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correlation_models.py
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# -*- coding: utf-8 -*-
# Author: Vincent Dubourg <vincent.dubourg@gmail.com>
# (mostly translation, see implementation details)
# License: BSD 3 clause
"""
The built-in correlation models submodule for the gaussian_process module.
"""
import numpy as np
from ..utils import deprecated
@deprecated("The function absolute_exponential of correlation_models is "
"deprecated in version 0.19.1 and will be removed in 0.22.")
def absolute_exponential(theta, d):
"""
Absolute exponential autocorrelation model.
(Ornstein-Uhlenbeck stochastic process)::
n
theta, d --> r(theta, d) = exp( sum - theta_i * |d_i| )
i = 1
Parameters
----------
theta : array_like
An array with shape 1 (isotropic) or n (anisotropic) giving the
autocorrelation parameter(s).
d : array_like
An array with shape (n_eval, n_features) giving the componentwise
distances between locations x and x' at which the correlation model
should be evaluated.
Returns
-------
r : array_like
An array with shape (n_eval, ) containing the values of the
autocorrelation model.
"""
theta = np.asarray(theta, dtype=np.float64)
d = np.abs(np.asarray(d, dtype=np.float64))
if d.ndim > 1:
n_features = d.shape[1]
else:
n_features = 1
if theta.size == 1:
return np.exp(- theta[0] * np.sum(d, axis=1))
elif theta.size != n_features:
raise ValueError("Length of theta must be 1 or %s" % n_features)
else:
return np.exp(- np.sum(theta.reshape(1, n_features) * d, axis=1))
@deprecated("The function squared_exponential of correlation_models is "
"deprecated in version 0.19.1 and will be removed in 0.22.")
def squared_exponential(theta, d):
"""
Squared exponential correlation model (Radial Basis Function).
(Infinitely differentiable stochastic process, very smooth)::
n
theta, d --> r(theta, d) = exp( sum - theta_i * (d_i)^2 )
i = 1
Parameters
----------
theta : array_like
An array with shape 1 (isotropic) or n (anisotropic) giving the
autocorrelation parameter(s).
d : array_like
An array with shape (n_eval, n_features) giving the componentwise
distances between locations x and x' at which the correlation model
should be evaluated.
Returns
-------
r : array_like
An array with shape (n_eval, ) containing the values of the
autocorrelation model.
"""
theta = np.asarray(theta, dtype=np.float64)
d = np.asarray(d, dtype=np.float64)
if d.ndim > 1:
n_features = d.shape[1]
else:
n_features = 1
if theta.size == 1:
return np.exp(-theta[0] * np.sum(d ** 2, axis=1))
elif theta.size != n_features:
raise ValueError("Length of theta must be 1 or %s" % n_features)
else:
return np.exp(-np.sum(theta.reshape(1, n_features) * d ** 2, axis=1))
@deprecated("The function generalized_exponential of correlation_models is "
"deprecated in version 0.19.1 and will be removed in 0.22.")
def generalized_exponential(theta, d):
"""
Generalized exponential correlation model.
(Useful when one does not know the smoothness of the function to be
predicted.)::
n
theta, d --> r(theta, d) = exp( sum - theta_i * |d_i|^p )
i = 1
Parameters
----------
theta : array_like
An array with shape 1+1 (isotropic) or n+1 (anisotropic) giving the
autocorrelation parameter(s) (theta, p).
d : array_like
An array with shape (n_eval, n_features) giving the componentwise
distances between locations x and x' at which the correlation model
should be evaluated.
Returns
-------
r : array_like
An array with shape (n_eval, ) with the values of the autocorrelation
model.
"""
theta = np.asarray(theta, dtype=np.float64)
d = np.asarray(d, dtype=np.float64)
if d.ndim > 1:
n_features = d.shape[1]
else:
n_features = 1
lth = theta.size
if n_features > 1 and lth == 2:
theta = np.hstack([np.repeat(theta[0], n_features), theta[1]])
elif lth != n_features + 1:
raise Exception("Length of theta must be 2 or %s" % (n_features + 1))
else:
theta = theta.reshape(1, lth)
td = theta[:, 0:-1].reshape(1, n_features) * np.abs(d) ** theta[:, -1]
r = np.exp(- np.sum(td, 1))
return r
@deprecated("The function pure_nugget of correlation_models is "
"deprecated in version 0.19.1 and will be removed in 0.22.")
def pure_nugget(theta, d):
"""
Spatial independence correlation model (pure nugget).
(Useful when one wants to solve an ordinary least squares problem!)::
n
theta, d --> r(theta, d) = 1 if sum |d_i| == 0
i = 1
0 otherwise
Parameters
----------
theta : array_like
None.
d : array_like
An array with shape (n_eval, n_features) giving the componentwise
distances between locations x and x' at which the correlation model
should be evaluated.
Returns
-------
r : array_like
An array with shape (n_eval, ) with the values of the autocorrelation
model.
"""
theta = np.asarray(theta, dtype=np.float64)
d = np.asarray(d, dtype=np.float64)
n_eval = d.shape[0]
r = np.zeros(n_eval)
r[np.all(d == 0., axis=1)] = 1.
return r
@deprecated("The function cubic of correlation_models is "
"deprecated in version 0.19.1 and will be removed in 0.22.")
def cubic(theta, d):
"""
Cubic correlation model::
theta, d --> r(theta, d) =
n
prod max(0, 1 - 3(theta_j*d_ij)^2 + 2(theta_j*d_ij)^3) , i = 1,...,m
j = 1
Parameters
----------
theta : array_like
An array with shape 1 (isotropic) or n (anisotropic) giving the
autocorrelation parameter(s).
d : array_like
An array with shape (n_eval, n_features) giving the componentwise
distances between locations x and x' at which the correlation model
should be evaluated.
Returns
-------
r : array_like
An array with shape (n_eval, ) with the values of the autocorrelation
model.
"""
theta = np.asarray(theta, dtype=np.float64)
d = np.asarray(d, dtype=np.float64)
if d.ndim > 1:
n_features = d.shape[1]
else:
n_features = 1
lth = theta.size
if lth == 1:
td = np.abs(d) * theta
elif lth != n_features:
raise Exception("Length of theta must be 1 or " + str(n_features))
else:
td = np.abs(d) * theta.reshape(1, n_features)
td[td > 1.] = 1.
ss = 1. - td ** 2. * (3. - 2. * td)
r = np.prod(ss, 1)
return r
@deprecated("The function linear of correlation_models is "
"deprecated in version 0.19.1 and will be removed in 0.22.")
def linear(theta, d):
"""
Linear correlation model::
theta, d --> r(theta, d) =
n
prod max(0, 1 - theta_j*d_ij) , i = 1,...,m
j = 1
Parameters
----------
theta : array_like
An array with shape 1 (isotropic) or n (anisotropic) giving the
autocorrelation parameter(s).
d : array_like
An array with shape (n_eval, n_features) giving the componentwise
distances between locations x and x' at which the correlation model
should be evaluated.
Returns
-------
r : array_like
An array with shape (n_eval, ) with the values of the autocorrelation
model.
"""
theta = np.asarray(theta, dtype=np.float64)
d = np.asarray(d, dtype=np.float64)
if d.ndim > 1:
n_features = d.shape[1]
else:
n_features = 1
lth = theta.size
if lth == 1:
td = np.abs(d) * theta
elif lth != n_features:
raise Exception("Length of theta must be 1 or %s" % n_features)
else:
td = np.abs(d) * theta.reshape(1, n_features)
td[td > 1.] = 1.
ss = 1. - td
r = np.prod(ss, 1)
return r