# -*- coding: utf-8 -*-
# Copyright (c) 2015, Alex Grigorevskiy, Arno Solin
# Licensed under the BSD 3-clause license (see LICENSE.txt)
"""
Classes in this module enhance Matern covariance functions with the
Stochastic Differential Equation (SDE) functionality.
"""
from .stationary import Matern32
from .stationary import Matern52
import numpy as np
[docs]class sde_Matern32(Matern32):
"""
Class provide extra functionality to transfer this covariance function into
SDE forrm.
Matern 3/2 kernel:
.. math::
k(r) = \sigma^2 (1 + \sqrt{3} r) \exp(- \sqrt{3} r) \\ \\ \\ \\ \text{ where } r = \sqrt{\sum_{i=1}^{input dim} \frac{(x_i-y_i)^2}{\ell_i^2} }
"""
[docs] def sde_update_gradient_full(self, gradients):
"""
Update gradient in the order in which parameters are represented in the
kernel
"""
self.variance.gradient = gradients[0]
self.lengthscale.gradient = gradients[1]
[docs] def sde(self):
"""
Return the state space representation of the covariance.
"""
variance = float(self.variance.values)
lengthscale = float(self.lengthscale.values)
foo = np.sqrt(3.)/lengthscale
F = np.array(((0, 1.0), (-foo**2, -2*foo)))
L = np.array(( (0,), (1.0,) ))
Qc = np.array(((12.*np.sqrt(3) / lengthscale**3 * variance,),))
H = np.array(((1.0, 0),))
Pinf = np.array(((variance, 0.0), (0.0, 3.*variance/(lengthscale**2))))
P0 = Pinf.copy()
# Allocate space for the derivatives
dF = np.empty([F.shape[0],F.shape[1],2])
dQc = np.empty([Qc.shape[0],Qc.shape[1],2])
dPinf = np.empty([Pinf.shape[0],Pinf.shape[1],2])
# The partial derivatives
dFvariance = np.zeros((2,2))
dFlengthscale = np.array(((0,0), (6./lengthscale**3,2*np.sqrt(3)/lengthscale**2)))
dQcvariance = np.array((12.*np.sqrt(3)/lengthscale**3))
dQclengthscale = np.array((-3*12*np.sqrt(3)/lengthscale**4*variance))
dPinfvariance = np.array(((1,0),(0,3./lengthscale**2)))
dPinflengthscale = np.array(((0,0), (0,-6*variance/lengthscale**3)))
# Combine the derivatives
dF[:,:,0] = dFvariance
dF[:,:,1] = dFlengthscale
dQc[:,:,0] = dQcvariance
dQc[:,:,1] = dQclengthscale
dPinf[:,:,0] = dPinfvariance
dPinf[:,:,1] = dPinflengthscale
dP0 = dPinf.copy()
return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)
[docs]class sde_Matern52(Matern52):
"""
Class provide extra functionality to transfer this covariance function into
SDE forrm.
Matern 5/2 kernel:
.. math::
k(r) = \sigma^2 (1 + \sqrt{5} r + \frac{5}{3}r^2) \exp(- \sqrt{5} r) \\ \\ \\ \\ \text{ where } r = \sqrt{\sum_{i=1}^{input dim} \frac{(x_i-y_i)^2}{\ell_i^2} }
"""
[docs] def sde_update_gradient_full(self, gradients):
"""
Update gradient in the order in which parameters are represented in the
kernel
"""
self.variance.gradient = gradients[0]
self.lengthscale.gradient = gradients[1]
[docs] def sde(self):
"""
Return the state space representation of the covariance.
"""
variance = float(self.variance.values)
lengthscale = float(self.lengthscale.values)
lamda = np.sqrt(5.0)/lengthscale
kappa = 5.0/3.0*variance/lengthscale**2
F = np.array(((0, 1,0), (0, 0, 1), (-lamda**3, -3.0*lamda**2, -3*lamda)))
L = np.array(((0,),(0,),(1,)))
Qc = np.array((((variance*400.0*np.sqrt(5.0)/3.0/lengthscale**5),),))
H = np.array(((1,0,0),))
Pinf = np.array(((variance,0,-kappa), (0, kappa, 0), (-kappa, 0, 25.0*variance/lengthscale**4)))
P0 = Pinf.copy()
# Allocate space for the derivatives
dF = np.empty((3,3,2))
dQc = np.empty((1,1,2))
dPinf = np.empty((3,3,2))
# The partial derivatives
dFvariance = np.zeros((3,3))
dFlengthscale = np.array(((0,0,0),(0,0,0),(15.0*np.sqrt(5.0)/lengthscale**4,
30.0/lengthscale**3, 3*np.sqrt(5.0)/lengthscale**2)))
dQcvariance = np.array((((400*np.sqrt(5)/3/lengthscale**5,),)))
dQclengthscale = np.array((((-variance*2000*np.sqrt(5)/3/lengthscale**6,),)))
dPinf_variance = Pinf/variance
kappa2 = -2.0*kappa/lengthscale
dPinf_lengthscale = np.array(((0,0,-kappa2),(0,kappa2,0),(-kappa2,
0,-100*variance/lengthscale**5)))
# Combine the derivatives
dF[:,:,0] = dFvariance
dF[:,:,1] = dFlengthscale
dQc[:,:,0] = dQcvariance
dQc[:,:,1] = dQclengthscale
dPinf[:,:,0] = dPinf_variance
dPinf[:,:,1] = dPinf_lengthscale
dP0 = dPinf.copy()
return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)