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https://github.com/SheffieldML/GPy.git
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some tidying in the likelihood classes
This commit is contained in:
James Hensman
parent
3a558d8244
commit
7dfbcebb87
6 changed files with 364 additions and 369 deletions
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@ -18,12 +18,10 @@ class EP:
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self.likelihood_function = likelihood_function
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self.epsilon = epsilon
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self.eta, self.delta = power_ep
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self.jitter = 1e-12 # TODO: is this needed?
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self.is_heteroscedastic = True
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"""
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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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"""
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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.tau_tilde = np.zeros(self.N)
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self.v_tilde = np.zeros(self.N)
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@ -32,8 +30,11 @@ class EP:
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mu_tilde = self.v_tilde/self.tau_tilde #When calling EP, this variable is used instead of Y in the GP model
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sigma_sum = 1./self.tau_ + 1./self.tau_tilde
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mu_diff_2 = (self.v_/self.tau_ - mu_tilde)**2
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Z_ep = np.sum(np.log(self.Z_hat)) + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum) #Normalization constant
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self.Y, self.beta, self.Z = self.tau_tilde[:,None], mu_tilde[:,None], Z_ep
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self.Z = np.sum(np.log(self.Z_hat)) + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum) #Normalization constant, aka Z_ep
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self.Y = mu_tilde[:,None]
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self.precsion = self.tau_tilde
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self.covariance_matrix = np.diag(1./self.precision)
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def fit_full(self,K):
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"""
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@ -2,6 +2,7 @@ import numpy as np
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class Gaussian:
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def __init__(self,data,variance=1.,normalize=False):
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self.is_heteroscedastic = False
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self.data = data
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self.N,D = data.shape
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self.Z = 0. # a correction factor which accounts for the approximation made
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@ -19,6 +20,7 @@ class Gaussian:
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self.YYT = np.dot(self.Y,self.Y.T)
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self._set_params(np.asarray(variance))
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def _get_params(self):
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return np.asarray(self._variance)
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@ -27,7 +29,8 @@ class Gaussian:
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def _set_params(self,x):
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self._variance = x
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self.variance = np.eye(self.N)*self._variance
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self.covariance_matrix = np.eye(self.N)*self._variance
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self.precision = 1./self._variance
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def fit(self):
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"""
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@ -8,18 +8,18 @@ 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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class likelihood_function:
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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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..Note:: Y values allowed depend on the likelihood_function used
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"""
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def __init__(self,location=0,scale=1):
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self.location = location
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self.scale = scale
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class probit(likelihood):
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class probit(likelihood_function):
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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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@ -29,7 +29,7 @@ class probit(likelihood):
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$$
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"""
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def __init__(self,location=0,scale=1):
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likelihood.__init__(self,Y,location,scale)
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likelihood_function.__init__(self,Y,location,scale)
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def moments_match(self,data_i,tau_i,v_i):
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"""
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@ -64,7 +64,7 @@ class probit(likelihood):
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def _log_likelihood_gradients():
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return np.zeros(0) # there are no parameters of whcih to compute the gradients
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class poisson(likelihood):
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class poisson(likelihood_function):
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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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@ -75,7 +75,7 @@ class poisson(likelihood):
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"""
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def __init__(self,Y,location=0,scale=1):
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assert len(Y[Y<0]) == 0, "Output cannot have negative values"
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likelihood.__init__(self,Y,location,scale)
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likelihood_function.__init__(self,Y,location,scale)
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def moments_match(self,i,tau_i,v_i):
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"""
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@ -160,7 +160,7 @@ class poisson(likelihood):
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if Z is not None:
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pb.plot(Z,Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
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class gaussian(likelihood):
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class gaussian(likelihood_function):
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"""
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Gaussian likelihood
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Y is expected to take values in (-inf,inf)
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@ -6,37 +6,36 @@ import pylab as pb
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from ..util.linalg import mdot, jitchol, chol_inv, pdinv
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from ..util.plot import gpplot
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from .. import kern
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from ..inference.likelihoods import likelihood
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from GP import GP
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#Still TODO:
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# make use of slices properly (kernel can now do this)
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# enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array)
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class sparse_GP(GP):
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"""
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Variational sparse GP model (Regression)
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Variational sparse GP model
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:param X: inputs
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:type X: np.ndarray (N x Q)
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:param Y: observed data
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:type Y: np.ndarray of observations (N x D)
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:param likelihood: a likelihood instance, containing the observed data
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:type likelihood: GPy.likelihood.(Gaussian | EP)
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:param kernel : the kernel/covariance function. See link kernels
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:type kernel: a GPy kernel
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:param Z: inducing inputs (optional, see note)
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:type Z: np.ndarray (M x Q) | None
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:param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance)
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:type X_uncertainty: np.ndarray (N x Q) | None
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:param Z: inducing inputs (optional, see note)
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:type Z: np.ndarray (M x Q) | None
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:param Zslices: slices for the inducing inputs (see slicing TODO: link)
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:param M : Number of inducing points (optional, default 10. Ignored if Z is not None)
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:type M: int
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:param beta: noise precision. TODO> ignore beta if doing EP
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:type beta: float
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:param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales)
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:type normalize_(X|Y): bool
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"""
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def __init__(self,X,Y=None,kernel=None,X_uncertainty=None,beta=100.,Z=None,Zslices=None,M=10,normalize_X=False,normalize_Y=False,likelihood=None,method_ep='DTC',epsilon_ep=1e-3,power_ep=[1.,1.]):
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def __init__(self,X,likelihood,kernel, X_uncertainty=None, Z=None,Zslices=None,M=10,normalize_X=False):
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self.scale_factor = 1000.0# a scaling factor to help keep the algorithm stable
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if Z is None:
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self.Z = np.random.permutation(X.copy())[:M]
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@ -52,140 +51,91 @@ class sparse_GP(GP):
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self.has_uncertain_inputs=True
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self.X_uncertainty = X_uncertainty
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GP.__init__(self, X=X, Y=Y, kernel=kernel, normalize_X=normalize_X, normalize_Y=normalize_Y,likelihood=likelihood,epsilon_ep=epsilon_ep,power_ep=power_ep)
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GP.__init__(self, X, Y, kernel=kernel, normalize_X=normalize_X, Xslices=Xslices)
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#normalise X uncertainty also
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if self.has_uncertain_inputs:
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self.X_uncertainty /= np.square(self._Xstd)
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if not self.EP:
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self.trYYT = np.sum(np.square(self.Y))
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else:
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self.method_ep = method_ep
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#normalise X uncertainty also
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if self.has_uncertain_inputs:
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self.X_uncertainty /= np.square(self._Xstd)
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def _set_params(self, p):
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self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
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if not self.EP:
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self.beta = p[self.M*self.Q]
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self.kern._set_params(p[self.Z.size + 1:])
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else:
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self.kern._set_params(p[self.Z.size:])
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if self.Y is None:
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self.Y = np.ones([self.N,1])
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self._compute_kernel_matrices()
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self._computations()
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def _get_params(self):
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if not self.EP:
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return np.hstack([self.Z.flatten(),self.beta,self.kern._get_params_transformed()])
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else:
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return np.hstack([self.Z.flatten(),self.kern._get_params_transformed()])
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def _get_param_names(self):
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if not self.EP:
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return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + ['noise_precision']+self.kern._get_param_names_transformed()
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else:
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return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + self.kern._get_param_names_transformed()
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def _compute_kernel_matrices(self):
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# kernel computations, using BGPLVM notation
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#TODO: slices for psi statistics (easy enough)
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self.Kmm = self.kern.K(self.Z)
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if self.has_uncertain_inputs:
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if not self.EP:
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self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty)#.sum() NOTE psi0 is now a vector
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self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
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self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
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#self.psi2_beta_scaled = ?
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else:
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raise NotImplementedError, "uncertain_inputs not yet supported for EP"
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else:
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self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices)#.sum()
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self.psi1 = self.kern.K(self.Z,self.X)
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self.psi2 = np.dot(self.psi1,self.psi1.T)
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self.psi2_beta_scaled = np.dot(self.psi1,self.beta*self.psi1.T)
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def _computations(self):
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# TODO find routine to multiply triangular matrices
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self.V = self.beta*self.Y
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#TODO: slices for psi statistics (easy enough)
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sf = self.scale_factor
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sf2 = sf**2
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# kernel computations, using BGPLVM notation
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self.Kmm = self.kern.K(self.Z)
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if self.has_uncertain_inputs:
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self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
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self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
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self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
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self.psi2_beta_scaled = (self.psi2*(self.beta/sf2)).sum(0)
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else:
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self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices).sum()
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self.psi1 = self.kern.K(self.Z,self.X)
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tmp = self.psi1*(np.sqrt(self.likelihood.beta)/sf)
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self.psi2_beta_scaled = np.dot(tmp,tmp.T)
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self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)#+np.eye(self.M)*1e-3)
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self.V = (self.likelihood.beta/self.scale_factor)*self.Y
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self.A = mdot(self.Lmi, self.psi2_beta_scaled, self.Lmi.T)
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self.B = np.eye(self.M)/sf2 + self.A
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self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
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self.psi1V = np.dot(self.psi1, self.V)
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self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
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self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)
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self.A = mdot(self.Lmi, self.psi2_beta_scaled, self.Lmi.T)
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self.B = np.eye(self.M) + self.A
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self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
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self.LLambdai = np.dot(self.LBi, self.Lmi)
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self.LBL_inv = mdot(self.Lmi.T, self.Bi, self.Lmi)
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self.C = mdot(self.LLambdai, self.psi1V)
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self.G = mdot(self.LBL_inv, self.psi1VVpsi1, self.LBL_inv.T)
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self.trace_K_beta_scaled = (self.psi0*self.beta).sum() - np.trace(self.A)
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if not self.EP:
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self.trace_K = self.psi0.sum() - np.trace(self.A)/self.beta
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self.C = mdot(self.Lmi.T, self.Bi, self.Lmi)
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self.E = mdot(self.C, self.psi1VVpsi1/sf2, self.C.T)
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# Compute dL_dpsi
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self.dL_dpsi1 = mdot(self.LLambdai.T,self.C,self.V.T)
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if not self.EP:
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self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
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if self.has_uncertain_inputs:
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self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
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else:
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self.dL_dpsi2_ = - 0.5 * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
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else:
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self.dL_dpsi0 = - 0.5 * self.D * self.beta.flatten()
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if not self.has_uncertain_inputs:
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self.dL_dpsi2_ = - 0.5 * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
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# Compute dL_dpsi # FIXME
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self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
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self.dL_dpsi1 = mdot(self.V, self.psi1V.T,self.C).T
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self.dL_dpsi2 = 0.5 * self.beta * self.D * self.Kmmi[None,:,:] # dB
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self.dL_dpsi2 += - 0.5 * self.beta/sf2 * self.D * self.C[None,:,:] # dC
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self.dL_dpsi2 += - 0.5 * self.beta * self.E[None,:,:] # dD
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# Compute dL_dKmm
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self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi) # dB
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self.dL_dKmm += -0.5 * self.D * (- self.LBL_inv - 2.*mdot(self.LBL_inv, self.psi2_beta_scaled, self.Kmmi) + self.Kmmi) # dC
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self.dL_dKmm += np.dot(np.dot(self.G,self.psi2_beta_scaled) - np.dot(self.LBL_inv, self.psi1VVpsi1), self.Kmmi) + 0.5*self.G # dE
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self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi)*sf2 # dB
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self.dL_dKmm += -0.5 * self.D * (- self.C/sf2 - 2.*mdot(self.C, self.psi2_beta_scaled, self.Kmmi) + self.Kmmi) # dC
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self.dL_dKmm += np.dot(np.dot(self.E*sf2, self.psi2_beta_scaled) - np.dot(self.C, self.psi1VVpsi1), self.Kmmi) + 0.5*self.E # dD
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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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if self.method_ep == 'DTC':
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self.ep_approx = DTC(self.Kmm,self.likelihood,self.psi1,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
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elif self.method_ep == 'FITC':
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self.ep_approx = FITC(self.Kmm,self.likelihood,self.psi1,self.psi0,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
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else:
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self.ep_approx = Full(self.X,self.likelihood,self.kernel,inducing=None,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
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self.beta, self.Y, self.Z_ep = self.ep_approx.fit_EP()
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self.trbetaYYT = np.sum(np.square(self.Y)*self.beta)
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def _set_params(self, p):
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self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
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self.beta = p[self.M*self.Q] # FIXME
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self.kern._set_params(p[self.Z.size + 1:])
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self._computations()
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def _get_params(self):
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return np.hstack([self.Z.flatten(),GP._get_params(self))
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def _get_param_names(self):
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return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + GP._get_param_names(self)
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def log_likelihood(self):
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"""
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Compute the (lower bound on the) log marginal likelihood
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"""
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if not self.EP:
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A = -0.5*self.N*self.D*(np.log(2.*np.pi) - np.log(self.beta))
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D = -0.5*self.beta*self.trYYT
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else:
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A = -0.5*self.D*(self.N*np.log(2.*np.pi) - np.sum(np.log(self.beta)))
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D = -0.5*self.trbetaYYT
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B = -0.5*self.D*self.trace_K_beta_scaled
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C = -0.5*self.D * self.B_logdet
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E = +0.5*np.sum(self.psi1VVpsi1 * self.LBL_inv)
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return A+B+C+D+E
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""" Compute the (lower bound on the) log marginal likelihood """
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sf2 = self.scale_factor**2
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A = -0.5*self.N*self.D*(np.log(2.*np.pi) - np.log(self.beta)) -0.5*self.beta*self.trYYT # FIXME
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B = -0.5*self.D*(self.beta*self.psi0-np.trace(self.A)*sf2)# FIXME
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C = -0.5*self.D * (self.B_logdet + self.M*np.log(sf2))
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D = +0.5*np.sum(self.psi1VVpsi1 * self.C)
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return A+B+C+D
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def _log_likelihood_gradients(self):
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return np.hstack([self.dL_dZ().flatten(), GP._log_likelihood_gradients(self)])
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||||
# FIXME: move this into the lieklihood class
|
||||
def dL_dbeta(self):
|
||||
"""
|
||||
Compute the gradient of the log likelihood wrt beta.
|
||||
"""
|
||||
#TODO: suport heteroscedatic noise
|
||||
dA_dbeta = 0.5 * self.N*self.D/self.beta
|
||||
dB_dbeta = - 0.5 * self.D * self.trace_K
|
||||
sf2 = self.scale_factor**2
|
||||
dA_dbeta = 0.5 * self.N*self.D/self.beta - 0.5 * self.trYYT
|
||||
dB_dbeta = - 0.5 * self.D * (self.psi0 - np.trace(self.A)/self.beta*sf2)
|
||||
dC_dbeta = - 0.5 * self.D * np.sum(self.Bi*self.A)/self.beta
|
||||
dD_dbeta = - 0.5 * self.trYYT
|
||||
tmp = mdot(self.LBi.T, self.LLambdai, self.psi1V)
|
||||
dE_dbeta = (np.sum(np.square(self.C)) - 0.5 * np.sum(self.A * np.dot(tmp, tmp.T)))/self.beta
|
||||
dD_dbeta = np.sum((self.C - 0.5 * mdot(self.C,self.psi2_beta_scaled,self.C) ) * self.psi1VVpsi1 )/self.beta
|
||||
|
||||
return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta + dE_dbeta)
|
||||
return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta)
|
||||
|
||||
def dL_dtheta(self):
|
||||
"""
|
||||
|
|
@ -195,10 +145,10 @@ class sparse_GP(GP):
|
|||
if self.has_uncertain_inputs:
|
||||
dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z,self.X,self.X_uncertainty)
|
||||
dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
|
||||
dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # for multiple_beta, dL_dpsi2 will be a different shape
|
||||
dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2,self.dL_dpsi1.T, self.Z,self.X, self.X_uncertainty)
|
||||
else:
|
||||
#re-cast computations in psi2 back to psi1:
|
||||
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2_,self.beta.T*self.psi1) #dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1)
|
||||
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2.sum(0),self.psi1)
|
||||
dL_dtheta += self.kern.dK_dtheta(dL_dpsi1,self.Z,self.X)
|
||||
dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X)
|
||||
|
||||
|
|
@ -208,48 +158,36 @@ class sparse_GP(GP):
|
|||
"""
|
||||
The derivative of the bound wrt the inducing inputs Z
|
||||
"""
|
||||
dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z,)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
|
||||
dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
|
||||
if self.has_uncertain_inputs:
|
||||
dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
|
||||
dL_dZ += self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty)
|
||||
dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1,self.Z,self.X, self.X_uncertainty)
|
||||
dL_dZ += 2.*self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # 'stripes'
|
||||
else:
|
||||
#re-cast computations in psi2 back to psi1:
|
||||
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2_,self.beta.T*self.psi1)#dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1)
|
||||
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2.sum(0),self.psi1)
|
||||
dL_dZ += self.kern.dK_dX(dL_dpsi1,self.Z,self.X)
|
||||
return dL_dZ
|
||||
|
||||
def _log_likelihood_gradients(self):
|
||||
if not self.EP:
|
||||
return np.hstack([self.dL_dZ().flatten(), self.dL_dbeta(), self.dL_dtheta()])
|
||||
else:
|
||||
return np.hstack([self.dL_dZ().flatten(), self.dL_dtheta()])
|
||||
|
||||
def _raw_predict(self, Xnew, slices, full_cov=False):
|
||||
"""Internal helper function for making predictions, does not account for normalisation"""
|
||||
|
||||
Kx = self.kern.K(self.Z, Xnew)
|
||||
mu = mdot(Kx.T, self.LBL_inv, self.psi1V)
|
||||
phi = None
|
||||
mu = mdot(Kx.T, self.C/self.scale_factor, self.psi1V)
|
||||
|
||||
if full_cov:
|
||||
Kxx = self.kern.K(Xnew)
|
||||
var = Kxx - mdot(Kx.T, (self.Kmmi - self.LBL_inv), Kx)
|
||||
if not self.EP:
|
||||
var += np.eye(Xnew.shape[0])/self.beta
|
||||
else:
|
||||
raise NotImplementedError, "full_cov = True not implemented for EP"
|
||||
var = Kxx - mdot(Kx.T, (self.Kmmi - self.C/self.scale_factor**2), Kx)
|
||||
else:
|
||||
Kxx = self.kern.Kdiag(Xnew)
|
||||
var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.LBL_inv, Kx),0)
|
||||
if not self.EP:
|
||||
var += 1./self.beta
|
||||
else:
|
||||
phi = self.likelihood.predictive_mean(mu,var)
|
||||
return mu,var,phi
|
||||
var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.C/self.scale_factor**2, Kx),0)
|
||||
|
||||
return mu,var
|
||||
|
||||
def plot(self, *args, **kwargs):
|
||||
"""
|
||||
Plot the fitted model: just call the GP_regression plot function and then add inducing inputs
|
||||
"""
|
||||
GP.plot(self,*args,**kwargs)
|
||||
GP_regression.plot(self,*args,**kwargs)
|
||||
if self.Q==1:
|
||||
pb.plot(self.Z,self.Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
|
||||
if self.has_uncertain_inputs:
|
||||
|
|
|
|||
258
GPy/models/sparse_GP_old.py
Normal file
258
GPy/models/sparse_GP_old.py
Normal file
|
|
@ -0,0 +1,258 @@
|
|||
# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
|
||||
# Licensed under the BSD 3-clause license (see LICENSE.txt)
|
||||
|
||||
import numpy as np
|
||||
import pylab as pb
|
||||
from ..util.linalg import mdot, jitchol, chol_inv, pdinv
|
||||
from ..util.plot import gpplot
|
||||
from .. import kern
|
||||
from GP import GP
|
||||
|
||||
|
||||
#Still TODO:
|
||||
# make use of slices properly (kernel can now do this)
|
||||
# enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array)
|
||||
|
||||
class sparse_GP(GP):
|
||||
"""
|
||||
Variational sparse GP model (Regression)
|
||||
|
||||
:param X: inputs
|
||||
:type X: np.ndarray (N x Q)
|
||||
:param Y: observed data
|
||||
:type Y: np.ndarray of observations (N x D)
|
||||
:param kernel : the kernel/covariance function. See link kernels
|
||||
:type kernel: a GPy kernel
|
||||
:param Z: inducing inputs (optional, see note)
|
||||
:type Z: np.ndarray (M x Q) | None
|
||||
:param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance)
|
||||
:type X_uncertainty: np.ndarray (N x Q) | None
|
||||
:param Zslices: slices for the inducing inputs (see slicing TODO: link)
|
||||
:param M : Number of inducing points (optional, default 10. Ignored if Z is not None)
|
||||
:type M: int
|
||||
:param beta: noise precision. TODO> ignore beta if doing EP
|
||||
:type beta: float
|
||||
:param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales)
|
||||
:type normalize_(X|Y): bool
|
||||
"""
|
||||
|
||||
def __init__(self,X,Y=None,kernel=None,X_uncertainty=None,beta=100.,Z=None,Zslices=None,M=10,normalize_X=False,normalize_Y=False,likelihood=None,method_ep='DTC',epsilon_ep=1e-3,power_ep=[1.,1.]):
|
||||
|
||||
if Z is None:
|
||||
self.Z = np.random.permutation(X.copy())[:M]
|
||||
self.M = M
|
||||
else:
|
||||
assert Z.shape[1]==X.shape[1]
|
||||
self.Z = Z
|
||||
self.M = Z.shape[0]
|
||||
if X_uncertainty is None:
|
||||
self.has_uncertain_inputs=False
|
||||
else:
|
||||
assert X_uncertainty.shape==X.shape
|
||||
self.has_uncertain_inputs=True
|
||||
self.X_uncertainty = X_uncertainty
|
||||
|
||||
GP.__init__(self, X=X, Y=Y, kernel=kernel, normalize_X=normalize_X, normalize_Y=normalize_Y,likelihood=likelihood,epsilon_ep=epsilon_ep,power_ep=power_ep)
|
||||
|
||||
#normalise X uncertainty also
|
||||
if self.has_uncertain_inputs:
|
||||
self.X_uncertainty /= np.square(self._Xstd)
|
||||
|
||||
if not self.EP:
|
||||
self.trYYT = np.sum(np.square(self.Y))
|
||||
else:
|
||||
self.method_ep = method_ep
|
||||
|
||||
#normalise X uncertainty also
|
||||
if self.has_uncertain_inputs:
|
||||
self.X_uncertainty /= np.square(self._Xstd)
|
||||
|
||||
def _set_params(self, p):
|
||||
self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
|
||||
if not self.EP:
|
||||
self.beta = p[self.M*self.Q]
|
||||
self.kern._set_params(p[self.Z.size + 1:])
|
||||
else:
|
||||
self.kern._set_params(p[self.Z.size:])
|
||||
if self.Y is None:
|
||||
self.Y = np.ones([self.N,1])
|
||||
self._compute_kernel_matrices()
|
||||
self._computations()
|
||||
|
||||
def _get_params(self):
|
||||
if not self.EP:
|
||||
return np.hstack([self.Z.flatten(),self.beta,self.kern._get_params_transformed()])
|
||||
else:
|
||||
return np.hstack([self.Z.flatten(),self.kern._get_params_transformed()])
|
||||
|
||||
def _get_param_names(self):
|
||||
if not self.EP:
|
||||
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + ['noise_precision']+self.kern._get_param_names_transformed()
|
||||
else:
|
||||
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + self.kern._get_param_names_transformed()
|
||||
|
||||
|
||||
def _compute_kernel_matrices(self):
|
||||
# kernel computations, using BGPLVM notation
|
||||
#TODO: slices for psi statistics (easy enough)
|
||||
|
||||
self.Kmm = self.kern.K(self.Z)
|
||||
if self.has_uncertain_inputs:
|
||||
if not self.EP:
|
||||
self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty)#.sum() NOTE psi0 is now a vector
|
||||
self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
|
||||
self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
|
||||
#self.psi2_beta_scaled = ?
|
||||
else:
|
||||
raise NotImplementedError, "uncertain_inputs not yet supported for EP"
|
||||
else:
|
||||
self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices)#.sum()
|
||||
self.psi1 = self.kern.K(self.Z,self.X)
|
||||
self.psi2 = np.dot(self.psi1,self.psi1.T)
|
||||
self.psi2_beta_scaled = np.dot(self.psi1,self.beta*self.psi1.T)
|
||||
|
||||
def _computations(self):
|
||||
# TODO find routine to multiply triangular matrices
|
||||
self.V = self.beta*self.Y
|
||||
self.psi1V = np.dot(self.psi1, self.V)
|
||||
self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
|
||||
self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)
|
||||
self.A = mdot(self.Lmi, self.psi2_beta_scaled, self.Lmi.T)
|
||||
self.B = np.eye(self.M) + self.A
|
||||
self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
|
||||
self.LLambdai = np.dot(self.LBi, self.Lmi)
|
||||
self.LBL_inv = mdot(self.Lmi.T, self.Bi, self.Lmi)
|
||||
self.C = mdot(self.LLambdai, self.psi1V)
|
||||
self.G = mdot(self.LBL_inv, self.psi1VVpsi1, self.LBL_inv.T)
|
||||
self.trace_K_beta_scaled = (self.psi0*self.beta).sum() - np.trace(self.A)
|
||||
if not self.EP:
|
||||
self.trace_K = self.psi0.sum() - np.trace(self.A)/self.beta
|
||||
|
||||
# Compute dL_dpsi
|
||||
self.dL_dpsi1 = mdot(self.LLambdai.T,self.C,self.V.T)
|
||||
if not self.EP:
|
||||
self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
|
||||
if self.has_uncertain_inputs:
|
||||
self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
|
||||
else:
|
||||
self.dL_dpsi2_ = - 0.5 * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
|
||||
else:
|
||||
self.dL_dpsi0 = - 0.5 * self.D * self.beta.flatten()
|
||||
if not self.has_uncertain_inputs:
|
||||
self.dL_dpsi2_ = - 0.5 * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
|
||||
|
||||
# Compute dL_dKmm
|
||||
self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi) # dB
|
||||
self.dL_dKmm += -0.5 * self.D * (- self.LBL_inv - 2.*mdot(self.LBL_inv, self.psi2_beta_scaled, self.Kmmi) + self.Kmmi) # dC
|
||||
self.dL_dKmm += np.dot(np.dot(self.G,self.psi2_beta_scaled) - np.dot(self.LBL_inv, self.psi1VVpsi1), self.Kmmi) + 0.5*self.G # dE
|
||||
|
||||
def approximate_likelihood(self):
|
||||
assert not isinstance(self.likelihood, gaussian), "EP is only available for non-gaussian likelihoods"
|
||||
if self.method_ep == 'DTC':
|
||||
self.ep_approx = DTC(self.Kmm,self.likelihood,self.psi1,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
|
||||
elif self.method_ep == 'FITC':
|
||||
self.ep_approx = FITC(self.Kmm,self.likelihood,self.psi1,self.psi0,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
|
||||
else:
|
||||
self.ep_approx = Full(self.X,self.likelihood,self.kernel,inducing=None,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
|
||||
self.beta, self.Y, self.Z_ep = self.ep_approx.fit_EP()
|
||||
self.trbetaYYT = np.sum(np.square(self.Y)*self.beta)
|
||||
self._computations()
|
||||
|
||||
def log_likelihood(self):
|
||||
"""
|
||||
Compute the (lower bound on the) log marginal likelihood
|
||||
"""
|
||||
if not self.EP:
|
||||
A = -0.5*self.N*self.D*(np.log(2.*np.pi) - np.log(self.beta))
|
||||
D = -0.5*self.beta*self.trYYT
|
||||
else:
|
||||
A = -0.5*self.D*(self.N*np.log(2.*np.pi) - np.sum(np.log(self.beta)))
|
||||
D = -0.5*self.trbetaYYT
|
||||
B = -0.5*self.D*self.trace_K_beta_scaled
|
||||
C = -0.5*self.D * self.B_logdet
|
||||
E = +0.5*np.sum(self.psi1VVpsi1 * self.LBL_inv)
|
||||
return A+B+C+D+E
|
||||
|
||||
def dL_dbeta(self):
|
||||
"""
|
||||
Compute the gradient of the log likelihood wrt beta.
|
||||
"""
|
||||
#TODO: suport heteroscedatic noise
|
||||
dA_dbeta = 0.5 * self.N*self.D/self.beta
|
||||
dB_dbeta = - 0.5 * self.D * self.trace_K
|
||||
dC_dbeta = - 0.5 * self.D * np.sum(self.Bi*self.A)/self.beta
|
||||
dD_dbeta = - 0.5 * self.trYYT
|
||||
tmp = mdot(self.LBi.T, self.LLambdai, self.psi1V)
|
||||
dE_dbeta = (np.sum(np.square(self.C)) - 0.5 * np.sum(self.A * np.dot(tmp, tmp.T)))/self.beta
|
||||
|
||||
return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta + dE_dbeta)
|
||||
|
||||
def dL_dtheta(self):
|
||||
"""
|
||||
Compute and return the derivative of the log marginal likelihood wrt the parameters of the kernel
|
||||
"""
|
||||
dL_dtheta = self.kern.dK_dtheta(self.dL_dKmm,self.Z)
|
||||
if self.has_uncertain_inputs:
|
||||
dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z,self.X,self.X_uncertainty)
|
||||
dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
|
||||
dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # for multiple_beta, dL_dpsi2 will be a different shape
|
||||
else:
|
||||
#re-cast computations in psi2 back to psi1:
|
||||
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2_,self.beta.T*self.psi1) #dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1)
|
||||
dL_dtheta += self.kern.dK_dtheta(dL_dpsi1,self.Z,self.X)
|
||||
dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X)
|
||||
|
||||
return dL_dtheta
|
||||
|
||||
def dL_dZ(self):
|
||||
"""
|
||||
The derivative of the bound wrt the inducing inputs Z
|
||||
"""
|
||||
dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z,)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
|
||||
if self.has_uncertain_inputs:
|
||||
dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
|
||||
dL_dZ += self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty)
|
||||
else:
|
||||
#re-cast computations in psi2 back to psi1:
|
||||
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2_,self.beta.T*self.psi1)#dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1)
|
||||
dL_dZ += self.kern.dK_dX(dL_dpsi1,self.Z,self.X)
|
||||
return dL_dZ
|
||||
|
||||
def _log_likelihood_gradients(self):
|
||||
if not self.EP:
|
||||
return np.hstack([self.dL_dZ().flatten(), self.dL_dbeta(), self.dL_dtheta()])
|
||||
else:
|
||||
return np.hstack([self.dL_dZ().flatten(), self.dL_dtheta()])
|
||||
|
||||
def _raw_predict(self, Xnew, slices, full_cov=False):
|
||||
"""Internal helper function for making predictions, does not account for normalisation"""
|
||||
Kx = self.kern.K(self.Z, Xnew)
|
||||
mu = mdot(Kx.T, self.LBL_inv, self.psi1V)
|
||||
phi = None
|
||||
if full_cov:
|
||||
Kxx = self.kern.K(Xnew)
|
||||
var = Kxx - mdot(Kx.T, (self.Kmmi - self.LBL_inv), Kx)
|
||||
if not self.EP:
|
||||
var += np.eye(Xnew.shape[0])/self.beta
|
||||
else:
|
||||
raise NotImplementedError, "full_cov = True not implemented for EP"
|
||||
else:
|
||||
Kxx = self.kern.Kdiag(Xnew)
|
||||
var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.LBL_inv, Kx),0)
|
||||
if not self.EP:
|
||||
var += 1./self.beta
|
||||
else:
|
||||
phi = self.likelihood.predictive_mean(mu,var)
|
||||
return mu,var,phi
|
||||
|
||||
def plot(self, *args, **kwargs):
|
||||
"""
|
||||
Plot the fitted model: just call the GP_regression plot function and then add inducing inputs
|
||||
"""
|
||||
GP.plot(self,*args,**kwargs)
|
||||
if self.Q==1:
|
||||
pb.plot(self.Z,self.Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
|
||||
if self.has_uncertain_inputs:
|
||||
pb.errorbar(self.X[:,0], pb.ylim()[0]+np.zeros(self.N), xerr=2*np.sqrt(self.X_uncertainty.flatten()))
|
||||
if self.Q==2:
|
||||
pb.plot(self.Z[:,0],self.Z[:,1],'wo')
|
||||
|
|
@ -1,205 +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 ..util.linalg import mdot, jitchol, chol_inv, pdinv
|
||||
from ..util.plot import gpplot
|
||||
from .. import kern
|
||||
from ..inference.likelihoods import likelihood
|
||||
from GP_regression import GP_regression
|
||||
|
||||
#Still TODO:
|
||||
# make use of slices properly (kernel can now do this)
|
||||
# enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array)
|
||||
|
||||
class sparse_GP_regression(GP_regression):
|
||||
"""
|
||||
Variational sparse GP model (Regression)
|
||||
|
||||
:param X: inputs
|
||||
:type X: np.ndarray (N x Q)
|
||||
:param Y: observed data
|
||||
:type Y: np.ndarray of observations (N x D)
|
||||
:param kernel : the kernel/covariance function. See link kernels
|
||||
:type kernel: a GPy kernel
|
||||
:param Z: inducing inputs (optional, see note)
|
||||
:type Z: np.ndarray (M x Q) | None
|
||||
:param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance)
|
||||
:type X_uncertainty: np.ndarray (N x Q) | None
|
||||
:param Zslices: slices for the inducing inputs (see slicing TODO: link)
|
||||
:param M : Number of inducing points (optional, default 10. Ignored if Z is not None)
|
||||
:type M: int
|
||||
:param beta: noise precision. TODO> ignore beta if doing EP
|
||||
:type beta: float
|
||||
:param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales)
|
||||
:type normalize_(X|Y): bool
|
||||
"""
|
||||
|
||||
def __init__(self,X,Y,kernel=None, X_uncertainty=None, beta=100., Z=None,Zslices=None,M=10,normalize_X=False,normalize_Y=False):
|
||||
self.scale_factor = 1000.0
|
||||
self.beta = beta
|
||||
if Z is None:
|
||||
self.Z = np.random.permutation(X.copy())[:M]
|
||||
self.M = M
|
||||
else:
|
||||
assert Z.shape[1]==X.shape[1]
|
||||
self.Z = Z
|
||||
self.M = Z.shape[0]
|
||||
if X_uncertainty is None:
|
||||
self.has_uncertain_inputs=False
|
||||
else:
|
||||
assert X_uncertainty.shape==X.shape
|
||||
self.has_uncertain_inputs=True
|
||||
self.X_uncertainty = X_uncertainty
|
||||
|
||||
GP_regression.__init__(self, X, Y, kernel=kernel, normalize_X=normalize_X, normalize_Y=normalize_Y)
|
||||
self.trYYT = np.sum(np.square(self.Y))
|
||||
|
||||
#normalise X uncertainty also
|
||||
if self.has_uncertain_inputs:
|
||||
self.X_uncertainty /= np.square(self._Xstd)
|
||||
|
||||
def _computations(self):
|
||||
# TODO find routine to multiply triangular matrices
|
||||
#TODO: slices for psi statistics (easy enough)
|
||||
|
||||
# kernel computations, using BGPLVM notation
|
||||
self.Kmm = self.kern.K(self.Z)
|
||||
if self.has_uncertain_inputs:
|
||||
self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
|
||||
self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
|
||||
self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
|
||||
self.psi2_beta_scaled = (self.psi2*(self.beta/self.scale_factor**2)).sum(0)
|
||||
else:
|
||||
self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices).sum()
|
||||
self.psi1 = self.kern.K(self.Z,self.X)
|
||||
#self.psi2 = np.dot(self.psi1,self.psi1.T)
|
||||
#self.psi2 = self.psi1.T[:,:,None]*self.psi1.T[:,None,:]
|
||||
tmp = self.psi1/(self.scale_factor/np.sqrt(self.beta))
|
||||
self.psi2_beta_scaled = np.dot(tmp,tmp.T)
|
||||
|
||||
sf = self.scale_factor
|
||||
sf2 = sf**2
|
||||
|
||||
self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)#+np.eye(self.M)*1e-3)
|
||||
|
||||
self.V = (self.beta/self.scale_factor)*self.Y
|
||||
self.A = mdot(self.Lmi, self.psi2_beta_scaled, self.Lmi.T)
|
||||
self.B = np.eye(self.M)/sf2 + self.A
|
||||
|
||||
self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
|
||||
|
||||
self.psi1V = np.dot(self.psi1, self.V)
|
||||
self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
|
||||
self.C = mdot(self.Lmi.T, self.Bi, self.Lmi)
|
||||
self.E = mdot(self.C, self.psi1VVpsi1/sf2, self.C.T)
|
||||
|
||||
# Compute dL_dpsi
|
||||
self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
|
||||
self.dL_dpsi1 = mdot(self.V, self.psi1V.T,self.C).T
|
||||
self.dL_dpsi2 = 0.5 * self.beta * self.D * self.Kmmi[None,:,:] # dB
|
||||
self.dL_dpsi2 += - 0.5 * self.beta/sf2 * self.D * self.C[None,:,:] # dC
|
||||
self.dL_dpsi2 += - 0.5 * self.beta * self.E[None,:,:] # dD
|
||||
|
||||
# Compute dL_dKmm
|
||||
self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi)*sf2 # dB
|
||||
self.dL_dKmm += -0.5 * self.D * (- self.C/sf2 - 2.*mdot(self.C, self.psi2_beta_scaled, self.Kmmi) + self.Kmmi) # dC
|
||||
self.dL_dKmm += np.dot(np.dot(self.E*sf2, self.psi2_beta_scaled) - np.dot(self.C, self.psi1VVpsi1), self.Kmmi) + 0.5*self.E # dD
|
||||
|
||||
|
||||
def _set_params(self, p):
|
||||
self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
|
||||
self.beta = p[self.M*self.Q]
|
||||
self.kern._set_params(p[self.Z.size + 1:])
|
||||
self._computations()
|
||||
|
||||
def _get_params(self):
|
||||
return np.hstack([self.Z.flatten(),self.beta,self.kern._get_params_transformed()])
|
||||
|
||||
def _get_param_names(self):
|
||||
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + ['noise_precision']+self.kern._get_param_names_transformed()
|
||||
|
||||
|
||||
def log_likelihood(self):
|
||||
""" Compute the (lower bound on the) log marginal likelihood """
|
||||
sf2 = self.scale_factor**2
|
||||
A = -0.5*self.N*self.D*(np.log(2.*np.pi) - np.log(self.beta)) -0.5*self.beta*self.trYYT
|
||||
B = -0.5*self.D*(self.beta*self.psi0-np.trace(self.A)*sf2)
|
||||
C = -0.5*self.D * (self.B_logdet + self.M*np.log(sf2))
|
||||
D = +0.5*np.sum(self.psi1VVpsi1 * self.C)
|
||||
return A+B+C+D
|
||||
|
||||
def _log_likelihood_gradients(self):
|
||||
return np.hstack([self.dL_dZ().flatten(), self.dL_dbeta(), self.dL_dtheta()])
|
||||
|
||||
def dL_dbeta(self):
|
||||
"""
|
||||
Compute the gradient of the log likelihood wrt beta.
|
||||
"""
|
||||
#TODO: suport heteroscedatic noise
|
||||
sf2 = self.scale_factor**2
|
||||
dA_dbeta = 0.5 * self.N*self.D/self.beta - 0.5 * self.trYYT
|
||||
dB_dbeta = - 0.5 * self.D * (self.psi0 - np.trace(self.A)/self.beta*sf2)
|
||||
dC_dbeta = - 0.5 * self.D * np.sum(self.Bi*self.A)/self.beta
|
||||
dD_dbeta = np.sum((self.C - 0.5 * mdot(self.C,self.psi2_beta_scaled,self.C) ) * self.psi1VVpsi1 )/self.beta
|
||||
|
||||
return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta)
|
||||
|
||||
def dL_dtheta(self):
|
||||
"""
|
||||
Compute and return the derivative of the log marginal likelihood wrt the parameters of the kernel
|
||||
"""
|
||||
dL_dtheta = self.kern.dK_dtheta(self.dL_dKmm,self.Z)
|
||||
if self.has_uncertain_inputs:
|
||||
dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z,self.X,self.X_uncertainty)
|
||||
dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
|
||||
dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2,self.dL_dpsi1.T, self.Z,self.X, self.X_uncertainty) # for multiple_beta, dL_dpsi2 will be a different shape
|
||||
else:
|
||||
#re-cast computations in psi2 back to psi1:
|
||||
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2.sum(0),self.psi1)
|
||||
dL_dtheta += self.kern.dK_dtheta(dL_dpsi1,self.Z,self.X)
|
||||
dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X)
|
||||
|
||||
return dL_dtheta
|
||||
|
||||
def dL_dZ(self):
|
||||
"""
|
||||
The derivative of the bound wrt the inducing inputs Z
|
||||
"""
|
||||
dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
|
||||
if self.has_uncertain_inputs:
|
||||
dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1,self.Z,self.X, self.X_uncertainty)
|
||||
dL_dZ += 2.*self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # 'stripes'
|
||||
else:
|
||||
#re-cast computations in psi2 back to psi1:
|
||||
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2.sum(0),self.psi1)
|
||||
dL_dZ += self.kern.dK_dX(dL_dpsi1,self.Z,self.X)
|
||||
return dL_dZ
|
||||
|
||||
def _raw_predict(self, Xnew, slices, full_cov=False):
|
||||
"""Internal helper function for making predictions, does not account for normalisation"""
|
||||
|
||||
Kx = self.kern.K(self.Z, Xnew)
|
||||
mu = mdot(Kx.T, self.C/self.scale_factor, self.psi1V)
|
||||
|
||||
if full_cov:
|
||||
Kxx = self.kern.K(Xnew)
|
||||
var = Kxx - mdot(Kx.T, (self.Kmmi - self.C/self.scale_factor**2), Kx) + np.eye(Xnew.shape[0])/self.beta # TODO: This beta doesn't belong here in the EP case.
|
||||
else:
|
||||
Kxx = self.kern.Kdiag(Xnew)
|
||||
var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.C/self.scale_factor**2, Kx),0) + 1./self.beta # TODO: This beta doesn't belong here in the EP case.
|
||||
|
||||
return mu,var
|
||||
|
||||
def plot(self, *args, **kwargs):
|
||||
"""
|
||||
Plot the fitted model: just call the GP_regression plot function and then add inducing inputs
|
||||
"""
|
||||
GP_regression.plot(self,*args,**kwargs)
|
||||
if self.Q==1:
|
||||
pb.plot(self.Z,self.Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
|
||||
if self.has_uncertain_inputs:
|
||||
pb.errorbar(self.X[:,0], pb.ylim()[0]+np.zeros(self.N), xerr=2*np.sqrt(self.X_uncertainty.flatten()))
|
||||
if self.Q==2:
|
||||
pb.plot(self.Z[:,0],self.Z[:,1],'wo')
|
||||
Loading…
Add table
Add a link
Reference in a new issue