Computing the correlation coefficient between two multi-dimensional arrays

I have two arrays that have the shapes N X T and M X T. I’d like to compute the correlation coefficient across T between every possible pair of rows n and m (from N and M, respectively).

What’s the fastest, most pythonic way to do this? (Looping over N and M would seem to me to be neither fast nor pythonic.) I’m expecting the answer to involve numpy and/or scipy. Right now my arrays are numpy arrays, but I’m open to converting them to a different type.

I’m expecting my output to be an array with the shape N X M.

N.B. When I say “correlation coefficient,” I mean the Pearson product-moment correlation coefficient.

Here are some things to note:

• The numpy function correlate requires input arrays to be one-dimensional.
• The numpy function corrcoef accepts two-dimensional arrays, but they must have the same shape.
• The scipy.stats function pearsonr requires input arrays to be one-dimensional.

Answer

Correlation (default ‘valid’ case) between two 2D arrays:

You can simply use matrix-multiplication np.dot like so –

out = np.dot(arr_one,arr_two.T)

Correlation with the default "valid" case between each pairwise row combinations (row1,row2) of the two input arrays would correspond to multiplication result at each (row1,row2) position.

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Row-wise Correlation Coefficient calculation for two 2D arrays:

def corr2_coeff(A, B):
    # Rowwise mean of input arrays & subtract from input arrays themeselves
    A_mA = A - A.mean(1)[:, None]
    B_mB = B - B.mean(1)[:, None]

    # Sum of squares across rows
    ssA = (A_mA**2).sum(1)
    ssB = (B_mB**2).sum(1)

    # Finally get corr coeff
    return np.dot(A_mA, B_mB.T) / np.sqrt(np.dot(ssA[:, None],ssB[None]))

This is based upon this solution to How to apply corr2 functions in Multidimentional arrays in MATLAB

Benchmarking

This section compares runtime performance with the proposed approach against generate_correlation_map & loopy pearsonr based approach listed in the other answer.(taken from the function test_generate_correlation_map() without the value correctness verification code at the end of it). Please note the timings for the proposed approach also include a check at the start to check for equal number of columns in the two input arrays, as also done in that other answer. The runtimes are listed next.

Case #1:

In [106]: A = np.random.rand(1000, 100)

In [107]: B = np.random.rand(1000, 100)

In [108]: %timeit corr2_coeff(A, B)
100 loops, best of 3: 15 ms per loop

In [109]: %timeit generate_correlation_map(A, B)
100 loops, best of 3: 19.6 ms per loop

Case #2:

In [110]: A = np.random.rand(5000, 100)

In [111]: B = np.random.rand(5000, 100)

In [112]: %timeit corr2_coeff(A, B)
1 loops, best of 3: 368 ms per loop

In [113]: %timeit generate_correlation_map(A, B)
1 loops, best of 3: 493 ms per loop

Case #3:

In [114]: A = np.random.rand(10000, 10)

In [115]: B = np.random.rand(10000, 10)

In [116]: %timeit corr2_coeff(A, B)
1 loops, best of 3: 1.29 s per loop

In [117]: %timeit generate_correlation_map(A, B)
1 loops, best of 3: 1.83 s per loop

The other loopy pearsonr based approach seemed too slow, but here are the runtimes for one small datasize –

In [118]: A = np.random.rand(1000, 100)

In [119]: B = np.random.rand(1000, 100)

In [120]: %timeit corr2_coeff(A, B)
100 loops, best of 3: 15.3 ms per loop

In [121]: %timeit generate_correlation_map(A, B)
100 loops, best of 3: 19.7 ms per loop

In [122]: %timeit pearsonr_based(A, B)
1 loops, best of 3: 33 s per loop