apunkt/GPy
1
0
Fork 0
mirror of https://github.com/SheffieldML/GPy.git synced 2026-05-27 14:25:16 +02:00
Code Issues Projects Releases Packages Wiki Activity Actions

Tidying up a lot, works for 1D, need to check for more dimensions

Browse source
This commit is contained in:
Alan Saul 2013-10-04 14:44:50 +01:00
parent da67e39e50
commit 2acf931482
11 changed files with 192 additions and 498 deletions
Download patch file Download diff file Expand all files Collapse all files

105
GPy/likelihoods/noise_models/student_t_noise.py
View file

@ -15,10 +15,8 @@ class StudentT(NoiseDistribution):
For nomanclature see Bayesian Data Analysis 2003 p576
$$\ln p(y_{i}|f_{i}) = \ln \Gamma(\frac{v+1}{2}) - \ln \Gamma(\frac{v}{2})\sqrt{v \pi}\sigma - \frac{v+1}{2}\ln (1 + \frac{1}{v}\left(\frac{y_{i} - f_{i}}{\sigma}\right)^2)$$
.. math::
Fill in maths
\\ln p(y_{i}|f_{i}) = \\ln \\Gamma(\\frac{v+1}{2}) - \\ln \\Gamma(\\frac{v}{2})\\sqrt{v \\pi}\\sigma - \\frac{v+1}{2}\\ln (1 + \\frac{1}{v}\\left(\\frac{y_{i} - f_{i}}{\\sigma}\\right)^2)
"""
def __init__(self,gp_link=None,analytical_mean=True,analytical_variance=True, deg_free=5, sigma2=2):
@ -42,16 +40,20 @@ class StudentT(NoiseDistribution):
def variance(self, extra_data=None):
return (self.v / float(self.v - 2)) * self.sigma2
def link_function(self, y, f, extra_data=None):
"""link_function $\ln p(y|f)$
$$\ln p(y_{i}|f_{i}) = \ln \Gamma(\frac{v+1}{2}) - \ln \Gamma(\frac{v}{2})\sqrt{v \pi}\sigma - \frac{v+1}{2}\ln (1 + \frac{1}{v}\left(\frac{y_{i} - f_{i}}{\sigma}\right)^2$$
def lik_function(self, y, f, extra_data=None):
"""
Log Likelihood Function
For wolfram alpha import parts for derivative of sigma are -log(sqrt(v*pi)*s) -(1/2)*(v + 1)*log(1 + (1/v)*((y-f)/(s))^2))
.. math::
\\ln p(y_{i}|f_{i}) = \\ln \\Gamma(\\frac{v+1}{2}) - \\ln \\Gamma(\\frac{v}{2})\\sqrt{v \\pi}\sigma - \\frac{v+1}{2}\\ln (1 + \\frac{1}{v}\\left(\\frac{y_{i} - f_{i}}{\\sigma}\\right)^2
:y: data
:f: latent variables f
:extra_data: extra_data which is not used in student t distribution
:returns: float(likelihood evaluated for this point)
:param y: data
:type y: NxD matrix
:param f: latent variables f
:type f: NxD matrix
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: likelihood evaluated for this point
:rtype: float
"""
assert y.shape == f.shape
@ -65,14 +67,18 @@ class StudentT(NoiseDistribution):
def dlik_df(self, y, f, extra_data=None):
"""
Gradient of the link function at y, given f w.r.t f
Gradient of the log likelihood function at y, given f w.r.t f
$$\frac{dp(y_{i}|f_{i})}{df} = \frac{(v+1)(y_{i}-f_{i})}{(y_{i}-f_{i})^{2} + \sigma^{2}v}$$
.. math::
\\frac{d \\ln p(y_{i}|f_{i})}{df} = \\frac{(v+1)(y_{i}-f_{i})}{(y_{i}-f_{i})^{2} + \\sigma^{2}v}
:y: data
:f: latent variables f
:extra_data: extra_data which is not used in student t distribution
:param y: data
:type y: NxD matrix
:param f: latent variables f
:type f: NxD matrix
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: gradient of likelihood evaluated at points
:rtype: 1xN array
"""
assert y.shape == f.shape
@ -82,18 +88,23 @@ class StudentT(NoiseDistribution):
def d2lik_d2f(self, y, f, extra_data=None):
"""
Hessian at this point (if we are only looking at the link function not the prior) the hessian will be 0 unless i == j
i.e. second derivative link_function at y given f f_j w.r.t f and f_j
Hessian at y, given f, w.r.t f the hessian will be 0 unless i == j
i.e. second derivative lik_function at y given f_{i} f_{j} w.r.t f_{i} and f_{j}
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 f_{i} not on f_{j!=i}
.. math::
\\frac{d^{2} \\ln p(y_{i}|f_{i})}{d^{2}f} = \\frac{(v+1)((y_{i}-f_{i})^{2} - \\sigma^{2}v)}{((y_{i}-f_{i})^{2} + \\sigma^{2}v)^{2}}
$$\frac{d^{2}p(y_{i}|f_{i})}{d^{3}f} = \frac{(v+1)((y_{i}-f_{i})^{2} - \sigma^{2}v)}{((y_{i}-f_{i})^{2} + \sigma^{2}v)^{2}}$$
:param y: data
:type y: NxD matrix
:param f: latent variables f
:type f: NxD matrix
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)
:rtype: 1xN array
:y: data
:f: latent variables f
:extra_data: extra_data which is not used in student t distribution
:returns: array which is diagonal of covariance matrix (second derivative of likelihood evaluated at points)
.. 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 f_{i} not on f_{j!=i}
"""
assert y.shape == f.shape
e = y - f
@ -102,9 +113,18 @@ class StudentT(NoiseDistribution):
def d3lik_d3f(self, y, f, extra_data=None):
"""
Third order derivative link_function (log-likelihood ) at y given f f_j w.r.t f and f_j
Third order derivative log-likelihood function at y given f w.r.t f
$$\frac{d^{3}p(y_{i}|f_{i})}{d^{3}f} = \frac{-2(v+1)((y_{i} - f_{i})^3 - 3(y_{i} - f_{i}) \sigma^{2} v))}{((y_{i} - f_{i}) + \sigma^{2} v)^3}$$
.. math::
\\frac{d^{3} \\ln p(y_{i}|f_{i})}{d^{3}f} = \\frac{-2(v+1)((y_{i} - f_{i})^3 - 3(y_{i} - f_{i}) \\sigma^{2} v))}{((y_{i} - f_{i}) + \\sigma^{2} v)^3}
:param y: data
:type y: NxD matrix
:param f: latent variables f
:type f: NxD matrix
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: third derivative of likelihood evaluated at points f
:rtype: 1xN array
"""
assert y.shape == f.shape
e = y - f
@ -115,23 +135,39 @@ class StudentT(NoiseDistribution):
def dlik_dvar(self, y, f, extra_data=None):
"""
Gradient of the likelihood (lik) w.r.t sigma parameter (standard deviation)
Gradient of the log-likelihood function at y given f, w.r.t variance parameter (t_noise)
Terms relavent to derivatives wrt sigma are:
-log(sqrt(v*pi)*s) -(1/2)*(v + 1)*log(1 + (1/v)*((y-f)/(s))^2))
.. math::
\\frac{d \\ln p(y_{i}|f_{i})}{d\\sigma^{2}} = -\\frac{1}{\\sigma} + \\frac{(1+v)(y_{i}-f_{i})^2}{\\sigma^3 v(1 + \\frac{1}{v}(\\frac{(y_{i} - f_{i})}{\\sigma^2})^2)}
$$\frac{dp(y_{i}|f_{i})}{d\sigma} = -\frac{1}{\sigma} + \frac{(1+v)(y_{i}-f_{i})^2}{\sigma^3 v(1 + \frac{1}{v}(\frac{(y_{i} - f_{i})}{\sigma^2})^2)}$$
:param y: data
:type y: NxD matrix
:param f: latent variables f
:type f: NxD matrix
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: derivative of likelihood evaluated at points f w.r.t variance parameter
:rtype: 1x1 array
"""
assert y.shape == f.shape
e = y - f
dlik_dvar = self.v*(e**2 - self.sigma2)/(2*self.sigma2*(self.sigma2*self.v + e**2))
return np.sum(dlik_dvar) #May not want to sum over all dimensions if using many D?
#FIXME: May not want to sum over all dimensions if using many D?
return np.sum(dlik_dvar)
def dlik_df_dvar(self, y, f, extra_data=None):
"""
Gradient of the dlik_df w.r.t sigma parameter (standard deviation)
Derivative of the dlik_df w.r.t variance parameter (t_noise)
$$\frac{d}{d\sigma}(\frac{dp(y_{i}|f_{i})}{df}) = \frac{-2\sigma v(v + 1)(y_{i}-f_{i})}{(y_{i}-f_{i})^2 + \sigma^2 v)^2}$$
.. math::
\\frac{d}{d\\sigma^{2}}(\\frac{d \\ln p(y_{i}|f_{i})}{df}) = \\frac{-2\\sigma v(v + 1)(y_{i}-f_{i})}{(y_{i}-f_{i})^2 + \\sigma^2 v)^2}
:param y: data
:type y: NxD matrix
:param f: latent variables f
:type f: NxD matrix
:param extra_data: extra_data which is not used in student t distribution - not used
:returns: derivative of likelihood evaluated at points f w.r.t variance parameter
:rtype: 1xN array
"""
assert y.shape == f.shape
e = y - f
@ -180,6 +216,7 @@ class StudentT(NoiseDistribution):
#However the variance of the student t distribution is not dependent on f, only on sigma and the degrees of freedom
true_var = sigma**2 + self.variance
print true_var
return true_var
def _predictive_mean_analytical(self, mu, var):