pyphysim.subspace package¶

Submodules¶

pyphysim.subspace.metrics module¶

Implement several metrics for subspaces.

pyphysim.subspace.metrics.calc_chordal_distance(matrix1: numpy.ndarray, matrix2: numpy.ndarray) → float [source]¶

Calculates the chordal distance between the two matrices

Parameters

matrix1 (np.ndarray) – A 2D numpy array.
matrix2 (np.ndarray) – A 2D numpy array.

Returns

chord_dist – The chordal distance.

Return type

float

Notes

Same as calc_chordal_distance_2(), but implemented differently.

Examples

>>> A = np.arange(1, 9.)
>>> A.shape = (4, 2)
>>> B = np.array([[1.2, 2.1], [2.9, 4.3], [5.2, 6.1], [6.8, 8.1]])
>>> print(round(calc_chordal_distance(A, B), 8))
0.47386786

pyphysim.subspace.metrics.calc_chordal_distance_2(matrix1: numpy.ndarray, matrix2: numpy.ndarray) → float [source]¶

Calculates the chordal distance between the two matrices

Parameters

matrix1 (np.ndarray) – A 2D numpy array.
matrix2 (np.ndarray) – A 2D numpy array.

Returns

chord_dist – The chordal distance.

Return type

float

Notes

Same as calc_chordal_distance(), but implemented differently.

Examples

>>> A = np.arange(1, 9.)
>>> A.shape = (4, 2)
>>> B = np.array([[1.2, 2.1], [2.9, 4.3], [5.2, 6.1], [6.8, 8.1]])
>>> print(round(calc_chordal_distance_2(A, B), 8))
0.47386786

pyphysim.subspace.metrics.calc_chordal_distance_from_principal_angles(principalAngles: numpy.ndarray) → float [source]¶

Calculates the chordal distance from the principal angles.

It is given by the square root of the sum of the squares of the sin of the principal angles.

Parameters: principalAngles (np.ndarray) – Numpy array with the principal angles. This is a 1D numpy array.
Returns: chord_dist – The chordal distance.
Return type: float

Examples

>>> A = np.arange(1, 9.)
>>> A.shape = (4, 2)
>>> B = np.array([[1.2, 2.1], [2.9, 4.3], [5.2, 6.1], [6.8, 8.1]])
>>> princ_angles = calc_principal_angles(A, B)
>>> print(round(calc_chordal_distance_from_principal_angles(princ_angles), 8))
0.47386786

pyphysim.subspace.metrics.calc_principal_angles(matrix1: numpy.ndarray, matrix2: numpy.ndarray) → numpy.ndarray [source]¶

Calculates the principal angles between matrix1 and matrix2.

Parameters

matrix1 (np.ndarray) – A 2D numpy array.
matrix2 (np.ndarray) – A 2D numpy array.

Returns

The principal angles between matrix1 and matrix2. This is a 1D numpy array.

Return type

np.ndarray

Examples

>>> A = np.array([[1, 2], [3, 4], [5, 6]])
>>> B = np.array([[1, 5], [3, 7], [5, -1]])
>>> print(calc_principal_angles(A, B))
[0.         0.54312217]

pyphysim.subspace.projections module¶

Module related to subspace projection.

class pyphysim.subspace.projections.Projection(A: numpy.ndarray)[source]¶

Bases: object

Class to calculate the projection, orthogonal projection and reflection of a given matrix in a Subspace S spanned by the columns of a matrix A.

The matrix A is provided in the constructor and after that the functions project, oProject and reflect can be called with M as an argument.

Parameters: A (np.ndarray) – The matrix whose columns form a basis for the projected subspace.

Examples

>>> A = np.array([[1 + 1j, 2 - 2j], [3 - 2j, 0], [-1 - 1j, 2 - 3j]])
>>> v = np.array([1, 2, 3])
>>> P = Projection(A)
>>> P.project(v)
array([1.69577465+0.87887324j, 1.33802817+0.41408451j,
       2.32957746-0.56901408j])
>>> P.oProject(v)
array([-0.69577465-0.87887324j,  0.66197183-0.41408451j,
        0.67042254+0.56901408j])
>>> P.reflect(v)
array([-2.3915493 -1.75774648j, -0.67605634-0.82816901j,
       -1.65915493+1.13802817j])

static calcOrthogonalProjectionMatrix(A: numpy.ndarray) → numpy.ndarray [source]¶

Calculates the projection matrix that projects a vector (or a matrix) into the signal space orthogonal to the signal space spanned by the columns of M.

Parameters: A (np.ndarray) – A matrix whose columns form a basis for the “desired subspace”.
Returns: The projection matrix that can be used to project a vector or a matrix into the subspace orthogonal to the subspace spanned by the columns of A
Return type: np.ndarray

See also

calcProjectionMatrix()

Examples

>>> A = np.array([[1, 2], [2, 2], [4, 3]])
>>> # Matrix that projects into the subspace orthogonal to the
>>> # subspace spanned by the columns of A
>>> oQ = calcOrthogonalProjectionMatrix(A)
>>> print(oQ)
[[ 0.12121212 -0.3030303   0.12121212]
 [-0.3030303   0.75757576 -0.3030303 ]
 [ 0.12121212 -0.3030303   0.12121212]]

static calcProjectionMatrix(A: numpy.ndarray) → numpy.ndarray [source]¶

Calculates the projection matrix that projects a vector (or a matrix) into the signal space spanned by the columns of A.

Parameters: A (np.ndarray) – A matrix whose columns form a basis for the desired subspace.
Returns: The projection matrix that can be used to project a vector or a matrix into the subspace spanned by the columns of A
Return type: np.ndarray

See also

calcOrthogonalProjectionMatrix()

Examples

>>> A = np.array([[1 + 1j, 2 - 2j], [3 - 2j, 0],                           [-1 - 1j, 2 - 3j]])
>>> # Matrix that projects into the subspace spanned by the columns
>>> # of A
>>> Q = calcProjectionMatrix(A)
>>> np.allclose(Q.round(4), np.array(           [[ 0.5239+0.j, 0.0366+0.3296j, 0.3662+0.0732j],            [ 0.0366-0.3296j, 0.7690+0.j, -0.0789+0.2479j],            [ 0.3662-0.0732j, -0.0789-0.2479j, 0.7070-0.j]]))
True

oProject(M: numpy.ndarray) → numpy.ndarray [source]¶

Project the matrix (or vector) M the subspace ORTHOGONAL to the subspace projected with project.

Parameters: M (np.ndarray) – The matrix to be projected.
Returns: The projection of M into the orthogonal subspace.
Return type: np.ndarray

project(M: numpy.ndarray) → numpy.ndarray [source]¶

Project the matrix (or vector) M in the desired subspace.

Parameters: M (np.ndarray) – The matrix to be projected.
Returns: The projection of M into the desired subspace.
Return type: np.ndarray

reflect(M: numpy.ndarray) → numpy.ndarray [source]¶

Find the reflection of the matrix in the subspace spanned by the columns of A

Parameters: M (np.ndarray) – The matrix to be projected.
Returns: The reflection of M in the subspace.
Return type: np.ndarray

pyphysim.subspace.projections.calcOrthogonalProjectionMatrix(A: numpy.ndarray) → numpy.ndarray ¶

Calculates the projection matrix that projects a vector (or a matrix) into the signal space orthogonal to the signal space spanned by the columns of M.

Parameters: A (np.ndarray) – A matrix whose columns form a basis for the “desired subspace”.
Returns: The projection matrix that can be used to project a vector or a matrix into the subspace orthogonal to the subspace spanned by the columns of A
Return type: np.ndarray

See also

calcProjectionMatrix()

Examples

>>> A = np.array([[1, 2], [2, 2], [4, 3]])
>>> # Matrix that projects into the subspace orthogonal to the
>>> # subspace spanned by the columns of A
>>> oQ = calcOrthogonalProjectionMatrix(A)
>>> print(oQ)
[[ 0.12121212 -0.3030303   0.12121212]
 [-0.3030303   0.75757576 -0.3030303 ]
 [ 0.12121212 -0.3030303   0.12121212]]

pyphysim.subspace.projections.calcProjectionMatrix(A: numpy.ndarray) → numpy.ndarray ¶

Calculates the projection matrix that projects a vector (or a matrix) into the signal space spanned by the columns of A.

Parameters: A (np.ndarray) – A matrix whose columns form a basis for the desired subspace.
Returns: The projection matrix that can be used to project a vector or a matrix into the subspace spanned by the columns of A
Return type: np.ndarray

See also

calcOrthogonalProjectionMatrix()

Examples

>>> A = np.array([[1 + 1j, 2 - 2j], [3 - 2j, 0],                           [-1 - 1j, 2 - 3j]])
>>> # Matrix that projects into the subspace spanned by the columns
>>> # of A
>>> Q = calcProjectionMatrix(A)
>>> np.allclose(Q.round(4), np.array(           [[ 0.5239+0.j, 0.0366+0.3296j, 0.3662+0.0732j],            [ 0.0366-0.3296j, 0.7690+0.j, -0.0789+0.2479j],            [ 0.3662-0.0732j, -0.0789-0.2479j, 0.7070-0.j]]))
True

Module contents¶

Subspace related stuff.

This includes projections and metrics for subspaces.