# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
import numpy as np
from .stationary import Stationary
from .psi_comp import PSICOMP_RBF, PSICOMP_RBF_GPU
from ...core import Param
from paramz.caching import Cache_this
from paramz.transformations import Logexp
from .grid_kerns import GridRBF
[docs]class RBF(Stationary):
"""
Radial Basis Function kernel, aka squared-exponential, exponentiated quadratic or Gaussian kernel:
.. math::
k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r^2 \\bigg)
"""
_support_GPU = True
def __init__(self, input_dim, variance=1., lengthscale=None, ARD=False, active_dims=None, name='rbf', useGPU=False, inv_l=False):
super(RBF, self).__init__(input_dim, variance, lengthscale, ARD, active_dims, name, useGPU=useGPU)
if self.useGPU:
self.psicomp = PSICOMP_RBF_GPU()
else:
self.psicomp = PSICOMP_RBF()
self.use_invLengthscale = inv_l
if inv_l:
self.unlink_parameter(self.lengthscale)
self.inv_l = Param('inv_lengthscale',1./self.lengthscale**2, Logexp())
self.link_parameter(self.inv_l)
[docs] def to_dict(self):
"""
Convert the object into a json serializable dictionary.
Note: It uses the private method _save_to_input_dict of the parent.
:return dict: json serializable dictionary containing the needed information to instantiate the object
"""
input_dict = super(RBF, self)._save_to_input_dict()
input_dict["class"] = "GPy.kern.RBF"
input_dict["inv_l"] = self.use_invLengthscale
if input_dict["inv_l"] == True:
input_dict["lengthscale"] = np.sqrt(1 / float(self.inv_l))
return input_dict
[docs] def K_of_r(self, r):
return self.variance * np.exp(-0.5 * r**2)
[docs] @Cache_this(limit=3, ignore_args=())
def dK_dX(self, X, X2, dimX):
r = self._scaled_dist(X, X2)
K = self.K_of_r(r)
dist = X[:,None,dimX]-X2[None,:,dimX]
lengthscale2inv = (np.ones((X.shape[1]))/(self.lengthscale**2))[dimX]
return -1.*K*dist*lengthscale2inv
[docs] @Cache_this(limit=3, ignore_args=())
def dK_dX2(self, X, X2, dimX2):
return -self.dK_dX(X,X2, dimX2)
[docs] @Cache_this(limit=3, ignore_args=())
def dK2_dXdX2(self, X, X2, dimX, dimX2):
r = self._scaled_dist(X, X2)
K = self.K_of_r(r)
if X2 is None:
X2=X
dist = X[:,None,:]-X2[None,:,:]
lengthscale2inv = np.ones((X.shape[1]))/(self.lengthscale**2)
return -1.*K*dist[:,:,dimX]*dist[:,:,dimX2]*lengthscale2inv[dimX]*lengthscale2inv[dimX2] + (dimX==dimX2)*K*lengthscale2inv[dimX]
[docs] def dK_dr(self, r):
return -r*self.K_of_r(r)
[docs] def dK2_drdr(self, r):
return (r**2-1)*self.K_of_r(r)
[docs] def dK2_drdr_diag(self):
return -self.variance # as the diagonal of r is always filled with zeros
[docs] @Cache_this(limit=3, ignore_args=())
def dK_dvariance(self,X,X2):
return self.K(X,X2)/self.variance
[docs] @Cache_this(limit=3, ignore_args=())
def dK2_dvariancedX(self, X, X2, dim):
return self.dK_dX(X,X2, dim)/self.variance
[docs] @Cache_this(limit=3, ignore_args=())
def dK2_dvariancedX2(self, X, X2, dim):
return self.dK_dX2(X,X2, dim)/self.variance
[docs] @Cache_this(limit=3, ignore_args=())
def dK3_dvariancedXdX2(self, X, X2, dim, dimX2):
return self.dK2_dXdX2(X, X2, dim, dimX2)/self.variance
[docs] @Cache_this(limit=3, ignore_args=())
def dK2_dlengthscaledX(self, X, X2, dimX):
r = self._scaled_dist(X, X2)
K = self.K_of_r(r)
if X2 is None:
X2=X
dist = X[:,None,:]-X2[None,:,:]
lengthscaleinv = np.ones((X.shape[1]))/(self.lengthscale)
if self.ARD:
g = []
for diml in range(X.shape[1]):
g += [-1.*K*dist[:,:,dimX]*(dist[:,:,diml]**2)*(lengthscaleinv[dimX]**2)*(lengthscaleinv[diml]**3) + 2.*dist[:,:,dimX]*(lengthscaleinv[diml]**3)*K*(dimX == diml)]
else:
g = -1.*K*dist[:,:,dimX]*np.sum(dist**2, axis=2)*(lengthscaleinv[dimX]**5) + 2.*dist[:,:,dimX]*(lengthscaleinv[dimX]**3)*K
return g
[docs] @Cache_this(limit=3, ignore_args=())
def dK2_dlengthscaledX2(self, X, X2, dimX2):
tmp = self.dK2_dlengthscaledX(X, X2, dimX2)
if self.ARD:
return [-1.*g for g in tmp]
else:
return -1*tmp
[docs] @Cache_this(limit=3, ignore_args=())
def dK3_dlengthscaledXdX2(self, X, X2, dimX, dimX2):
r = self._scaled_dist(X, X2)
K = self.K_of_r(r)
if X2 is None:
X2=X
dist = X[:,None,:]-X2[None,:,:]
lengthscaleinv = np.ones((X.shape[1]))/(self.lengthscale)
lengthscale2inv = lengthscaleinv**2
if self.ARD:
g = []
for diml in range(X.shape[1]):
tmp = -1.*K*dist[:,:,dimX]*dist[:,:,dimX2]*(dist[:,:,diml]**2)*lengthscale2inv[dimX]*lengthscale2inv[dimX2]*(lengthscaleinv[diml]**3)
if dimX == dimX2:
tmp += K*lengthscale2inv[dimX]*(lengthscaleinv[diml]**3)*(dist[:,:,diml]**2)
if diml == dimX:
tmp += 2.*K*dist[:,:,dimX]*dist[:,:,dimX2]*lengthscale2inv[dimX2]*(lengthscaleinv[dimX]**3)
if diml == dimX2:
tmp += 2.*K*dist[:,:,dimX]*dist[:,:,dimX2]*lengthscale2inv[dimX]*(lengthscaleinv[dimX2]**3)
if dimX == dimX2:
tmp += -2.*K*(lengthscaleinv[dimX]**3)
g += [tmp]
else:
g = -1.*K*dist[:,:,dimX]*dist[:,:,dimX2]*np.sum(dist**2, axis=2)*(lengthscaleinv[dimX]**7) +4*K*dist[:,:,dimX]*dist[:,:,dimX2]*(lengthscaleinv[dimX]**5)
if dimX == dimX2:
g += -2.*K*(lengthscaleinv[dimX]**3) + K*(lengthscaleinv[dimX]**5)*np.sum(dist**2, axis=2)
return g
def __getstate__(self):
dc = super(RBF, self).__getstate__()
if self.useGPU:
dc['psicomp'] = PSICOMP_RBF()
dc['useGPU'] = False
return dc
def __setstate__(self, state):
self.use_invLengthscale = False
return super(RBF, self).__setstate__(state)
[docs] def spectrum(self, omega):
assert self.input_dim == 1 #TODO: higher dim spectra?
return self.variance*np.sqrt(2*np.pi)*self.lengthscale*np.exp(-self.lengthscale*2*omega**2/2)
[docs] def parameters_changed(self):
if self.use_invLengthscale: self.lengthscale[:] = 1./np.sqrt(self.inv_l+1e-200)
super(RBF,self).parameters_changed()
[docs] def get_one_dimensional_kernel(self, dim):
"""
Specially intended for Grid regression.
"""
oneDkernel = GridRBF(input_dim=1, variance=self.variance.copy(), originalDimensions=dim)
return oneDkernel
#---------------------------------------#
# PSI statistics #
#---------------------------------------#
[docs] def psi0(self, Z, variational_posterior):
return self.psicomp.psicomputations(self, Z, variational_posterior)[0]
[docs] def psi1(self, Z, variational_posterior):
return self.psicomp.psicomputations(self, Z, variational_posterior)[1]
[docs] def psi2(self, Z, variational_posterior):
return self.psicomp.psicomputations(self, Z, variational_posterior, return_psi2_n=False)[2]
[docs] def psi2n(self, Z, variational_posterior):
return self.psicomp.psicomputations(self, Z, variational_posterior, return_psi2_n=True)[2]
[docs] def update_gradients_expectations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior):
dL_dvar, dL_dlengscale = self.psicomp.psiDerivativecomputations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior)[:2]
self.variance.gradient = dL_dvar
self.lengthscale.gradient = dL_dlengscale
if self.use_invLengthscale:
self.inv_l.gradient = dL_dlengscale*(self.lengthscale**3/-2.)
[docs] def gradients_Z_expectations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior):
return self.psicomp.psiDerivativecomputations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior)[2]
[docs] def gradients_qX_expectations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior):
return self.psicomp.psiDerivativecomputations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior)[3:]
[docs] def update_gradients_diag(self, dL_dKdiag, X):
super(RBF,self).update_gradients_diag(dL_dKdiag, X)
if self.use_invLengthscale: self.inv_l.gradient =self.lengthscale.gradient*(self.lengthscale**3/-2.)
[docs] def update_gradients_full(self, dL_dK, X, X2=None):
super(RBF,self).update_gradients_full(dL_dK, X, X2)
if self.use_invLengthscale: self.inv_l.gradient =self.lengthscale.gradient*(self.lengthscale**3/-2.)