# -*- coding: utf-8 -*-
# Copyright (c) 2015, Alex Grigorevskiy
# Licensed under the BSD 3-clause license (see LICENSE.txt)
"""
The standard periodic kernel which mentioned in:
[1] Gaussian Processes for Machine Learning, C. E. Rasmussen, C. K. I. Williams.
The MIT Press, 2005.
[2] Introduction to Gaussian processes. D. J. C. MacKay. In C. M. Bishop, editor,
Neural Networks and Machine Learning, pages 133-165. Springer, 1998.
"""
from .kern import Kern
from ...core.parameterization import Param
from paramz.transformations import Logexp
import numpy as np
[docs]class StdPeriodic(Kern):
"""
Standart periodic kernel
.. math::
k(x,y) = \theta_1 \exp \left[ - \frac{1}{2} \sum_{i=1}^{input\_dim}
\left( \frac{\sin(\frac{\pi}{T_i} (x_i - y_i) )}{l_i} \right)^2 \right] }
:param input_dim: the number of input dimensions
:type input_dim: int
:param variance: the variance :math:`\theta_1` in the formula above
:type variance: float
:param period: the vector of periods :math:`\T_i`. If None then 1.0 is assumed.
:type period: array or list of the appropriate size (or float if there is only one period parameter)
:param lengthscale: the vector of lengthscale :math:`\l_i`. If None then 1.0 is assumed.
:type lengthscale: array or list of the appropriate size (or float if there is only one lengthscale parameter)
:param ARD1: Auto Relevance Determination with respect to period.
If equal to "False" one single period parameter :math:`\T_i` for
each dimension is assumed, otherwise there is one lengthscale
parameter per dimension.
:type ARD1: Boolean
:param ARD2: Auto Relevance Determination with respect to lengthscale.
If equal to "False" one single lengthscale parameter :math:`l_i` for
each dimension is assumed, otherwise there is one lengthscale
parameter per dimension.
:type ARD2: Boolean
:param active_dims: indices of dimensions which are used in the computation of the kernel
:type active_dims: array or list of the appropriate size
:param name: Name of the kernel for output
:type String
:param useGPU: whether of not use GPU
:type Boolean
"""
def __init__(self, input_dim, variance=1., period=None, lengthscale=None, ARD1=False, ARD2=False, active_dims=None, name='std_periodic',useGPU=False):
super(StdPeriodic, self).__init__(input_dim, active_dims, name, useGPU=useGPU)
self.ARD1 = ARD1 # correspond to periods
self.ARD2 = ARD2 # correspond to lengthscales
self.name = name
if self.ARD1 == False:
if period is not None:
period = np.asarray(period)
assert period.size == 1, "Only one period needed for non-ARD kernel"
else:
period = np.ones(1)
else:
if period is not None:
period = np.asarray(period)
assert period.size == input_dim, "bad number of periods"
else:
period = np.ones(input_dim)
if self.ARD2 == False:
if lengthscale is not None:
lengthscale = np.asarray(lengthscale)
assert lengthscale.size == 1, "Only one lengthscale needed for non-ARD kernel"
else:
lengthscale = np.ones(1)
else:
if lengthscale is not None:
lengthscale = np.asarray(lengthscale)
assert lengthscale.size == input_dim, "bad number of lengthscales"
else:
lengthscale = np.ones(input_dim)
self.variance = Param('variance', variance, Logexp())
assert self.variance.size==1, "Variance size must be one"
self.period = Param('period', period, Logexp())
self.lengthscale = Param('lengthscale', lengthscale, Logexp())
self.link_parameters(self.variance, self.period, self.lengthscale)
[docs] def to_dict(self):
"""
Convert the object into a json serializable dictionary.
Note: It uses the private method _save_to_input_dict of the parent.
:return dict: json serializable dictionary containing the needed information to instantiate the object
"""
input_dict = super(StdPeriodic, self)._save_to_input_dict()
input_dict["class"] = "GPy.kern.StdPeriodic"
input_dict["variance"] = self.variance.values.tolist()
input_dict["period"] = self.period.values.tolist()
input_dict["lengthscale"] = self.lengthscale.values.tolist()
input_dict["ARD1"] = self.ARD1
input_dict["ARD2"] = self.ARD2
return input_dict
[docs] def parameters_changed(self):
"""
This functions deals as a callback for each optimization iteration.
If one optimization step was successfull and the parameters
this callback function will be called to be able to update any
precomputations for the kernel.
"""
pass
[docs] def K(self, X, X2=None):
"""Compute the covariance matrix between X and X2."""
if X2 is None:
X2 = X
base = np.pi * (X[:, None, :] - X2[None, :, :]) / self.period
exp_dist = np.exp( -0.5* np.sum( np.square( np.sin( base ) / self.lengthscale ), axis = -1 ) )
return self.variance * exp_dist
[docs] def Kdiag(self, X):
"""Compute the diagonal of the covariance matrix associated to X."""
ret = np.empty(X.shape[0])
ret[:] = self.variance
return ret
[docs] def update_gradients_full(self, dL_dK, X, X2=None):
"""derivative of the covariance matrix with respect to the parameters."""
if X2 is None:
X2 = X
base = np.pi * (X[:, None, :] - X2[None, :, :]) / self.period
sin_base = np.sin( base )
exp_dist = np.exp( -0.5* np.sum( np.square( sin_base / self.lengthscale ), axis = -1 ) )
dwl = self.variance * (1.0/np.square(self.lengthscale)) * sin_base*np.cos(base) * (base / self.period)
dl = self.variance * np.square( sin_base) / np.power( self.lengthscale, 3)
self.variance.gradient = np.sum(exp_dist * dL_dK)
#target[0] += np.sum( exp_dist * dL_dK)
if self.ARD1: # different periods
self.period.gradient = (dwl * exp_dist[:,:,None] * dL_dK[:, :, None]).sum(0).sum(0)
else: # same period
self.period.gradient = np.sum(dwl.sum(-1) * exp_dist * dL_dK)
if self.ARD2: # different lengthscales
self.lengthscale.gradient = (dl * exp_dist[:,:,None] * dL_dK[:, :, None]).sum(0).sum(0)
else: # same lengthscales
self.lengthscale.gradient = np.sum(dl.sum(-1) * exp_dist * dL_dK)
[docs] def update_gradients_diag(self, dL_dKdiag, X):
"""derivative of the diagonal of the covariance matrix with respect to the parameters."""
self.variance.gradient = np.sum(dL_dKdiag)
self.period.gradient = 0
self.lengthscale.gradient = 0
[docs] def gradients_X(self, dL_dK, X, X2=None):
K = self.K(X, X2)
if X2 is None:
dL_dK = dL_dK+dL_dK.T
X2 = X
dX = -np.pi*((dL_dK*K)[:,:,None]*np.sin(2*np.pi/self.period*(X[:,None,:] - X2[None,:,:]))/(2.*np.square(self.lengthscale)*self.period)).sum(1)
return dX
[docs] def gradients_X_diag(self, dL_dKdiag, X):
return np.zeros(X.shape)