# Copyright (c) 2012, GPy authors (see AUTHORS.txt). # Licensed under the BSD 3-clause license (see LICENSE.txt) from kernpart import Kernpart import numpy as np class ODE_1(Kernpart): """ kernel resultiong from a first order ODE with OU driving GP :param input_dim: the number of input dimension, has to be equal to one :type input_dim: int :param varianceU: variance of the driving GP :type varianceU: float :param lengthscaleU: lengthscale of the driving GP (sqrt(3)/lengthscaleU) :type lengthscaleU: float :param varianceY: 'variance' of the transfer function :type varianceY: float :param lengthscaleY: 'lengthscale' of the transfer function (1/lengthscaleY) :type lengthscaleY: float :rtype: kernel object """ def __init__(self, input_dim=1, varianceU=1., varianceY=1., lengthscaleU=None, lengthscaleY=None): assert input_dim==1, "Only defined for input_dim = 1" self.input_dim = input_dim self.num_params = 4 self.name = 'ODE_1' if lengthscaleU is not None: lengthscaleU = np.asarray(lengthscaleU) assert lengthscaleU.size == 1, "lengthscaleU should be one dimensional" else: lengthscaleU = np.ones(1) if lengthscaleY is not None: lengthscaleY = np.asarray(lengthscaleY) assert lengthscaleY.size == 1, "lengthscaleY should be one dimensional" else: lengthscaleY = np.ones(1) #lengthscaleY = 0.5 self._set_params(np.hstack((varianceU, varianceY, lengthscaleU,lengthscaleY))) def _get_params(self): """return the value of the parameters.""" return np.hstack((self.varianceU,self.varianceY, self.lengthscaleU,self.lengthscaleY)) def _set_params(self, x): """set the value of the parameters.""" assert x.size == self.num_params self.varianceU = x[0] self.varianceY = x[1] self.lengthscaleU = x[2] self.lengthscaleY = x[3] def _get_param_names(self): """return parameter names.""" return ['varianceU','varianceY', 'lengthscaleU', 'lengthscaleY'] def K(self, X, X2, target): """Compute the covariance matrix between X and X2.""" if X2 is None: X2 = X # i1 = X[:,1] # i2 = X2[:,1] # X = X[:,0].reshape(-1,1) # X2 = X2[:,0].reshape(-1,1) dist = np.abs(X - X2.T) ly=1/self.lengthscaleY lu=np.sqrt(3)/self.lengthscaleU #ly=self.lengthscaleY #lu=self.lengthscaleU k1 = np.exp(-ly*dist)*(2*lu+ly)/(lu+ly)**2 k2 = (np.exp(-lu*dist)*(ly-2*lu+lu*ly*dist-lu**2*dist) + np.exp(-ly*dist)*(2*lu-ly) ) / (ly-lu)**2 k3 = np.exp(-lu*dist) * ( (1+lu*dist)/(lu+ly) + (lu)/(lu+ly)**2 ) np.add(self.varianceU*self.varianceY*(k1+k2+k3), target, target) def Kdiag(self, X, target): """Compute the diagonal of the covariance matrix associated to X.""" ly=1/self.lengthscaleY lu=np.sqrt(3)/self.lengthscaleU #ly=self.lengthscaleY #lu=self.lengthscaleU k1 = (2*lu+ly)/(lu+ly)**2 k2 = (ly-2*lu + 2*lu-ly ) / (ly-lu)**2 k3 = 1/(lu+ly) + (lu)/(lu+ly)**2 np.add(self.varianceU*self.varianceY*(k1+k2+k3), target, target) def dK_dtheta(self, dL_dK, X, X2, target): """derivative of the covariance matrix with respect to the parameters.""" if X2 is None: X2 = X dist = np.abs(X - X2.T) ly=1/self.lengthscaleY lu=np.sqrt(3)/self.lengthscaleU #ly=self.lengthscaleY #lu=self.lengthscaleU dk1theta1 = np.exp(-ly*dist)*2*(-lu)/(lu+ly)**3 #c=np.sqrt(3) #t1=c/lu #t2=1/ly #dk1theta1=np.exp(-dist*ly)*t2*( (2*c*t2+2*t1)/(c*t2+t1)**2 -2*(2*c*t2*t1+t1**2)/(c*t2+t1)**3 ) dk2theta1 = 1*( np.exp(-lu*dist)*dist*(-ly+2*lu-lu*ly*dist+dist*lu**2)*(ly-lu)**(-2) + np.exp(-lu*dist)*(-2+ly*dist-2*dist*lu)*(ly-lu)**(-2) +np.exp(-dist*lu)*(ly-2*lu+ly*lu*dist-dist*lu**2)*2*(ly-lu)**(-3) +np.exp(-dist*ly)*2*(ly-lu)**(-2) +np.exp(-dist*ly)*2*(2*lu-ly)*(ly-lu)**(-3) ) dk3theta1 = np.exp(-dist*lu)*(lu+ly)**(-2)*((2*lu+ly+dist*lu**2+lu*ly*dist)*(-dist-2/(lu+ly))+2+2*lu*dist+ly*dist) dktheta1 = self.varianceU*self.varianceY*(dk1theta1+dk2theta1+dk3theta1) dk1theta2 = np.exp(-ly*dist) * ((lu+ly)**(-2)) * ( (-dist)*(2*lu+ly) + 1 + (-2)*(2*lu+ly)/(lu+ly) ) dk2theta2 = 1*( np.exp(-dist*lu)*(ly-lu)**(-2) * ( 1+lu*dist+(-2)*(ly-2*lu+lu*ly*dist-dist*lu**2)*(ly-lu)**(-1) ) +np.exp(-dist*ly)*(ly-lu)**(-2) * ( (-dist)*(2*lu-ly) -1+(2*lu-ly)*(-2)*(ly-lu)**(-1) ) ) dk3theta2 = np.exp(-dist*lu) * (-3*lu-ly-dist*lu**2-lu*ly*dist)/(lu+ly)**3 dktheta2 = self.varianceU*self.varianceY*(dk1theta2 + dk2theta2 +dk3theta2) k1 = np.exp(-ly*dist)*(2*lu+ly)/(lu+ly)**2 k2 = (np.exp(-lu*dist)*(ly-2*lu+lu*ly*dist-lu**2*dist) + np.exp(-ly*dist)*(2*lu-ly) ) / (ly-lu)**2 k3 = np.exp(-lu*dist) * ( (1+lu*dist)/(lu+ly) + (lu)/(lu+ly)**2 ) dkdvar = k1+k2+k3 target[0] += np.sum(self.varianceY*dkdvar * dL_dK) target[1] += np.sum(self.varianceU*dkdvar * dL_dK) target[2] += np.sum(dktheta1*(-np.sqrt(3)*self.lengthscaleU**(-2)) * dL_dK) target[3] += np.sum(dktheta2*(-self.lengthscaleY**(-2)) * dL_dK) # def dKdiag_dtheta(self, dL_dKdiag, X, target): # """derivative of the diagonal of the covariance matrix with respect to the parameters.""" # # NB: derivative of diagonal elements wrt lengthscale is 0 # target[0] += np.sum(dL_dKdiag) # def dK_dX(self, dL_dK, X, X2, target): # """derivative of the covariance matrix with respect to X.""" # if X2 is None: X2 = X # dist = np.sqrt(np.sum(np.square((X[:, None, :] - X2[None, :, :]) / self.lengthscale), -1))[:, :, None] # ddist_dX = (X[:, None, :] - X2[None, :, :]) / self.lengthscale ** 2 / np.where(dist != 0., dist, np.inf) # dK_dX = -np.transpose(self.variance * np.exp(-dist) * ddist_dX, (1, 0, 2)) # target += np.sum(dK_dX * dL_dK.T[:, :, None], 0) # def dKdiag_dX(self, dL_dKdiag, X, target): # pass