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317 lines
14 KiB
Python
317 lines
14 KiB
Python
# 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 linalg
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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, tdot
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from ..util.plot import gpplot, x_frame1D, x_frame2D, Tango
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from ..likelihoods import EP, Laplace
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class GP(model):
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"""
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Gaussian Process model for regression and EP
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:param X: input observations
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:param kernel: a GPy kernel, defaults to rbf+white
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:parm likelihood: a GPy likelihood
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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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:rtype: model object
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:param epsilon_ep: convergence criterion for the Expectation Propagation algorithm, defaults to 0.1
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:param powerep: power-EP parameters [$\eta$,$\delta$], defaults to [1.,1.]
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:type powerep: list
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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, likelihood, kernel, normalize_X=False):
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self.has_uncertain_inputs=False
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# parse arguments
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self.X = X
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assert len(self.X.shape) == 2
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self.N, self.Q = self.X.shape
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assert isinstance(kernel, kern.kern)
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self.kern = kernel
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self.likelihood = likelihood
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assert self.X.shape[0] == self.likelihood.data.shape[0]
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self.N, self.D = self.likelihood.data.shape
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# here's some simple normalization for the inputs
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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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if not hasattr(self,'has_uncertain_inputs'):
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self.has_uncertain_inputs = False
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model.__init__(self)
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def dL_dZ(self):
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"""
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TODO: one day we might like to learn Z by gradient methods?
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"""
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#FIXME: this doesn;t live here.
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return np.zeros_like(self.Z)
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def _set_params(self, p):
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self.kern._set_params_transformed(p[:self.kern.Nparam_transformed()])
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# self.likelihood._set_params(p[self.kern.Nparam:]) # test by Nicolas
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self.likelihood._set_params(p[self.kern.Nparam_transformed():]) # test by Nicolas
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if isinstance(self.likelihood, Laplace):
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print "Updating approx: ", p
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self.likelihood.fit_full(self.kern.K(self.X))
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self.likelihood._set_params(self.likelihood._get_params())
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self.K = self.kern.K(self.X)
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self.K += self.likelihood.covariance_matrix
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self.Ki, self.L, self.Li, self.K_logdet = pdinv(self.K)
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# the gradient of the likelihood wrt the covariance matrix
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if self.likelihood.YYT is None:
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#alpha = np.dot(self.Ki, self.likelihood.Y)
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alpha,_ = linalg.lapack.flapack.dpotrs(self.L, self.likelihood.Y,lower=1)
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self.dL_dK = 0.5 * (tdot(alpha) - self.D * self.Ki)
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else:
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#tmp = mdot(self.Ki, self.likelihood.YYT, self.Ki)
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tmp, _ = linalg.lapack.flapack.dpotrs(self.L, np.asfortranarray(self.likelihood.YYT), lower=1)
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tmp, _ = linalg.lapack.flapack.dpotrs(self.L, np.asfortranarray(tmp.T), lower=1)
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self.dL_dK = 0.5 * (tmp - self.D * self.Ki)
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def _get_params(self):
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return np.hstack((self.kern._get_params_transformed(), self.likelihood._get_params()))
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def _get_param_names(self):
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return self.kern._get_param_names_transformed() + self.likelihood._get_param_names()
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def _update_params_callback(self, p):
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#parameters will be in transformed space
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self.kern._set_params_transformed(p[:self.kern.Nparam_transformed()])
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#set_params_transformed for likelihood doesn't exist?
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self.likelihood._set_params(p[self.kern.Nparam_transformed():])
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#update the likelihood approximation within the optimisation with the current parameters
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self.update_likelihood_approximation()
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def update_likelihood_approximation(self):
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"""
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Approximates a non-gaussian likelihood using Expectation Propagation
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For a Gaussian likelihood, no iteration is required:
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this function does nothing
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"""
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self.likelihood.fit_full(self.kern.K(self.X))
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self._set_params(self._get_params()) # update the GP
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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.likelihood.YYT is None:
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tmp, _ = linalg.lapack.flapack.dtrtrs(self.L, np.asfortranarray(self.likelihood.Y), lower=1)
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return -0.5 * np.sum(np.square(tmp))
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#return -0.5 * np.sum(np.square(np.dot(self.Li, self.likelihood.Y)))
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else:
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return -0.5 * np.sum(np.multiply(self.Ki, self.likelihood.YYT))
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def log_likelihood(self):
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"""
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The log marginal likelihood of the GP.
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For an EP model, can be written as the log likelihood of a regression
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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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l = -0.5 * self.D * self.K_logdet + self._model_fit_term() + self.likelihood.Z
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print "Log likelihood: ", l
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return l
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def _log_likelihood_gradients(self):
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"""
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The gradient of all parameters.
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Note, we use the chain rule: dL_dtheta = dL_dK * d_K_dtheta
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"""
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dL_dthetaK = self.kern.dK_dtheta(dL_dK=self.dL_dK, X=self.X)
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if isinstance(self.likelihood, Laplace):
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dL_dthetaK_explicit = dL_dthetaK
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#Need to pass in a matrix of ones to get access to raw dK_dthetaK values without being chained
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fake_dL_dKs = np.eye(self.dL_dK.shape[0]) #FIXME: Check this is right...
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dK_dthetaK = self.kern.dK_dtheta(dL_dK=fake_dL_dKs, X=self.X)
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dL_dthetaK = self.likelihood._Kgradients(dL_d_K_Sigma=self.dL_dK, dK_dthetaK=dK_dthetaK)
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dL_dthetaL = self.likelihood._gradients(partial=np.diag(self.dL_dK))
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print "Stacked dL_dthetaK, dL_dthetaL: ", np.hstack((dL_dthetaK, dL_dthetaL))
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else:
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dL_dthetaL = self.likelihood._gradients(partial=np.diag(self.dL_dK))
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print "Stacked dL_dthetaK, dL_dthetaL: ", np.hstack((dL_dthetaK, dL_dthetaL))
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return np.hstack((dL_dthetaK, dL_dthetaL))
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#return np.hstack((self.kern.dK_dtheta(dL_dK=self.dL_dK, X=self.X), self.likelihood._gradients(partial=np.diag(self.dL_dK))))
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def _raw_predict(self, _Xnew, which_parts='all', full_cov=False,stop=False):
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"""
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Internal helper function for making predictions, does not account
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for normalization or likelihood
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"""
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Kx = self.kern.K(_Xnew,self.X,which_parts=which_parts).T
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#KiKx = np.dot(self.Ki, Kx)
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KiKx, _ = linalg.lapack.flapack.dpotrs(self.L, np.asfortranarray(Kx), lower=1)
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mu = np.dot(KiKx.T, self.likelihood.Y)
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if full_cov:
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Kxx = self.kern.K(_Xnew, which_parts=which_parts)
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var = Kxx - np.dot(KiKx.T, Kx)
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else:
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Kxx = self.kern.Kdiag(_Xnew, which_parts=which_parts)
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var = Kxx - np.sum(np.multiply(KiKx, Kx), 0)
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var = var[:, None]
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if stop:
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debug_this
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return mu, var
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def predict(self, Xnew, which_parts='all', 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 which_parts: specifies which outputs kernel(s) to use in prediction
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:type which_parts: ('all', list of bools)
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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 1 if full_cov=False, Nnew x Nnew otherwise
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:rtype: lower and upper boundaries of the 95% confidence intervals, Numpy arrays, Nnew x self.D
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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 normalizations of the output dimensions.
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"""
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# normalize X values
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Xnew = (Xnew.copy() - self._Xmean) / self._Xstd
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mu, var = self._raw_predict(Xnew, which_parts, full_cov)
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# now push through likelihood
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mean, var, _025pm, _975pm = self.likelihood.predictive_values(mu, var, full_cov)
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return mean, var, _025pm, _975pm
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def plot_f(self, samples=0, plot_limits=None, which_data='all', which_parts='all', resolution=None, full_cov=False):
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"""
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Plot the GP's view of the world, where the data is normalized and the
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likelihood is Gaussian.
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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_parts: which of the kernel functions to plot (additively)
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:type which_parts: 'all', or 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
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using which_data and which_functions
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"""
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if which_data == 'all':
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which_data = slice(None)
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if self.X.shape[1] == 1:
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Xnew, xmin, xmax = x_frame1D(self.X, plot_limits=plot_limits)
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if samples == 0:
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m, v = self._raw_predict(Xnew, which_parts=which_parts)
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gpplot(Xnew, m, m - 2 * np.sqrt(v), m + 2 * np.sqrt(v))
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pb.plot(self.X[which_data], self.likelihood.Y[which_data], 'kx', mew=1.5)
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else:
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m, v = self._raw_predict(Xnew, which_parts=which_parts, full_cov=True)
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Ysim = np.random.multivariate_normal(m.flatten(), v, samples)
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gpplot(Xnew, m, m - 2 * np.sqrt(np.diag(v)[:, None]), m + 2 * np.sqrt(np.diag(v))[:, None])
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for i in range(samples):
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pb.plot(Xnew, Ysim[i, :], Tango.colorsHex['darkBlue'], linewidth=0.25)
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pb.plot(self.X[which_data], self.likelihood.Y[which_data], 'kx', mew=1.5)
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pb.xlim(xmin, xmax)
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ymin, ymax = min(np.append(self.likelihood.Y, m - 2 * np.sqrt(np.diag(v)[:, None]))), max(np.append(self.likelihood.Y, m + 2 * np.sqrt(np.diag(v)[:, None])))
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ymin, ymax = ymin - 0.1 * (ymax - ymin), ymax + 0.1 * (ymax - ymin)
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pb.ylim(ymin, ymax)
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if hasattr(self, 'Z'):
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pb.plot(self.Z, self.Z * 0 + pb.ylim()[0], 'r|', mew=1.5, markersize=12)
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elif self.X.shape[1] == 2:
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resolution = resolution or 50
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Xnew, xmin, xmax, xx, yy = x_frame2D(self.X, plot_limits, resolution)
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m, v = self._raw_predict(Xnew, which_parts=which_parts)
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m = m.reshape(resolution, resolution).T
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pb.contour(xx, yy, m, vmin=m.min(), vmax=m.max(), cmap=pb.cm.jet)
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pb.scatter(Xorig[:, 0], Xorig[:, 1], 40, Yorig, linewidth=0, cmap=pb.cm.jet, vmin=m.min(), vmax=m.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 define a frame with more than two input dimensions"
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def plot(self, samples=0, plot_limits=None, which_data='all', which_parts='all', resolution=None, levels=20):
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"""
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TODO: Docstrings!
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:param levels: for 2D plotting, the number of contour levels to use
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"""
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# TODO include samples
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if which_data == 'all':
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which_data = slice(None)
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if self.X.shape[1] == 1:
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Xu = self.X * self._Xstd + self._Xmean # NOTE self.X are the normalized values now
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Xnew, xmin, xmax = x_frame1D(Xu, plot_limits=plot_limits)
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m, var, lower, upper = self.predict(Xnew, which_parts=which_parts)
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gpplot(Xnew, m, lower, upper)
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pb.plot(Xu[which_data], self.likelihood.data[which_data], 'kx', mew=1.5)
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if self.has_uncertain_inputs:
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pb.errorbar(Xu[which_data, 0], self.likelihood.data[which_data, 0],
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xerr=2 * np.sqrt(self.X_variance[which_data, 0]),
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ecolor='k', fmt=None, elinewidth=.5, alpha=.5)
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ymin, ymax = min(np.append(self.likelihood.data, lower)), max(np.append(self.likelihood.data, upper))
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ymin, ymax = ymin - 0.1 * (ymax - ymin), ymax + 0.1 * (ymax - ymin)
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pb.xlim(xmin, xmax)
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pb.ylim(ymin, ymax)
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if hasattr(self, 'Z'):
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Zu = self.Z * self._Xstd + self._Xmean
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pb.plot(Zu, Zu * 0 + pb.ylim()[0], 'r|', mew=1.5, markersize=12)
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# pb.errorbar(self.X[:,0], pb.ylim()[0]+np.zeros(self.N), xerr=2*np.sqrt(self.X_variance.flatten()))
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elif self.X.shape[1] == 2: # FIXME
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resolution = resolution or 50
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Xnew, xx, yy, xmin, xmax = x_frame2D(self.X, plot_limits, resolution)
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x, y = np.linspace(xmin[0], xmax[0], resolution), np.linspace(xmin[1], xmax[1], resolution)
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m, var, lower, upper = self.predict(Xnew, which_parts=which_parts)
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m = m.reshape(resolution, resolution).T
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pb.contour(x, y, m, levels, vmin=m.min(), vmax=m.max(), cmap=pb.cm.jet)
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Yf = self.likelihood.Y.flatten()
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pb.scatter(self.X[:, 0], self.X[:, 1], 40, Yf, cmap=pb.cm.jet, vmin=m.min(), vmax=m.max(), linewidth=0.)
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pb.xlim(xmin[0], xmax[0])
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pb.ylim(xmin[1], xmax[1])
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if hasattr(self, 'Z'):
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pb.plot(self.Z[:, 0], self.Z[:, 1], 'wo')
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else:
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raise NotImplementedError, "Cannot define a frame with more than two input dimensions"
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