mirror of
https://github.com/SheffieldML/GPy.git
synced 2026-07-05 16:02:13 +02:00
merged with master
This commit is contained in:
Nicolo Fusi
commit
2abba8cf14
59 changed files with 2845 additions and 1608 deletions
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@ -1,240 +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 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, epsiolon %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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@ -1,219 +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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from scipy import stats
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import scipy as sp
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import pylab as pb
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from ..util.plot import gpplot
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class likelihood:
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def __init__(self,Y,location=0,scale=1):
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"""
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Likelihood class for doing Expectation propagation
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:param Y: observed output (Nx1 numpy.darray)
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..Note:: Y values allowed depend on the likelihood used
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"""
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self.Y = Y
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self.N = self.Y.shape[0]
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self.location = location
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self.scale = scale
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def plot1Da(self,X_new,Mean_new,Var_new,X_u,Mean_u,Var_u):
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"""
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Plot the predictive distribution of the GP model for 1-dimensional inputs
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:param X_new: The points at which to make a prediction
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:param Mean_new: mean values at X_new
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:param Var_new: variance values at X_new
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:param X_u: input (inducing) points used to train the model
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:param Mean_u: mean values at X_u
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:param Var_new: variance values at X_u
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"""
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assert X_new.shape[1] == 1, 'Number of dimensions must be 1'
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gpplot(X_new,Mean_new,Var_new)
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pb.errorbar(X_u,Mean_u,2*np.sqrt(Var_u),fmt='r+')
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pb.plot(X_u,Mean_u,'ro')
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def plot2D(self,X,X_new,F_new,U=None):
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"""
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Predictive distribution of the fitted GP model for 2-dimensional inputs
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:param X_new: The points at which to make a prediction
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:param Mean_new: mean values at X_new
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:param Var_new: variance values at X_new
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:param X_u: input points used to train the model
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:param Mean_u: mean values at X_u
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:param Var_new: variance values at X_u
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"""
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N,D = X_new.shape
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assert D == 2, 'Number of dimensions must be 2'
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n = np.sqrt(N)
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x1min = X_new[:,0].min()
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x1max = X_new[:,0].max()
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x2min = X_new[:,1].min()
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x2max = X_new[:,1].max()
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pb.imshow(F_new.reshape(n,n),extent=(x1min,x1max,x2max,x2min),vmin=0,vmax=1)
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pb.colorbar()
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C1 = np.arange(self.N)[self.Y.flatten()==1]
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C2 = np.arange(self.N)[self.Y.flatten()==-1]
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[pb.plot(X[i,0],X[i,1],'ro') for i in C1]
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[pb.plot(X[i,0],X[i,1],'bo') for i in C2]
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pb.xlim(x1min,x1max)
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pb.ylim(x2min,x2max)
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if U is not None:
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[pb.plot(a,b,'wo') for a,b in U]
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class probit(likelihood):
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"""
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Probit likelihood
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Y is expected to take values in {-1,1}
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-----
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$$
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L(x) = \\Phi (Y_i*f_i)
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$$
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"""
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def moments_match(self,i,tau_i,v_i):
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"""
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Moments match of the marginal approximation in EP algorithm
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:param i: number of observation (int)
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:param tau_i: precision of the cavity distribution (float)
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:param v_i: mean/variance of the cavity distribution (float)
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"""
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z = self.Y[i]*v_i/np.sqrt(tau_i**2 + tau_i)
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Z_hat = stats.norm.cdf(z)
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phi = stats.norm.pdf(z)
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mu_hat = v_i/tau_i + self.Y[i]*phi/(Z_hat*np.sqrt(tau_i**2 + tau_i))
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sigma2_hat = 1./tau_i - (phi/((tau_i**2+tau_i)*Z_hat))*(z+phi/Z_hat)
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return Z_hat, mu_hat, sigma2_hat
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def plot1Db(self,X,X_new,F_new,U=None):
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assert X.shape[1] == 1, 'Number of dimensions must be 1'
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gpplot(X_new,F_new,np.zeros(X_new.shape[0]))
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pb.plot(X,(self.Y+1)/2,'kx',mew=1.5)
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pb.ylim(-0.2,1.2)
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if U is not None:
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pb.plot(U,U*0+.5,'r|',mew=1.5,markersize=12)
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def predictive_mean(self,mu,variance):
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return stats.norm.cdf(mu/np.sqrt(1+variance))
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def _log_likelihood_gradients():
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raise NotImplementedError
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class poisson(likelihood):
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"""
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Poisson likelihood
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Y is expected to take values in {0,1,2,...}
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-----
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$$
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L(x) = \exp(\lambda) * \lambda**Y_i / Y_i!
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$$
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"""
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def moments_match(self,i,tau_i,v_i):
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"""
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Moments match of the marginal approximation in EP algorithm
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:param i: number of observation (int)
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:param tau_i: precision of the cavity distribution (float)
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:param v_i: mean/variance of the cavity distribution (float)
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"""
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mu = v_i/tau_i
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sigma = np.sqrt(1./tau_i)
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def poisson_norm(f):
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"""
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Product of the likelihood and the cavity distribution
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"""
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pdf_norm_f = stats.norm.pdf(f,loc=mu,scale=sigma)
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rate = np.exp( (f*self.scale)+self.location)
|
||||
poisson = stats.poisson.pmf(float(self.Y[i]),rate)
|
||||
return pdf_norm_f*poisson
|
||||
|
||||
def log_pnm(f):
|
||||
"""
|
||||
Log of poisson_norm
|
||||
"""
|
||||
return -(-.5*(f-mu)**2/sigma**2 - np.exp( (f*self.scale)+self.location) + ( (f*self.scale)+self.location)*self.Y[i])
|
||||
|
||||
"""
|
||||
Golden Search and Simpson's Rule
|
||||
--------------------------------
|
||||
Simpson's Rule is used to calculate the moments mumerically, it needs a grid of points as input.
|
||||
Golden Search is used to find the mode in the poisson_norm distribution and define around it the grid for Simpson's Rule
|
||||
"""
|
||||
#TODO golden search & simpson's rule can be defined in the general likelihood class, rather than in each specific case.
|
||||
|
||||
#Golden search
|
||||
golden_A = -1 if self.Y[i] == 0 else np.array([np.log(self.Y[i]),mu]).min() #Lower limit
|
||||
golden_B = np.array([np.log(self.Y[i]),mu]).max() #Upper limit
|
||||
golden_A = (golden_A - self.location)/self.scale
|
||||
golden_B = (golden_B - self.location)/self.scale
|
||||
opt = sp.optimize.golden(log_pnm,brack=(golden_A,golden_B)) #Better to work with log_pnm than with poisson_norm
|
||||
|
||||
# Simpson's approximation
|
||||
width = 3./np.log(max(self.Y[i],2))
|
||||
A = opt - width #Lower limit
|
||||
B = opt + width #Upper limit
|
||||
K = 10*int(np.log(max(self.Y[i],150))) #Number of points in the grid, we DON'T want K to be the same number for every case
|
||||
h = (B-A)/K # length of the intervals
|
||||
grid_x = np.hstack([np.linspace(opt-width,opt,K/2+1)[1:-1], np.linspace(opt,opt+width,K/2+1)]) # grid of points (X axis)
|
||||
x = np.hstack([A,B,grid_x[range(1,K,2)],grid_x[range(2,K-1,2)]]) # grid_x rearranged, just to make Simpson's algorithm easier
|
||||
zeroth = np.hstack([poisson_norm(A),poisson_norm(B),[4*poisson_norm(f) for f in grid_x[range(1,K,2)]],[2*poisson_norm(f) for f in grid_x[range(2,K-1,2)]]]) # grid of points (Y axis) rearranged like x
|
||||
first = zeroth*x
|
||||
second = first*x
|
||||
Z_hat = sum(zeroth)*h/3 # Zero-th moment
|
||||
mu_hat = sum(first)*h/(3*Z_hat) # First moment
|
||||
m2 = sum(second)*h/(3*Z_hat) # Second moment
|
||||
sigma2_hat = m2 - mu_hat**2 # Second central moment
|
||||
return float(Z_hat), float(mu_hat), float(sigma2_hat)
|
||||
|
||||
def plot1Db(self,X,X_new,F_new,F2_new=None,U=None):
|
||||
pb.subplot(212)
|
||||
#gpplot(X_new,F_new,np.sqrt(F2_new))
|
||||
pb.plot(X_new,F_new)#,np.sqrt(F2_new)) #FIXME
|
||||
pb.plot(X,self.Y,'kx',mew=1.5)
|
||||
if U is not None:
|
||||
pb.plot(U,np.ones(U.shape[0])*self.Y.min()*.8,'r|',mew=1.5,markersize=12)
|
||||
def predictive_mean(self,mu,variance):
|
||||
return np.exp(mu*self.scale + self.location)
|
||||
def predictive_variance(self,mu,variance):
|
||||
return mu
|
||||
def _log_likelihood_gradients():
|
||||
raise NotImplementedError
|
||||
|
||||
class gaussian(likelihood):
|
||||
"""
|
||||
Gaussian likelihood
|
||||
Y is expected to take values in (-inf,inf)
|
||||
"""
|
||||
def moments_match(self,i,tau_i,v_i):
|
||||
"""
|
||||
Moments match of the marginal approximation in EP algorithm
|
||||
|
||||
:param i: number of observation (int)
|
||||
:param tau_i: precision of the cavity distribution (float)
|
||||
:param v_i: mean/variance of the cavity distribution (float)
|
||||
"""
|
||||
mu = v_i/tau_i
|
||||
sigma = np.sqrt(1./tau_i)
|
||||
s = 1. if self.Y[i] == 0 else 1./self.Y[i]
|
||||
sigma2_hat = 1./(1./sigma**2 + 1./s**2)
|
||||
mu_hat = sigma2_hat*(mu/sigma**2 + self.Y[i]/s**2)
|
||||
Z_hat = 1./np.sqrt(2*np.pi) * 1./np.sqrt(sigma**2+s**2) * np.exp(-.5*(mu-self.Y[i])**2/(sigma**2 + s**2))
|
||||
return Z_hat, mu_hat, sigma2_hat
|
||||
|
||||
def plot1Db(self,X,X_new,F_new,U=None):
|
||||
assert X.shape[1] == 1, 'Number of dimensions must be 1'
|
||||
gpplot(X_new,F_new,np.zeros(X_new.shape[0]))
|
||||
pb.plot(X,self.Y,'kx',mew=1.5)
|
||||
if U is not None:
|
||||
pb.plot(U,np.ones(U.shape[0])*self.Y.min()*.8,'r|',mew=1.5,markersize=12)
|
||||
|
||||
def predictive_mean(self,mu,Sigma):
|
||||
return mu
|
||||
|
||||
def _log_likelihood_gradients():
|
||||
raise NotImplementedError
|
||||
Loading…
Add table
Add a link
Reference in a new issue