GPy/GPy/likelihoods/noise_models/exponential_noise.py

# Copyright (c) 2012, 2013 Ricardo Andrade
# Licensed under the BSD 3-clause license (see LICENSE.txt)


import numpy as np
from scipy import stats,special
import scipy as sp
from GPy.util.univariate_Gaussian import std_norm_pdf,std_norm_cdf
import gp_transformations
from noise_distributions import NoiseDistribution

class Exponential(NoiseDistribution):
    """
    Expoential likelihood
    Y is expected to take values in {0,1,2,...}
    -----
    $$
    L(x) = \exp(\lambda) * \lambda**Y_i / Y_i!
    $$
    """
    def __init__(self,gp_link=None,analytical_mean=False,analytical_variance=False):
        super(Exponential, self).__init__(gp_link,analytical_mean,analytical_variance)

    def _preprocess_values(self,Y):
        return Y

    def pdf_link(self, link_f, y, extra_data=None):
        """
        Likelihood function given link(f)

        .. math::
            p(y_{i}|\\lambda(f_{i})) = \\lambda(f_{i})\\exp (-y\\lambda(f_{i}))

        :param link_f: latent variables link(f)
        :type link_f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param extra_data: extra_data which is not used in exponential distribution
        :returns: likelihood evaluated for this point
        :rtype: float
        """
        assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
        log_objective = link_f*np.exp(-y*link_f)
        return np.exp(np.sum(np.log(log_objective)))
        #return np.exp(np.sum(-y/link_f - np.log(link_f) ))

    def logpdf_link(self, link_f, y, extra_data=None):
        """
        Log Likelihood Function given link(f)

        .. math::
            \\ln p(y_{i}|\lambda(f_{i})) = \\ln \\lambda(f_{i}) - y_{i}\\lambda(f_{i})

        :param link_f: latent variables (link(f))
        :type link_f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param extra_data: extra_data which is not used in exponential distribution
        :returns: likelihood evaluated for this point
        :rtype: float

        """
        assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
        log_objective = np.log(link_f) - y*link_f
        #logpdf_link = np.sum(-np.log(link_f) - y/link_f)
        return np.sum(log_objective)

    def dlogpdf_dlink(self, link_f, y, extra_data=None):
        """
        Gradient of the log likelihood function at y, given link(f) w.r.t link(f)

        .. math::
            \\frac{d \\ln p(y_{i}|\lambda(f_{i}))}{d\\lambda(f)} = \\frac{1}{\\lambda(f)} - y_{i}

        :param link_f: latent variables (f)
        :type link_f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param extra_data: extra_data which is not used in exponential distribution
        :returns: gradient of likelihood evaluated at points
        :rtype: Nx1 array

        """
        assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
        grad = 1./link_f - y
        #grad = y/(link_f**2) - 1./link_f
        return grad

    def d2logpdf_dlink2(self, link_f, y, extra_data=None):
        """
        Hessian at y, given link(f), w.r.t link(f)
        i.e. second derivative logpdf at y given link(f_i) and link(f_j)  w.r.t link(f_i) and link(f_j)
        The hessian will be 0 unless i == j

        .. math::
            \\frac{d^{2} \\ln p(y_{i}|\lambda(f_{i}))}{d^{2}\\lambda(f)} = -\\frac{1}{\\lambda(f_{i})^{2}}

        :param link_f: latent variables link(f)
        :type link_f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param extra_data: extra_data which is not used in exponential distribution
        :returns: Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)
        :rtype: Nx1 array

        .. Note::
            Will return diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
            (the distribution for y_i depends only on link(f_i) not on link(f_(j!=i))
        """
        assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
        hess = -1./(link_f**2)
        #hess = -2*y/(link_f**3) + 1/(link_f**2)
        return hess

    def d3logpdf_dlink3(self, link_f, y, extra_data=None):
        """
        Third order derivative log-likelihood function at y given link(f) w.r.t link(f)

        .. math::
            \\frac{d^{3} \\ln p(y_{i}|\lambda(f_{i}))}{d^{3}\\lambda(f)} = \\frac{2}{\\lambda(f_{i})^{3}}

        :param link_f: latent variables link(f)
        :type link_f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param extra_data: extra_data which is not used in exponential distribution
        :returns: third derivative of likelihood evaluated at points f
        :rtype: Nx1 array
        """
        assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
        d3lik_dlink3 = 2./(link_f**3)
        #d3lik_dlink3 = 6*y/(link_f**4) - 2./(link_f**3)
        return d3lik_dlink3

    def _mean(self,gp):
        """
        Mass (or density) function
        """
        return self.gp_link.transf(gp)

    def _variance(self,gp):
        """
        Mass (or density) function
        """
        return self.gp_link.transf(gp)**2

    def samples(self, gp):
        """
        Returns a set of samples of observations based on a given value of the latent variable.

        :param gp: latent variable
        """
        orig_shape = gp.shape
        gp = gp.flatten()
        Ysim = np.random.exponential(1.0/self.gp_link.transf(gp))
        return Ysim.reshape(orig_shape)