u_product

u_product(a, b)

Inner product in the Hilbert space of \(U\)-centered distance matrices.

This inner product is defined as

\[\frac{1}{n(n-3)} \sum_{i,j=1}^n a_{i, j} b_{i, j}\]

Parameters

• a (Array) – First input array to be multiplied.

• b (Array) – Second input array to be multiplied.

Returns

Inner product.

Return type

Array

See also

mean_product

Examples

>>> import numpy as np
>>> import dcor
>>> a = np.array([[  0.,   3.,  11.,   6.],
...               [  3.,   0.,   8.,   3.],
...               [ 11.,   8.,   0.,   5.],
...               [  6.,   3.,   5.,   0.]])
>>> b = np.array([[  0.,  13.,  11.,   3.],
...               [ 13.,   0.,   2.,  10.],
...               [ 11.,   2.,   0.,   8.],
...               [  3.,  10.,   8.,   0.]])
>>> u_a = dcor.u_centered(a)
>>> u_a
array([[ 0., -2.,  1.,  1.],
       [-2.,  0.,  1.,  1.],
       [ 1.,  1.,  0., -2.],
       [ 1.,  1., -2.,  0.]])
>>> u_b = dcor.u_centered(b)
>>> u_b
array([[ 0.        ,  2.66666667,  2.66666667, -5.33333333],
       [ 2.66666667,  0.        , -5.33333333,  2.66666667],
       [ 2.66666667, -5.33333333,  0.        ,  2.66666667],
       [-5.33333333,  2.66666667,  2.66666667,  0.        ]])
>>> dcor.u_product(u_a, u_a)
6.0
>>> dcor.u_product(u_a, u_b)
-8.0

Note that the formula is well defined as long as the matrices involved are square and have the same dimensions, even if they are not in the Hilbert space of \(U\)-centered distance matrices

>>> dcor.u_product(a, a)
132.0

Also the formula produces a division by 0 for 3x3 matrices

>>> import warnings
>>> b = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> with warnings.catch_warnings():
...     warnings.simplefilter("ignore")
...     dcor.u_product(b, b)
inf