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Nicolo Fusi
commit
2abba8cf14
59 changed files with 2845 additions and 1608 deletions
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# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
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# Copyright (c) 2012, James Hensman
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# Licensed under the BSD 3-clause license (see LICENSE.txt)
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import numpy as np
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import pylab as pb
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from ..util.linalg import mdot, jitchol, chol_inv, pdinv
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from ..util.plot import gpplot
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from sparse_GP import sparse_GP
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from .. import likelihoods
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from .. import kern
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from ..inference.likelihoods import likelihood
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from GP_regression import GP_regression
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#Still TODO:
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# make use of slices properly (kernel can now do this)
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# enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array)
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class sparse_GP_regression(GP_regression):
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class sparse_GP_regression(sparse_GP):
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"""
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Variational sparse GP model (Regression)
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Gaussian Process model for regression
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This is a thin wrapper around the GP class, with a set of sensible defalts
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:param X: input observations
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:param Y: observed values
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:param kernel: a GPy kernel, defaults to rbf+white
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:param normalize_X: whether to normalize the input data before computing (predictions will be in original scales)
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:type normalize_X: False|True
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:param normalize_Y: whether to normalize the input data before computing (predictions will be in original scales)
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:type normalize_Y: False|True
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:param Xslices: how the X,Y data co-vary in the kernel (i.e. which "outputs" they correspond to). See (link:slicing)
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:rtype: model object
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.. Note:: Multiple independent outputs are allowed using columns of Y
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:param X: inputs
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:type X: np.ndarray (N x Q)
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:param Y: observed data
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:type Y: np.ndarray of observations (N x D)
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:param kernel : the kernel/covariance function. See link kernels
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:type kernel: a GPy kernel
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:param Z: inducing inputs (optional, see note)
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:type Z: np.ndarray (M x Q) | None
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:param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance)
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:type X_uncertainty: np.ndarray (N x Q) | None
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:param Zslices: slices for the inducing inputs (see slicing TODO: link)
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:param M : Number of inducing points (optional, default 10. Ignored if Z is not None)
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:type M: int
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:param beta: noise precision. TODO> ignore beta if doing EP
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:type beta: float
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:param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales)
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:type normalize_(X|Y): bool
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"""
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def __init__(self,X,Y,kernel=None, X_uncertainty=None, beta=100., Z=None,Zslices=None,M=10,normalize_X=False,normalize_Y=False):
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self.scale_factor = 100.0
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self.beta = beta
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def __init__(self,X,Y,kernel=None,normalize_X=False,normalize_Y=False, Xslices=None,Z=None, M=10):
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#kern defaults to rbf
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if kernel is None:
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kernel = kern.rbf(X.shape[1]) + kern.white(X.shape[1],1e-3)
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#Z defaults to a subset of the data
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if Z is None:
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self.Z = np.random.permutation(X.copy())[:M]
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self.M = M
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Z = np.random.permutation(X.copy())[:M]
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else:
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assert Z.shape[1]==X.shape[1]
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self.Z = Z
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self.M = Z.shape[0]
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if X_uncertainty is None:
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self.has_uncertain_inputs=False
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else:
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assert X_uncertainty.shape==X.shape
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self.has_uncertain_inputs=True
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self.X_uncertainty = X_uncertainty
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GP_regression.__init__(self, X, Y, kernel=kernel, normalize_X=normalize_X, normalize_Y=normalize_Y)
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self.trYYT = np.sum(np.square(self.Y))
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#likelihood defaults to Gaussian
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likelihood = likelihoods.Gaussian(Y,normalize=normalize_Y)
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#normalise X uncertainty also
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if self.has_uncertain_inputs:
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self.X_uncertainty /= np.square(self._Xstd)
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def _computations(self):
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# TODO find routine to multiply triangular matrices
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#TODO: slices for psi statistics (easy enough)
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# kernel computations, using BGPLVM notation
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self.Kmm = self.kern.K(self.Z)
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if self.has_uncertain_inputs:
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self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
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self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
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self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
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self.psi2_beta_scaled = (self.psi2*(self.beta/self.scale_factor**2)).sum(0)
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else:
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self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices).sum()
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self.psi1 = self.kern.K(self.Z,self.X)
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#self.psi2 = np.dot(self.psi1,self.psi1.T)
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#self.psi2 = self.psi1.T[:,:,None]*self.psi1.T[:,None,:]
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tmp = self.psi1/(self.scale_factor/np.sqrt(self.beta))
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self.psi2_beta_scaled = np.dot(tmp,tmp.T)
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sf = self.scale_factor
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sf2 = sf**2
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self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)#+np.eye(self.M)*1e-3)
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self.V = (self.beta/self.scale_factor)*self.Y
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self.A = mdot(self.Lmi, self.psi2_beta_scaled, self.Lmi.T)
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self.B = np.eye(self.M)/sf2 + self.A
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self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
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self.psi1V = np.dot(self.psi1, self.V)
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self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
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self.C = mdot(self.Lmi.T, self.Bi, self.Lmi)
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self.E = mdot(self.C, self.psi1VVpsi1/sf2, self.C.T)
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# Compute dL_dpsi
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self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
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self.dL_dpsi1 = mdot(self.V, self.psi1V.T,self.C).T
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self.dL_dpsi2 = 0.5 * self.beta * self.D * self.Kmmi[None,:,:] # dB
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self.dL_dpsi2 += - 0.5 * self.beta/sf2 * self.D * self.C[None,:,:] # dC
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self.dL_dpsi2 += - 0.5 * self.beta * self.E[None,:,:] # dD
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# Compute dL_dKmm
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self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi)*sf2 # dB
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self.dL_dKmm += -0.5 * self.D * (- self.C/sf2 - 2.*mdot(self.C, self.psi2_beta_scaled, self.Kmmi) + self.Kmmi) # dC
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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
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def _set_params(self, p):
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self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
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self.beta = p[self.M*self.Q]
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self.kern._set_params(p[self.Z.size + 1:])
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self._computations()
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def _get_params(self):
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return np.hstack([self.Z.flatten(),self.beta,self.kern._get_params_transformed()])
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def _get_param_names(self):
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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()
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def log_likelihood(self):
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""" Compute the (lower bound on the) log marginal likelihood """
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sf2 = self.scale_factor**2
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A = -0.5*self.N*self.D*(np.log(2.*np.pi) - np.log(self.beta)) -0.5*self.beta*self.trYYT
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B = -0.5*self.D*(self.beta*self.psi0-np.trace(self.A)*sf2)
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C = -0.5*self.D * (self.B_logdet + self.M*np.log(sf2))
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D = +0.5*np.sum(self.psi1VVpsi1 * self.C)
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return A+B+C+D
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def _log_likelihood_gradients(self):
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return np.hstack([self.dL_dZ().flatten(), self.dL_dbeta(), self.dL_dtheta()])
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def dL_dbeta(self):
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"""
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Compute the gradient of the log likelihood wrt beta.
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"""
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#TODO: suport heteroscedatic noise
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sf2 = self.scale_factor**2
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dA_dbeta = 0.5 * self.N*self.D/self.beta - 0.5 * self.trYYT
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dB_dbeta = - 0.5 * self.D * (self.psi0 - np.trace(self.A)/self.beta*sf2)
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dC_dbeta = - 0.5 * self.D * np.sum(self.Bi*self.A)/self.beta
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dD_dbeta = np.sum((self.C - 0.5 * mdot(self.C,self.psi2_beta_scaled,self.C) ) * self.psi1VVpsi1 )/self.beta
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return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta)
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def dL_dtheta(self):
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"""
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Compute and return the derivative of the log marginal likelihood wrt the parameters of the kernel
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"""
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dL_dtheta = self.kern.dK_dtheta(self.dL_dKmm,self.Z)
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if self.has_uncertain_inputs:
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dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z,self.X,self.X_uncertainty)
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dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
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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
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else:
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#re-cast computations in psi2 back to psi1:
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dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2.sum(0),self.psi1)
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dL_dtheta += self.kern.dK_dtheta(dL_dpsi1,self.Z,self.X)
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dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X)
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return dL_dtheta
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def dL_dZ(self):
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"""
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The derivative of the bound wrt the inducing inputs Z
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"""
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dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
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if self.has_uncertain_inputs:
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dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1,self.Z,self.X, self.X_uncertainty)
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dL_dZ += 2.*self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # 'stripes'
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else:
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#re-cast computations in psi2 back to psi1:
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dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2.sum(0),self.psi1)
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dL_dZ += self.kern.dK_dX(dL_dpsi1,self.Z,self.X)
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return dL_dZ
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def _raw_predict(self, Xnew, slices, full_cov=False):
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"""Internal helper function for making predictions, does not account for normalisation"""
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Kx = self.kern.K(self.Z, Xnew)
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mu = mdot(Kx.T, self.C/self.scale_factor, self.psi1V)
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if full_cov:
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Kxx = self.kern.K(Xnew)
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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.
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else:
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Kxx = self.kern.Kdiag(Xnew)
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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.
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return mu,var
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def plot(self, *args, **kwargs):
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"""
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Plot the fitted model: just call the GP_regression plot function and then add inducing inputs
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"""
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GP_regression.plot(self,*args,**kwargs)
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if self.Q==1:
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pb.plot(self.Z,self.Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
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if self.has_uncertain_inputs:
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pb.errorbar(self.X[:,0], pb.ylim()[0]+np.zeros(self.N), xerr=2*np.sqrt(self.X_uncertainty.flatten()))
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if self.Q==2:
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pb.plot(self.Z[:,0],self.Z[:,1],'wo')
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sparse_GP.__init__(self, X, likelihood, kernel, Z, normalize_X=normalize_X, Xslices=Xslices)
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