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No more GP_EP stuff
This commit is contained in:
Ricardo Andrade
parent
6a2e0a1fe5
commit
738ca78dac
5 changed files with 2 additions and 682 deletions
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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 random
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from scipy import stats, linalg
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from .likelihoods import likelihood
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from ..core import model
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from ..util.linalg import pdinv,mdot,jitchol
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from ..util.plot import gpplot
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from .. import kern
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class EP_base:
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"""
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Expectation Propagation.
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This is just the base class for expectation propagation. We'll extend it for full and sparse EP.
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"""
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def __init__(self,likelihood,epsilon=1e-3,powerep=[1.,1.]):
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self.likelihood = likelihood
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self.epsilon = epsilon
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self.eta, self.delta = powerep
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self.jitter = 1e-12
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#Initial values - Likelihood approximation parameters:
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#p(y|f) = t(f|tau_tilde,v_tilde)
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self.restart_EP()
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def restart_EP(self):
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"""
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Set the EP approximation to initial state
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"""
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self.tau_tilde = np.zeros(self.N)
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self.v_tilde = np.zeros(self.N)
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self.mu = np.zeros(self.N)
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class Full(EP_base):
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"""
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:param likelihood: Output's likelihood (e.g. probit)
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:type likelihood: GPy.inference.likelihood instance
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:param K: prior covariance matrix
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:type K: np.ndarray (N x N)
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:param likelihood: Output's likelihood (e.g. probit)
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:type likelihood: GPy.inference.likelihood instance
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:param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
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:param powerep: Power-EP parameters (eta,delta) - 2x1 numpy array (floats)
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"""
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def __init__(self,K,likelihood,*args,**kwargs):
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assert K.shape[0] == K.shape[1]
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self.K = K
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self.N = self.K.shape[0]
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EP_base.__init__(self,likelihood,*args,**kwargs)
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def fit_EP(self,messages=False):
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"""
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The expectation-propagation algorithm.
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For nomenclature see Rasmussen & Williams 2006 (pag. 52-60)
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"""
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#Prior distribution parameters: p(f|X) = N(f|0,K)
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#self.K = self.kernel.K(self.X,self.X)
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#Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma)
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self.mu=np.zeros(self.N)
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self.Sigma=self.K.copy()
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"""
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Initial values - Cavity distribution parameters:
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q_(f|mu_,sigma2_) = Product{q_i(f|mu_i,sigma2_i)}
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sigma_ = 1./tau_
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mu_ = v_/tau_
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"""
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self.tau_ = np.empty(self.N,dtype=np.float64)
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self.v_ = np.empty(self.N,dtype=np.float64)
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#Initial values - Marginal moments
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z = np.empty(self.N,dtype=np.float64)
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self.Z_hat = np.empty(self.N,dtype=np.float64)
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phi = np.empty(self.N,dtype=np.float64)
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mu_hat = np.empty(self.N,dtype=np.float64)
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sigma2_hat = np.empty(self.N,dtype=np.float64)
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#Approximation
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epsilon_np1 = self.epsilon + 1.
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epsilon_np2 = self.epsilon + 1.
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self.iterations = 0
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self.np1 = [self.tau_tilde.copy()]
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self.np2 = [self.v_tilde.copy()]
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while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
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update_order = np.random.permutation(self.N)
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for i in update_order:
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#Cavity distribution parameters
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self.tau_[i] = 1./self.Sigma[i,i] - self.eta*self.tau_tilde[i]
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self.v_[i] = self.mu[i]/self.Sigma[i,i] - self.eta*self.v_tilde[i]
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#Marginal moments
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self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.likelihood.moments_match(i,self.tau_[i],self.v_[i])
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#Site parameters update
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Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./self.Sigma[i,i])
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Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - self.mu[i]/self.Sigma[i,i])
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self.tau_tilde[i] = self.tau_tilde[i] + Delta_tau
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self.v_tilde[i] = self.v_tilde[i] + Delta_v
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#Posterior distribution parameters update
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si=self.Sigma[:,i].reshape(self.N,1)
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self.Sigma = self.Sigma - Delta_tau/(1.+ Delta_tau*self.Sigma[i,i])*np.dot(si,si.T)
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self.mu = np.dot(self.Sigma,self.v_tilde)
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self.iterations += 1
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#Sigma recomptutation with Cholesky decompositon
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Sroot_tilde_K = np.sqrt(self.tau_tilde)[:,None]*(self.K)
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B = np.eye(self.N) + np.sqrt(self.tau_tilde)[None,:]*Sroot_tilde_K
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L = jitchol(B)
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V,info = linalg.flapack.dtrtrs(L,Sroot_tilde_K,lower=1)
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self.Sigma = self.K - np.dot(V.T,V)
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self.mu = np.dot(self.Sigma,self.v_tilde)
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epsilon_np1 = np.mean(self.tau_tilde-self.np1[-1]**2)
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epsilon_np2 = np.mean(self.v_tilde-self.np2[-1]**2)
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self.np1.append(self.tau_tilde.copy())
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self.np2.append(self.v_tilde.copy())
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if messages:
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print "EP iteration %i, epsilon %d"%(self.iterations,epsilon_np1)
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class FITC(EP_base):
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"""
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:param likelihood: Output's likelihood (e.g. probit)
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:type likelihood: GPy.inference.likelihood instance
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:param Knn_diag: The diagonal elements of Knn is a 1D vector
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:param Kmn: The 'cross' variance between inducing inputs and data
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:param Kmm: the covariance matrix of the inducing inputs
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:param likelihood: Output's likelihood (e.g. probit)
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:type likelihood: GPy.inference.likelihood instance
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:param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
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:param powerep: Power-EP parameters (eta,delta) - 2x1 numpy array (floats)
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"""
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def __init__(self,likelihood,Knn_diag,Kmn,Kmm,*args,**kwargs):
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self.Knn_diag = Knn_diag
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self.Kmn = Kmn
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self.Kmm = Kmm
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self.M = self.Kmn.shape[0]
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self.N = self.Kmn.shape[1]
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assert self.M <= self.N, 'The number of inducing inputs must be smaller than the number of observations'
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assert len(Knn_diag) == self.N, 'Knn_diagonal has size different from N'
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EP_base.__init__(self,likelihood,*args,**kwargs)
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def fit_EP(self):
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"""
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The expectation-propagation algorithm with sparse pseudo-input.
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For nomenclature see Naish-Guzman and Holden, 2008.
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"""
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"""
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Prior approximation parameters:
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q(f|X) = int_{df}{N(f|KfuKuu_invu,diag(Kff-Qff)*N(u|0,Kuu)} = N(f|0,Sigma0)
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Sigma0 = diag(Knn-Qnn) + Qnn, Qnn = Knm*Kmmi*Kmn
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"""
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self.Kmmi, self.Kmm_hld = pdinv(self.Kmm)
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self.P0 = self.Kmn.T
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self.KmnKnm = np.dot(self.P0.T, self.P0)
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self.KmmiKmn = np.dot(self.Kmmi,self.P0.T)
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self.Qnn_diag = np.sum(self.P0.T*self.KmmiKmn,-2)
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self.Diag0 = self.Knn_diag - self.Qnn_diag
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self.R0 = jitchol(self.Kmmi).T
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"""
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Posterior approximation: q(f|y) = N(f| mu, Sigma)
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Sigma = Diag + P*R.T*R*P.T + K
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mu = w + P*gamma
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"""
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self.w = np.zeros(self.N)
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self.gamma = np.zeros(self.M)
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self.mu = np.zeros(self.N)
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self.P = self.P0.copy()
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self.R = self.R0.copy()
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self.Diag = self.Diag0.copy()
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self.Sigma_diag = self.Knn_diag
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"""
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Initial values - Cavity distribution parameters:
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q_(g|mu_,sigma2_) = Product{q_i(g|mu_i,sigma2_i)}
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sigma_ = 1./tau_
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mu_ = v_/tau_
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"""
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self.tau_ = np.empty(self.N,dtype=np.float64)
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self.v_ = np.empty(self.N,dtype=np.float64)
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#Initial values - Marginal moments
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z = np.empty(self.N,dtype=np.float64)
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self.Z_hat = np.empty(self.N,dtype=np.float64)
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phi = np.empty(self.N,dtype=np.float64)
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mu_hat = np.empty(self.N,dtype=np.float64)
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sigma2_hat = np.empty(self.N,dtype=np.float64)
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#Approximation
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epsilon_np1 = 1
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epsilon_np2 = 1
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self.iterations = 0
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self.np1 = [self.tau_tilde.copy()]
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self.np2 = [self.v_tilde.copy()]
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while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
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update_order = np.arange(self.N)
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random.shuffle(update_order)
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for i in update_order:
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#Cavity distribution parameters
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self.tau_[i] = 1./self.Sigma_diag[i] - self.eta*self.tau_tilde[i]
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self.v_[i] = self.mu[i]/self.Sigma_diag[i] - self.eta*self.v_tilde[i]
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#Marginal moments
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self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.likelihood.moments_match(i,self.tau_[i],self.v_[i])
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#Site parameters update
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Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./self.Sigma_diag[i])
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Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - self.mu[i]/self.Sigma_diag[i])
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self.tau_tilde[i] = self.tau_tilde[i] + Delta_tau
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self.v_tilde[i] = self.v_tilde[i] + Delta_v
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#Posterior distribution parameters update
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dtd1 = Delta_tau*self.Diag[i] + 1.
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dii = self.Diag[i]
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self.Diag[i] = dii - (Delta_tau * dii**2.)/dtd1
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pi_ = self.P[i,:].reshape(1,self.M)
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self.P[i,:] = pi_ - (Delta_tau*dii)/dtd1 * pi_
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Rp_i = np.dot(self.R,pi_.T)
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RTR = np.dot(self.R.T,np.dot(np.eye(self.M) - Delta_tau/(1.+Delta_tau*self.Sigma_diag[i]) * np.dot(Rp_i,Rp_i.T),self.R))
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self.R = jitchol(RTR).T
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self.w[i] = self.w[i] + (Delta_v - Delta_tau*self.w[i])*dii/dtd1
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self.gamma = self.gamma + (Delta_v - Delta_tau*self.mu[i])*np.dot(RTR,self.P[i,:].T)
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self.RPT = np.dot(self.R,self.P.T)
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self.Sigma_diag = self.Diag + np.sum(self.RPT.T*self.RPT.T,-1)
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self.mu = self.w + np.dot(self.P,self.gamma)
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self.iterations += 1
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#Sigma recomptutation with Cholesky decompositon
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self.Diag = self.Diag0/(1.+ self.Diag0 * self.tau_tilde)
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self.P = (self.Diag / self.Diag0)[:,None] * self.P0
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self.RPT0 = np.dot(self.R0,self.P0.T)
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L = jitchol(np.eye(self.M) + np.dot(self.RPT0,(1./self.Diag0 - self.Diag/(self.Diag0**2))[:,None]*self.RPT0.T))
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self.R,info = linalg.flapack.dtrtrs(L,self.R0,lower=1)
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self.RPT = np.dot(self.R,self.P.T)
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self.Sigma_diag = self.Diag + np.sum(self.RPT.T*self.RPT.T,-1)
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self.w = self.Diag * self.v_tilde
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self.gamma = np.dot(self.R.T, np.dot(self.RPT,self.v_tilde))
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self.mu = self.w + np.dot(self.P,self.gamma)
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epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.N
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epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.N
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self.np1.append(self.tau_tilde.copy())
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self.np2.append(self.v_tilde.copy())
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@ -1,160 +0,0 @@
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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 scipy import stats, linalg
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from .. import kern
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from ..inference.Expectation_Propagation import Full
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from ..inference.likelihoods import likelihood,probit#,poisson,gaussian
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from ..core import model
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from ..util.linalg import pdinv,jitchol
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from ..util.plot import gpplot
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class GP_EP(model):
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def __init__(self,X,likelihood,kernel=None,epsilon_ep=1e-3,epsion_em=.1,powerep=[1.,1.]):
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"""
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Simple Gaussian Process with Non-Gaussian likelihood
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Arguments
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---------
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:param X: input observations (NxD numpy.darray)
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:param likelihood: a GPy likelihood (likelihood class)
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:param kernel: a GPy kernel (kern class)
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:param epsilon_ep: convergence criterion for the Expectation Propagation algorithm, defaults to 0.1 (float)
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:param powerep: power-EP parameters [$\eta$,$\delta$], defaults to [1.,1.] (list)
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:rtype: GPy model class.
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"""
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if kernel is None:
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kernel = kern.rbf(X.shape[1]) + kern.bias(X.shape[1]) + kern.white(X.shape[1])
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assert isinstance(kernel,kern.kern), 'kernel is not a kern instance'
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self.likelihood = likelihood
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self.Y = self.likelihood.Y
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self.kernel = kernel
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self.X = X
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self.N, self.D = self.X.shape
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self.eta,self.delta = powerep
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self.epsilon_ep = epsilon_ep
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self.jitter = 1e-12
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self.K = self.kernel.K(self.X)
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model.__init__(self)
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def _set_params(self,p):
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self.kernel._set_params_transformed(p)
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def _get_params(self):
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return self.kernel._get_params_transformed()
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def _get_param_names(self):
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return self.kernel._get_param_names_transformed()
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def approximate_likelihood(self):
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self.ep_approx = Full(self.K,self.likelihood,epsilon=self.epsilon_ep,powerep=[self.eta,self.delta])
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self.ep_approx.fit_EP()
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def posterior_param(self):
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self.K = self.kernel.K(self.X)
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self.Sroot_tilde_K = np.sqrt(self.ep_approx.tau_tilde)[:,None]*self.K
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B = np.eye(self.N) + np.sqrt(self.ep_approx.tau_tilde)*self.Sroot_tilde_K
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#self.L = np.linalg.cholesky(B)
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self.L = jitchol(B)
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V,info = linalg.flapack.dtrtrs(self.L,self.Sroot_tilde_K,lower=1)
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self.Sigma = self.K - np.dot(V.T,V)
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self.mu = np.dot(self.Sigma,self.ep_approx.v_tilde) * self.Z_hat
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def log_likelihood(self):
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"""
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Returns
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-------
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The EP approximation to the log-marginal likelihood
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"""
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self.posterior_param()
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mu_ = self.ep_approx.v_/self.ep_approx.tau_
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L1 =.5*sum(np.log(1+self.ep_approx.tau_tilde*1./self.ep_approx.tau_))-sum(np.log(np.diag(self.L)))
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L2A =.5*np.sum((self.Sigma-np.diag(1./(self.ep_approx.tau_+self.ep_approx.tau_tilde))) * np.dot(self.ep_approx.v_tilde[:,None],self.ep_approx.v_tilde[None,:]))
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L2B = .5*np.dot(mu_*(self.ep_approx.tau_/(self.ep_approx.tau_tilde+self.ep_approx.tau_)),self.ep_approx.tau_tilde*mu_ - 2*self.ep_approx.v_tilde)
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L3 = sum(np.log(self.ep_approx.Z_hat))
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return L1 + L2A + L2B + L3
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def _log_likelihood_gradients(self):
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dK_dp = self.kernel.dK_dtheta(self.X)
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self.dK_dp = dK_dp
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aux1,info_1 = linalg.flapack.dtrtrs(self.L,np.dot(self.Sroot_tilde_K,self.ep_approx.v_tilde),lower=1)
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b = self.ep_approx.v_tilde - np.sqrt(self.ep_approx.tau_tilde)*linalg.flapack.dtrtrs(self.L.T,aux1)[0]
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U,info_u = linalg.flapack.dtrtrs(self.L,np.diag(np.sqrt(self.ep_approx.tau_tilde)),lower=1)
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dL_dK = 0.5*(np.outer(b,b)-np.dot(U.T,U))
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self.dL_dK = dL_dK
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return np.array([np.sum(dK_dpi*dL_dK) for dK_dpi in dK_dp.T])
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def predict(self,X):
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#TODO: check output dimensions
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self.posterior_param()
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K_x = self.kernel.K(self.X,X)
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Kxx = self.kernel.K(X)
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aux1,info = linalg.flapack.dtrtrs(self.L,np.dot(self.Sroot_tilde_K,self.ep_approx.v_tilde),lower=1)
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aux2,info = linalg.flapack.dtrtrs(self.L.T, aux1,lower=0)
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zeta = np.sqrt(self.ep_approx.tau_tilde)*aux2
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f = np.dot(K_x.T,self.ep_approx.v_tilde-zeta)
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v,info = linalg.flapack.dtrtrs(self.L,np.sqrt(self.ep_approx.tau_tilde)[:,None]*K_x,lower=1)
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variance = Kxx - np.dot(v.T,v)
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vdiag = np.diag(variance)
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y=self.likelihood.predictive_mean(f,vdiag)
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return f,vdiag,y
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def plot(self):
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"""
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Plot the fitted model: training function values, inducing points used, mean estimate and confidence intervals.
|
||||
"""
|
||||
if self.X.shape[1]==1:
|
||||
pb.figure()
|
||||
xmin,xmax = self.X.min(),self.X.max()
|
||||
xmin, xmax = xmin-0.2*(xmax-xmin), xmax+0.2*(xmax-xmin)
|
||||
Xnew = np.linspace(xmin,xmax,100)[:,None]
|
||||
mu_f, var_f, mu_phi = self.predict(Xnew)
|
||||
pb.subplot(211)
|
||||
self.likelihood.plot1Da(X_new=Xnew,Mean_new=mu_f,Var_new=var_f,X_u=self.X,Mean_u=self.mu,Var_u=np.diag(self.Sigma))
|
||||
pb.subplot(212)
|
||||
self.likelihood.plot1Db(self.X,Xnew,mu_phi)
|
||||
elif self.X.shape[1]==2:
|
||||
pb.figure()
|
||||
x1min,x1max = self.X[:,0].min(0),self.X[:,0].max(0)
|
||||
x2min,x2max = self.X[:,1].min(0),self.X[:,1].max(0)
|
||||
x1min, x1max = x1min-0.2*(x1max-x1min), x1max+0.2*(x1max-x1min)
|
||||
x2min, x2max = x2min-0.2*(x2max-x2min), x2max+0.2*(x1max-x1min)
|
||||
axis1 = np.linspace(x1min,x1max,50)
|
||||
axis2 = np.linspace(x2min,x2max,50)
|
||||
XX1, XX2 = [e.flatten() for e in np.meshgrid(axis1,axis2)]
|
||||
Xnew = np.c_[XX1.flatten(),XX2.flatten()]
|
||||
f,v,p = self.predict(Xnew)
|
||||
self.likelihood.plot2D(self.X,Xnew,p)
|
||||
else:
|
||||
raise NotImplementedError, "Cannot plot GPs with more than two input dimensions"
|
||||
|
||||
def em(self,max_f_eval=1e4,epsilon=.1,plot_all=False): #TODO check this makes sense
|
||||
"""
|
||||
Fits sparse_EP and optimizes the hyperparametes iteratively until convergence is achieved.
|
||||
"""
|
||||
self.epsilon_em = epsilon
|
||||
log_likelihood_change = self.epsilon_em + 1.
|
||||
self.parameters_path = [self.kernel._get_params()]
|
||||
self.approximate_likelihood()
|
||||
self.site_approximations_path = [[self.ep_approx.tau_tilde,self.ep_approx.v_tilde]]
|
||||
self.log_likelihood_path = [self.log_likelihood()]
|
||||
iteration = 0
|
||||
while log_likelihood_change > self.epsilon_em:
|
||||
print 'EM iteration', iteration
|
||||
self.optimize(max_f_eval = max_f_eval)
|
||||
log_likelihood_new = self.log_likelihood()
|
||||
log_likelihood_change = log_likelihood_new - self.log_likelihood_path[-1]
|
||||
if log_likelihood_change < 0:
|
||||
print 'log_likelihood decrement'
|
||||
self.kernel._set_params_transformed(self.parameters_path[-1])
|
||||
self.kernM._set_params_transformed(self.parameters_path[-1])
|
||||
else:
|
||||
self.approximate_likelihood()
|
||||
self.log_likelihood_path.append(self.log_likelihood())
|
||||
self.parameters_path.append(self.kernel._get_params())
|
||||
self.site_approximations_path.append([self.ep_approx.tau_tilde,self.ep_approx.v_tilde])
|
||||
iteration += 1
|
||||
|
|
@ -1,279 +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 scipy import stats, linalg
|
||||
from .. import kern
|
||||
from ..inference.EP import Full
|
||||
from ..inference.likelihoods import likelihood,probit,poisson,gaussian
|
||||
from ..core import model
|
||||
from ..util.linalg import pdinv,mdot #,jitchol
|
||||
from ..util.plot import gpplot, Tango
|
||||
|
||||
class GP_EP2(model):
|
||||
def __init__(self,X,likelihood,kernel=None,normalize_X=False,Xslices=None,epsilon_ep=1e-3,epsion_em=.1,powerep=[1.,1.]):
|
||||
"""
|
||||
Simple Gaussian Process with Non-Gaussian likelihood
|
||||
|
||||
Arguments
|
||||
---------
|
||||
:param X: input observations (NxD numpy.darray)
|
||||
:param likelihood: a GPy likelihood (likelihood class)
|
||||
:param kernel: a GPy kernel, defaults to rbf+white
|
||||
:param normalize_X: whether to normalize the input data before computing (predictions will be in original scales)
|
||||
:type normalize_X: False|True
|
||||
:param epsilon_ep: convergence criterion for the Expectation Propagation algorithm, defaults to 1e-3
|
||||
:param powerep: power-EP parameters [$\eta$,$\delta$], defaults to [1.,1.] (list)
|
||||
:param Xslices: how the X,Y data co-vary in the kernel (i.e. which "outputs" they correspond to). See (link:slicing)
|
||||
:rtype: model object.
|
||||
"""
|
||||
#.. Note:: Multiple independent outputs are allowed using columns of Y #TODO add this note?
|
||||
if kernel is None:
|
||||
kernel = kern.rbf(X.shape[1]) + kern.bias(X.shape[1]) + kern.white(X.shape[1])
|
||||
|
||||
# parse arguments
|
||||
self.Xslices = Xslices
|
||||
assert isinstance(kernel, kern.kern)
|
||||
self.likelihood = likelihood
|
||||
self.kern = kernel
|
||||
self.X = X
|
||||
assert len(self.X.shape)==2
|
||||
assert self.X.shape[0] == self.likelihood.Y.shape[0]
|
||||
self.D = self.likelihood.Y.shape[1]
|
||||
self.N, self.Q = self.X.shape
|
||||
|
||||
#here's some simple normalisation
|
||||
if normalize_X:
|
||||
self._Xmean = X.mean(0)[None,:]
|
||||
self._Xstd = X.std(0)[None,:]
|
||||
self.X = (X.copy() - self._Xmean) / self._Xstd
|
||||
if hasattr(self,'Z'):
|
||||
self.Z = (self.Z - self._Xmean) / self._Xstd
|
||||
else:
|
||||
self._Xmean = np.zeros((1,self.X.shape[1]))
|
||||
self._Xstd = np.ones((1,self.X.shape[1]))
|
||||
|
||||
#THIS PART IS NOT NEEDED
|
||||
"""
|
||||
if normalize_Y:
|
||||
self._Ymean = Y.mean(0)[None,:]
|
||||
self._Ystd = Y.std(0)[None,:]
|
||||
self.Y = (Y.copy()- self._Ymean) / self._Ystd
|
||||
else:
|
||||
self._Ymean = np.zeros((1,self.Y.shape[1]))
|
||||
self._Ystd = np.ones((1,self.Y.shape[1]))
|
||||
|
||||
if self.D > self.N:
|
||||
# then it's more efficient to store YYT
|
||||
self.YYT = np.dot(self.Y, self.Y.T)
|
||||
else:
|
||||
self.YYT = None
|
||||
"""
|
||||
self.eta,self.delta = powerep
|
||||
self.epsilon_ep = epsilon_ep
|
||||
self.tau_tilde = np.ones([self.N,self.D])
|
||||
self.v_tilde = np.zeros([self.N,self.D])
|
||||
self.tau_ = np.ones([self.N,self.D])
|
||||
self.v_ = np.zeros([self.N,self.D])
|
||||
self.Z_hat = np.ones([self.N,self.D])
|
||||
model.__init__(self)
|
||||
|
||||
def _set_params(self,p):
|
||||
self.kern._set_params_transformed(p)
|
||||
self.K = self.kern.K(self.X,slices1=self.Xslices)
|
||||
self._ep_params()
|
||||
|
||||
def _get_params(self):
|
||||
return self.kern._get_params_transformed()
|
||||
|
||||
def _get_param_names(self):
|
||||
return self.kern._get_param_names_transformed()
|
||||
|
||||
def approximate_likelihood(self):
|
||||
self.ep_approx = Full(self.K,self.likelihood,epsilon=self.epsilon_ep,powerep=[self.eta,self.delta])
|
||||
self.tau_tilde, self.v_tilde, self.Z_hat, self.tau_, self.v_=self.ep_approx.fit_EP()
|
||||
self._ep_params()
|
||||
|
||||
def _ep_params(self):
|
||||
# Posterior mean and Variance computation
|
||||
self.Sroot_tilde_K = np.sqrt(self.tau_tilde)*self.K
|
||||
B = np.eye(self.N) + np.sqrt(self.tau_tilde.flatten())[None,:]*self.Sroot_tilde_K
|
||||
self.Bi,self.L,self.Li,B_logdet = pdinv(B)
|
||||
V = np.dot(self.Li,self.Sroot_tilde_K)
|
||||
self.Sigma = self.K - np.dot(V.T,V) #posterior variance
|
||||
self.mu = np.dot(self.Sigma,self.v_tilde) #posterior mean
|
||||
# Kernel plus noise variance term
|
||||
self.Kplus = self.K + np.diag(1./self.tau_tilde.flatten())
|
||||
self.Kplusi,self.Lplus,self.Lplusi,self.Kplus_logdet = pdinv(self.Kplus)
|
||||
# Y: EP likelihood is defined as a regression model for mu_tilde
|
||||
self.Y = self.v_tilde/self.tau_tilde
|
||||
self._Ymean = np.zeros((1,self.Y.shape[1]))
|
||||
self._Ystd = np.ones((1,self.Y.shape[1]))
|
||||
self.YYT = None #np.dot(self.Y, self.Y.T)
|
||||
self.mu_ = self.v_/self.tau_
|
||||
|
||||
def _model_fit_term(self):
|
||||
"""
|
||||
Computes the model fit using YYT if it's available
|
||||
"""
|
||||
if self.YYT is None:
|
||||
return -0.5*np.sum(np.square(np.dot(self.Lplusi,self.Y)))
|
||||
else:
|
||||
return -0.5*np.sum(np.multiply(self.Kplusi, self.YYT))
|
||||
|
||||
def _normalization_term(self):
|
||||
"""
|
||||
Computes the marginal likelihood normalization constants
|
||||
"""
|
||||
sigma_sum = 1./self.tau_ + 1./self.tau_tilde
|
||||
mu_diff_2 = (self.mu_ - self.Y)**2
|
||||
penalty_term = np.sum(np.log(self.Z_hat))
|
||||
return penalty_term + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum)
|
||||
|
||||
def log_likelihood(self):
|
||||
"""
|
||||
The log marginal likelihood for an EP model can be written as the log likelihood of
|
||||
a regression model for a new variable Y* = v_tilde/tau_tilde, with a covariance
|
||||
matrix K* = K + diag(1./tau_tilde) plus a normalization term.
|
||||
"""
|
||||
complexity_term = -0.5*self.D*self.Kplus_logdet
|
||||
return complexity_term + self._model_fit_term() + self._normalization_term()
|
||||
|
||||
def dL_dK(self):
|
||||
if self.YYT is None:
|
||||
alpha = np.dot(self.Kplusi,self.Y)
|
||||
dL_dK = 0.5*(np.dot(alpha,alpha.T)-self.D*self.Kplusi)
|
||||
else:
|
||||
dL_dK = 0.5*(mdot(self.Kplusi, self.YYT, self.Kplusi) - self.D*self.Kplusi)
|
||||
return dL_dK
|
||||
|
||||
def _log_likelihood_gradients(self):
|
||||
return self.kern.dK_dtheta(partial=self.dL_dK(),X=self.X)
|
||||
|
||||
def predict(self,Xnew, slices=None, full_cov=False):
|
||||
"""
|
||||
|
||||
Predict the function(s) at the new point(s) Xnew.
|
||||
|
||||
Arguments
|
||||
---------
|
||||
:param Xnew: The points at which to make a prediction
|
||||
:type Xnew: np.ndarray, Nnew x self.Q
|
||||
:param slices: specifies which outputs kernel(s) the Xnew correspond to (see below)
|
||||
:type slices: (None, list of slice objects, list of ints)
|
||||
:param full_cov: whether to return the folll covariance matrix, or just the diagonal
|
||||
:type full_cov: bool
|
||||
:rtype: posterior mean, a Numpy array, Nnew x self.D
|
||||
:rtype: posterior variance, a Numpy array, Nnew x Nnew x (self.D)
|
||||
|
||||
.. Note:: "slices" specifies how the the points X_new co-vary wich the training points.
|
||||
|
||||
- If None, the new points covary throigh every kernel part (default)
|
||||
- If a list of slices, the i^th slice specifies which data are affected by the i^th kernel part
|
||||
- If a list of booleans, specifying which kernel parts are active
|
||||
|
||||
If full_cov and self.D > 1, the return shape of var is Nnew x Nnew x self.D. If self.D == 1, the return shape is Nnew x Nnew.
|
||||
This is to allow for different normalisations of the output dimensions.
|
||||
|
||||
|
||||
"""
|
||||
|
||||
#normalise X values
|
||||
Xnew = (Xnew.copy() - self._Xmean) / self._Xstd
|
||||
mu, var, phi = self._raw_predict(Xnew, slices, full_cov)
|
||||
|
||||
#un-normalise
|
||||
mu = mu*self._Ystd + self._Ymean
|
||||
if full_cov:
|
||||
if self.D==1:
|
||||
var *= np.square(self._Ystd)
|
||||
else:
|
||||
var = var[:,:,None] * np.square(self._Ystd)
|
||||
else:
|
||||
if self.D==1:
|
||||
var *= np.square(np.squeeze(self._Ystd))
|
||||
else:
|
||||
var = var[:,None] * np.square(self._Ystd)
|
||||
|
||||
return mu,var,phi
|
||||
|
||||
def _raw_predict(self,_Xnew,slices, full_cov=False):
|
||||
"""Internal helper function for making predictions, does not account for normalisation"""
|
||||
K_x = self.kern.K(self.X,_Xnew,slices1=self.Xslices,slices2=slices)
|
||||
aux2 = mdot(self.Bi,self.Sroot_tilde_K,self.v_tilde)
|
||||
zeta = np.sqrt(self.tau_tilde)*aux2
|
||||
f = np.dot(K_x.T,self.v_tilde-zeta)
|
||||
v = mdot(self.Li,np.sqrt(self.tau_tilde)*K_x)
|
||||
if full_cov:
|
||||
Kxx = self.kern.K(_Xnew,slices1=slices,slices2=slices)
|
||||
var = Kxx - np.dot(v.T,v)
|
||||
var_diag = np.diag(var)[:,None]
|
||||
else:
|
||||
Kxx = self.kern.Kdiag(_Xnew, slices=slices)
|
||||
var_diag = (Kxx - np.sum(v**2,-2))[:,None]
|
||||
phi = self.likelihood.predictive_mean(f,var_diag)
|
||||
return f, var_diag, phi
|
||||
|
||||
def plot(self,samples=0,plot_limits=None,which_data='all',which_functions='all',resolution=None):
|
||||
"""
|
||||
:param samples: the number of a posteriori samples to plot
|
||||
:param which_data: which if the training data to plot (default all)
|
||||
:type which_data: 'all' or a slice object to slice self.X, self.Y
|
||||
:param plot_limits: The limits of the plot. If 1D [xmin,xmax], if 2D [[xmin,ymin],[xmax,ymax]]. Defaluts to data limits
|
||||
:param which_functions: which of the kernel functions to plot (additively)
|
||||
:type which_functions: list of bools
|
||||
:param resolution: the number of intervals to sample the GP on. Defaults to 200 in 1D and 50 (a 50x50 grid) in 2D
|
||||
|
||||
Plot the posterior of the GP.
|
||||
- In one dimension, the function is plotted with a shaded region identifying two standard deviations.
|
||||
- In two dimsensions, a contour-plot shows the mean predicted function
|
||||
- In higher dimensions, we've no implemented this yet !TODO!
|
||||
|
||||
Can plot only part of the data and part of the posterior functions using which_data and which_functions
|
||||
"""
|
||||
if which_functions=='all':
|
||||
which_functions = [True]*self.kern.Nparts
|
||||
if which_data=='all':
|
||||
which_data = slice(None)
|
||||
|
||||
X = self.X[which_data,:]
|
||||
Y = self.Y[which_data,:]
|
||||
|
||||
Xorig = X*self._Xstd + self._Xmean
|
||||
Yorig = Y*self._Ystd + self._Ymean
|
||||
if plot_limits is None:
|
||||
xmin,xmax = Xorig.min(0),Xorig.max(0)
|
||||
xmin, xmax = xmin-0.2*(xmax-xmin), xmax+0.2*(xmax-xmin)
|
||||
elif len(plot_limits)==2:
|
||||
xmin, xmax = plot_limits
|
||||
else:
|
||||
raise ValueError, "Bad limits for plotting"
|
||||
|
||||
if self.X.shape[1]==1:
|
||||
Xnew = np.linspace(xmin,xmax,resolution or 200)[:,None]
|
||||
#m,v,phi = self.predict(Xnew,slices=which_functions)
|
||||
#gpplot(Xnew,m,v)
|
||||
mu_f, var_f, phi_f = self.predict(Xnew,slices=which_functions)
|
||||
pb.subplot(211)
|
||||
self.likelihood.plot1Da(X=Xnew,mean=mu_f,var=var_f,Z=self.X,mean_Z=self.mu,var_Z=np.diag(self.Sigma))
|
||||
if samples:
|
||||
s = np.random.multivariate_normal(m.flatten(),v,samples)
|
||||
pb.plot(Xnew.flatten(),s.T, alpha = 0.4, c='#3465a4', linewidth = 0.8)
|
||||
pb.xlim(xmin,xmax)
|
||||
pb.subplot(212)
|
||||
self.likelihood.plot1Db(self.X,Xnew,phi_f)
|
||||
|
||||
elif self.X.shape[1]==2:
|
||||
resolution = 50 or resolution
|
||||
xx,yy = np.mgrid[xmin[0]:xmax[0]:1j*resolution,xmin[1]:xmax[1]:1j*resolution]
|
||||
Xtest = np.vstack((xx.flatten(),yy.flatten())).T
|
||||
zz,vv = self.predict(Xtest,slices=which_functions)
|
||||
zz = zz.reshape(resolution,resolution)
|
||||
pb.contour(xx,yy,zz,vmin=zz.min(),vmax=zz.max(),cmap=pb.cm.jet)
|
||||
pb.scatter(Xorig[:,0],Xorig[:,1],40,Yorig,linewidth=0,cmap=pb.cm.jet,vmin=zz.min(),vmax=zz.max())
|
||||
pb.xlim(xmin[0],xmax[0])
|
||||
pb.ylim(xmin[1],xmax[1])
|
||||
|
||||
else:
|
||||
raise NotImplementedError, "Cannot plot GPs with more than two input dimensions"
|
||||
|
|
@ -6,8 +6,6 @@ from GP_regression import GP_regression
|
|||
from sparse_GP_regression import sparse_GP_regression
|
||||
from GPLVM import GPLVM
|
||||
from warped_GP import warpedGP
|
||||
from GP_EP import GP_EP
|
||||
from GP_EP2 import GP_EP2
|
||||
from generalized_FITC import generalized_FITC
|
||||
from sparse_GPLVM import sparse_GPLVM
|
||||
from uncollapsed_sparse_GP import uncollapsed_sparse_GP
|
||||
|
|
|
|||
|
|
@ -9,7 +9,8 @@ from .. import kern
|
|||
from ..core import model
|
||||
from ..util.linalg import pdinv,mdot
|
||||
from ..util.plot import gpplot
|
||||
from ..inference.Expectation_Propagation import FITC
|
||||
#from ..inference.Expectation_Propagation import FITC
|
||||
from ..inference.EP import FITC
|
||||
from ..inference.likelihoods import likelihood,probit
|
||||
|
||||
class generalized_FITC(model):
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue