Efficient pairwise correlation for two matrices of features

Question

In Python I need to find the pairwise correlation between all features in a matrix A and all features in a matrix B. In particular, I am interestin

Answer 1

Adding on to the above answer from personal experience,

p1 = N*np.dot(B.T,A)

worked way faster for me when compared to

p1 = N*np.einsum('ij,ik->kj',A,B)

This was especially true when A and B are large dimensional matrices.

Answer 2

Seems scipy.stats.pearsonr follows this definition of Pearson Correlation Coefficient Formula applied on column-wise pairs from A & B -

Based on that formula, you can vectorized easily as the pairwise computations of columns from A and B are independent of each other. Here's one vectorized solution using broadcasting -

# Get number of rows in either A or B
N = B.shape[0]

# Store columnw-wise in A and B, as they would be used at few places
sA = A.sum(0)
sB = B.sum(0)

# Basically there are four parts in the formula. We would compute them one-by-one
p1 = N*np.einsum('ij,ik->kj',A,B)
p2 = sA*sB[:,None]
p3 = N*((B**2).sum(0)) - (sB**2)
p4 = N*((A**2).sum(0)) - (sA**2)

# Finally compute Pearson Correlation Coefficient as 2D array 
pcorr = ((p1 - p2)/np.sqrt(p4*p3[:,None]))

# Get the element corresponding to absolute argmax along the columns 
out = pcorr[np.nanargmax(np.abs(pcorr),axis=0),np.arange(pcorr.shape[1])]

Sample run -

1) Inputs :

In [12]: A
Out[12]: 
array([[ 10.  ,   8.04,   9.14,   7.46],
       [  8.  ,   6.95,   8.14,   6.77],
       [ 13.  ,   7.58,   8.74,  12.74],
       [  9.  ,   8.81,   8.77,   7.11],
       [ 11.  ,   8.33,   9.26,   7.81]])

In [13]: B
Out[13]: 
array([[-14.  ,  -9.96,   8.1 ,   8.84,   8.  ,   7.04],
       [ -6.  ,  -7.24,   6.13,   6.08,   5.  ,   5.25],
       [ -4.  ,  -4.26,   3.1 ,   5.39,   8.  ,   5.56],
       [-12.  , -10.84,   9.13,   8.15,   5.  ,   7.91],
       [ -7.  ,  -4.82,   7.26,   6.42,   8.  ,   6.89]])

2) Original loopy code run -

In [14]: high_corr_out = np.zeros(A.shape[1])
    ...: for A_col in range(A.shape[1]): 
    ...:     high_corr = 0
    ...:     for B_col in range(B.shape[1]):
    ...:         corr,_ = pearsonr(A[:,A_col], B[:,B_col])
    ...:         high_corr = max_absolute(high_corr, corr)
    ...:     high_corr_out[A_col] = high_corr
    ...:     

In [15]: high_corr_out
Out[15]: array([ 0.8067843 ,  0.95678152,  0.74016181, -0.85127779])

3) Proposed code run -

In [16]: N = B.shape[0]
    ...: sA = A.sum(0)
    ...: sB = B.sum(0)
    ...: p1 = N*np.einsum('ij,ik->kj',A,B)
    ...: p2 = sA*sB[:,None]
    ...: p3 = N*((B**2).sum(0)) - (sB**2)
    ...: p4 = N*((A**2).sum(0)) - (sA**2)
    ...: pcorr = ((p1 - p2)/np.sqrt(p4*p3[:,None]))
    ...: out = pcorr[np.nanargmax(np.abs(pcorr),axis=0),np.arange(pcorr.shape[1])]
    ...: 

In [17]: pcorr # Pearson Correlation Coefficient array
Out[17]: 
array([[ 0.41895565, -0.5910935 , -0.40465987,  0.5818286 ],
       [ 0.66609445, -0.41950457,  0.02450215,  0.64028344],
       [-0.64953314,  0.65669916,  0.30836196, -0.85127779],
       [-0.41917583,  0.59043266,  0.40364532, -0.58144102],
       [ 0.8067843 ,  0.07947386,  0.74016181,  0.53165395],
       [-0.1613146 ,  0.95678152,  0.62107101, -0.4215393 ]])

In [18]: out # elements corresponding to absolute argmax along columns
Out[18]: array([ 0.8067843 ,  0.95678152,  0.74016181, -0.85127779])

Runtime tests -

In [36]: A = np.random.rand(4000,40)

In [37]: B = np.random.rand(4000,144)

In [38]: np.allclose(org_app(A,B),proposed_app(A,B))
Out[38]: True

In [39]: %timeit org_app(A,B) # Original approach
1 loops, best of 3: 1.35 s per loop

In [40]: %timeit proposed_app(A,B) # Proposed vectorized approach
10 loops, best of 3: 39.1 ms per loop