为何numpy.correlate与numpy.corrcoef对归一化向量计算结果存在差异？

我原本认为，对齐的归一化向量下，numpy.correlate和numpy.corrcoef应该返回相同结果，但实际测试发现两个反例，两者数值存在明显差异。

测试代码 #

from math import isclose as near
import numpy as np


def normalizedCrossCorrelation(a, b):
    assert len(a) == len(b)
    normalized_a = [aa / np.linalg.norm(a) for aa in a]
    normalized_b = [bb / np.linalg.norm(b) for bb in b]
    return np.correlate(normalized_a, normalized_b)[0]


def test_normalizedCrossCorrelationOfSimilarVectorsRegression0():
    v0 = [1, 2, 3, 2, 1, 0, -2, -1, 0]
    v1 = [1, 1.9, 2.8, 2, 1.1, 0, -2.2, -0.9, 0.2]
    assert near(normalizedCrossCorrelation(v0, v1), 0.9969260391224474)
    print(f"{np.corrcoef(v0, v1)=}")
    assert near(normalizedCrossCorrelation(v0, v1), np.corrcoef(v0, v1)[0, 1])


def test_normalizedCrossCorrelationOfSimilarVectorsRegression1():
    v0 = [1, 2, 3, 2, 1, 0, -2, -1, 0]
    v1 = [0.8, 1.9, 2.5, 2.1, 1.2, -0.3, -2.4, -1.4, 0.4]
    assert near(normalizedCrossCorrelation(v0, v1), 0.9809817769512982)
    print(f"{np.corrcoef(v0, v1)=}")
    assert near(normalizedCrossCorrelation(v0, v1), np.corrcoef(v0, v1)[0, 1])

Pytest失败输出 #

E       assert False
E        +  where False = near(0.9969260391224474, 0.9963146417122921)
E        +    where 0.9969260391224474 = normalizedCrossCorrelation([1, 2, 3, 2, 1, 0, ...], [1, 1.9, 2.8, 2, 1.1, 0, ...])


E       assert False
E        +  where False = near(0.9809817769512982, 0.9826738919606931)
E        +    where 0.9809817769512982 = normalizedCrossCorrelation([1, 2, 3, 2, 1, 0, ...], [0.8, 1.9, 2.5, 2.1, 1.2, -0.3, ...])

差异原因 #

两者的核心区别在于归一化的前提：

• 自定义的normalizedCrossCorrelation函数是对原始向量做L2归一化（直接除以向量的L2范数），再计算点积，对应归一化互相关的结果。
• numpy.corrcoef计算的是皮尔逊相关系数，它会先对向量减去各自的均值（去中心化），再对去均值后的向量做L2归一化，最后计算点积。

当测试向量的均值不为0时，两种归一化方式的结果自然会产生差异。比如测试用的v0均值约为0.6667，v1的均值也不为0，这就导致最终的相关值出现明显偏差。

内容的提问来源于stack exchange，提问作者user2183336