"""Total Sobol sensitivity index."""
import numpy
import numpoly
from ..conditional import E_cond
from ..variance import Var
[docs]def Sens_t(poly, dist, **kws):
"""
Variance-based decomposition
AKA Sobol' indices
Total effect sensitivity index
Args:
poly (numpoly.ndpoly):
Polynomial to find first order Sobol indices on.
dist (Distribution):
The distributions of the input used in ``poly``.
Returns:
(numpy.ndarray) :
First order sensitivity indices for each parameters in ``poly``,
with shape ``(len(dist),) + poly.shape``.
Examples:
>>> q0, q1 = chaospy.variable(2)
>>> poly = chaospy.polynomial([1, q0, q1, 10*q0*q1-1])
>>> dist = chaospy.Iid(chaospy.Uniform(0, 1), 2)
>>> chaospy.Sens_t(poly, dist)
array([[0. , 1. , 0. , 0.57142857],
[0. , 0. , 1. , 0.57142857]])
"""
dim = len(dist)
poly = numpoly.set_dimensions(poly, dim)
out = numpy.zeros((dim,) + poly.shape, dtype=float)
variance = Var(poly, dist, **kws)
valids = variance != 0
if not numpy.all(valids):
out[:, valids] = Sens_t(poly[valids], dist, **kws)
return out
out[:] = variance
for idx, unit_vec in enumerate(numpy.eye(dim, dtype=int)):
conditional = E_cond(poly, 1 - unit_vec, dist, **kws)
out[idx] -= Var(conditional, dist, **kws)
out[idx] /= variance
return out
def TotalOrderSobol(
expansion,
coefficients,
):
"""
Total Sobel indices.
Args:
expansion (numpoly.ndpoly):
The polynomial expansion used as basis when creating a chaos
expansion.
coefficients (numpy.ndarray):
The Fourier coefficients generated whent fitting the chaos
expansion. Typically retrieved by passing ``retall=True`` to
``chaospy.fit_regression`` or ``chaospy.fit_quadrature``.
Examples:
>>> q0, q1 = chaospy.variable(2)
>>> expansion = chaospy.polynomial([1, q0, q1, 10*q0*q1-1])
>>> coeffs = [1, 2, 2, 4]
>>> chaospy.TotalOrderSobol(expansion, coeffs)
array([0.83333333, 0.83333333])
"""
dic = expansion.todict()
alphas = []
for idx in range(len(expansion)):
expons = numpy.array([key for key, value in dic.items() if value[idx]])
alphas.append(tuple(expons[numpy.argmax(expons.sum(1))]))
coefficients = numpy.asfarray(coefficients)
index = numpy.array([any(alpha) for alpha in alphas])
variance = numpy.sum(coefficients[index] ** 2, axis=0)
sens = []
for idx in range(len(alphas[0])):
index = numpy.array([alpha[idx] > 0 for alpha in alphas])
sens.append(numpy.sum(coefficients[index] ** 2, axis=0) / variance)
return numpy.array(sens)