (9 * np.median(np.abs(b)))) ** 2
a = (v_a > 0) * a * v_a ** 2
b = (v_b > 0) * b * v_b ** 2
s_ab = s_aa = s_bb = 0
for i in range(n):
s_ab += a[i] * b[i]
s_aa += a[i] * a[i]
s_bb += b[i] * b[i]
return s_ab / np.sqrt(s_aa) / np.sqrt(s_bb)
但是没有避免执行预计算的简单方法 2 * comb(n, 2) 次，而不是n次.
故事的另一面是，此类方法需要较少的内存，因为每次迭代仅考虑两个一维数组.
But there is no simple way of avoiding to perform the pre-computations 2 * comb(n, 2) times instead of n times.
The other side of the story is that this class of approaches requires less memory as only two 1D array are considered at each iteration.
对于建议的输入:
import pandas as pd
df = pd.DataFrame({'sepal_length': {'iris_0': 5.1, 'iris_1': 4.9, 'iris_2': 4.7, 'iris_3': 4.6, 'iris_4': 5.0, 'iris_5': 5.4, 'iris_6': 4.6, 'iris_7': 5.0, 'iris_8': 4.4, 'iris_9': 4.9, 'iris_10': 5.4, 'iris_11': 4.8, 'iris_12': 4.8, 'iris_13': 4.3, 'iris_14': 5.8, 'iris_15': 5.7, 'iris_16': 5.4, 'iris_17': 5.1, 'iris_18': 5.7, 'iris_19': 5.1, 'iris_20': 5.4, 'iris_21': 5.1, 'iris_22': 4.6, 'iris_23': 5.1, 'iris_24': 4.8, 'iris_25': 5.0, 'iris_26': 5.0, 'iris_27': 5.2, 'iris_28': 5.2, 'iris_29': 4.7, 'iris_30': 4.8, 'iris_31': 5.4, 'iris_32': 5.2, 'iris_33': 5.5, 'iris_34': 4.9, 'iris_35': 5.0, 'iris_36': 5.5, 'iris_37': 4.9, 'iris_38': 4.4, 'iris_39': 5.1, 'iris_40': 5.0, 'iris_41': 4.5, 'iris_42': 4.4, 'iris_43': 5.0, 'iris_44': 5.1, 'iris_45': 4.8, 'iris_46': 5.1, 'iris_47': 4.6, 'iris_48': 5.3, 'iris_49': 5.0, 'iris_50': 7.0, 'iris_51': 6.4, 'iris_52': 6.9, 'iris_53': 5.5, 'iris_54': 6.5, 'iris_55': 5.7, 'iris_56': 6.3, 'iris_57': 4.9, 'iris_58': 6.6, 'iris_59': 5.2, 'iris_60': 5.0, 'iris_61': 5.9, 'iris_62': 6.0, 'iris_63': 6.1, 'iris_64': 5.6, 'iris_65': 6.7, 'iris_66': 5.6, 'iris_67': 5.8, 'iris_68': 6.2, 'iris_69': 5.6, 'iris_70': 5.9, 'iris_71': 6.1, 'iris_72': 6.3, 'iris_73': 6.1, 'iris_74': 6.4, 'iris_75': 6.6, 'iris_76': 6.8, 'iris_77': 6.7, 'iris_78': 6.0, 'iris_79': 5.7, 'iris_80': 5.5, 'iris_81': 5.5, 'iris_82': 5.8, 'iris_83': 6.0, 'iris_84': 5.4, 'iris_85': 6.0, 'iris_86': 6.7, 'iris_87': 6.3, 'iris_88': 5.6, 'iris_89': 5.5, 'iris_90': 5.5, 'iris_91': 6.1, 'iris_92': 5.8, 'iris_93': 5.0, 'iris_94': 5.6, 'iris_95': 5.7, 'iris_96': 5.7, 'iris_97': 6.2, 'iris_98': 5.1, 'iris_99': 5.7, 'iris_100': 6.3, 'iris_101': 5.8, 'iris_102': 7.1, 'iris_103': 6.3, 'iris_104': 6.5, 'iris_105': 7.6, 'iris_106': 4.9, 'iris_107': 7.3, 'iris_108': 6.7, 'iris_109': 7.2, 'iris_110': 6.5, 'iris_111': 6.4, 'iris_112': 6.8, 'iris_113': 5.7, 'iris_114': 5.8, 'iris_115': 6.4, 'iris_116': 6.5, 'iris_117': 7.7, 'iris_118': 7.7, 'iris_119': 6.0, 'iris_120': 6.9, 'iris_121': 5.6, 'iris_122': 7.7, 'iris_123': 6.3, 'iris_124': 6.7, 'iris_125': 7.2, 'iris_126': 6.2, 'iris_127': 6.1, 'iris_128': 6.4, 'iris_129': 7.2, 'iris_130': 7.4, 'iris_131': 7.9, 'iris_132': 6.4, 'iris_133': 6.3, 'iris_134': 6.1, 'iris_135': 7.7, 'iris_136': 6.3, 'iris_137': 6.4, 'iris_138': 6.0, 'iris_139': 6.9, 'iris_140': 6.7, 'iris_141': 6.9, 'iris_142': 5.8, 'iris_143': 6.8, 'iris_144': 6.7, 'iris_145': 6.7, 'iris_146': 6.3, 'iris_147': 6.5, 'iris_148': 6.2, 'iris_149': 5.9}, 'sepal_width': {'iris_0': 3.5, 'iris_1': 3.0, 'iris_2': 3.2, 'iris_3': 3.1, 'iris_4': 3.6, 'iris_5': 3.9, 'iris_6': 3.4, 'iris_7': 3.4, 'iris_8': 2.9, 'iris_9': 3.1, 'iris_10': 3.7, 'iris_11': 3.4, 'iris_12': 3.0, 'iris_13': 3.0, 'iris_14': 4.0, 'iris_15': 4.4, 'iris_16': 3.9, 'iris_17': 3.5, 'iris_18': 3.8, 'iris_19': 3.8, 'iris_20': 3.4, 'iris_21': 3.7, 'iris_22': 3.6, 'iris_23': 3.3, 'iris_24': 3.4, 'iris_25': 3.0, 'iris_26': 3.4, 'iris_27': 3.5, 'iris_28': 3.4, 'iris_29': 3.2, 'iris_30': 3.1, 'iris_31': 3.4, 'iris_32': 4.1, 'iris_33': 4.2, 'iris_34': 3.1, 'iris_35': 3.2, 'iris_36': 3.5, 'iris_37': 3.6, 'iris_38': 3.0, 'iris_39': 3.4, 'iris_40': 3.5, 'iris_41': 2.3, 'iris_42': 3.2, 'iris_43': 3.5, 'iris_44': 3.8, 'iris_45': 3.0, 'iris_46': 3.8, 'iris_47': 3.2, 'iris_48': 3.7, 'iris_49': 3.3, 'iris_50': 3.2, 'iris_51': 3.2, 'iris_52': 3.1, 'iris_53': 2.3, 'iris_54': 2.8, 'iris_55': 2.8, 'iris_56': 3.3, 'iris_57': 2.4, 'iris_58': 2.9, 'iris_59': 2.7, 'iris_60': 2.0, 'iris_61': 3.0, 'iris_62': 2.2, 'iris_63': 2.9, 'iris_64': 2.9, 'iris_65': 3.1, 'iris_66': 3.0, 'iris_67': 2.7, 'iris_68': 2.2, 'iris_69': 2.5, 'iris_70': 3.2, 'iris_71': 2.8, 'iris_72': 2.5, 'iris_73': 2.8, 'iris_74': 2.9, 'iris_75': 3.0, 'iris_76': 2.8, 'iris_77': 3.0, 'iris_78': 2.9, 'iris_79': 2.6, 'iris_80': 2.4, 'iris_81': 2.4, 'iris_82': 2.7, 'iris_83': 2.7, 'iris_84': 3.0, 'iris_85': 3.4, 'iris_86': 3.1, 'iris_87': 2.3, 'iris_88': 3.0, 'iris_89': 2.5, 'iris_90': 2.6, 'iris_91': 3.0, 'iris_92': 2.6, 'iris_93': 2.3, 'iris_94': 2.7, 'iris_95': 3.0, 'iris_96': 2.9, 'iris_97': 2.9, 'iris_98': 2.5, 'iris_99': 2.8, 'iris_100': 3.3, 'iris_101': 2.7, 'iris_102': 3.0, 'iris_103': 2.9, 'iris_104': 3.0, 'iris_105': 3.0, 'iris_106': 2.5, 'iris_107': 2.9, 'iris_108': 2.5, 'iris_109': 3.6, 'iris_110': 3.2, 'iris_111': 2.7, 'iris_112': 3.0, 'iris_113': 2.5, 'iris_114': 2.8, 'iris_115': 3.2, 'iris_116': 3.0, 'iris_117': 3.8, 'iris_118': 2.6, 'iris_119': 2.2, 'iris_120': 3.2, 'iris_121': 2.8, 'iris_122': 2.8, 'iris_123': 2.7, 'iris_124': 3.3, 'iris_125': 3.2, 'iris_126': 2.8, 'iris_127': 3.0, 'iris_128': 2.8, 'iris_129': 3.0, 'iris_130': 2.8, 'iris_131': 3.8, 'iris_132': 2.8, 'iris_133': 2.8, 'iris_134': 2.6, 'iris_135': 3.0, 'iris_136': 3.4, 'iris_137': 3.1, 'iris_138': 3.0, 'iris_139': 3.1, 'iris_140': 3.1, 'iris_141': 3.1, 'iris_142': 2.7, 'iris_143': 3.2, 'iris_144': 3.3, 'iris_145': 3.0, 'iris_146': 2.5, 'iris_147': 3.0, 'iris_148': 3.4, 'iris_149': 3.0}, 'petal_length': {'iris_0': 1.4, 'iris_1': 1.4, 'iris_2': 1.3, 'iris_3': 1.5, 'iris_4': 1.4, 'iris_5': 1.7, 'iris_6': 1.4, 'iris_7': 1.5, 'iris_8': 1.4, 'iris_9': 1.5, 'iris_10': 1.5, 'iris_11': 1.6, 'iris_12': 1.4, 'iris_13': 1.1, 'iris_14': 1.2, 'iris_15': 1.5, 'iris_16': 1.3, 'iris_17': 1.4, 'iris_18': 1.7, 'iris_19': 1.5, 'iris_20': 1.7, 'iris_21': 1.5, 'iris_22': 1.0, 'iris_23': 1.7, 'iris_24': 1.9, 'iris_25': 1.6, 'iris_26': 1.6, 'iris_27': 1.5, 'iris_28': 1.4, 'iris_29': 1.6, 'iris_30': 1.6, 'iris_31': 1.5, 'iris_32': 1.5, 'iris_33': 1.4, 'iris_34': 1.5, 'iris_35': 1.2, 'iris_36': 1.3, 'iris_37': 1.4, 'iris_38': 1.3, 'iris_39': 1.5, 'iris_40': 1.3, 'iris_41': 1.3, 'iris_42': 1.3, 'iris_43': 1.6, 'iris_44': 1.9, 'iris_45': 1.4, 'iris_46': 1.6, 'iris_47': 1.4, 'iris_48': 1.5, 'iris_49': 1.4, 'iris_50': 4.7, 'iris_51': 4.5, 'iris_52': 4.9, 'iris_53': 4.0, 'iris_54': 4.6, 'iris_55': 4.5, 'iris_56': 4.7, 'iris_57': 3.3, 'iris_58': 4.6, 'iris_59': 3.9, 'iris_60': 3.5, 'iris_61': 4.2, 'iris_62': 4.0, 'iris_63': 4.7, 'iris_64': 3.6, 'iris_65': 4.4, 'iris_66': 4.5, 'iris_67': 4.1, 'iris_68': 4.5, 'iris_69': 3.9, 'iris_70': 4.8, 'iris_71': 4.0, 'iris_72': 4.9, 'iris_73': 4.7, 'iris_74': 4.3, 'iris_75': 4.4, 'iris_76': 4.8, 'iris_77': 5.0, 'iris_78': 4.5, 'iris_79': 3.5, 'iris_80': 3.8, 'iris_81': 3.7, 'iris_82': 3.9, 'iris_83': 5.1, 'iris_84': 4.5, 'iris_85': 4.5, 'iris_86': 4.7, 'iris_87': 4.4, 'iris_88': 4.1, 'iris_89': 4.0, 'iris_90': 4.4, 'iris_91': 4.6, 'iris_92': 4.0, 'iris_93': 3.3, 'iris_94': 4.2, 'iris_95': 4.2, 'iris_96': 4.2, 'iris_97': 4.3, 'iris_98': 3.0, 'iris_99': 4.1, 'iris_100': 6.0, 'iris_101': 5.1, 'iris_102': 5.9, 'iris_103': 5.6, 'iris_104': 5.8, 'iris_105': 6.6, 'iris_106': 4.5, 'iris_107': 6.3, 'iris_108': 5.8, 'iris_109': 6.1, 'iris_110': 5.1, 'iris_111': 5.3, 'iris_112': 5.5, 'iris_113': 5.0, 'iris_114': 5.1, 'iris_115': 5.3, 'iris_116': 5.5, 'iris_117': 6.7, 'iris_118': 6.9, 'iris_119': 5.0, 'iris_120': 5.7, 'iris_121': 4.9, 'iris_122': 6.7, 'iris_123': 4.9, 'iris_124': 5.7, 'iris_125': 6.0, 'iris_126': 4.8, 'iris_127': 4.9, 'iris_128': 5.6, 'iris_129': 5.8, 'iris_130': 6.1, 'iris_131': 6.4, 'iris_132': 5.6, 'iris_133': 5.1, 'iris_134': 5.6, 'iris_135': 6.1, 'iris_136': 5.6, 'iris_137': 5.5, 'iris_138': 4.8, 'iris_139': 5.4, 'iris_140': 5.6, 'iris_141': 5.1, 'iris_142': 5.1, 'iris_143': 5.9, 'iris_144': 5.7, 'iris_145': 5.2, 'iris_146': 5.0, 'iris_147': 5.2, 'iris_148': 5.4, 'iris_149': 5.1}, 'petal_width': {'iris_0': 0.2, 'iris_1': 0.2, 'iris_2': 0.2, 'iris_3': 0.2, 'iris_4': 0.2, 'iris_5': 0.4, 'iris_6': 0.3, 'iris_7': 0.2, 'iris_8': 0.2, 'iris_9': 0.1, 'iris_10': 0.2, 'iris_11': 0.2, 'iris_12': 0.1, 'iris_13': 0.1, 'iris_14': 0.2, 'iris_15': 0.4, 'iris_16': 0.4, 'iris_17': 0.3, 'iris_18': 0.3, 'iris_19': 0.3, 'iris_20': 0.2, 'iris_21': 0.4, 'iris_22': 0.2, 'iris_23': 0.5, 'iris_24': 0.2, 'iris_25': 0.2, 'iris_26': 0.4, 'iris_27': 0.2, 'iris_28': 0.2, 'iris_29': 0.2, 'iris_30': 0.2, 'iris_31': 0.4, 'iris_32': 0.1, 'iris_33': 0.2, 'iris_34': 0.2, 'iris_35': 0.2, 'iris_36': 0.2, 'iris_37': 0.1, 'iris_38': 0.2, 'iris_39': 0.2, 'iris_40': 0.3, 'iris_41': 0.3, 'iris_42': 0.2, 'iris_43': 0.6, 'iris_44': 0.4, 'iris_45': 0.3, 'iris_46': 0.2, 'iris_47': 0.2, 'iris_48': 0.2, 'iris_49': 0.2, 'iris_50': 1.4, 'iris_51': 1.5, 'iris_52': 1.5, 'iris_53': 1.3, 'iris_54': 1.5, 'iris_55': 1.3, 'iris_56': 1.6, 'iris_57': 1.0, 'iris_58': 1.3, 'iris_59': 1.4, 'iris_60': 1.0, 'iris_61': 1.5, 'iris_62': 1.0, 'iris_63': 1.4, 'iris_64': 1.3, 'iris_65': 1.4, 'iris_66': 1.5, 'iris_67': 1.0, 'iris_68': 1.5, 'iris_69': 1.1, 'iris_70': 1.8, 'iris_71': 1.3, 'iris_72': 1.5, 'iris_73': 1.2, 'iris_74': 1.3, 'iris_75': 1.4, 'iris_76': 1.4, 'iris_77': 1.7, 'iris_78': 1.5, 'iris_79': 1.0, 'iris_80': 1.1, 'iris_81': 1.0, 'iris_82': 1.2, 'iris_83': 1.6, 'iris_84': 1.5, 'iris_85': 1.6, 'iris_86': 1.5, 'iris_87': 1.3, 'iris_88': 1.3, 'iris_89': 1.3, 'iris_90': 1.2, 'iris_91': 1.4, 'iris_92': 1.2, 'iris_93': 1.0, 'iris_94': 1.3, 'iris_95': 1.2, 'iris_96': 1.3, 'iris_97': 1.3, 'iris_98': 1.1, 'iris_99': 1.3, 'iris_100': 2.5, 'iris_101': 1.9, 'iris_102': 2.1, 'iris_103': 1.8, 'iris_104': 2.2, 'iris_105': 2.1, 'iris_106': 1.7, 'iris_107': 1.8, 'iris_108': 1.8, 'iris_109': 2.5, 'iris_110': 2.0, 'iris_111': 1.9, 'iris_112': 2.1, 'iris_113': 2.0, 'iris_114': 2.4, 'iris_115': 2.3, 'iris_116': 1.8, 'iris_117': 2.2, 'iris_118': 2.3, 'iris_119': 1.5, 'iris_120': 2.3, 'iris_121': 2.0, 'iris_122': 2.0, 'iris_123': 1.8, 'iris_124': 2.1, 'iris_125': 1.8, 'iris_126': 1.8, 'iris_127': 1.8, 'iris_128': 2.1, 'iris_129': 1.6, 'iris_130': 1.9, 'iris_131': 2.0, 'iris_132': 2.2, 'iris_133': 1.5, 'iris_134': 1.4, 'iris_135': 2.3, 'iris_136': 2.4, 'iris_137': 1.8, 'iris_138': 1.8, 'iris_139': 2.1, 'iris_140': 2.4, 'iris_141': 2.3, 'iris_142': 1.9, 'iris_143': 2.3, 'iris_144': 2.5, 'iris_145': 2.3, 'iris_146': 1.9, 'iris_147': 2.0, 'iris_148': 2.3, 'iris_149': 1.8}})
我们获得:
print(np.all(np.isclose(biweight_midcorrelation_pd_OP(df), result)))
# True
print(np.all(np.isclose(corr_np2pd(df, biweight_midcorrelation_OP), result)))
# True
print(np.all(np.isclose(corr_np2pd(df, biweight_midcorrelation_np), result)))
# True
print(np.all(np.isclose(corr_np2pd(df, biweight_midcorrelation_npv), result)))
# True
print(np.all(np.isclose(corr_np2pd(df, biweight_midcorrelation_nb), result)))
# True
print(np.all(np.isclose(df.corr(method=pairwise_biweight_midcorrelation_OP), result)))
# True
print(np.all(np.isclose(df.corr(method=pairwise_biweight_midcorrelation_opt), result)))
# True
print(np.all(np.isclose(df.corr(method=pairwise_biweight_midcorrelation_nb), result)))
# True
基准
%timeit biweight_midcorrelation_pd_OP(df)
# 10 loops, best of 3: 22.1 ms per loop
%timeit corr_np2pd(df, biweight_midcorrelation_OP)
# 1000 loops, best of 3: 682 µs per loop
%timeit corr_np2pd(df, biweight_midcorrelation_np)
# 1000 loops, best of 3: 422 µs per loop
%timeit corr_np2pd(df, biweight_midcorrelation_npv)
# 1000 loops, best of 3: 341 µs per loop
%timeit corr_np2pd(df, biweight_midcorrelation_nb)
# 1000 loops, best of 3: 325 µs per loop
%timeit df.corr(method=pairwise_biweight_midcorrelation_OP)
# 100 loops, best of 3: 1.96 ms per loop
%timeit df.corr(method=pairwise_biweight_midcorrelation_opt)
# 100 loops, best of 3: 1.83 ms per loop
%timeit df.corr(method=pairwise_biweight_midcorrelation_nb)
# 1000 loops, best of 3: 506 µs per loop
这些结果表明基于Numba的方法是最快的方法，紧随其后的是原始方法的NumPy矢量化版本.
These results would indicate the Numba-based approach to be the fastest, closely followed by the NumPy-vectorized version of your original approach.
请注意，从基于Pandas的计算过渡到基于NumPy的纯方法(甚至具有显式循环)，我们得到的速度因子几乎是30倍.
对两个for循环进行矢量化处理后，我们可以得到另一个近似值. 2倍.
Note that going from a Pandas-based computation to a pure NumPy-based approach (even with explicit looping) we get almost 30x speed factor.
And vectorizing the two for loops buys us another approx. 2x factor.
基于pd.DataFrame.corr()的方法在不使用Numba时约为比用NumPy重写的原始方法慢4倍，因此即使看不到显式循环也要小心！
Numba加速pairwise_biweight_midcorrelation_nb()大大促进了这一系列方法的发展，但它可能无法避免预计算的开销.
The pd.DataFrame.corr() based approach(es) are, when not using Numba, approx. 4x slower than your original approach rewritten in NumPy, so be careful even if you do not see explicit looping!
The Numba accelerated pairwise_biweight_midcorrelation_nb() gives a significant boost to this family of approaches, but it cannot possibly avoid the overhead of the pre-computations.
最终警告:所有这些基准测试都应加盐！
Final warning: all these benchmarks should be taken with a grain of salt!
(已编辑，包括基于Numba的方法与pd.DataFrame.corr()一起使用).
(EDITED to include a Numba-based approach to use with pd.DataFrame.corr()).
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