How can I use broadcasting with NumPy to speed up this correlation calculation?

问题

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.

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)

回答1:

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, :]

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)

whose usage is:

corr_df = corr_np2pd(df, biweight_midcorrelation)

This could be made even faster by implementing the last computation with Numba.

---

Introduction

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.

---

There are two practical approaches to your problem:

1. to manually compute the correlation matrix
2. to generate a correlation function to be passed to pd.DataFrame.corr()

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.

Let us see how can we push the performances in both cases.

Manually Computing the Correlation Matrix

---

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)

---

The function with explicit looping that is now in the comments belongs to this category and it is reported below as 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)

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

Now, this has some points of improvement:

• the intermediate computations can be reduced
• the final nested loop could be made more efficient

This last point could be improved with two ways:

• 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, :]

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

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)))

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)))

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)

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.

---

Testing

For the suggested input:

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}})

we obtain:

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

---

Benchmarks

%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

These results would indicate the Numba-based approach to be the fastest, closely followed by the NumPy-vectorized version of your original approach.

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.

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.

Final warning: all these benchmarks should be taken with a grain of salt!

(EDITED to include a Numba-based approach to use with pd.DataFrame.corr()).

回答2:

With a copy-n-paste of your X:

In [26]: X                                                                                     
Out[26]: 
          sepal_length  sepal_width  petal_length  petal_width
iris_0             5.1          3.5           1.4          0.2
iris_1             4.9          3.0           1.4          0.2
iris_2             4.7          3.2           1.3          0.2
iris_3             4.6          3.1           1.5          0.2
iris_4             5.0          3.6           1.4          0.2
...                ...          ...           ...          ...
iris_145           6.7          3.0           5.2          2.3
iris_146           6.3          2.5           5.0          1.9
iris_147           6.5          3.0           5.2          2.0
iris_148           6.2          3.4           5.4          2.3
iris_149           5.9          3.0           5.1          1.8

[150 rows x 4 columns]

and using it:

In [29]: X.corr(method=_biweight_midcorrelation)                                               
Out[29]: 
              sepal_length  sepal_width  petal_length  petal_width
sepal_length      1.000000    -0.134780      0.831958     0.818575
sepal_width      -0.134780     1.000000     -0.430312    -0.374034
petal_length      0.831958    -0.430312      1.000000     0.952285
petal_width       0.818575    -0.374034      0.952285     1.000000
In [30]: X.corr?                                                                               
In [31]: _biweight_midcorrelation(X['sepal_length'],X['sepal_width'])                          
Out[31]: -0.13477989268659313
In [32]: _biweight_midcorrelation(X['sepal_length'],X['petal_length'])                         
Out[32]: 0.831958204443503

In _biweight_midcorrelation(a, b), a and b are Series, the same size. So all their derived arrays have the same shape, and (a_item * b_item) works just (by broadcasting - the rules of broadcasting apply to 2 1d arrays). I don't see any need for 'outer products'.

来源：https://stackoverflow.com/questions/61090539/how-can-i-use-broadcasting-with-numpy-to-speed-up-this-correlation-calculation