def monomial_variance_uniform_variables(indices, coeffs):
mean = monomial_mean_uniform_variables(indices, coeffs)
squared_indices, squared_coeffs = multiply_multivariate_polynomials(
indices, coeffs, indices, coeffs)
variance = monomial_mean_uniform_variables(
squared_indices, squared_coeffs)-mean**2
return variance
def coeffs_of_active_subspace_polynomials(W1_trans, as_poly_indices):
num_active_vars, num_vars = W1_trans.shape
num_as_moments = as_poly_indices.shape[1]
moments = np.zeros((num_as_moments), float)
monomial_power_indices = []
monomial_power_coeffs = []
for var_num in range(num_active_vars):
monomial_power_indices.append([])
monomial_power_coeffs.append([])
for degree in range(as_poly_indices[var_num, :].max() + 1):
coeffs, indices = coeffs_of_power_of_nd_linear_polynomial(
num_vars, degree, W1_trans[var_num, :])
monomial_power_coeffs[var_num].append(coeffs)
monomial_power_indices[var_num].append(indices)
indices_list = []
coeffs_list = []
for ii in range(as_poly_indices.shape[1]):
as_index = as_poly_indices[:, ii]
activated_vars = np.where(as_index > 0)[0]
num_activated_vars = activated_vars.shape[0]
if num_activated_vars > 0:
var_num = activated_vars[0]
degree = as_index[var_num]
else:
degree = 0
var_num = 0
coeffs = monomial_power_coeffs[var_num][degree]
indices = monomial_power_indices[var_num][degree]
for dd in range(1, num_activated_vars):
var_num = activated_vars[dd]
degree = as_index[var_num]
coeffs_dd = monomial_power_coeffs[var_num][degree]
indices_dd = monomial_power_indices[var_num][degree]
# TODO Extension does not groud like terms,but pure python function
# multiply_multivariate_polynomials does.
# This is only a valid substitution if this function is
# only called by intergrate moments
try:
from pyapprox.cython.manipulate_polynomials import \
multiply_multivariate_polynomials_pyx
indices, coeffs = multiply_multivariate_polynomials_pyx(
indices, coeffs, indices_dd, coeffs_dd)
except:
print('multiply_multivariate_polynomials extension failed')
indices, coeffs = multiply_multivariate_polynomials(
indices, coeffs, indices_dd, coeffs_dd)
indices_list.append(indices)
coeffs_list.append(coeffs)
return indices_list, coeffs_list
def coeffs_of_active_subspace_polynomial(W1_trans, as_index):
"""
Evaluate the coefficients of a polynomial in an
active subspace in terms of a polynomial in the original variable space.
In active subspace with a monomial basis we may have a polynomial with
terms
[1,y1,y2,y1**2,y1*y2,y2**2,y1**3,y1**2*y2,y1*y2**2,y2**3]
We want to compute a polynomial in terms of the original variables x. E.g.
y1**2*y2 = W_11*x1+W_12*x2+W_13*x3)**2*(W_21*x1+W_22*x2+W_23*x3)
where the native function space (x) has dimension num_vars=3
Parameters
----------
W1_trans : np.ndarray (num_active_vars, num_vars)
The active subspace active variable transformation such that
the active variable y is obtained from the full dimension variable
x via y = W1_trans*x
as_index : np.ndarray (num_active_vars)
The multivariate index of the polynomial in terms of the active
subspace variables y
Returns
-------
coeff : np.ndarray (num_moments)
The coefficients of the polynomial in terms of the original variables x
indices : np.ndarray (num_terms)
The set of multivariate indices that define equivalent polynomial
(to the active subapce polynomial) in terms of the original variables x
"""
num_vars = W1_trans.shape[1]
activated_vars = np.where(as_index > 0)[0]
num_activated_vars = activated_vars.shape[0]
if num_activated_vars > 0:
var_num = activated_vars[0]
degree = as_index[var_num]
else:
degree = 0
var_num = 0
coeffs, indices = coeffs_of_power_of_nd_linear_polynomial(
num_vars, degree, W1_trans[var_num, :])
for dd in range(1, num_activated_vars):
var_num = activated_vars[dd]
degree = as_index[var_num]
coeffs_d, indices_d = coeffs_of_power_of_nd_linear_polynomial(
num_vars, degree, W1_trans[var_num, :])
indices, coeffs = multiply_multivariate_polynomials(
indices, coeffs, indices_d, coeffs_d)
return coeffs, indices
def coeffs_of_active_subspace_polynomials(W1_trans, as_poly_indices):
num_active_vars, num_vars = W1_trans.shape
monomial_power_indices = [[] for ii in range(num_active_vars)]
monomial_power_coeffs = [[] for ii in range(num_active_vars)]
for var_num in range(num_active_vars):
for degree in range(as_poly_indices[var_num, :].max()+1):
coeffs, indices = coeffs_of_power_of_nd_linear_monomial(
num_vars, degree, W1_trans[var_num, :])
monomial_power_coeffs[var_num].append(coeffs[:, None].copy())
monomial_power_indices[var_num].append(indices)
indices_list = []
coeffs_list = []
for ii in range(as_poly_indices.shape[1]):
as_index = as_poly_indices[:, ii]
activated_vars = np.where(as_index > 0)[0]
num_activated_vars = activated_vars.shape[0]
if num_activated_vars > 0:
var_num = activated_vars[0]
degree = as_index[var_num]
else:
degree = 0
var_num = 0
coeffs = monomial_power_coeffs[var_num][degree]
indices = monomial_power_indices[var_num][degree]
for dd in range(1, num_activated_vars):
var_num = activated_vars[dd]
degree = as_index[var_num]
coeffs_dd = monomial_power_coeffs[var_num][degree]
indices_dd = monomial_power_indices[var_num][degree]
# # TODO Extension does not group like terms, but pure python function
# # multiply_multivariate_polynomials does.
# # This is only a valid substitution if this function is
# # only called by intergrate moments
# try:
# from pyapprox.cython.manipulate_polynomials import \
# multiply_multivariate_polynomials_pyx
# indices, coeffs = multiply_multivariate_polynomials_pyx(
# indices, coeffs, indices_dd, coeffs_dd)
# except (ImportError, TypeError) as e:
# from pyapprox.sys_utilities import trace_error_with_msg
# trace_error_with_msg('multiply_multivariate_polynomials extension failed', e)
# print(coeffs.shape, coeffs_dd.shape)
indices, coeffs = multiply_multivariate_polynomials(
indices, coeffs, indices_dd, coeffs_dd)
indices_list.append(indices)
coeffs_list.append(coeffs[:, 0])
return indices_list, coeffs_list
