def norm_factor_gradient(self, variances):
"""Compute normalization factor and its gradient.
Compute normalization factor given current variance
and dimensionality.
Parameters
----------
variances : array-like, shape=[n]
Value of variance.
Returns
-------
norm_factor : array-like, shape=[n]
Normalisation factor.
norm_factor_gradient : array-like, shape=[n]
Gradient of the normalization factor.
"""
variances = gs.transpose(gs.to_ndarray(variances, to_ndim=2))
dim_range = gs.arange(0, self.dim, 1.0)
alpha = self._compute_alpha(dim_range)
binomial_coefficient = gs.ones(self.dim)
binomial_coefficient[1:] = (self.dim - 1 + 1 - dim_range[1:]) / dim_range[1:]
binomial_coefficient = gs.cumprod(binomial_coefficient)
beta = ((-gs.ones(self.dim)) ** dim_range) * binomial_coefficient
sigma_repeated = gs.repeat(variances, self.dim, -1)
prod_alpha_sigma = gs.einsum("ij,j->ij", sigma_repeated, alpha)
term_2 = gs.exp((prod_alpha_sigma) ** 2) * (1 + gs.erf(prod_alpha_sigma))
term_1 = gs.sqrt(gs.pi / 2.0) * (1.0 / (2 ** (self.dim - 1)))
term_2 = gs.einsum("ij,j->ij", term_2, beta)
norm_factor = term_1 * variances * gs.sum(term_2, axis=-1, keepdims=True)
grad_term_1 = 1 / variances
grad_term_21 = 1 / gs.sum(term_2, axis=-1, keepdims=True)
grad_term_211 = (
gs.exp((prod_alpha_sigma) ** 2)
* (1 + gs.erf(prod_alpha_sigma))
* gs.einsum("ij,j->ij", sigma_repeated, alpha**2)
* 2
)
grad_term_212 = gs.repeat(
gs.expand_dims((2 / gs.sqrt(gs.pi)) * alpha, axis=0),
variances.shape[0],
axis=0,
)
grad_term_22 = grad_term_211 + grad_term_212
grad_term_22 = gs.einsum("ij, j->ij", grad_term_22, beta)
grad_term_22 = gs.sum(grad_term_22, axis=-1, keepdims=True)
norm_factor_gradient = grad_term_1 + (grad_term_21 * grad_term_22)
return gs.squeeze(norm_factor), gs.squeeze(norm_factor_gradient)
def summand(k):
exp_arg = -((dim - 1 - 2 * k)**2) / 2 / variances
erf_arg_1 = (dim - 1 - 2 * k) * 1j / gs.sqrt(2 * variances)
erf_arg_2 = (gs.pi * variances -
(dim - 1 - 2 * k) * 1j) / gs.sqrt(2 * variances)
sign = (-1.0)**k
comb = gs.comb(dim - 1, k)
erf_terms = gs.imag(gs.erf(erf_arg_2) + gs.erf(erf_arg_1))
return sign * comb * gs.exp(exp_arg) * erf_terms
def summand(k):
exp_arg = -((dim - 1 - 2 * k)**2) / 2 / variances
erf_arg_2 = (gs.pi * variances -
(dim - 1 - 2 * k) * 1j) / gs.sqrt(2 * variances)
sign = (-1.0)**k
comb_2 = gs.comb(k, dim - 1)
return sign * comb_2 * gs.exp(exp_arg) * gs.real(gs.erf(erf_arg_2))
def _normalization_factor_odd_dim(self, variances):
"""Compute the normalization factor - odd dimension."""
dim = self.dim
half_dim = int((dim + 1) / 2)
area = 2 * gs.pi**half_dim / math.factorial(half_dim - 1)
comb = gs.comb(dim - 1, half_dim - 1)
erf_arg = gs.sqrt(variances / 2) * gs.pi
first_term = (area / (2**dim - 1) * comb *
gs.sqrt(gs.pi / (2 * variances)) * gs.erf(erf_arg))
def summand(k):
exp_arg = -((dim - 1 - 2 * k)**2) / 2 / variances
erf_arg_2 = (gs.pi * variances -
(dim - 1 - 2 * k) * 1j) / gs.sqrt(2 * variances)
sign = (-1.0)**k
comb_2 = gs.comb(k, dim - 1)
return sign * comb_2 * gs.exp(exp_arg) * gs.real(gs.erf(erf_arg_2))
if half_dim > 2:
sum_term = gs.sum(
gs.stack([summand(k)] for k in range(half_dim - 2)))
else:
sum_term = summand(0)
coef = area / 2 / erf_arg * gs.pi**0.5 * (-1.0)**(half_dim - 1)
return first_term + coef / 2**(dim - 2) * sum_term
def normalization_factor(self, variances):
"""Return normalization factor.
Parameters
----------
variances : array-like, shape=[n,]
Array of equally distant values of the
variance precision time.
Returns
-------
norm_func : array-like, shape=[n,]
Normalisation factor for all given variances.
"""
binomial_coefficient = None
n_samples = variances.shape[0]
expand_variances = gs.expand_dims(variances, axis=0)
expand_variances = gs.repeat(expand_variances, self.dim, axis=0)
if binomial_coefficient is None:
dim_range = gs.arange(self.dim)
dim_range[0] = 1
n_fact = dim_range.prod()
k_fact = gs.concatenate([
gs.expand_dims(dim_range[:i].prod(), 0)
for i in range(1, dim_range.shape[0] + 1)
], 0)
nmk_fact = gs.flip(k_fact, 0)
binomial_coefficient = n_fact / (k_fact * nmk_fact)
binomial_coefficient = gs.expand_dims(binomial_coefficient, -1)
binomial_coefficient = gs.repeat(binomial_coefficient,
n_samples,
axis=1)
range_ = gs.expand_dims(gs.arange(self.dim), -1)
range_ = gs.repeat(range_, n_samples, axis=1)
ones_ = gs.expand_dims(gs.ones(self.dim), -1)
ones_ = gs.repeat(ones_, n_samples, axis=1)
alternate_neg = (-ones_)**(range_)
erf_arg = ((
(self.dim - 1) - 2 * range_) * expand_variances) / gs.sqrt(2)
exp_arg = ((((self.dim - 1) - 2 * range_) * expand_variances) /
gs.sqrt(2))**2
norm_func_1 = (1 + gs.erf(erf_arg)) * gs.exp(exp_arg)
norm_func_2 = binomial_coefficient * norm_func_1
norm_func_3 = alternate_neg * norm_func_2
norm_func = NORMALIZATION_FACTOR_CST * variances * \
norm_func_3.sum(0) * (1 / (2 ** (self.dim - 1)))
return norm_func
def weighted_gmm_pdf(mixture_coefficients,
mesh_data,
means,
variances,
metric):
"""Return the probability density function of a GMM.
Parameters
----------
mixture_coefficients : array-like, shape=[n_gaussians,]
Coefficients of the Gaussian mixture model.
mesh_data : array-like, shape=[n_precision, dim]
Points at which the GMM probability density is computed.
means : array-like, shape=[n_gaussians, dim]
Means of each component of the GMM.
variances : array-like, shape=[n_gaussians,]
Variances of each component of the GMM.
metric : function
Distance function associated with the used metric.
Returns
-------
weighted_pdf : array-like, shape=[n_precision, n_gaussians,]
Probability density function computed for each point of
the mesh data, for each component of the GMM.
"""
distance_to_mean = metric.dist_broadcast(mesh_data, means)
variances_units = gs.expand_dims(variances, 0)
variances_units = gs.repeat(
variances_units, distance_to_mean.shape[0], axis=0)
distribution_normal = gs.exp(
-(distance_to_mean ** 2) / (2 * variances_units ** 2))
zeta_sigma = PI_2_3 * variances
zeta_sigma = zeta_sigma * gs.exp(
(variances ** 2 / 2) * gs.erf(variances / gs.sqrt(2)))
result_num = gs.expand_dims(mixture_coefficients, 0)
result_num = gs.repeat(
result_num, len(distribution_normal), axis=0)
result_num = result_num * distribution_normal
result_denum = gs.expand_dims(zeta_sigma, 0)
result_denum = gs.repeat(
result_denum, len(distribution_normal), axis=0)
weighted_pdf = result_num / result_denum
return weighted_pdf