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GPy/GPy/kern/_src/ODE_st.py

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st
2014-04-29 18:08:25 +01:00
# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
from kern import Kern
from ...core.parameterization import Param
from ...core.parameterization.transformations import Logexp
import numpy as np
from independent_outputs import index_to_slices
class ODE_st(Kern):
"""
kernel resultiong from a first order ODE with OU driving GP
:param input_dim: the number of input dimension, has to be equal to one
:type input_dim: int
:param varianceU: variance of the driving GP
:type varianceU: float
:param lengthscaleU: lengthscale of the driving GP (sqrt(3)/lengthscaleU)
:type lengthscaleU: float
:param varianceY: 'variance' of the transfer function
:type varianceY: float
:param lengthscaleY: 'lengthscale' of the transfer function (1/lengthscaleY)
:type lengthscaleY: float
:rtype: kernel object
"""
def __init__(self, input_dim, a=1.,b=1., c=1.,variance_Yx=3.,variance_Yt=1.5, lengthscale_Yx=1.5, lengthscale_Yt=1.5, active_dims=None, name='ode_st'):
assert input_dim ==3, "only defined for 3 input dims"
super(ODE_st, self).__init__(input_dim, active_dims, name)
self.variance_Yt = Param('variance_Yt', variance_Yt, Logexp())
self.variance_Yx = Param('variance_Yx', variance_Yx, Logexp())
self.lengthscale_Yt = Param('lengthscale_Yt', lengthscale_Yt, Logexp())
self.lengthscale_Yx = Param('lengthscale_Yx', lengthscale_Yx, Logexp())
self.a= Param('a', a, Logexp())
self.b = Param('b', b, Logexp())
self.c = Param('c', c, Logexp())
self.add_parameters(self.a, self.b, self.c, self.variance_Yt, self.variance_Yx, self.lengthscale_Yt,self.lengthscale_Yx)
def K(self, X, X2=None):
# model : -a d^2y/dx^2 + b dy/dt + c * y = U
# kernel Kyy rbf spatiol temporal
# vyt Y temporal variance vyx Y spatiol variance lyt Y temporal lengthscale lyx Y spatiol lengthscale
# kernel Kuu doper( doper(Kyy))
# a b c lyt lyx vyx*vyt
"""Compute the covariance matrix between X and X2."""
X,slices = X[:,:-1],index_to_slices(X[:,-1])
if X2 is None:
X2,slices2 = X,slices
K = np.zeros((X.shape[0], X.shape[0]))
else:
X2,slices2 = X2[:,:-1],index_to_slices(X2[:,-1])
K = np.zeros((X.shape[0], X2.shape[0]))
tdist = (X[:,0][:,None] - X2[:,0][None,:])**2
xdist = (X[:,1][:,None] - X2[:,1][None,:])**2
ttdist = (X[:,0][:,None] - X2[:,0][None,:])
#rdist = [tdist,xdist]
#dist = np.abs(X - X2.T)
vyt = self.variance_Yt
vyx = self.variance_Yx
lyt=1/(2*self.lengthscale_Yt)
lyx=1/(2*self.lengthscale_Yx)
a = self.a ## -a is used in the model, negtive diffusion
b = self.b
c = self.c
kyy = lambda tdist,xdist: np.exp(-lyt*(tdist) -lyx*(xdist))
k1 = lambda tdist: (2*lyt - 4*lyt**2 * (tdist) )
k2 = lambda xdist: ( 4*lyx**2 * (xdist) - 2*lyx )
k3 = lambda xdist: ( 3*4*lyx**2 - 6*8*xdist*lyx**3 + 16*xdist**2*lyx**4 )
k4 = lambda ttdist: 2*lyt*(ttdist)
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i==0 and j==0:
K[ss1,ss2] = vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
elif i==0 and j==1:
K[ss1,ss2] = (-a*k2(xdist[ss1,ss2]) + b*k4(ttdist[ss1,ss2]) + c)*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
#K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kuyp(np.abs(rdist[ss1,ss2])), kuyn(np.abs(rdist[ss1,ss2]) ) )
#K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kuyp(rdist[ss1,ss2]), kuyn(rdist[ss1,ss2] ) )
elif i==1 and j==1:
K[ss1,ss2] = ( b**2*k1(tdist[ss1,ss2]) - 2*a*c*k2(xdist[ss1,ss2]) + a**2*k3(xdist[ss1,ss2]) + c**2 )* vyt*vyx* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
else:
K[ss1,ss2] = (-a*k2(xdist[ss1,ss2]) - b*k4(ttdist[ss1,ss2]) + c)*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
#K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kyup(np.abs(rdist[ss1,ss2])), kyun(np.abs(rdist[ss1,ss2]) ) )
#K[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , kyup(rdist[ss1,ss2]), kyun(rdist[ss1,ss2] ) )
#stop
return K
def Kdiag(self, X):
"""Compute the diagonal of the covariance matrix associated to X."""
vyt = self.variance_Yt
vyx = self.variance_Yx
lyt = 1./(2*self.lengthscale_Yt)
lyx = 1./(2*self.lengthscale_Yx)
a = self.a
b = self.b
c = self.c
## dk^2/dtdt'
k1 = (2*lyt )*vyt*vyx
## dk^2/dx^2
k2 = ( - 2*lyx )*vyt*vyx
## dk^4/dx^2dx'^2
k3 = ( 4*3*lyx**2 )*vyt*vyx
Kdiag = np.zeros(X.shape[0])
slices = index_to_slices(X[:,-1])
for i, ss1 in enumerate(slices):
for s1 in ss1:
if i==0:
Kdiag[s1]+= vyt*vyx
elif i==1:
#i=1
Kdiag[s1]+= b**2*k1 - 2*a*c*k2 + a**2*k3 + c**2*vyt*vyx
#Kdiag[s1]+= Vu*Vy*(k1+k2+k3)
else:
raise ValueError, "invalid input/output index"
return Kdiag
def update_gradients_full(self, dL_dK, X, X2=None):
#def dK_dtheta(self, dL_dK, X, X2, target):
"""derivative of the covariance matrix with respect to the parameters."""
X,slices = X[:,:-1],index_to_slices(X[:,-1])
if X2 is None:
X2,slices2 = X,slices
K = np.zeros((X.shape[0], X.shape[0]))
else:
X2,slices2 = X2[:,:-1],index_to_slices(X2[:,-1])
vyt = self.variance_Yt
vyx = self.variance_Yx
lyt = 1./(2*self.lengthscale_Yt)
lyx = 1./(2*self.lengthscale_Yx)
a = self.a
b = self.b
c = self.c
tdist = (X[:,0][:,None] - X2[:,0][None,:])**2
xdist = (X[:,1][:,None] - X2[:,1][None,:])**2
#rdist = [tdist,xdist]
ttdist = (X[:,0][:,None] - X2[:,0][None,:])
rd=tdist.shape[0]
dka = np.zeros([rd,rd])
dkb = np.zeros([rd,rd])
dkc = np.zeros([rd,rd])
dkYdvart = np.zeros([rd,rd])
dkYdvarx = np.zeros([rd,rd])
dkYdlent = np.zeros([rd,rd])
dkYdlenx = np.zeros([rd,rd])
kyy = lambda tdist,xdist: np.exp(-lyt*(tdist) -lyx*(xdist))
#k1 = lambda tdist: (lyt - lyt**2 * (tdist) )
#k2 = lambda xdist: ( lyx**2 * (xdist) - lyx )
#k3 = lambda xdist: ( 3*lyx**2 - 6*xdist*lyx**3 + xdist**2*lyx**4 )
#k4 = lambda tdist: -lyt*np.sqrt(tdist)
k1 = lambda tdist: (2*lyt - 4*lyt**2 * (tdist) )
k2 = lambda xdist: ( 4*lyx**2 * (xdist) - 2*lyx )
k3 = lambda xdist: ( 3*4*lyx**2 - 6*8*xdist*lyx**3 + 16*xdist**2*lyx**4 )
k4 = lambda ttdist: 2*lyt*(ttdist)
dkyydlyx = lambda tdist,xdist: kyy(tdist,xdist)*(-xdist)
dkyydlyt = lambda tdist,xdist: kyy(tdist,xdist)*(-tdist)
dk1dlyt = lambda tdist: 2. - 4*2.*lyt*tdist
dk2dlyx = lambda xdist: (4.*2.*lyx*xdist -2.)
dk3dlyx = lambda xdist: (6.*4.*lyx - 18.*8*xdist*lyx**2 + 4*16*xdist**2*lyx**3)
dk4dlyt = lambda ttdist: 2*(ttdist)
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i==0 and j==0:
dka[ss1,ss2] = 0
dkb[ss1,ss2] = 0
dkc[ss1,ss2] = 0
dkYdvart[ss1,ss2] = vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdvarx[ss1,ss2] = vyt*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdlenx[ss1,ss2] = vyt*vyx*dkyydlyx(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*vyx*dkyydlyt(tdist[ss1,ss2],xdist[ss1,ss2])
elif i==0 and j==1:
dka[ss1,ss2] = -k2(xdist[ss1,ss2])*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkb[ss1,ss2] = k4(ttdist[ss1,ss2])*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkc[ss1,ss2] = vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
#dkYdvart[ss1,ss2] = 0
#dkYdvarx[ss1,ss2] = 0
#dkYdlent[ss1,ss2] = 0
#dkYdlenx[ss1,ss2] = 0
dkYdvart[ss1,ss2] = (-a*k2(xdist[ss1,ss2])+b*k4(ttdist[ss1,ss2])+c)*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdvarx[ss1,ss2] = (-a*k2(xdist[ss1,ss2])+b*k4(ttdist[ss1,ss2])+c)*vyt*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*vyx*dkyydlyt(tdist[ss1,ss2],xdist[ss1,ss2])* (-a*k2(xdist[ss1,ss2])+b*k4(ttdist[ss1,ss2])+c)+\
vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])*b*dk4dlyt(ttdist[ss1,ss2])
dkYdlenx[ss1,ss2] = vyt*vyx*dkyydlyx(tdist[ss1,ss2],xdist[ss1,ss2])*(-a*k2(xdist[ss1,ss2])+b*k4(ttdist[ss1,ss2])+c)+\
vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])*(-a*dk2dlyx(xdist[ss1,ss2]))
elif i==1 and j==1:
dka[ss1,ss2] = (2*a*k3(xdist[ss1,ss2]) - 2*c*k2(xdist[ss1,ss2]))*vyt*vyx* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkb[ss1,ss2] = 2*b*k1(tdist[ss1,ss2])*vyt*vyx* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkc[ss1,ss2] = (-2*a*k2(xdist[ss1,ss2]) + 2*c )*vyt*vyx* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdvart[ss1,ss2] = ( b**2*k1(tdist[ss1,ss2]) - 2*a*c*k2(xdist[ss1,ss2]) + a**2*k3(xdist[ss1,ss2]) + c**2 )*vyx* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdvarx[ss1,ss2] = ( b**2*k1(tdist[ss1,ss2]) - 2*a*c*k2(xdist[ss1,ss2]) + a**2*k3(xdist[ss1,ss2]) + c**2 )*vyt* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*vyx*dkyydlyt(tdist[ss1,ss2],xdist[ss1,ss2])*( b**2*k1(tdist[ss1,ss2]) - 2*a*c*k2(xdist[ss1,ss2]) + a**2*k3(xdist[ss1,ss2]) + c**2 ) +\
vyx*vyt*kyy(tdist[ss1,ss2],xdist[ss1,ss2])*b**2*dk1dlyt(tdist[ss1,ss2])
dkYdlenx[ss1,ss2] = vyt*vyx*dkyydlyx(tdist[ss1,ss2],xdist[ss1,ss2])*( b**2*k1(tdist[ss1,ss2]) - 2*a*c*k2(xdist[ss1,ss2]) + a**2*k3(xdist[ss1,ss2]) + c**2 ) +\
vyx*vyt*kyy(tdist[ss1,ss2],xdist[ss1,ss2])* (-2*a*c*dk2dlyx(xdist[ss1,ss2]) + a**2*dk3dlyx(xdist[ss1,ss2]) )
else:
dka[ss1,ss2] = -k2(xdist[ss1,ss2])*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkb[ss1,ss2] = -k4(ttdist[ss1,ss2])*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkc[ss1,ss2] = vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
#dkYdvart[ss1,ss2] = 0
#dkYdvarx[ss1,ss2] = 0
#dkYdlent[ss1,ss2] = 0
#dkYdlenx[ss1,ss2] = 0
dkYdvart[ss1,ss2] = (-a*k2(xdist[ss1,ss2])-b*k4(ttdist[ss1,ss2])+c)*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdvarx[ss1,ss2] = (-a*k2(xdist[ss1,ss2])-b*k4(ttdist[ss1,ss2])+c)*vyt*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*vyx*dkyydlyt(tdist[ss1,ss2],xdist[ss1,ss2])* (-a*k2(xdist[ss1,ss2])-b*k4(ttdist[ss1,ss2])+c)+\
vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])*(-1)*b*dk4dlyt(ttdist[ss1,ss2])
dkYdlenx[ss1,ss2] = vyt*vyx*dkyydlyx(tdist[ss1,ss2],xdist[ss1,ss2])*(-a*k2(xdist[ss1,ss2])-b*k4(ttdist[ss1,ss2])+c)+\
vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])*(-a*dk2dlyx(xdist[ss1,ss2]))
self.a.gradient = np.sum(dka * dL_dK)
self.b.gradient = np.sum(dkb * dL_dK)
self.c.gradient = np.sum(dkc * dL_dK)
self.variance_Yt.gradient = np.sum(dkYdvart * dL_dK) # Vy
self.variance_Yx.gradient = np.sum(dkYdvarx * dL_dK)
self.lengthscale_Yt.gradient = np.sum(dkYdlent*(-0.5*self.lengthscale_Yt**(-2)) * dL_dK) #ly np.sum(dktheta2*(-self.lengthscale_Y**(-2)) * dL_dK)
self.lengthscale_Yx.gradient = np.sum(dkYdlenx*(-0.5*self.lengthscale_Yx**(-2)) * dL_dK)
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