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fixing EP and merging it with GP_regression
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Ricardo Andrade
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7 changed files with 403 additions and 93 deletions
312
GPy/models/GP.py
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312
GPy/models/GP.py
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# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
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# Licensed under the BSD 3-clause license (see LICENSE.txt)
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import numpy as np
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import pylab as pb
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from .. import kern
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from ..core import model
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from ..util.linalg import pdinv,mdot
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from ..util.plot import gpplot, Tango
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from ..inference.EP import Full
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from ..inference.likelihoods import likelihood,probit,poisson,gaussian
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class GP(model):
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"""
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Gaussian Process model for regression
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:param X: input observations
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:param Y: observed values
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:param kernel: a GPy kernel, defaults to rbf+white
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:param normalize_X: whether to normalize the input data before computing (predictions will be in original scales)
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:type normalize_X: False|True
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:param normalize_Y: whether to normalize the input data before computing (predictions will be in original scales)
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:type normalize_Y: False|True
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:param Xslices: how the X,Y data co-vary in the kernel (i.e. which "outputs" they correspond to). See (link:slicing)
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:rtype: model object
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.. Note:: Multiple independent outputs are allowed using columns of Y
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"""
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def __init__(self,X,Y=None,kernel=None,normalize_X=False,normalize_Y=False, Xslices=None,likelihood=None,epsilon_ep=1e-3,epsion_em=.1,powerep=[1.,1.]):
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#TODO: specify beta parameter explicitely
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# parse arguments
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self.Xslices = Xslices
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self.X = X
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self.N, self.Q = self.X.shape
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assert len(self.X.shape)==2
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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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else:
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assert isinstance(kernel, kern.kern)
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self.kern = kernel
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#here's some simple normalisation
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if normalize_X:
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self._Xmean = X.mean(0)[None,:]
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self._Xstd = X.std(0)[None,:]
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self.X = (X.copy() - self._Xmean) / self._Xstd
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if hasattr(self,'Z'):
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self.Z = (self.Z - self._Xmean) / self._Xstd
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else:
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self._Xmean = np.zeros((1,self.X.shape[1]))
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self._Xstd = np.ones((1,self.X.shape[1]))
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# Y - likelihood related variables, these might change whether using EP or not
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if likelihood is None:
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assert Y is not None, "Either Y or likelihood must be defined"
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self.likelihood = gaussian(Y)
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else:
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self.likelihood = likelihood
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assert len(self.likelihood.Y.shape)==2
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assert self.X.shape[0] == self.likelihood.Y.shape[0]
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self.N, self.D = self.likelihood.Y.shape
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if isinstance(self.likelihood,gaussian):
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self.EP = False
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self.Y = Y
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#here's some simple normalisation
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if normalize_Y:
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self._Ymean = Y.mean(0)[None,:]
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self._Ystd = Y.std(0)[None,:]
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self.Y = (Y.copy()- self._Ymean) / self._Ystd
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else:
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self._Ymean = np.zeros((1,self.Y.shape[1]))
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self._Ystd = np.ones((1,self.Y.shape[1]))
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if self.D > self.N:
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# then it's more efficient to store YYT
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self.YYT = np.dot(self.Y, self.Y.T)
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else:
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self.YYT = None
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else:
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# Y is defined after approximating the likelihood
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self.EP = True
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self.eta,self.delta = powerep
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self.epsilon_ep = epsilon_ep
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self.tau_tilde = np.ones([self.N,self.D])
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self.v_tilde = np.zeros([self.N,self.D])
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self.tau_ = np.ones([self.N,self.D])
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self.v_ = np.zeros([self.N,self.D])
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self.Z_hat = np.ones([self.N,self.D])
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model.__init__(self)
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def _set_params(self,p):
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# TODO: remove beta when using EP
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self.kern._set_params_transformed(p)
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if not self.EP:
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self.K = self.kern.K(self.X,slices1=self.Xslices)
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self.Ki, self.L, self.Li, self.K_logdet = pdinv(self.K)
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else:
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self._ep_covariance()
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def _get_params(self):
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# TODO: remove beta when using EP
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return self.kern._get_params_transformed()
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def _get_param_names(self):
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# TODO: remove beta when using EP
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return self.kern._get_param_names_transformed()
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def approximate_likelihood(self):
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assert not isinstance(self.likelihood, gaussian), "EP is only available for non-gaussian likelihoods"
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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.tau_tilde, self.v_tilde, self.Z_hat, self.tau_, self.v_=self.ep_approx.fit_EP()
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# Y: EP likelihood is defined as a regression model for mu_tilde
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self.Y = self.v_tilde/self.tau_tilde
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self._Ymean = np.zeros((1,self.Y.shape[1]))
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self._Ystd = np.ones((1,self.Y.shape[1]))
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if self.D > self.N:
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# then it's more efficient to store YYT
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self.YYT = np.dot(self.Y, self.Y.T)
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else:
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self.YYT = None
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self.mu_ = self.v_/self.tau_
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self._ep_covariance()
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def _ep_covariance(self):
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# Kernel plus noise variance term
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self.K = self.kern.K(self.X,slices1=self.Xslices) + np.diag(1./self.tau_tilde.flatten())
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self.Ki, self.L, self.Li, self.K_logdet = pdinv(self.K)
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def _model_fit_term(self):
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"""
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Computes the model fit using YYT if it's available
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"""
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if self.YYT is None:
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return -0.5*np.sum(np.square(np.dot(self.Li,self.Y)))
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else:
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return -0.5*np.sum(np.multiply(self.Ki, self.YYT))
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def _normalization_term(self):
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"""
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Computes the marginal likelihood normalization constants
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"""
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sigma_sum = 1./self.tau_ + 1./self.tau_tilde
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mu_diff_2 = (self.mu_ - self.Y)**2
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penalty_term = np.sum(np.log(self.Z_hat))
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return penalty_term + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum)
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def log_likelihood(self):
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"""
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The log marginal likelihood for an EP model can be written as the log likelihood of
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a regression model for a new variable Y* = v_tilde/tau_tilde, with a covariance
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matrix K* = K + diag(1./tau_tilde) plus a normalization term.
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"""
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complexity_term = -0.5*self.D*self.Kplus_logdet
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normalization_term = 0 if self.EP == False else self.normalization_term()
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return complexity_term + normalization_term + self._model_fit_term()
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def log_likelihood(self):
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complexity_term = -0.5*self.N*self.D*np.log(2.*np.pi) - 0.5*self.D*self.K_logdet
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return complexity_term + self._model_fit_term()
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def dL_dK(self):
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if self.YYT is None:
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alpha = np.dot(self.Ki,self.Y)
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dL_dK = 0.5*(np.dot(alpha,alpha.T)-self.D*self.Ki)
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else:
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dL_dK = 0.5*(mdot(self.Ki, self.YYT, self.Ki) - self.D*self.Ki)
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return dL_dK
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def _log_likelihood_gradients(self):
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return self.kern.dK_dtheta(partial=self.dL_dK(),X=self.X)
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def predict(self,Xnew, slices=None, full_cov=False):
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"""
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Predict the function(s) at the new point(s) Xnew.
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Arguments
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---------
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:param Xnew: The points at which to make a prediction
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:type Xnew: np.ndarray, Nnew x self.Q
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:param slices: specifies which outputs kernel(s) the Xnew correspond to (see below)
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:type slices: (None, list of slice objects, list of ints)
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:param full_cov: whether to return the folll covariance matrix, or just the diagonal
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:type full_cov: bool
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:rtype: posterior mean, a Numpy array, Nnew x self.D
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:rtype: posterior variance, a Numpy array, Nnew x Nnew x (self.D)
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.. Note:: "slices" specifies how the the points X_new co-vary wich the training points.
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- If None, the new points covary throigh every kernel part (default)
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- If a list of slices, the i^th slice specifies which data are affected by the i^th kernel part
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- If a list of booleans, specifying which kernel parts are active
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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.
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This is to allow for different normalisations of the output dimensions.
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"""
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#normalise X values
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Xnew = (Xnew.copy() - self._Xmean) / self._Xstd
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mu, var, phi = self._raw_predict(Xnew, slices, full_cov)
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#un-normalise
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mu = mu*self._Ystd + self._Ymean
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if full_cov:
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if self.D==1:
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var *= np.square(self._Ystd)
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else:
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var = var[:,:,None] * np.square(self._Ystd)
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else:
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if self.D==1:
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var *= np.square(np.squeeze(self._Ystd))
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else:
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var = var[:,None] * np.square(self._Ystd)
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return mu,var,phi
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def _raw_predict(self,_Xnew,slices, full_cov=False):
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"""Internal helper function for making predictions, does not account for normalisation"""
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Kx = self.kern.K(self.X,_Xnew, slices1=self.Xslices,slices2=slices)
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mu = np.dot(np.dot(Kx.T,self.Ki),self.Y)
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KiKx = np.dot(self.Ki,Kx)
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if full_cov:
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Kxx = self.kern.K(_Xnew, slices1=slices,slices2=slices)
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var = Kxx - np.dot(KiKx.T,Kx)
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else:
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Kxx = self.kern.Kdiag(_Xnew, slices=slices)
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var = Kxx - np.sum(np.multiply(KiKx,Kx),0)
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phi = None if not self.EP else self.likelihood.predictive_mean(mu,var)
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return mu, var, phi
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def plot(self,samples=0,plot_limits=None,which_data='all',which_functions='all',resolution=None):
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"""
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:param samples: the number of a posteriori samples to plot
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:param which_data: which if the training data to plot (default all)
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:type which_data: 'all' or a slice object to slice self.X, self.Y
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:param plot_limits: The limits of the plot. If 1D [xmin,xmax], if 2D [[xmin,ymin],[xmax,ymax]]. Defaluts to data limits
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:param which_functions: which of the kernel functions to plot (additively)
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:type which_functions: list of bools
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:param resolution: the number of intervals to sample the GP on. Defaults to 200 in 1D and 50 (a 50x50 grid) in 2D
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Plot the posterior of the GP.
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- In one dimension, the function is plotted with a shaded region identifying two standard deviations.
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- In two dimsensions, a contour-plot shows the mean predicted function
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- In higher dimensions, we've no implemented this yet !TODO!
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Can plot only part of the data and part of the posterior functions using which_data and which_functions
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"""
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if which_functions=='all':
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which_functions = [True]*self.kern.Nparts
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if which_data=='all':
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which_data = slice(None)
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X = self.X[which_data,:]
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Y = self.Y[which_data,:]
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Xorig = X*self._Xstd + self._Xmean
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Yorig = Y*self._Ystd + self._Ymean if not self.EP else self.likelihood.Y
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if plot_limits is None:
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xmin,xmax = Xorig.min(0),Xorig.max(0)
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xmin, xmax = xmin-0.2*(xmax-xmin), xmax+0.2*(xmax-xmin)
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elif len(plot_limits)==2:
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xmin, xmax = plot_limits
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else:
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raise ValueError, "Bad limits for plotting"
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if self.X.shape[1]==1:
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Xnew = np.linspace(xmin,xmax,resolution or 200)[:,None]
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m,v,phi = self.predict(Xnew,slices=which_functions)
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if self.EP:
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pb.subplot(211)
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gpplot(Xnew,m,v)
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if samples:
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s = np.random.multivariate_normal(m.flatten(),v,samples)
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pb.plot(Xnew.flatten(),s.T, alpha = 0.4, c='#3465a4', linewidth = 0.8)
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if not self.EP:
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pb.plot(Xorig,Yorig,'kx',mew=1.5)
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pb.xlim(xmin,xmax)
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else:
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pb.xlim(xmin,xmax)
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pb.subplot(212)
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self.likelihood.plot1Db(self.X,Xnew,phi)
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pb.xlim(xmin,xmax)
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elif self.X.shape[1]==2:
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resolution = 50 or resolution
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xx,yy = np.mgrid[xmin[0]:xmax[0]:1j*resolution,xmin[1]:xmax[1]:1j*resolution]
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Xtest = np.vstack((xx.flatten(),yy.flatten())).T
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zz,vv,phi = self.predict(Xtest,slices=which_functions)
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zz = zz.reshape(resolution,resolution)
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pb.contour(xx,yy,zz,vmin=zz.min(),vmax=zz.max(),cmap=pb.cm.jet)
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pb.scatter(Xorig[:,0],Xorig[:,1],40,Yorig,linewidth=0,cmap=pb.cm.jet,vmin=zz.min(),vmax=zz.max())
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pb.xlim(xmin[0],xmax[0])
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pb.ylim(xmin[1],xmax[1])
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else:
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raise NotImplementedError, "Cannot plot GPs with more than two input dimensions"
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