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some tidying in the likelihood classes

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This commit is contained in:
James Hensman 2013-02-01 09:47:30 +00:00
parent 3a558d8244
commit 7dfbcebb87
6 changed files with 364 additions and 369 deletions
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15
GPy/likelihoods/EP.py
View file

@ -18,12 +18,10 @@ class EP:
self.likelihood_function = likelihood_function self.likelihood_function = likelihood_function
self.epsilon = epsilon self.epsilon = epsilon
self.eta, self.delta = power_ep self.eta, self.delta = power_ep
self.jitter = 1e-12 # TODO: is this needed? self.is_heteroscedastic = True
""" #Initial values - Likelihood approximation parameters:
Initial values - Likelihood approximation parameters: #p(y|f) = t(f|tau_tilde,v_tilde)
p(y|f) = t(f|tau_tilde,v_tilde)
"""
self.tau_tilde = np.zeros(self.N) self.tau_tilde = np.zeros(self.N)
self.v_tilde = np.zeros(self.N) self.v_tilde = np.zeros(self.N)
@ -32,8 +30,11 @@ class EP:
mu_tilde = self.v_tilde/self.tau_tilde #When calling EP, this variable is used instead of Y in the GP model mu_tilde = self.v_tilde/self.tau_tilde #When calling EP, this variable is used instead of Y in the GP model
sigma_sum = 1./self.tau_ + 1./self.tau_tilde sigma_sum = 1./self.tau_ + 1./self.tau_tilde
mu_diff_2 = (self.v_/self.tau_ - mu_tilde)**2 mu_diff_2 = (self.v_/self.tau_ - mu_tilde)**2
Z_ep = np.sum(np.log(self.Z_hat)) + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum) #Normalization constant self.Z = np.sum(np.log(self.Z_hat)) + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum) #Normalization constant, aka Z_ep
self.Y, self.beta, self.Z = self.tau_tilde[:,None], mu_tilde[:,None], Z_ep
self.Y = mu_tilde[:,None]
self.precsion = self.tau_tilde
self.covariance_matrix = np.diag(1./self.precision)
def fit_full(self,K): def fit_full(self,K):
""" """

5
GPy/likelihoods/Gaussian.py
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@ -2,6 +2,7 @@ import numpy as np
class Gaussian: class Gaussian:
def __init__(self,data,variance=1.,normalize=False): def __init__(self,data,variance=1.,normalize=False):
self.is_heteroscedastic = False
self.data = data self.data = data
self.N,D = data.shape self.N,D = data.shape
self.Z = 0. # a correction factor which accounts for the approximation made self.Z = 0. # a correction factor which accounts for the approximation made
@ -19,6 +20,7 @@ class Gaussian:
self.YYT = np.dot(self.Y,self.Y.T) self.YYT = np.dot(self.Y,self.Y.T)
self._set_params(np.asarray(variance)) self._set_params(np.asarray(variance))
def _get_params(self): def _get_params(self):
return np.asarray(self._variance) return np.asarray(self._variance)
@ -27,7 +29,8 @@ class Gaussian:
def _set_params(self,x): def _set_params(self,x):
self._variance = x self._variance = x
self.variance = np.eye(self.N)*self._variance self.covariance_matrix = np.eye(self.N)*self._variance
self.precision = 1./self._variance
def fit(self): def fit(self):
""" """

14
GPy/likelihoods/likelihood_functions.py
View file

@ -8,18 +8,18 @@ import scipy as sp
import pylab as pb import pylab as pb
from ..util.plot import gpplot from ..util.plot import gpplot
class likelihood: class likelihood_function:
""" """
Likelihood class for doing Expectation propagation Likelihood class for doing Expectation propagation
:param Y: observed output (Nx1 numpy.darray) :param Y: observed output (Nx1 numpy.darray)
..Note:: Y values allowed depend on the likelihood used ..Note:: Y values allowed depend on the likelihood_function used
""" """
def __init__(self,location=0,scale=1): def __init__(self,location=0,scale=1):
self.location = location self.location = location
self.scale = scale self.scale = scale
class probit(likelihood): class probit(likelihood_function):
""" """
Probit likelihood Probit likelihood
Y is expected to take values in {-1,1} Y is expected to take values in {-1,1}
@ -29,7 +29,7 @@ class probit(likelihood):
$$ $$
""" """
def __init__(self,location=0,scale=1): def __init__(self,location=0,scale=1):
likelihood.__init__(self,Y,location,scale) likelihood_function.__init__(self,Y,location,scale)
def moments_match(self,data_i,tau_i,v_i): def moments_match(self,data_i,tau_i,v_i):
""" """
@ -64,7 +64,7 @@ class probit(likelihood):
def _log_likelihood_gradients(): def _log_likelihood_gradients():
return np.zeros(0) # there are no parameters of whcih to compute the gradients return np.zeros(0) # there are no parameters of whcih to compute the gradients
class poisson(likelihood): class poisson(likelihood_function):
""" """
Poisson likelihood Poisson likelihood
Y is expected to take values in {0,1,2,...} Y is expected to take values in {0,1,2,...}
@ -75,7 +75,7 @@ class poisson(likelihood):
""" """
def __init__(self,Y,location=0,scale=1): def __init__(self,Y,location=0,scale=1):
assert len(Y[Y<0]) == 0, "Output cannot have negative values" assert len(Y[Y<0]) == 0, "Output cannot have negative values"
likelihood.__init__(self,Y,location,scale) likelihood_function.__init__(self,Y,location,scale)
def moments_match(self,i,tau_i,v_i): def moments_match(self,i,tau_i,v_i):
""" """
@ -160,7 +160,7 @@ class poisson(likelihood):
if Z is not None: if Z is not None:
pb.plot(Z,Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12) pb.plot(Z,Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
class gaussian(likelihood): class gaussian(likelihood_function):
""" """
Gaussian likelihood Gaussian likelihood
Y is expected to take values in (-inf,inf) Y is expected to take values in (-inf,inf)

236
GPy/models/sparse_GP.py
View file

@ -6,37 +6,36 @@ import pylab as pb
from ..util.linalg import mdot, jitchol, chol_inv, pdinv from ..util.linalg import mdot, jitchol, chol_inv, pdinv
from ..util.plot import gpplot from ..util.plot import gpplot
from .. import kern from .. import kern
from ..inference.likelihoods import likelihood
from GP import GP from GP import GP
#Still TODO: #Still TODO:
# make use of slices properly (kernel can now do this) # make use of slices properly (kernel can now do this)
# enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array) # enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array)
class sparse_GP(GP): class sparse_GP(GP):
""" """
Variational sparse GP model (Regression) Variational sparse GP model
:param X: inputs :param X: inputs
:type X: np.ndarray (N x Q) :type X: np.ndarray (N x Q)
:param Y: observed data :param likelihood: a likelihood instance, containing the observed data
:type Y: np.ndarray of observations (N x D) :type likelihood: GPy.likelihood.(Gaussian | EP)
:param kernel : the kernel/covariance function. See link kernels :param kernel : the kernel/covariance function. See link kernels
:type kernel: a GPy kernel :type kernel: a GPy kernel
:param Z: inducing inputs (optional, see note)
:type Z: np.ndarray (M x Q) | None
:param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance) :param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance)
:type X_uncertainty: np.ndarray (N x Q) | None :type X_uncertainty: np.ndarray (N x Q) | None
:param Z: inducing inputs (optional, see note)
:type Z: np.ndarray (M x Q) | None
:param Zslices: slices for the inducing inputs (see slicing TODO: link) :param Zslices: slices for the inducing inputs (see slicing TODO: link)
:param M : Number of inducing points (optional, default 10. Ignored if Z is not None) :param M : Number of inducing points (optional, default 10. Ignored if Z is not None)
:type M: int :type M: int
:param beta: noise precision. TODO> ignore beta if doing EP
:type beta: float
:param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales) :param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales)
:type normalize_(X|Y): bool :type normalize_(X|Y): bool
""" """
def __init__(self,X,Y=None,kernel=None,X_uncertainty=None,beta=100.,Z=None,Zslices=None,M=10,normalize_X=False,normalize_Y=False,likelihood=None,method_ep='DTC',epsilon_ep=1e-3,power_ep=[1.,1.]): def __init__(self,X,likelihood,kernel, X_uncertainty=None, Z=None,Zslices=None,M=10,normalize_X=False):
self.scale_factor = 1000.0# a scaling factor to help keep the algorithm stable
if Z is None: if Z is None:
self.Z = np.random.permutation(X.copy())[:M] self.Z = np.random.permutation(X.copy())[:M]
@ -52,140 +51,91 @@ class sparse_GP(GP):
self.has_uncertain_inputs=True self.has_uncertain_inputs=True
self.X_uncertainty = X_uncertainty self.X_uncertainty = X_uncertainty
GP.__init__(self, X=X, Y=Y, kernel=kernel, normalize_X=normalize_X, normalize_Y=normalize_Y,likelihood=likelihood,epsilon_ep=epsilon_ep,power_ep=power_ep) GP.__init__(self, X, Y, kernel=kernel, normalize_X=normalize_X, Xslices=Xslices)
#normalise X uncertainty also #normalise X uncertainty also
if self.has_uncertain_inputs: if self.has_uncertain_inputs:
self.X_uncertainty /= np.square(self._Xstd) self.X_uncertainty /= np.square(self._Xstd)
if not self.EP:
self.trYYT = np.sum(np.square(self.Y))
else:
self.method_ep = method_ep
#normalise X uncertainty also
if self.has_uncertain_inputs:
self.X_uncertainty /= np.square(self._Xstd)
def _set_params(self, p):
self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
if not self.EP:
self.beta = p[self.M*self.Q]
self.kern._set_params(p[self.Z.size + 1:])
else:
self.kern._set_params(p[self.Z.size:])
if self.Y is None:
self.Y = np.ones([self.N,1])
self._compute_kernel_matrices()
self._computations()
def _get_params(self):
if not self.EP:
return np.hstack([self.Z.flatten(),self.beta,self.kern._get_params_transformed()])
else:
return np.hstack([self.Z.flatten(),self.kern._get_params_transformed()])
def _get_param_names(self):
if not self.EP:
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + ['noise_precision']+self.kern._get_param_names_transformed()
else:
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + self.kern._get_param_names_transformed()
def _compute_kernel_matrices(self):
# kernel computations, using BGPLVM notation
#TODO: slices for psi statistics (easy enough)
self.Kmm = self.kern.K(self.Z)
if self.has_uncertain_inputs:
if not self.EP:
self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty)#.sum() NOTE psi0 is now a vector
self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
#self.psi2_beta_scaled = ?
else:
raise NotImplementedError, "uncertain_inputs not yet supported for EP"
else:
self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices)#.sum()
self.psi1 = self.kern.K(self.Z,self.X)
self.psi2 = np.dot(self.psi1,self.psi1.T)
self.psi2_beta_scaled = np.dot(self.psi1,self.beta*self.psi1.T)
def _computations(self): def _computations(self):
# TODO find routine to multiply triangular matrices # TODO find routine to multiply triangular matrices
self.V = self.beta*self.Y #TODO: slices for psi statistics (easy enough)
sf = self.scale_factor
sf2 = sf**2
# kernel computations, using BGPLVM notation
self.Kmm = self.kern.K(self.Z)
if self.has_uncertain_inputs:
self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
self.psi2_beta_scaled = (self.psi2*(self.beta/sf2)).sum(0)
else:
self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices).sum()
self.psi1 = self.kern.K(self.Z,self.X)
tmp = self.psi1*(np.sqrt(self.likelihood.beta)/sf)
self.psi2_beta_scaled = np.dot(tmp,tmp.T)
self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)#+np.eye(self.M)*1e-3)
self.V = (self.likelihood.beta/self.scale_factor)*self.Y
self.A = mdot(self.Lmi, self.psi2_beta_scaled, self.Lmi.T)
self.B = np.eye(self.M)/sf2 + self.A
self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
self.psi1V = np.dot(self.psi1, self.V) self.psi1V = np.dot(self.psi1, self.V)
self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T) self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm) self.C = mdot(self.Lmi.T, self.Bi, self.Lmi)
self.A = mdot(self.Lmi, self.psi2_beta_scaled, self.Lmi.T) self.E = mdot(self.C, self.psi1VVpsi1/sf2, self.C.T)
self.B = np.eye(self.M) + self.A
self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
self.LLambdai = np.dot(self.LBi, self.Lmi)
self.LBL_inv = mdot(self.Lmi.T, self.Bi, self.Lmi)
self.C = mdot(self.LLambdai, self.psi1V)
self.G = mdot(self.LBL_inv, self.psi1VVpsi1, self.LBL_inv.T)
self.trace_K_beta_scaled = (self.psi0*self.beta).sum() - np.trace(self.A)
if not self.EP:
self.trace_K = self.psi0.sum() - np.trace(self.A)/self.beta
# Compute dL_dpsi # Compute dL_dpsi # FIXME
self.dL_dpsi1 = mdot(self.LLambdai.T,self.C,self.V.T) self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
if not self.EP: self.dL_dpsi1 = mdot(self.V, self.psi1V.T,self.C).T
self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N) self.dL_dpsi2 = 0.5 * self.beta * self.D * self.Kmmi[None,:,:] # dB
if self.has_uncertain_inputs: self.dL_dpsi2 += - 0.5 * self.beta/sf2 * self.D * self.C[None,:,:] # dC
self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G) self.dL_dpsi2 += - 0.5 * self.beta * self.E[None,:,:] # dD
else:
self.dL_dpsi2_ = - 0.5 * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
else:
self.dL_dpsi0 = - 0.5 * self.D * self.beta.flatten()
if not self.has_uncertain_inputs:
self.dL_dpsi2_ = - 0.5 * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
# Compute dL_dKmm # Compute dL_dKmm
self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi) # dB self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi)*sf2 # dB
self.dL_dKmm += -0.5 * self.D * (- self.LBL_inv - 2.*mdot(self.LBL_inv, self.psi2_beta_scaled, self.Kmmi) + self.Kmmi) # dC self.dL_dKmm += -0.5 * self.D * (- self.C/sf2 - 2.*mdot(self.C, self.psi2_beta_scaled, self.Kmmi) + self.Kmmi) # dC
self.dL_dKmm += np.dot(np.dot(self.G,self.psi2_beta_scaled) - np.dot(self.LBL_inv, self.psi1VVpsi1), self.Kmmi) + 0.5*self.G # dE self.dL_dKmm += np.dot(np.dot(self.E*sf2, self.psi2_beta_scaled) - np.dot(self.C, self.psi1VVpsi1), self.Kmmi) + 0.5*self.E # dD
def approximate_likelihood(self):
assert not isinstance(self.likelihood, gaussian), "EP is only available for non-gaussian likelihoods" def _set_params(self, p):
if self.method_ep == 'DTC': self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
self.ep_approx = DTC(self.Kmm,self.likelihood,self.psi1,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta]) self.beta = p[self.M*self.Q] # FIXME
elif self.method_ep == 'FITC': self.kern._set_params(p[self.Z.size + 1:])
self.ep_approx = FITC(self.Kmm,self.likelihood,self.psi1,self.psi0,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
else:
self.ep_approx = Full(self.X,self.likelihood,self.kernel,inducing=None,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
self.beta, self.Y, self.Z_ep = self.ep_approx.fit_EP()
self.trbetaYYT = np.sum(np.square(self.Y)*self.beta)
self._computations() self._computations()
def _get_params(self):
return np.hstack([self.Z.flatten(),GP._get_params(self))
def _get_param_names(self):
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + GP._get_param_names(self)
def log_likelihood(self): def log_likelihood(self):
""" """ Compute the (lower bound on the) log marginal likelihood """
Compute the (lower bound on the) log marginal likelihood sf2 = self.scale_factor**2
""" A = -0.5*self.N*self.D*(np.log(2.*np.pi) - np.log(self.beta)) -0.5*self.beta*self.trYYT # FIXME
if not self.EP: B = -0.5*self.D*(self.beta*self.psi0-np.trace(self.A)*sf2)# FIXME
A = -0.5*self.N*self.D*(np.log(2.*np.pi) - np.log(self.beta)) C = -0.5*self.D * (self.B_logdet + self.M*np.log(sf2))
D = -0.5*self.beta*self.trYYT D = +0.5*np.sum(self.psi1VVpsi1 * self.C)
else: return A+B+C+D
A = -0.5*self.D*(self.N*np.log(2.*np.pi) - np.sum(np.log(self.beta)))
D = -0.5*self.trbetaYYT
B = -0.5*self.D*self.trace_K_beta_scaled
C = -0.5*self.D * self.B_logdet
E = +0.5*np.sum(self.psi1VVpsi1 * self.LBL_inv)
return A+B+C+D+E
def _log_likelihood_gradients(self):
return np.hstack([self.dL_dZ().flatten(), GP._log_likelihood_gradients(self)])
# FIXME: move this into the lieklihood class
def dL_dbeta(self): def dL_dbeta(self):
""" sf2 = self.scale_factor**2
Compute the gradient of the log likelihood wrt beta. dA_dbeta = 0.5 * self.N*self.D/self.beta - 0.5 * self.trYYT
""" dB_dbeta = - 0.5 * self.D * (self.psi0 - np.trace(self.A)/self.beta*sf2)
#TODO: suport heteroscedatic noise
dA_dbeta = 0.5 * self.N*self.D/self.beta
dB_dbeta = - 0.5 * self.D * self.trace_K
dC_dbeta = - 0.5 * self.D * np.sum(self.Bi*self.A)/self.beta dC_dbeta = - 0.5 * self.D * np.sum(self.Bi*self.A)/self.beta
dD_dbeta = - 0.5 * self.trYYT dD_dbeta = np.sum((self.C - 0.5 * mdot(self.C,self.psi2_beta_scaled,self.C) ) * self.psi1VVpsi1 )/self.beta
tmp = mdot(self.LBi.T, self.LLambdai, self.psi1V)
dE_dbeta = (np.sum(np.square(self.C)) - 0.5 * np.sum(self.A * np.dot(tmp, tmp.T)))/self.beta
return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta + dE_dbeta) return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta)
def dL_dtheta(self): def dL_dtheta(self):
""" """
@ -195,10 +145,10 @@ class sparse_GP(GP):
if self.has_uncertain_inputs: if self.has_uncertain_inputs:
dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z,self.X,self.X_uncertainty) dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z,self.X,self.X_uncertainty)
dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty) dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # for multiple_beta, dL_dpsi2 will be a different shape dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2,self.dL_dpsi1.T, self.Z,self.X, self.X_uncertainty)
else: else:
#re-cast computations in psi2 back to psi1: #re-cast computations in psi2 back to psi1:
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2_,self.beta.T*self.psi1) #dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1) dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2.sum(0),self.psi1)
dL_dtheta += self.kern.dK_dtheta(dL_dpsi1,self.Z,self.X) dL_dtheta += self.kern.dK_dtheta(dL_dpsi1,self.Z,self.X)
dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X) dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X)
@ -208,48 +158,36 @@ class sparse_GP(GP):
""" """
The derivative of the bound wrt the inducing inputs Z The derivative of the bound wrt the inducing inputs Z
""" """
dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z,)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
if self.has_uncertain_inputs: if self.has_uncertain_inputs:
dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty) dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1,self.Z,self.X, self.X_uncertainty)
dL_dZ += self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) dL_dZ += 2.*self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # 'stripes'
else: else:
#re-cast computations in psi2 back to psi1: #re-cast computations in psi2 back to psi1:
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2_,self.beta.T*self.psi1)#dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1) dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2.sum(0),self.psi1)
dL_dZ += self.kern.dK_dX(dL_dpsi1,self.Z,self.X) dL_dZ += self.kern.dK_dX(dL_dpsi1,self.Z,self.X)
return dL_dZ return dL_dZ
def _log_likelihood_gradients(self):
if not self.EP:
return np.hstack([self.dL_dZ().flatten(), self.dL_dbeta(), self.dL_dtheta()])
else:
return np.hstack([self.dL_dZ().flatten(), self.dL_dtheta()])
def _raw_predict(self, Xnew, slices, full_cov=False): def _raw_predict(self, Xnew, slices, full_cov=False):
"""Internal helper function for making predictions, does not account for normalisation""" """Internal helper function for making predictions, does not account for normalisation"""
Kx = self.kern.K(self.Z, Xnew) Kx = self.kern.K(self.Z, Xnew)
mu = mdot(Kx.T, self.LBL_inv, self.psi1V) mu = mdot(Kx.T, self.C/self.scale_factor, self.psi1V)
phi = None
if full_cov: if full_cov:
Kxx = self.kern.K(Xnew) Kxx = self.kern.K(Xnew)
var = Kxx - mdot(Kx.T, (self.Kmmi - self.LBL_inv), Kx) var = Kxx - mdot(Kx.T, (self.Kmmi - self.C/self.scale_factor**2), Kx)
if not self.EP:
var += np.eye(Xnew.shape[0])/self.beta
else:
raise NotImplementedError, "full_cov = True not implemented for EP"
else: else:
Kxx = self.kern.Kdiag(Xnew) Kxx = self.kern.Kdiag(Xnew)
var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.LBL_inv, Kx),0) var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.C/self.scale_factor**2, Kx),0)
if not self.EP:
var += 1./self.beta return mu,var
else:
phi = self.likelihood.predictive_mean(mu,var)
return mu,var,phi
def plot(self, *args, **kwargs): def plot(self, *args, **kwargs):
""" """
Plot the fitted model: just call the GP_regression plot function and then add inducing inputs Plot the fitted model: just call the GP_regression plot function and then add inducing inputs
""" """
GP.plot(self,*args,**kwargs) GP_regression.plot(self,*args,**kwargs)
if self.Q==1: if self.Q==1:
pb.plot(self.Z,self.Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12) pb.plot(self.Z,self.Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
if self.has_uncertain_inputs: if self.has_uncertain_inputs:

258
GPy/models/sparse_GP_old.py Normal file
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@ -0,0 +1,258 @@
# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
import numpy as np
import pylab as pb
from ..util.linalg import mdot, jitchol, chol_inv, pdinv
from ..util.plot import gpplot
from .. import kern
from GP import GP
#Still TODO:
# make use of slices properly (kernel can now do this)
# enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array)
class sparse_GP(GP):
"""
Variational sparse GP model (Regression)
:param X: inputs
:type X: np.ndarray (N x Q)
:param Y: observed data
:type Y: np.ndarray of observations (N x D)
:param kernel : the kernel/covariance function. See link kernels
:type kernel: a GPy kernel
:param Z: inducing inputs (optional, see note)
:type Z: np.ndarray (M x Q) | None
:param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance)
:type X_uncertainty: np.ndarray (N x Q) | None
:param Zslices: slices for the inducing inputs (see slicing TODO: link)
:param M : Number of inducing points (optional, default 10. Ignored if Z is not None)
:type M: int
:param beta: noise precision. TODO> ignore beta if doing EP
:type beta: float
:param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales)
:type normalize_(X|Y): bool
"""
def __init__(self,X,Y=None,kernel=None,X_uncertainty=None,beta=100.,Z=None,Zslices=None,M=10,normalize_X=False,normalize_Y=False,likelihood=None,method_ep='DTC',epsilon_ep=1e-3,power_ep=[1.,1.]):
if Z is None:
self.Z = np.random.permutation(X.copy())[:M]
self.M = M
else:
assert Z.shape[1]==X.shape[1]
self.Z = Z
self.M = Z.shape[0]
if X_uncertainty is None:
self.has_uncertain_inputs=False
else:
assert X_uncertainty.shape==X.shape
self.has_uncertain_inputs=True
self.X_uncertainty = X_uncertainty
GP.__init__(self, X=X, Y=Y, kernel=kernel, normalize_X=normalize_X, normalize_Y=normalize_Y,likelihood=likelihood,epsilon_ep=epsilon_ep,power_ep=power_ep)
#normalise X uncertainty also
if self.has_uncertain_inputs:
self.X_uncertainty /= np.square(self._Xstd)
if not self.EP:
self.trYYT = np.sum(np.square(self.Y))
else:
self.method_ep = method_ep
#normalise X uncertainty also
if self.has_uncertain_inputs:
self.X_uncertainty /= np.square(self._Xstd)
def _set_params(self, p):
self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
if not self.EP:
self.beta = p[self.M*self.Q]
self.kern._set_params(p[self.Z.size + 1:])
else:
self.kern._set_params(p[self.Z.size:])
if self.Y is None:
self.Y = np.ones([self.N,1])
self._compute_kernel_matrices()
self._computations()
def _get_params(self):
if not self.EP:
return np.hstack([self.Z.flatten(),self.beta,self.kern._get_params_transformed()])
else:
return np.hstack([self.Z.flatten(),self.kern._get_params_transformed()])
def _get_param_names(self):
if not self.EP:
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + ['noise_precision']+self.kern._get_param_names_transformed()
else:
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + self.kern._get_param_names_transformed()
def _compute_kernel_matrices(self):
# kernel computations, using BGPLVM notation
#TODO: slices for psi statistics (easy enough)
self.Kmm = self.kern.K(self.Z)
if self.has_uncertain_inputs:
if not self.EP:
self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty)#.sum() NOTE psi0 is now a vector
self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
#self.psi2_beta_scaled = ?
else:
raise NotImplementedError, "uncertain_inputs not yet supported for EP"
else:
self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices)#.sum()
self.psi1 = self.kern.K(self.Z,self.X)
self.psi2 = np.dot(self.psi1,self.psi1.T)
self.psi2_beta_scaled = np.dot(self.psi1,self.beta*self.psi1.T)
def _computations(self):
# TODO find routine to multiply triangular matrices
self.V = self.beta*self.Y
self.psi1V = np.dot(self.psi1, self.V)
self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)
self.A = mdot(self.Lmi, self.psi2_beta_scaled, self.Lmi.T)
self.B = np.eye(self.M) + self.A
self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
self.LLambdai = np.dot(self.LBi, self.Lmi)
self.LBL_inv = mdot(self.Lmi.T, self.Bi, self.Lmi)
self.C = mdot(self.LLambdai, self.psi1V)
self.G = mdot(self.LBL_inv, self.psi1VVpsi1, self.LBL_inv.T)
self.trace_K_beta_scaled = (self.psi0*self.beta).sum() - np.trace(self.A)
if not self.EP:
self.trace_K = self.psi0.sum() - np.trace(self.A)/self.beta
# Compute dL_dpsi
self.dL_dpsi1 = mdot(self.LLambdai.T,self.C,self.V.T)
if not self.EP:
self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
if self.has_uncertain_inputs:
self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
else:
self.dL_dpsi2_ = - 0.5 * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
else:
self.dL_dpsi0 = - 0.5 * self.D * self.beta.flatten()
if not self.has_uncertain_inputs:
self.dL_dpsi2_ = - 0.5 * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
# Compute dL_dKmm
self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi) # dB
self.dL_dKmm += -0.5 * self.D * (- self.LBL_inv - 2.*mdot(self.LBL_inv, self.psi2_beta_scaled, self.Kmmi) + self.Kmmi) # dC
self.dL_dKmm += np.dot(np.dot(self.G,self.psi2_beta_scaled) - np.dot(self.LBL_inv, self.psi1VVpsi1), self.Kmmi) + 0.5*self.G # dE
def approximate_likelihood(self):
assert not isinstance(self.likelihood, gaussian), "EP is only available for non-gaussian likelihoods"
if self.method_ep == 'DTC':
self.ep_approx = DTC(self.Kmm,self.likelihood,self.psi1,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
elif self.method_ep == 'FITC':
self.ep_approx = FITC(self.Kmm,self.likelihood,self.psi1,self.psi0,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
else:
self.ep_approx = Full(self.X,self.likelihood,self.kernel,inducing=None,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
self.beta, self.Y, self.Z_ep = self.ep_approx.fit_EP()
self.trbetaYYT = np.sum(np.square(self.Y)*self.beta)
self._computations()
def log_likelihood(self):
"""
Compute the (lower bound on the) log marginal likelihood
"""
if not self.EP:
A = -0.5*self.N*self.D*(np.log(2.*np.pi) - np.log(self.beta))
D = -0.5*self.beta*self.trYYT
else:
A = -0.5*self.D*(self.N*np.log(2.*np.pi) - np.sum(np.log(self.beta)))
D = -0.5*self.trbetaYYT
B = -0.5*self.D*self.trace_K_beta_scaled
C = -0.5*self.D * self.B_logdet
E = +0.5*np.sum(self.psi1VVpsi1 * self.LBL_inv)
return A+B+C+D+E
def dL_dbeta(self):
"""
Compute the gradient of the log likelihood wrt beta.
"""
#TODO: suport heteroscedatic noise
dA_dbeta = 0.5 * self.N*self.D/self.beta
dB_dbeta = - 0.5 * self.D * self.trace_K
dC_dbeta = - 0.5 * self.D * np.sum(self.Bi*self.A)/self.beta
dD_dbeta = - 0.5 * self.trYYT
tmp = mdot(self.LBi.T, self.LLambdai, self.psi1V)
dE_dbeta = (np.sum(np.square(self.C)) - 0.5 * np.sum(self.A * np.dot(tmp, tmp.T)))/self.beta
return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta + dE_dbeta)
def dL_dtheta(self):
"""
Compute and return the derivative of the log marginal likelihood wrt the parameters of the kernel
"""
dL_dtheta = self.kern.dK_dtheta(self.dL_dKmm,self.Z)
if self.has_uncertain_inputs:
dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z,self.X,self.X_uncertainty)
dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # for multiple_beta, dL_dpsi2 will be a different shape
else:
#re-cast computations in psi2 back to psi1:
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2_,self.beta.T*self.psi1) #dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1)
dL_dtheta += self.kern.dK_dtheta(dL_dpsi1,self.Z,self.X)
dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X)
return dL_dtheta
def dL_dZ(self):
"""
The derivative of the bound wrt the inducing inputs Z
"""
dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z,)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
if self.has_uncertain_inputs:
dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
dL_dZ += self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty)
else:
#re-cast computations in psi2 back to psi1:
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2_,self.beta.T*self.psi1)#dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1)
dL_dZ += self.kern.dK_dX(dL_dpsi1,self.Z,self.X)
return dL_dZ
def _log_likelihood_gradients(self):
if not self.EP:
return np.hstack([self.dL_dZ().flatten(), self.dL_dbeta(), self.dL_dtheta()])
else:
return np.hstack([self.dL_dZ().flatten(), self.dL_dtheta()])
def _raw_predict(self, Xnew, slices, full_cov=False):
"""Internal helper function for making predictions, does not account for normalisation"""
Kx = self.kern.K(self.Z, Xnew)
mu = mdot(Kx.T, self.LBL_inv, self.psi1V)
phi = None
if full_cov:
Kxx = self.kern.K(Xnew)
var = Kxx - mdot(Kx.T, (self.Kmmi - self.LBL_inv), Kx)
if not self.EP:
var += np.eye(Xnew.shape[0])/self.beta
else:
raise NotImplementedError, "full_cov = True not implemented for EP"
else:
Kxx = self.kern.Kdiag(Xnew)
var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.LBL_inv, Kx),0)
if not self.EP:
var += 1./self.beta
else:
phi = self.likelihood.predictive_mean(mu,var)
return mu,var,phi
def plot(self, *args, **kwargs):
"""
Plot the fitted model: just call the GP_regression plot function and then add inducing inputs
"""
GP.plot(self,*args,**kwargs)
if self.Q==1:
pb.plot(self.Z,self.Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
if self.has_uncertain_inputs:
pb.errorbar(self.X[:,0], pb.ylim()[0]+np.zeros(self.N), xerr=2*np.sqrt(self.X_uncertainty.flatten()))
if self.Q==2:
pb.plot(self.Z[:,0],self.Z[:,1],'wo')

205
GPy/models/sparse_GP_regression.py
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@ -1,205 +0,0 @@
# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
import numpy as np
import pylab as pb
from ..util.linalg import mdot, jitchol, chol_inv, pdinv
from ..util.plot import gpplot
from .. import kern
from ..inference.likelihoods import likelihood
from GP_regression import GP_regression
#Still TODO:
# make use of slices properly (kernel can now do this)
# enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array)
class sparse_GP_regression(GP_regression):
"""
Variational sparse GP model (Regression)
:param X: inputs
:type X: np.ndarray (N x Q)
:param Y: observed data
:type Y: np.ndarray of observations (N x D)
:param kernel : the kernel/covariance function. See link kernels
:type kernel: a GPy kernel
:param Z: inducing inputs (optional, see note)
:type Z: np.ndarray (M x Q) | None
:param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance)
:type X_uncertainty: np.ndarray (N x Q) | None
:param Zslices: slices for the inducing inputs (see slicing TODO: link)
:param M : Number of inducing points (optional, default 10. Ignored if Z is not None)
:type M: int
:param beta: noise precision. TODO> ignore beta if doing EP
:type beta: float
:param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales)
:type normalize_(X|Y): bool
"""
def __init__(self,X,Y,kernel=None, X_uncertainty=None, beta=100., Z=None,Zslices=None,M=10,normalize_X=False,normalize_Y=False):
self.scale_factor = 1000.0
self.beta = beta
if Z is None:
self.Z = np.random.permutation(X.copy())[:M]
self.M = M
else:
assert Z.shape[1]==X.shape[1]
self.Z = Z
self.M = Z.shape[0]
if X_uncertainty is None:
self.has_uncertain_inputs=False
else:
assert X_uncertainty.shape==X.shape
self.has_uncertain_inputs=True
self.X_uncertainty = X_uncertainty
GP_regression.__init__(self, X, Y, kernel=kernel, normalize_X=normalize_X, normalize_Y=normalize_Y)
self.trYYT = np.sum(np.square(self.Y))
#normalise X uncertainty also
if self.has_uncertain_inputs:
self.X_uncertainty /= np.square(self._Xstd)
def _computations(self):
# TODO find routine to multiply triangular matrices
#TODO: slices for psi statistics (easy enough)
# kernel computations, using BGPLVM notation
self.Kmm = self.kern.K(self.Z)
if self.has_uncertain_inputs:
self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
self.psi2_beta_scaled = (self.psi2*(self.beta/self.scale_factor**2)).sum(0)
else:
self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices).sum()
self.psi1 = self.kern.K(self.Z,self.X)
#self.psi2 = np.dot(self.psi1,self.psi1.T)
#self.psi2 = self.psi1.T[:,:,None]*self.psi1.T[:,None,:]
tmp = self.psi1/(self.scale_factor/np.sqrt(self.beta))
self.psi2_beta_scaled = np.dot(tmp,tmp.T)
sf = self.scale_factor
sf2 = sf**2
self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)#+np.eye(self.M)*1e-3)
self.V = (self.beta/self.scale_factor)*self.Y
self.A = mdot(self.Lmi, self.psi2_beta_scaled, self.Lmi.T)
self.B = np.eye(self.M)/sf2 + self.A
self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
self.psi1V = np.dot(self.psi1, self.V)
self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
self.C = mdot(self.Lmi.T, self.Bi, self.Lmi)
self.E = mdot(self.C, self.psi1VVpsi1/sf2, self.C.T)
# Compute dL_dpsi
self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
self.dL_dpsi1 = mdot(self.V, self.psi1V.T,self.C).T
self.dL_dpsi2 = 0.5 * self.beta * self.D * self.Kmmi[None,:,:] # dB
self.dL_dpsi2 += - 0.5 * self.beta/sf2 * self.D * self.C[None,:,:] # dC
self.dL_dpsi2 += - 0.5 * self.beta * self.E[None,:,:] # dD
# Compute dL_dKmm
self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi)*sf2 # dB
self.dL_dKmm += -0.5 * self.D * (- self.C/sf2 - 2.*mdot(self.C, self.psi2_beta_scaled, self.Kmmi) + self.Kmmi) # dC
self.dL_dKmm += np.dot(np.dot(self.E*sf2, self.psi2_beta_scaled) - np.dot(self.C, self.psi1VVpsi1), self.Kmmi) + 0.5*self.E # dD
def _set_params(self, p):
self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
self.beta = p[self.M*self.Q]
self.kern._set_params(p[self.Z.size + 1:])
self._computations()
def _get_params(self):
return np.hstack([self.Z.flatten(),self.beta,self.kern._get_params_transformed()])
def _get_param_names(self):
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + ['noise_precision']+self.kern._get_param_names_transformed()
def log_likelihood(self):
""" Compute the (lower bound on the) log marginal likelihood """
sf2 = self.scale_factor**2
A = -0.5*self.N*self.D*(np.log(2.*np.pi) - np.log(self.beta)) -0.5*self.beta*self.trYYT
B = -0.5*self.D*(self.beta*self.psi0-np.trace(self.A)*sf2)
C = -0.5*self.D * (self.B_logdet + self.M*np.log(sf2))
D = +0.5*np.sum(self.psi1VVpsi1 * self.C)
return A+B+C+D
def _log_likelihood_gradients(self):
return np.hstack([self.dL_dZ().flatten(), self.dL_dbeta(), self.dL_dtheta()])
def dL_dbeta(self):
"""
Compute the gradient of the log likelihood wrt beta.
"""
#TODO: suport heteroscedatic noise
sf2 = self.scale_factor**2
dA_dbeta = 0.5 * self.N*self.D/self.beta - 0.5 * self.trYYT
dB_dbeta = - 0.5 * self.D * (self.psi0 - np.trace(self.A)/self.beta*sf2)
dC_dbeta = - 0.5 * self.D * np.sum(self.Bi*self.A)/self.beta
dD_dbeta = np.sum((self.C - 0.5 * mdot(self.C,self.psi2_beta_scaled,self.C) ) * self.psi1VVpsi1 )/self.beta
return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta)
def dL_dtheta(self):
"""
Compute and return the derivative of the log marginal likelihood wrt the parameters of the kernel
"""
dL_dtheta = self.kern.dK_dtheta(self.dL_dKmm,self.Z)
if self.has_uncertain_inputs:
dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z,self.X,self.X_uncertainty)
dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2,self.dL_dpsi1.T, self.Z,self.X, self.X_uncertainty) # for multiple_beta, dL_dpsi2 will be a different shape
else:
#re-cast computations in psi2 back to psi1:
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2.sum(0),self.psi1)
dL_dtheta += self.kern.dK_dtheta(dL_dpsi1,self.Z,self.X)
dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X)
return dL_dtheta
def dL_dZ(self):
"""
The derivative of the bound wrt the inducing inputs Z
"""
dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
if self.has_uncertain_inputs:
dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1,self.Z,self.X, self.X_uncertainty)
dL_dZ += 2.*self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # 'stripes'
else:
#re-cast computations in psi2 back to psi1:
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2.sum(0),self.psi1)
dL_dZ += self.kern.dK_dX(dL_dpsi1,self.Z,self.X)
return dL_dZ
def _raw_predict(self, Xnew, slices, full_cov=False):
"""Internal helper function for making predictions, does not account for normalisation"""
Kx = self.kern.K(self.Z, Xnew)
mu = mdot(Kx.T, self.C/self.scale_factor, self.psi1V)
if full_cov:
Kxx = self.kern.K(Xnew)
var = Kxx - mdot(Kx.T, (self.Kmmi - self.C/self.scale_factor**2), Kx) + np.eye(Xnew.shape[0])/self.beta # TODO: This beta doesn't belong here in the EP case.
else:
Kxx = self.kern.Kdiag(Xnew)
var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.C/self.scale_factor**2, Kx),0) + 1./self.beta # TODO: This beta doesn't belong here in the EP case.
return mu,var
def plot(self, *args, **kwargs):
"""
Plot the fitted model: just call the GP_regression plot function and then add inducing inputs
"""
GP_regression.plot(self,*args,**kwargs)
if self.Q==1:
pb.plot(self.Z,self.Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
if self.has_uncertain_inputs:
pb.errorbar(self.X[:,0], pb.ylim()[0]+np.zeros(self.N), xerr=2*np.sqrt(self.X_uncertainty.flatten()))
if self.Q==2:
pb.plot(self.Z[:,0],self.Z[:,1],'wo')