import numpy as np
import warnings
from numpy.linalg import multi_dot
[docs]
def sort_eigenvalues_eigenvectors(eigenvalues, eigenvectors):
"""
Sort eigenvalues and eigenvectors in descending order of eigenvalues.
Parameters:
eigenvalues (ndarray): Array of eigenvalues.
eigenvectors (ndarray): Corresponding eigenvectors.
Returns:
tuple: A tuple containing:
- eigenvalues (ndarray): 1D array of eigenvalues sorted in descending order.
- eigenvectors (ndarray): 2D array of eigenvectors sorted to match the order of sorted_eigenvalues.
"""
eigenvalues = np.asarray(eigenvalues)
eigenvectors = np.asarray(eigenvectors)
# Get the indices that would sort the eigenvalues in descending order
idx = eigenvalues.argsort()[::-1]
# Sort the eigenvalues and eigenvectors
eigenvalues = eigenvalues[idx]
eigenvectors = eigenvectors[:, idx]
return eigenvalues, eigenvectors
[docs]
def sqrt_symmetric_matrix(A, inverse=False):
"""
Compute the square root or inverse square root of a symmetric matrix.
Parameters:
A (ndarray): Symmetric matrix to compute the square root or inverse square root of.
inverse (bool, default=False) If True, compute the inverse square root. Otherwise, compute the square root.
Returns:
ndarray: The (inverse) square root of the matrix.
"""
# Inputs validation
A = np.asarray(A)
if A.ndim != 2 or A.shape[0] != A.shape[1]:
raise ValueError("A must be a square 2D matrix.")
if not isinstance(inverse, bool):
raise TypeError("inverse must be a boolean.")
# Compute the eigenvalues and eigenvectors of the matrix
A_eigenval, A_eigenvect = np.linalg.eigh(A)
# # Checks
# if np.any(A_eigenval < -1e-12):
# raise ValueError("A must be positive semi-definite to compute a real square root.")
# if inverse and np.any(A_eigenval < 1e-12):
# raise np.linalg.LinAlgError("A is singular; cannot compute inverse square root.")
# Sort the eigenvalues and eigenvectors
A_eigenval, A_eigenvect = sort_eigenvalues_eigenvectors(A_eigenval, A_eigenvect)
# Compute the power for the eigenvalues (inverse square root if inverse is True, otherwise square root)
power = -0.5 if inverse else 0.5
# Compute the (inverse) square root matrix
A_sqrt = multi_dot([A_eigenvect, np.diag(A_eigenval ** power), A_eigenvect.T])
return A_sqrt
def _check_gen_kurtosis(gen_kurtosis):
"""
Check the gen_kurtosis array for NA, infinite, and complex values.
Parameters:
gen_kurtosis (ndarray): Array of kurtosis values.
"""
if not np.all(np.isfinite(gen_kurtosis)):
warnings.warn("Some generalized kurtosis values are infinite")
if np.any(np.iscomplex(gen_kurtosis)):
warnings.warn("Some generalized kurtosis values are complex")
if np.any(np.isnan(gen_kurtosis)):
warnings.warn("Some generalized kurtosis values are NA (Not Available)")
def _sign_max(row):
"""
Determine the sign of the maximum absolute value in a row.
This function checks if the maximum value in the row is the same as the maximum absolute value.
If it is, the function returns 1, indicating that the maximum value is positive.
Otherwise, it returns -1, indicating that the maximum absolute value is negative.
Parameters:
row(array-like): The input row of numerical values.
Returns:
int: 1 if the maximum value is positive, -1 if the maximum absolute value is negative.
Examples
--------
>>> _sign_max([1, -3, 2])
-1
>>> _sign_max([-1, -2, -3])
-1
>>> _sign_max([0, 2, -2])
1
"""
return 1 if max(row) == np.max(np.abs(row)) else -1