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refactoring files added
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Max Zwiessele
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252
GPy/models/fitc.py
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252
GPy/models/fitc.py
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# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
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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, tdot, symmetrify, pdinv
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from ..util.plot import gpplot
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from .. import kern
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from scipy import stats, linalg
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from GPy.core.sparse_gp import SparseGP
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def backsub_both_sides(L, X):
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""" Return L^-T * X * L^-1, assumuing X is symmetrical and L is lower cholesky"""
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tmp, _ = linalg.lapack.flapack.dtrtrs(L, np.asfortranarray(X), lower=1, trans=1)
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return linalg.lapack.flapack.dtrtrs(L, np.asfortranarray(tmp.T), lower=1, trans=1)[0].T
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class FITC(SparseGP):
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def __init__(self, X, likelihood, kernel, Z, X_variance=None, normalize_X=False):
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super(FITC, self).__init__(X, likelihood, kernel, normalize_X=normalize_X)
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def update_likelihood_approximation(self):
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"""
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Approximates a non-gaussian likelihood using Expectation Propagation
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For a Gaussian (or direct: TODO) likelihood, no iteration is required:
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this function does nothing
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Diag(Knn - Qnn) is added to the noise term to use the tools already implemented in SparseGP.
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The true precison is now 'true_precision' not 'precision'.
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"""
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if self.has_uncertain_inputs:
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raise NotImplementedError, "FITC approximation not implemented for uncertain inputs"
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else:
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self.likelihood.fit_FITC(self.Kmm, self.psi1, self.psi0)
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self._set_params(self._get_params()) # update the GP
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def _computations(self):
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# factor Kmm
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self.Lm = jitchol(self.Kmm)
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self.Lmi, info = linalg.lapack.flapack.dtrtrs(self.Lm, np.eye(self.num_inducing), lower=1)
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Lmipsi1 = np.dot(self.Lmi, self.psi1)
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self.Qnn = np.dot(Lmipsi1.T, Lmipsi1).copy()
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self.Diag0 = self.psi0 - np.diag(self.Qnn)
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self.beta_star = self.likelihood.precision / (1. + self.likelihood.precision * self.Diag0[:, None]) # Includes Diag0 in the precision
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self.V_star = self.beta_star * self.likelihood.Y
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# The rather complex computations of self.A
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if self.has_uncertain_inputs:
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raise NotImplementedError
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else:
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if self.likelihood.is_heteroscedastic:
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assert self.likelihood.input_dim == 1
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tmp = self.psi1 * (np.sqrt(self.beta_star.flatten().reshape(1, self.N)))
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tmp, _ = linalg.lapack.flapack.dtrtrs(self.Lm, np.asfortranarray(tmp), lower=1)
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self.A = tdot(tmp)
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# factor B
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self.B = np.eye(self.num_inducing) + self.A
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self.LB = jitchol(self.B)
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self.LBi = chol_inv(self.LB)
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self.psi1V = np.dot(self.psi1, self.V_star)
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Lmi_psi1V, info = linalg.lapack.flapack.dtrtrs(self.Lm, np.asfortranarray(self.psi1V), lower=1, trans=0)
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self._LBi_Lmi_psi1V, _ = linalg.lapack.flapack.dtrtrs(self.LB, np.asfortranarray(Lmi_psi1V), lower=1, trans=0)
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Kmmipsi1 = np.dot(self.Lmi.T, Lmipsi1)
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b_psi1_Ki = self.beta_star * Kmmipsi1.T
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Ki_pbp_Ki = np.dot(Kmmipsi1, b_psi1_Ki)
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Kmmi = np.dot(self.Lmi.T, self.Lmi)
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LBiLmi = np.dot(self.LBi, self.Lmi)
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LBL_inv = np.dot(LBiLmi.T, LBiLmi)
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VVT = np.outer(self.V_star, self.V_star)
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VV_p_Ki = np.dot(VVT, Kmmipsi1.T)
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Ki_pVVp_Ki = np.dot(Kmmipsi1, VV_p_Ki)
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psi1beta = self.psi1 * self.beta_star.T
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H = self.Kmm + mdot(self.psi1, psi1beta.T)
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LH = jitchol(H)
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LHi = chol_inv(LH)
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Hi = np.dot(LHi.T, LHi)
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betapsi1TLmiLBi = np.dot(psi1beta.T, LBiLmi.T)
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alpha = np.array([np.dot(a.T, a) for a in betapsi1TLmiLBi])[:, None]
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gamma_1 = mdot(VVT, self.psi1.T, Hi)
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pHip = mdot(self.psi1.T, Hi, self.psi1)
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gamma_2 = mdot(self.beta_star * pHip, self.V_star)
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gamma_3 = self.V_star * gamma_2
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self._dL_dpsi0 = -0.5 * self.beta_star # dA_dpsi0: logdet(self.beta_star)
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self._dL_dpsi0 += .5 * self.V_star ** 2 # dA_psi0: yT*beta_star*y
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self._dL_dpsi0 += .5 * alpha # dC_dpsi0
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self._dL_dpsi0 += 0.5 * mdot(self.beta_star * pHip, self.V_star) ** 2 - self.V_star * mdot(self.V_star.T, pHip * self.beta_star).T # dD_dpsi0
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self._dL_dpsi1 = b_psi1_Ki.copy() # dA_dpsi1: logdet(self.beta_star)
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self._dL_dpsi1 += -np.dot(psi1beta.T, LBL_inv) # dC_dpsi1
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self._dL_dpsi1 += gamma_1 - mdot(psi1beta.T, Hi, self.psi1, gamma_1) # dD_dpsi1
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self._dL_dKmm = -0.5 * np.dot(Kmmipsi1, b_psi1_Ki) # dA_dKmm: logdet(self.beta_star)
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self._dL_dKmm += .5 * (LBL_inv - Kmmi) + mdot(LBL_inv, psi1beta, Kmmipsi1.T) # dC_dKmm
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self._dL_dKmm += -.5 * mdot(Hi, self.psi1, gamma_1) # dD_dKmm
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self._dpsi1_dtheta = 0
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self._dpsi1_dX = 0
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self._dKmm_dtheta = 0
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self._dKmm_dX = 0
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self._dpsi1_dX_jkj = 0
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self._dpsi1_dtheta_jkj = 0
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for i, V_n, alpha_n, gamma_n, gamma_k in zip(range(self.N), self.V_star, alpha, gamma_2, gamma_3):
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K_pp_K = np.dot(Kmmipsi1[:, i:(i + 1)], Kmmipsi1[:, i:(i + 1)].T)
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# Diag_dpsi1 = Diag_dA_dpsi1: yT*beta_star*y + Diag_dC_dpsi1 +Diag_dD_dpsi1
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_dpsi1 = (-V_n ** 2 - alpha_n + 2.*gamma_k - gamma_n ** 2) * Kmmipsi1.T[i:(i + 1), :]
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# Diag_dKmm = Diag_dA_dKmm: yT*beta_star*y +Diag_dC_dKmm +Diag_dD_dKmm
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_dKmm = .5 * (V_n ** 2 + alpha_n + gamma_n ** 2 - 2.*gamma_k) * K_pp_K # Diag_dD_dKmm
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self._dpsi1_dtheta += self.kern.dK_dtheta(_dpsi1, self.X[i:i + 1, :], self.Z)
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self._dKmm_dtheta += self.kern.dK_dtheta(_dKmm, self.Z)
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self._dKmm_dX += 2.*self.kern.dK_dX(_dKmm , self.Z)
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self._dpsi1_dX += self.kern.dK_dX(_dpsi1.T, self.Z, self.X[i:i + 1, :])
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# the partial derivative vector for the likelihood
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if self.likelihood.Nparams == 0:
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# save computation here.
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self.partial_for_likelihood = None
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elif self.likelihood.is_heteroscedastic:
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raise NotImplementedError, "heteroscedatic derivates not implemented"
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else:
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# likelihood is not heterscedatic
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dbstar_dnoise = self.likelihood.precision * (self.beta_star ** 2 * self.Diag0[:, None] - self.beta_star)
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Lmi_psi1 = mdot(self.Lmi, self.psi1)
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LBiLmipsi1 = np.dot(self.LBi, Lmi_psi1)
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aux_0 = np.dot(self._LBi_Lmi_psi1V.T, LBiLmipsi1)
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aux_1 = self.likelihood.Y.T * np.dot(self._LBi_Lmi_psi1V.T, LBiLmipsi1)
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aux_2 = np.dot(LBiLmipsi1.T, self._LBi_Lmi_psi1V)
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dA_dnoise = 0.5 * self.input_dim * (dbstar_dnoise / self.beta_star).sum() - 0.5 * self.input_dim * np.sum(self.likelihood.Y ** 2 * dbstar_dnoise)
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dC_dnoise = -0.5 * np.sum(mdot(self.LBi.T, self.LBi, Lmi_psi1) * Lmi_psi1 * dbstar_dnoise.T)
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dC_dnoise = -0.5 * np.sum(mdot(self.LBi.T, self.LBi, Lmi_psi1) * Lmi_psi1 * dbstar_dnoise.T)
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dD_dnoise_1 = mdot(self.V_star * LBiLmipsi1.T, LBiLmipsi1 * dbstar_dnoise.T * self.likelihood.Y.T)
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alpha = mdot(LBiLmipsi1, self.V_star)
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alpha_ = mdot(LBiLmipsi1.T, alpha)
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dD_dnoise_2 = -0.5 * self.input_dim * np.sum(alpha_ ** 2 * dbstar_dnoise)
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dD_dnoise_1 = mdot(self.V_star.T, self.psi1.T, self.Lmi.T, self.LBi.T, self.LBi, self.Lmi, self.psi1, dbstar_dnoise * self.likelihood.Y)
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dD_dnoise_2 = 0.5 * mdot(self.V_star.T, self.psi1.T, Hi, self.psi1, dbstar_dnoise * self.psi1.T, Hi, self.psi1, self.V_star)
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dD_dnoise = dD_dnoise_1 + dD_dnoise_2
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self.partial_for_likelihood = dA_dnoise + dC_dnoise + dD_dnoise
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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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A = -0.5 * self.N * self.input_dim * np.log(2.*np.pi) + 0.5 * np.sum(np.log(self.beta_star)) - 0.5 * np.sum(self.V_star * self.likelihood.Y)
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C = -self.input_dim * (np.sum(np.log(np.diag(self.LB))))
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D = 0.5 * np.sum(np.square(self._LBi_Lmi_psi1V))
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return A + C + D
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def _log_likelihood_gradients(self):
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pass
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return np.hstack((self.dL_dZ().flatten(), self.dL_dtheta(), self.likelihood._gradients(partial=self.partial_for_likelihood)))
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def dL_dtheta(self):
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if self.has_uncertain_inputs:
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raise NotImplementedError, "FITC approximation not implemented for uncertain inputs"
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else:
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dL_dtheta = self.kern.dKdiag_dtheta(self._dL_dpsi0, self.X)
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dL_dtheta += self.kern.dK_dtheta(self._dL_dpsi1, self.X, self.Z)
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dL_dtheta += self.kern.dK_dtheta(self._dL_dKmm, X=self.Z)
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dL_dtheta += self._dKmm_dtheta
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dL_dtheta += self._dpsi1_dtheta
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return dL_dtheta
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def dL_dZ(self):
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if self.has_uncertain_inputs:
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raise NotImplementedError, "FITC approximation not implemented for uncertain inputs"
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else:
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dL_dZ = self.kern.dK_dX(self._dL_dpsi1.T, self.Z, self.X)
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dL_dZ += 2. * self.kern.dK_dX(self._dL_dKmm, X=self.Z)
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dL_dZ += self._dpsi1_dX
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dL_dZ += self._dKmm_dX
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return dL_dZ
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def _raw_predict(self, Xnew, which_parts, full_cov=False):
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if self.likelihood.is_heteroscedastic:
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Iplus_Dprod_i = 1. / (1. + self.Diag0 * self.likelihood.precision.flatten())
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self.Diag = self.Diag0 * Iplus_Dprod_i
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self.P = Iplus_Dprod_i[:, None] * self.psi1.T
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self.RPT0 = np.dot(self.Lmi, self.psi1)
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self.L = np.linalg.cholesky(np.eye(self.num_inducing) + np.dot(self.RPT0, ((1. - Iplus_Dprod_i) / self.Diag0)[:, None] * self.RPT0.T))
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self.R, info = linalg.flapack.dtrtrs(self.L, self.Lmi, lower=1)
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self.RPT = np.dot(self.R, self.P.T)
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self.Sigma = np.diag(self.Diag) + np.dot(self.RPT.T, self.RPT)
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self.w = self.Diag * self.likelihood.v_tilde
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self.Gamma = np.dot(self.R.T, np.dot(self.RPT, self.likelihood.v_tilde))
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self.mu = self.w + np.dot(self.P, self.Gamma)
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"""
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Make a prediction for the generalized FITC model
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Arguments
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---------
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X : Input prediction data - Nx1 numpy array (floats)
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"""
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# q(u|f) = N(u| R0i*mu_u*f, R0i*C*R0i.T)
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# Ci = I + (RPT0)Di(RPT0).T
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# C = I - [RPT0] * (input_dim+[RPT0].T*[RPT0])^-1*[RPT0].T
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# = I - [RPT0] * (input_dim + self.Qnn)^-1 * [RPT0].T
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# = I - [RPT0] * (U*U.T)^-1 * [RPT0].T
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# = I - V.T * V
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U = np.linalg.cholesky(np.diag(self.Diag0) + self.Qnn)
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V, info = linalg.flapack.dtrtrs(U, self.RPT0.T, lower=1)
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C = np.eye(self.num_inducing) - np.dot(V.T, V)
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mu_u = np.dot(C, self.RPT0) * (1. / self.Diag0[None, :])
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# self.C = C
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# self.RPT0 = np.dot(self.R0,self.Knm.T) P0.T
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# self.mu_u = mu_u
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# self.U = U
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# q(u|y) = N(u| R0i*mu_H,R0i*Sigma_H*R0i.T)
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mu_H = np.dot(mu_u, self.mu)
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self.mu_H = mu_H
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Sigma_H = C + np.dot(mu_u, np.dot(self.Sigma, mu_u.T))
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# q(f_star|y) = N(f_star|mu_star,sigma2_star)
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Kx = self.kern.K(self.Z, Xnew, which_parts=which_parts)
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KR0T = np.dot(Kx.T, self.Lmi.T)
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mu_star = np.dot(KR0T, mu_H)
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if full_cov:
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Kxx = self.kern.K(Xnew, which_parts=which_parts)
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var = Kxx + np.dot(KR0T, np.dot(Sigma_H - np.eye(self.num_inducing), KR0T.T))
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else:
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Kxx = self.kern.Kdiag(Xnew, which_parts=which_parts)
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var = (Kxx + np.sum(KR0T.T * np.dot(Sigma_H - np.eye(self.num_inducing), KR0T.T), 0))[:, None]
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return mu_star[:, None], var
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else:
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raise NotImplementedError, "homoscedastic fitc not implemented"
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"""
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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) #NOTE this won't work for plotting
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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)
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return mu,var[:,None]
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"""
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