u_complementary_projection

u_complementary_projection(a)

Return the orthogonal projection function over \(a^{\perp}\).

The function returned computes the orthogonal projection over \(a^{\perp}\) (the complementary projection over a) in the Hilbert space of \(U\)-centered distance matrices.

The projection of a matrix \(B\) over a matrix \(A^{\perp}\) is defined as

\[\text{proj}_{A^{\perp}}(B) = B - \text{proj}_A(B)\]

Parameters

a (Array) – \(U\)-centered distance matrix.

Returns

Function that receives a \(U\)-centered distance matrices and computes its orthogonal projection over \(a^{\perp}\).

Return type

Callable[[Array], Array]

See also

u_projection u_centered

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.        ]])
>>> proj_a = dcor.u_complementary_projection(u_a)
>>> proj_a(u_a)
array([[0., 0., 0., 0.],
       [0., 0., 0., 0.],
       [0., 0., 0., 0.],
       [0., 0., 0., 0.]])
>>> proj_a(u_b)
array([[ 0.0000000e+00, -4.4408921e-16,  4.0000000e+00, -4.0000000e+00],
       [-4.4408921e-16,  0.0000000e+00, -4.0000000e+00,  4.0000000e+00],
       [ 4.0000000e+00, -4.0000000e+00,  0.0000000e+00, -4.4408921e-16],
       [-4.0000000e+00,  4.0000000e+00, -4.4408921e-16,  0.0000000e+00]])
>>> proj_null = dcor.u_complementary_projection(np.zeros((4, 4)))
>>> proj_null(u_a)
array([[ 0., -2.,  1.,  1.],
       [-2.,  0.,  1.,  1.],
       [ 1.,  1.,  0., -2.],
       [ 1.,  1., -2.,  0.]])