# 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, trace_dot, tdot from ..util.plot import gpplot from .. import kern from GP import GP from scipy import linalg def backsub_both_sides(L,X): """ Return L^-T * X * L^-1, assumuing X is symmetrical and L is lower cholesky""" tmp,_ = linalg.lapack.flapack.dtrtrs(L,np.asfortranarray(X),lower=1,trans=1) return linalg.lapack.flapack.dtrtrs(L,np.asfortranarray(tmp.T),lower=1,trans=1)[0].T class sparse_GP(GP): """ Variational sparse GP model :param X: inputs :type X: np.ndarray (N x Q) :param likelihood: a likelihood instance, containing the observed data :type likelihood: GPy.likelihood.(Gaussian | EP) :param kernel : the kernel/covariance function. See link kernels :type kernel: a GPy kernel :param X_variance: The uncertainty in the measurements of X (Gaussian variance) :type X_variance: np.ndarray (N x Q) | None :param Z: inducing inputs (optional, see note) :type Z: np.ndarray (M x Q) | None :param M : Number of inducing points (optional, default 10. Ignored if Z is not None) :type M: int :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, likelihood, kernel, Z, X_variance=None, normalize_X=False): self.scale_factor = 100.0# a scaling factor to help keep the algorithm stable self.auto_scale_factor = False self.Z = Z self.M = Z.shape[0] self.likelihood = likelihood if X_variance is None: self.has_uncertain_inputs=False else: assert X_variance.shape==X.shape self.has_uncertain_inputs=True self.X_variance = X_variance GP.__init__(self, X, likelihood, kernel=kernel, normalize_X=normalize_X) #normalize X uncertainty also if self.has_uncertain_inputs: self.X_variance /= np.square(self._Xstd) def _compute_kernel_matrices(self): # 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_variance) self.psi1 = self.kern.psi1(self.Z,self.X, self.X_variance).T self.psi2 = self.kern.psi2(self.Z,self.X, self.X_variance) else: self.psi0 = self.kern.Kdiag(self.X) self.psi1 = self.kern.K(self.Z,self.X) self.psi2 = None def _computations(self): #TODO: find routine to multiply triangular matrices sf = self.scale_factor sf2 = sf**2 #invert Kmm self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm) #The rather complex computations of psi2_beta_scaled and self.A if self.likelihood.is_heteroscedastic: assert self.likelihood.D == 1 #TODO: what if the likelihood is heterscedatic and there are multiple independent outputs? if self.has_uncertain_inputs: self.psi2_beta_scaled = (self.psi2*(self.likelihood.precision.flatten().reshape(self.N,1,1)/sf2)).sum(0) evals, evecs = linalg.eigh(self.psi2_beta_scaled) clipped_evals = np.clip(evals,0.,1e6) # TODO: make clipping configurable if not np.allclose(evals, clipped_evals): print "Warning: clipping posterior eigenvalues" tmp = evecs*np.sqrt(clipped_evals) tmp, _ = linalg.lapack.flapack.dtrtrs(self.Lm,np.asfortranarray(tmp),lower=1) self.A = tdot(tmp) else: tmp = self.psi1*(np.sqrt(self.likelihood.precision.flatten().reshape(1,self.N))/sf) self.psi2_beta_scaled = tdot(tmp) tmp, _ = linalg.lapack.flapack.dtrtrs(self.Lm,np.asfortranarray(tmp),lower=1) self.A = tdot(tmp) else: if self.has_uncertain_inputs: self.psi2_beta_scaled = (self.psi2*(self.likelihood.precision/sf2)).sum(0) evals, evecs = linalg.eigh(self.psi2_beta_scaled) clipped_evals = np.clip(evals,0.,1e6) # TODO: make clipping configurable if not np.allclose(evals, clipped_evals): print "Warning: clipping posterior eigenvalues" tmp = evecs*np.sqrt(clipped_evals) self.psi2_beta_scaled = tdot(tmp) tmp, _ = linalg.lapack.flapack.dtrtrs(self.Lm,np.asfortranarray(tmp),lower=1) self.A = tdot(tmp) else: tmp = self.psi1*(np.sqrt(self.likelihood.precision)/sf) self.psi2_beta_scaled = tdot(tmp) tmp, _ = linalg.lapack.flapack.dtrtrs(self.Lm,np.asfortranarray(tmp),lower=1) self.A = tdot(tmp) #invert B and compute C. C is the posterior covariance of u self.B = np.eye(self.M)/sf2 + self.A self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B) tmp = linalg.lapack.flapack.dtrtrs(self.Lm,np.asfortranarray(self.Bi),lower=1,trans=1)[0] self.C = linalg.lapack.flapack.dtrtrs(self.Lm,np.asfortranarray(tmp.T),lower=1,trans=1)[0] self.V = (self.likelihood.precision/self.scale_factor)*self.likelihood.Y self.psi1V = np.dot(self.psi1, self.V) #back substutue C into psi1V tmp,info1 = linalg.lapack.flapack.dtrtrs(self.Lm,np.asfortranarray(self.psi1V),lower=1,trans=0) self._P = tdot(tmp) tmp,info2 = linalg.lapack.flapack.dpotrs(self.LB,tmp,lower=1) self.Cpsi1V,info3 = linalg.lapack.flapack.dtrtrs(self.Lm,tmp,lower=1,trans=1) #self.Cpsi1V = np.dot(self.C,self.psi1V) self.Cpsi1VVpsi1 = np.dot(self.Cpsi1V,self.psi1V.T) #TODO: this dot can be eliminated self.E = tdot(self.Cpsi1V/sf) # Compute dL_dpsi # FIXME: this is untested for the heterscedastic + uncertin inputs case self.dL_dpsi0 = - 0.5 * self.D * (self.likelihood.precision * np.ones([self.N,1])).flatten() self.dL_dpsi1 = np.dot(self.Cpsi1V,self.V.T) if self.likelihood.is_heteroscedastic: if self.has_uncertain_inputs: #self.dL_dpsi2 = 0.5 * self.likelihood.precision[:,None,None] * self.D * self.Kmmi[None,:,:] # dB #self.dL_dpsi2 += - 0.5 * self.likelihood.precision[:,None,None]/sf2 * self.D * self.C[None,:,:] # dC #self.dL_dpsi2 += - 0.5 * self.likelihood.precision[:,None,None]* self.E[None,:,:] # dD self.dL_dpsi2 = 0.5*self.likelihood.precision[:,None,None]*(self.D*(self.Kmmi - self.C/sf2) -self.E)[None,:,:] else: #self.dL_dpsi1 += mdot(self.Kmmi,self.psi1*self.likelihood.precision.flatten().reshape(1,self.N)) #dB #self.dL_dpsi1 += -mdot(self.C,self.psi1*self.likelihood.precision.flatten().reshape(1,self.N)/sf2) #dC #self.dL_dpsi1 += -mdot(self.E,self.psi1*self.likelihood.precision.flatten().reshape(1,self.N)) #dD self.dL_dpsi1 += np.dot(self.Kmmi - self.C/sf2 -self.E,self.psi1*self.likelihood.precision.reshape(1,self.N)) self.dL_dpsi2 = None else: self.dL_dpsi2 = 0.5*self.likelihood.precision*(self.D*(self.Kmmi - self.C/sf2) -self.E) if self.has_uncertain_inputs: #repeat for each of the N psi_2 matrices self.dL_dpsi2 = np.repeat(self.dL_dpsi2[None,:,:],self.N,axis=0) else: #subsume back into psi1 (==Kmn) self.dL_dpsi1 += 2.*np.dot(self.dL_dpsi2,self.psi1) self.dL_dpsi2 = None # 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) - self.Cpsi1VVpsi1, self.Kmmi) + 0.5*self.E # dD tmp = linalg.lapack.flapack.dtrtrs(self.Lm,np.asfortranarray(self.B),lower=1,trans=1)[0] self.dL_dKmm = -0.5*self.D*sf2*linalg.lapack.flapack.dtrtrs(self.Lm,np.asfortranarray(tmp.T),lower=1,trans=1)[0] tmp = np.dot(self.D*self.C + self.E*sf2,self.psi2_beta_scaled) - self.Cpsi1VVpsi1 tmp = linalg.lapack.flapack.dpotrs(self.Lm,np.asfortranarray(tmp.T),lower=1)[0].T self.dL_dKmm += 0.5*(self.D*self.C/sf2 + self.E) +tmp # d(C+D) #the partial derivative vector for the likelihood if self.likelihood.Nparams ==0: #save computation here. self.partial_for_likelihood = None elif self.likelihood.is_heteroscedastic: raise NotImplementedError, "heteroscedatic derivates not implemented" #self.partial_for_likelihood = - 0.5 * self.D*self.likelihood.precision + 0.5 * (self.likelihood.Y**2).sum(1)*self.likelihood.precision**2 #dA #self.partial_for_likelihood += 0.5 * self.D * (self.psi0*self.likelihood.precision**2 - (self.psi2*self.Kmmi[None,:,:]*self.likelihood.precision[:,None,None]**2).sum(1).sum(1)/sf2) #dB #self.partial_for_likelihood += 0.5 * self.D * np.sum(self.Bi*self.A)*self.likelihood.precision #dC #self.partial_for_likelihood += -np.diag(np.dot((self.C - 0.5 * mdot(self.C,self.psi2_beta_scaled,self.C) ) , self.psi1VVpsi1 ))*self.likelihood.precision #dD else: #likelihood is not heterscedatic self.partial_for_likelihood = - 0.5 * self.N*self.D*self.likelihood.precision + 0.5 * self.likelihood.trYYT*self.likelihood.precision**2 self.partial_for_likelihood += 0.5 * self.D * (self.psi0.sum()*self.likelihood.precision**2 - np.trace(self.A)*self.likelihood.precision*sf2) self.partial_for_likelihood += 0.5 * self.D * trace_dot(self.Bi,self.A)*self.likelihood.precision self.partial_for_likelihood += self.likelihood.precision*(0.5*trace_dot(self.psi2_beta_scaled,self.E*sf2) - np.trace(self.Cpsi1VVpsi1)) def log_likelihood(self): """ Compute the (lower bound on the) log marginal likelihood """ sf2 = self.scale_factor**2 if self.likelihood.is_heteroscedastic: A = -0.5*self.N*self.D*np.log(2.*np.pi) +0.5*np.sum(np.log(self.likelihood.precision)) -0.5*np.sum(self.V*self.likelihood.Y) B = -0.5*self.D*(np.sum(self.likelihood.precision.flatten()*self.psi0) - np.trace(self.A)*sf2) else: A = -0.5*self.N*self.D*(np.log(2.*np.pi) + np.log(self.likelihood._variance)) -0.5*self.likelihood.precision*self.likelihood.trYYT B = -0.5*self.D*(np.sum(self.likelihood.precision*self.psi0) - np.trace(self.A)*sf2) C = -0.5*self.D * (self.B_logdet + self.M*np.log(sf2)) D = 0.5*np.trace(self.Cpsi1VVpsi1) return A+B+C+D def _set_params(self, p): self.Z = p[:self.M*self.Q].reshape(self.M, self.Q) self.kern._set_params(p[self.Z.size:self.Z.size+self.kern.Nparam]) self.likelihood._set_params(p[self.Z.size+self.kern.Nparam:]) self._compute_kernel_matrices() #if self.auto_scale_factor: # self.scale_factor = np.sqrt(self.psi2.sum(0).mean()*self.likelihood.precision) if self.auto_scale_factor: if self.likelihood.is_heteroscedastic: self.scale_factor = max(100,np.sqrt(self.psi2_beta_scaled.sum(0).mean())) else: self.scale_factor = np.sqrt(self.psi2.sum(0).mean()*self.likelihood.precision) 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 j in range(self.Z.shape[1])] for i in range(self.Z.shape[0])],[]) + GP._get_param_names(self) def update_likelihood_approximation(self): """ Approximates a non-gaussian likelihood using Expectation Propagation For a Gaussian (or direct: TODO) likelihood, no iteration is required: this function does nothing """ if self.has_uncertain_inputs: raise NotImplementedError, "EP approximation not implemented for uncertain inputs" else: self.likelihood.fit_DTC(self.Kmm,self.psi1) #self.likelihood.fit_FITC(self.Kmm,self.psi1,self.psi0) self._set_params(self._get_params()) # update the GP def _log_likelihood_gradients(self): return np.hstack((self.dL_dZ().flatten(), self.dL_dtheta(), self.likelihood._gradients(partial=self.partial_for_likelihood))) 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_variance) dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_variance) dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2, self.Z,self.X, self.X_variance) else: dL_dtheta += self.kern.dK_dtheta(self.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_variance) dL_dZ += self.kern.dpsi2_dZ(self.dL_dpsi2, self.Z, self.X, self.X_variance) else: dL_dZ += self.kern.dK_dX(self.dL_dpsi1,self.Z,self.X) return dL_dZ def _raw_predict(self, Xnew, which_parts='all', full_cov=False): """Internal helper function for making predictions, does not account for normalization""" 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,which_parts=which_parts) var = Kxx - mdot(Kx.T, (self.Kmmi - self.C/self.scale_factor**2), Kx) #NOTE this won't work for plotting else: Kxx = self.kern.Kdiag(Xnew,which_parts=which_parts) var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.C/self.scale_factor**2, Kx),0) return mu,var[:,None]