# Source code for GPy.kern.src.integral

# Written by Mike Smith michaeltsmith.org.uk

from __future__ import division
import numpy as np
from .kern import Kern
from ...core.parameterization import Param
from paramz.transformations import Logexp
import math

[docs]class Integral(Kern): #todo do I need to inherit from Stationary
"""
Integral kernel between...
"""

def __init__(self, input_dim, variances=None, lengthscale=None, ARD=False, active_dims=None, name='integral'):
super(Integral, self).__init__(input_dim, active_dims, name)

if lengthscale is None:
lengthscale = np.ones(1)
else:
lengthscale = np.asarray(lengthscale)

self.lengthscale = Param('lengthscale', lengthscale, Logexp()) #Logexp - transforms to allow positive only values...
self.variances = Param('variances', variances, Logexp()) #and here.
self.link_parameters(self.variances, self.lengthscale) #this just takes a list of parameters we need to optimise.

[docs]    def h(self, z):
return 0.5 * z * np.sqrt(math.pi) * math.erf(z) + np.exp(-(z**2))

[docs]    def dk_dl(self, t, tprime, l): #derivative of the kernel wrt lengthscale
return l * ( self.h(t/l) - self.h((t - tprime)/l) + self.h(tprime/l) - 1)

[docs]    def update_gradients_full(self, dL_dK, X, X2=None):
if X2 is None:  #we're finding dK_xx/dTheta
dK_dl = np.zeros([X.shape[0],X.shape[0]])
dK_dv = np.zeros([X.shape[0],X.shape[0]])
for i,x in enumerate(X):
for j,x2 in enumerate(X):
dK_dl[i,j] = self.variances[0]*self.dk_dl(x[0],x2[0],self.lengthscale[0]) #TODO Multiple length scales
dK_dv[i,j] = self.k_xx(x[0],x2[0],self.lengthscale[0])  #the gradient wrt the variance is k_xx.
else:     #we're finding dK_xf/Dtheta
raise NotImplementedError("Currently this function only handles finding the gradient of a single vector of inputs (X) not a pair of vectors (X and X2)")

#useful little function to help calculate the covariances.
[docs]    def g(self,z):
return 1.0 * z * np.sqrt(math.pi) * math.erf(z) + np.exp(-(z**2))

#covariance between gradients (it's the gradients that we want out... maybe we should have a way of getting K_ff too? Currently you get the diag of K_ff from Kdiag)
[docs]    def k_xx(self,t,tprime,l):
return 0.5 * (l**2) * ( self.g(t/l) - self.g((t - tprime)/l) + self.g(tprime/l) - 1)

[docs]    def k_ff(self,t,tprime,l):
return np.exp(-((t-tprime)**2)/(l**2)) #rbf

#covariance between the gradient and the actual value
[docs]    def k_xf(self,t,tprime,l):
return 0.5 * np.sqrt(math.pi) * l * (math.erf((t-tprime)/l) + math.erf(tprime/l))

[docs]    def K(self, X, X2=None):
if X2 is None:
K_xx = np.zeros([X.shape[0],X.shape[0]])
for i,x in enumerate(X):
for j,x2 in enumerate(X):
K_xx[i,j] = self.k_xx(x[0],x2[0],self.lengthscale[0])
return K_xx * self.variances[0]
else:
K_xf = np.zeros([X.shape[0],X2.shape[0]])
for i,x in enumerate(X):
for j,x2 in enumerate(X2):
K_xf[i,j] = self.k_xf(x[0],x2[0],self.lengthscale[0])
return K_xf * self.variances[0]

[docs]    def Kdiag(self, X):
"""I've used the fact that we call this method for K_ff when finding the covariance as a hack so
I know if I should return K_ff or K_xx. In this case we're returning K_ff!!
$K_{ff}^{post} = K_{ff} - K_{fx} K_{xx}^{-1} K_{xf}$"""
K_ff = np.zeros(X.shape[0])
for i,x in enumerate(X):
K_ff[i] = self.k_ff(x[0],x[0],self.lengthscale[0])
return K_ff * self.variances[0]