New FITC model and other stuff

GPy/core/FITC.py

@@ -0,0 +1,312 @@
+# 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, tdot, symmetrify,pdinv
+from ..util.plot import gpplot
+from .. import kern
+from scipy import stats, linalg
+#from ..core import sparse_GP
+from gp_base import GPBase
+
+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 FITC(GPBase):
+    """
+    sparse FITC approximation
+
+    :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.kern.kern instance
+    :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, normalize_X=False):
+        GPBase.__init__(self, X, likelihood, kernel, normalize_X=normalize_X)
+
+        self.Z = Z
+        self.M = Z.shape[0]
+        self.likelihood = likelihood
+
+        X_variance = None
+        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
+
+        if normalize_X:
+            self.Z = (self.Z.copy() - self._Xmean) / self._Xstd
+
+        # normalize X uncertainty also
+        if self.has_uncertain_inputs:
+            self.X_variance /= np.square(self._Xstd)
+
+    def _set_params(self, p):
+        self.Z = p[:self.M * self.input_dim].reshape(self.M, self.input_dim)
+        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()
+        self._computations()
+
+    def _get_params(self):
+        return np.hstack([self.Z.flatten(), self.kern._get_params_transformed(), self.likelihood._get_params()])
+
+    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])],[])\
+            + self.kern._get_param_names_transformed() + self.likelihood._get_param_names()
+
+
+    def update_likelihood_approximation(self):
+        """
+        Approximates a non-Gaussian likelihood using Expectation Propagation
+
+        For a Gaussian likelihood, no iteration is required:
+        this function does nothing
+        """
+        if self.has_uncertain_inputs:
+            raise NotImplementedError, "FITC approximation not implemented for uncertain inputs"
+        else:
+            self.likelihood.fit_FITC(self.Kmm,self.psi1,self.psi0)
+            self._set_params(self._get_params()) # update the GP
+
+    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):
+        #factor Kmm
+        self.Lm = jitchol(self.Kmm)
+        self.Lmi,info = linalg.lapack.flapack.dtrtrs(self.Lm,np.eye(self.M),lower=1)
+        Lmipsi1 = np.dot(self.Lmi,self.psi1)
+        self.Qnn = np.dot(Lmipsi1.T,Lmipsi1).copy()
+        self.Diag0 = self.psi0 - np.diag(self.Qnn)
+        self.beta_star = self.likelihood.precision/(1. + self.likelihood.precision*self.Diag0[:,None]) #Includes Diag0 in the precision
+        self.V_star = self.beta_star * self.likelihood.Y
+
+        # The rather complex computations of self.A
+        if self.has_uncertain_inputs:
+                raise NotImplementedError
+        else:
+            if self.likelihood.is_heteroscedastic:
+                assert self.likelihood.D == 1
+            tmp = self.psi1 * (np.sqrt(self.beta_star.flatten().reshape(1, self.N)))
+            tmp, _ = linalg.lapack.flapack.dtrtrs(self.Lm, np.asfortranarray(tmp), lower=1)
+            self.A = tdot(tmp)
+
+        # factor B
+        self.B = np.eye(self.M) + self.A
+        self.LB = jitchol(self.B)
+        self.LBi = chol_inv(self.LB)
+        self.psi1V = np.dot(self.psi1, self.V_star)
+
+        Lmi_psi1V, info = linalg.lapack.flapack.dtrtrs(self.Lm, np.asfortranarray(self.psi1V), lower=1, trans=0)
+        self._LBi_Lmi_psi1V, _ = linalg.lapack.flapack.dtrtrs(self.LB, np.asfortranarray(Lmi_psi1V), lower=1, trans=0)
+
+        Kmmipsi1 = np.dot(self.Lmi.T,Lmipsi1)
+        b_psi1_Ki = self.beta_star * Kmmipsi1.T
+        Ki_pbp_Ki = np.dot(Kmmipsi1,b_psi1_Ki)
+        Kmmi = np.dot(self.Lmi.T,self.Lmi)
+        LBiLmi = np.dot(self.LBi,self.Lmi)
+        LBL_inv = np.dot(LBiLmi.T,LBiLmi)
+        VVT = np.outer(self.V_star,self.V_star)
+        VV_p_Ki = np.dot(VVT,Kmmipsi1.T)
+        Ki_pVVp_Ki = np.dot(Kmmipsi1,VV_p_Ki)
+        psi1beta = self.psi1*self.beta_star.T
+        H = self.Kmm + mdot(self.psi1,psi1beta.T)
+        LH = jitchol(H)
+        LHi = chol_inv(LH)
+        Hi = np.dot(LHi.T,LHi)
+
+        betapsi1TLmiLBi = np.dot(psi1beta.T,LBiLmi.T)
+        alpha = np.array([np.dot(a.T,a) for a in betapsi1TLmiLBi])[:,None]
+        gamma_1 = mdot(VVT,self.psi1.T,Hi)
+        pHip = mdot(self.psi1.T,Hi,self.psi1)
+        gamma_2 = mdot(self.beta_star*pHip,self.V_star)
+        gamma_3 = self.V_star * gamma_2
+
+        self._dL_dpsi0 = -0.5 * self.beta_star#dA_dpsi0: logdet(self.beta_star)
+        self._dL_dpsi0 += .5 * self.V_star**2 #dA_psi0: yT*beta_star*y
+        self._dL_dpsi0 += .5 *alpha #dC_dpsi0
+        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
+
+        self._dL_dpsi1 = b_psi1_Ki.copy() #dA_dpsi1: logdet(self.beta_star)
+        self._dL_dpsi1 += -np.dot(psi1beta.T,LBL_inv) #dC_dpsi1
+        self._dL_dpsi1 += gamma_1 - mdot(psi1beta.T,Hi,self.psi1,gamma_1) #dD_dpsi1
+
+        self._dL_dKmm = -0.5 * np.dot(Kmmipsi1,b_psi1_Ki) #dA_dKmm: logdet(self.beta_star)
+        self._dL_dKmm += .5*(LBL_inv - Kmmi) + mdot(LBL_inv,psi1beta,Kmmipsi1.T) #dC_dKmm
+        self._dL_dKmm += -.5 * mdot(Hi,self.psi1,gamma_1) #dD_dKmm
+
+        self._dpsi1_dtheta = 0
+        self._dpsi1_dX = 0
+        self._dKmm_dtheta = 0
+        self._dKmm_dX = 0
+
+        self._dpsi1_dX_jkj = 0
+        self._dpsi1_dtheta_jkj = 0
+
+        for i,V_n,alpha_n,gamma_n,gamma_k in zip(range(self.N),self.V_star,alpha,gamma_2,gamma_3):
+            K_pp_K = np.dot(Kmmipsi1[:,i:(i+1)],Kmmipsi1[:,i:(i+1)].T)
+
+            #Diag_dpsi1 = Diag_dA_dpsi1: yT*beta_star*y + Diag_dC_dpsi1 +Diag_dD_dpsi1
+            _dpsi1 = (-V_n**2 - alpha_n + 2.*gamma_k - gamma_n**2) * Kmmipsi1.T[i:(i+1),:]
+
+            #Diag_dKmm = Diag_dA_dKmm: yT*beta_star*y +Diag_dC_dKmm +Diag_dD_dKmm
+            _dKmm = .5*(V_n**2 + alpha_n + gamma_n**2 - 2.*gamma_k) * K_pp_K #Diag_dD_dKmm
+
+            self._dpsi1_dtheta += self.kern.dK_dtheta(_dpsi1,self.X[i:i+1,:],self.Z)
+            self._dKmm_dtheta += self.kern.dK_dtheta(_dKmm,self.Z)
+
+            self._dKmm_dX += 2.*self.kern.dK_dX(_dKmm ,self.Z)
+            self._dpsi1_dX += self.kern.dK_dX(_dpsi1.T,self.Z,self.X[i:i+1,:])
+
+        # 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"
+        else:
+            # likelihood is not heterscedatic
+            dbstar_dnoise = self.likelihood.precision * (self.beta_star**2 * self.Diag0[:,None] - self.beta_star)
+            Lmi_psi1 = mdot(self.Lmi,self.psi1)
+            LBiLmipsi1 = np.dot(self.LBi,Lmi_psi1)
+            aux_0 = np.dot(self._LBi_Lmi_psi1V.T,LBiLmipsi1)
+            aux_1 = self.likelihood.Y.T * np.dot(self._LBi_Lmi_psi1V.T,LBiLmipsi1)
+            aux_2 = np.dot(LBiLmipsi1.T,self._LBi_Lmi_psi1V)
+
+            dA_dnoise = 0.5 * self.D * (dbstar_dnoise/self.beta_star).sum() - 0.5 * self.D * np.sum(self.likelihood.Y**2 * dbstar_dnoise)
+            dC_dnoise = -0.5 * np.sum(mdot(self.LBi.T,self.LBi,Lmi_psi1) *  Lmi_psi1 * dbstar_dnoise.T)
+            dC_dnoise = -0.5 * np.sum(mdot(self.LBi.T,self.LBi,Lmi_psi1) *  Lmi_psi1 * dbstar_dnoise.T)
+
+            dD_dnoise_1 =  mdot(self.V_star*LBiLmipsi1.T,LBiLmipsi1*dbstar_dnoise.T*self.likelihood.Y.T)
+            alpha = mdot(LBiLmipsi1,self.V_star)
+            alpha_ = mdot(LBiLmipsi1.T,alpha)
+            dD_dnoise_2 = -0.5 * self.D * np.sum(alpha_**2 * dbstar_dnoise )
+
+            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)
+            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)
+            dD_dnoise = dD_dnoise_1 + dD_dnoise_2
+
+            self.partial_for_likelihood = dA_dnoise + dC_dnoise + dD_dnoise
+
+    def log_likelihood(self):
+        """ Compute the (lower bound on the) log marginal likelihood """
+        A = -0.5 * self.N * self.D * np.log(2.*np.pi) + 0.5 * np.sum(np.log(self.beta_star)) - 0.5 * np.sum(self.V_star * self.likelihood.Y)
+        C = -self.D * (np.sum(np.log(np.diag(self.LB))))
+        D = 0.5 * np.sum(np.square(self._LBi_Lmi_psi1V))
+        return A + C + D
+
+    def _log_likelihood_gradients(self):
+        pass
+        return np.hstack((self.dL_dZ().flatten(), self.dL_dtheta(), self.likelihood._gradients(partial=self.partial_for_likelihood)))
+
+    def dL_dtheta(self):
+        if self.has_uncertain_inputs:
+            raise NotImplementedError, "FITC approximation not implemented for uncertain inputs"
+        else:
+            dL_dtheta = self.kern.dKdiag_dtheta(self._dL_dpsi0,self.X)
+            dL_dtheta += self.kern.dK_dtheta(self._dL_dpsi1,self.X,self.Z)
+            dL_dtheta += self.kern.dK_dtheta(self._dL_dKmm,X=self.Z)
+            dL_dtheta += self._dKmm_dtheta
+            dL_dtheta += self._dpsi1_dtheta
+        return dL_dtheta
+
+    def dL_dZ(self):
+        if self.has_uncertain_inputs:
+            raise NotImplementedError, "FITC approximation not implemented for uncertain inputs"
+        else:
+            dL_dZ = self.kern.dK_dX(self._dL_dpsi1.T,self.Z,self.X)
+            dL_dZ += 2. * self.kern.dK_dX(self._dL_dKmm,X=self.Z)
+            dL_dZ += self._dpsi1_dX
+            dL_dZ += self._dKmm_dX
+        return dL_dZ
+
+    def _raw_predict(self, Xnew, which_parts, full_cov=False):
+
+        if self.likelihood.is_heteroscedastic:
+            Iplus_Dprod_i = 1./(1.+ self.Diag0 * self.likelihood.precision.flatten())
+            self.Diag = self.Diag0 * Iplus_Dprod_i
+            self.P = Iplus_Dprod_i[:,None] * self.psi1.T
+            self.RPT0 = np.dot(self.Lmi,self.psi1)
+            self.L = np.linalg.cholesky(np.eye(self.M) + np.dot(self.RPT0,((1. - Iplus_Dprod_i)/self.Diag0)[:,None]*self.RPT0.T))
+            self.R,info = linalg.flapack.dtrtrs(self.L,self.Lmi,lower=1)
+            self.RPT = np.dot(self.R,self.P.T)
+            self.Sigma = np.diag(self.Diag) + np.dot(self.RPT.T,self.RPT)
+            self.w = self.Diag * self.likelihood.v_tilde
+            self.Gamma = np.dot(self.R.T, np.dot(self.RPT,self.likelihood.v_tilde))
+            self.mu = self.w + np.dot(self.P,self.Gamma)
+
+            """
+            Make a prediction for the generalized FITC model
+
+            Arguments
+            ---------
+            X : Input prediction data - Nx1 numpy array (floats)
+            """
+            # q(u|f) = N(u| R0i*mu_u*f, R0i*C*R0i.T)
+
+            # Ci = I + (RPT0)Di(RPT0).T
+            # C = I - [RPT0] * (D+[RPT0].T*[RPT0])^-1*[RPT0].T
+            #   = I - [RPT0] * (D + self.Qnn)^-1 * [RPT0].T
+            #   = I - [RPT0] * (U*U.T)^-1 * [RPT0].T
+            #   = I - V.T * V
+            U = np.linalg.cholesky(np.diag(self.Diag0) + self.Qnn)
+            V,info = linalg.flapack.dtrtrs(U,self.RPT0.T,lower=1)
+            C = np.eye(self.M) - np.dot(V.T,V)
+            mu_u = np.dot(C,self.RPT0)*(1./self.Diag0[None,:])
+            #self.C = C
+            #self.RPT0 = np.dot(self.R0,self.Knm.T) P0.T
+            #self.mu_u = mu_u
+            #self.U = U
+            # q(u|y) = N(u| R0i*mu_H,R0i*Sigma_H*R0i.T)
+            mu_H = np.dot(mu_u,self.mu)
+            self.mu_H = mu_H
+            Sigma_H = C + np.dot(mu_u,np.dot(self.Sigma,mu_u.T))
+            # q(f_star|y) = N(f_star|mu_star,sigma2_star)
+            Kx = self.kern.K(self.Z, Xnew, which_parts=which_parts)
+            KR0T = np.dot(Kx.T,self.Lmi.T)
+            mu_star = np.dot(KR0T,mu_H)
+            if full_cov:
+                Kxx = self.kern.K(Xnew,which_parts=which_parts)
+                var = Kxx + np.dot(KR0T,np.dot(Sigma_H - np.eye(self.M),KR0T.T))
+            else:
+                Kxx = self.kern.Kdiag(Xnew,which_parts=which_parts)
+                var = (Kxx + np.sum(KR0T.T*np.dot(Sigma_H - np.eye(self.M),KR0T.T),0))[:,None]
+            return mu_star[:,None],var
+        else:
+            raise NotImplementedError, "homoscedastic FITC not implemented"
+            """
+            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) #NOTE this won't work for plotting
+            else:
+                Kxx = self.kern.Kdiag(Xnew)
+                var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.C/self.scale_factor**2, Kx),0)
+            return mu,var[:,None]
+            """

GPy/core/__init__.py

@@ -3,6 +3,7 @@
 
 from GP import GP
 from sparse_GP import sparse_GP
+from FITC import FITC
 from model import *
 from parameterised import *
 import priors

GPy/examples/__init__.py

@@ -4,5 +4,5 @@
 import classification
 import regression
 import dimensionality_reduction
-import non_gaussian
+import non_Gaussian
 import tutorials

GPy/examples/classification.py

@@ -135,3 +135,27 @@ def sparse_crescent_data(inducing=10, seed=default_seed):
     print(m)
     m.plot()
     return m
+
+def FITC_crescent_data(inducing=10, seed=default_seed):
+    """Run a Gaussian process classification on the crescent data. The demonstration calls the basic GP classification model and uses EP to approximate the likelihood.
+
+    :param model_type: type of model to fit ['Full', 'FITC', 'DTC'].
+    :param seed : seed value for data generation.
+    :type seed: int
+    :param inducing : number of inducing variables (only used for 'FITC' or 'DTC').
+    :type inducing: int
+    """
+
+    data = GPy.util.datasets.crescent_data(seed=seed)
+    Y = data['Y']
+    Y[Y.flatten()==-1]=0
+
+    m = GPy.models.FITC_classification(data['X'], Y)
+    m.ensure_default_constraints()
+    m['.*len'] = 10.
+    m.update_likelihood_approximation()
+    m.optimize()
+    print(m)
+    m.plot()
+    return m
+

GPy/models/GP_regression.py

@@ -11,7 +11,7 @@ class GP_regression(GP):
     """
     Gaussian Process model for regression
 
-    This is a thin wrapper around the models.GP class, with a set of sensible defalts
+    This is a thin wrapper around the models.GP class, with a set of sensible defaults
 
     :param X: input observations
     :param Y: observed values

GPy/models/__init__.py

@@ -6,10 +6,9 @@ from GP_regression import GP_regression
 from GP_classification import GP_classification
 from sparse_GP_regression import sparse_GP_regression
 from sparse_GP_classification import sparse_GP_classification
+from FITC_classification import FITC_classification
 from GPLVM import GPLVM
 from warped_GP import warpedGP
 from sparse_GPLVM import sparse_GPLVM
 from Bayesian_GPLVM import Bayesian_GPLVM
 from mrd import MRD
-from generalized_FITC import generalized_FITC
-from FITC import FITC

GPy/models/sparse_GP_classification.py

@@ -7,13 +7,12 @@ from ..core import sparse_GP
 from .. import likelihoods
 from .. import kern
 from ..likelihoods import likelihood
-from GP_regression import GP_regression
 
 class sparse_GP_classification(sparse_GP):
     """
     sparse Gaussian Process model for classification
 
-    This is a thin wrapper around the sparse_GP class, with a set of sensible defalts
+    This is a thin wrapper around the sparse_GP class, with a set of sensible defaults
 
     :param X: input observations
     :param Y: observed values
@@ -25,8 +24,6 @@ class sparse_GP_classification(sparse_GP):
     :type normalize_Y: False|True
     :rtype: model object
 
-    .. Note:: Multiple independent outputs are allowed using columns of Y
-
     """
 
     def __init__(self, X, Y=None, likelihood=None, kernel=None, normalize_X=False, normalize_Y=False, Z=None, M=10):