From 5fa15037ca3f5dda76a4d2c607f1dbc903b24a33 Mon Sep 17 00:00:00 2001 From: Michael T Smith Date: Thu, 9 Jun 2016 14:08:58 +0100 Subject: [PATCH] Integral kernels added --- GPy/kern/src/integral.py | 84 ++++++++++++ GPy/kern/src/integral_limits.py | 94 ++++++++++++++ .../src/multidimensional_integral_limits.py | 120 ++++++++++++++++++ 3 files changed, 298 insertions(+) create mode 100644 GPy/kern/src/integral.py create mode 100644 GPy/kern/src/integral_limits.py create mode 100644 GPy/kern/src/multidimensional_integral_limits.py diff --git a/GPy/kern/src/integral.py b/GPy/kern/src/integral.py new file mode 100644 index 00000000..971a48a8 --- /dev/null +++ b/GPy/kern/src/integral.py @@ -0,0 +1,84 @@ +# Written by Mike Smith michaeltsmith.org.uk + +import numpy as np +from .kern import Kern +from ...core.parameterization import Param +from paramz.transformations import Logexp +import math + +class Integral(Kern): #todo do I need to inherit from Stationary + """ + Integral kernel between... + """ + + def __init__(self, input_dim, variances=None, lengthscale=None, ARD=False, active_dims=None, name='integral'): + super(Integral, self).__init__(input_dim, active_dims, name) + + if lengthscale is None: + lengthscale = np.ones(1) + else: + lengthscale = np.asarray(lengthscale) + + self.lengthscale = Param('lengthscale', lengthscale, Logexp()) #Logexp - transforms to allow positive only values... + self.variances = Param('variances', variances, Logexp()) #and here. + self.link_parameters(self.variances, self.lengthscale) #this just takes a list of parameters we need to optimise. + + def h(self, z): + return 0.5 * z * np.sqrt(math.pi) * math.erf(z) + np.exp(-(z**2)) + + def dk_dl(self, t, tprime, l): #derivative of the kernel wrt lengthscale + return l * ( self.h(t/l) - self.h((t - tprime)/l) + self.h(tprime/l) - 1) + + def update_gradients_full(self, dL_dK, X, X2=None): + if X2 is None: #we're finding dK_xx/dTheta + dK_dl = np.zeros([X.shape[0],X.shape[0]]) + dK_dv = np.zeros([X.shape[0],X.shape[0]]) + for i,x in enumerate(X): + for j,x2 in enumerate(X): + dK_dl[i,j] = self.variances[0]*self.dk_dl(x[0],x2[0],self.lengthscale[0]) #TODO Multiple length scales + dK_dv[i,j] = self.k_xx(x[0],x2[0],self.lengthscale[0]) #the gradient wrt the variance is k_xx. + self.lengthscale.gradient = np.sum(dK_dl * dL_dK) + self.variances.gradient = np.sum(dK_dv * dL_dK) + #print "V%0.5f" % self.variances.gradient + #print "L%0.5f" % self.lengthscale.gradient + else: #we're finding dK_xf/Dtheta + print("NEED TO HANDLE TODO!") + + #useful little function to help calculate the covariances. + def g(self,z): + return 1.0 * z * np.sqrt(math.pi) * math.erf(z) + np.exp(-(z**2)) + + #covariance between gradients (it's the gradients that we want out... maybe we should have a way of getting K_ff too? Currently you get the diag of K_ff from Kdiag) + def k_xx(self,t,tprime,l): + return 0.5 * (l**2) * ( self.g(t/l) - self.g((t - tprime)/l) + self.g(tprime/l) - 1) + + def k_ff(self,t,tprime,l): + return np.exp(-((t-tprime)**2)/(l**2)) #rbf + + #covariance between the gradient and the actual value + def k_xf(self,t,tprime,l): + return 0.5 * np.sqrt(math.pi) * l * (math.erf((t-tprime)/l) + math.erf(tprime/l)) + + def K(self, X, X2=None): + if X2 is None: + K_xx = np.zeros([X.shape[0],X.shape[0]]) + for i,x in enumerate(X): + for j,x2 in enumerate(X): + K_xx[i,j] = self.k_xx(x[0],x2[0],self.lengthscale[0]) + return K_xx * self.variances[0] + else: + K_xf = np.zeros([X.shape[0],X2.shape[0]]) + for i,x in enumerate(X): + for j,x2 in enumerate(X2): + K_xf[i,j] = self.k_xf(x[0],x2[0],self.lengthscale[0]) + #print self.variances[0] + return K_xf * self.variances[0] + + def Kdiag(self, X): + """I've used the fact that we call this method for K_ff when finding the covariance as a hack so + I know if I should return K_ff or K_xx. In this case we're returning K_ff!! + $K_{ff}^{post} = K_{ff} - K_{fx} K_{xx}^{-1} K_{xf}$""" + K_ff = np.zeros(X.shape[0]) + for i,x in enumerate(X): + K_ff[i] = self.k_ff(x[0],x[0],self.lengthscale[0]) + return K_ff * self.variances[0] diff --git a/GPy/kern/src/integral_limits.py b/GPy/kern/src/integral_limits.py new file mode 100644 index 00000000..7006ee6f --- /dev/null +++ b/GPy/kern/src/integral_limits.py @@ -0,0 +1,94 @@ +# Written by Mike Smith michaeltsmith.org.uk + +import numpy as np +from .kern import Kern +from ...core.parameterization import Param +from paramz.transformations import Logexp +import math + +class Integral_Limits(Kern): #todo do I need to inherit from Stationary + """ + Integral kernel, can include limits on each integral value. + """ + + def __init__(self, input_dim, variances=None, lengthscale=None, ARD=False, active_dims=None, name='integral'): + super(Integral_Limits, self).__init__(input_dim, active_dims, name) + + if lengthscale is None: + lengthscale = np.ones(1) + else: + lengthscale = np.asarray(lengthscale) + + self.lengthscale = Param('lengthscale', lengthscale, Logexp()) #Logexp - transforms to allow positive only values... + self.variances = Param('variances', variances, Logexp()) #and here. + self.link_parameters(self.variances, self.lengthscale) #this just takes a list of parameters we need to optimise. + + def h(self, z): + return 0.5 * z * np.sqrt(math.pi) * math.erf(z) + np.exp(-(z**2)) + + def dk_dl(self, t, tprime, s, sprime, l): #derivative of the kernel wrt lengthscale + return l * ( self.h((t-sprime)/l) - self.h((t - tprime)/l) + self.h((tprime-s)/l) - self.h((s-sprime)/l)) + + def update_gradients_full(self, dL_dK, X, X2=None): + if X2 is None: #we're finding dK_xx/dTheta + dK_dl = np.zeros([X.shape[0],X.shape[0]]) + dK_dv = np.zeros([X.shape[0],X.shape[0]]) + for i,x in enumerate(X): + for j,x2 in enumerate(X): + dK_dl[i,j] = self.variances[0]*self.dk_dl(x[0],x2[0],x[1],x2[1],self.lengthscale[0]) + dK_dv[i,j] = self.k_xx(x[0],x2[0],x[1],x2[1],self.lengthscale[0]) #the gradient wrt the variance is k_xx. + self.lengthscale.gradient = np.sum(dK_dl * dL_dK) + self.variances.gradient = np.sum(dK_dv * dL_dK) + #print "V%0.5f" % self.variances.gradient + #print "L%0.5f" % self.lengthscale.gradient + else: #we're finding dK_xf/Dtheta + print("NEED TO HANDLE TODO!") + + #useful little function to help calculate the covariances. + def g(self,z): + return 1.0 * z * np.sqrt(math.pi) * math.erf(z) + np.exp(-(z**2)) + + def k_xx(self,t,tprime,s,sprime,l): + """Covariance between observed values. + + s and t are one domain of the integral (i.e. the integral between s and t) + sprime and tprime are another domain of the integral (i.e. the integral between sprime and tprime) + + We're interested in how correlated these two integrals are. + + Note: We've not multiplied by the variance, this is done in K.""" + return 0.5 * (l**2) * ( self.g((t-sprime)/l) + self.g((tprime-s)/l) - self.g((t - tprime)/l) - self.g((s-sprime)/l)) + + def k_ff(self,t,tprime,l): + """Doesn't need s or sprime as we're looking at the 'derivatives', so no domains over which to integrate are required""" + return np.exp(-((t-tprime)**2)/(l**2)) #rbf + + def k_xf(self,t,tprime,s,l): + """Covariance between the gradient (latent value) and the actual (observed) value. + + Note that sprime isn't actually used in this expression, presumably because the 'primes' are the gradient (latent) values which don't + involve an integration, and thus there is no domain over which they're integrated, just a single value that we want.""" + return 0.5 * np.sqrt(math.pi) * l * (math.erf((t-tprime)/l) + math.erf((tprime-s)/l)) + + def K(self, X, X2=None): + if X2 is None: + K_xx = np.zeros([X.shape[0],X.shape[0]]) + for i,x in enumerate(X): + for j,x2 in enumerate(X): + K_xx[i,j] = self.k_xx(x[0],x2[0],x[1],x2[1],self.lengthscale[0]) + return K_xx * self.variances[0] + else: + K_xf = np.zeros([X.shape[0],X2.shape[0]]) + for i,x in enumerate(X): + for j,x2 in enumerate(X2): + K_xf[i,j] = self.k_xf(x[0],x2[0],x[1],self.lengthscale[0]) #x2[1] unused, see k_xf docstring for explanation. + return K_xf * self.variances[0] + + def Kdiag(self, X): + """I've used the fact that we call this method for K_ff when finding the covariance as a hack so + I know if I should return K_ff or K_xx. In this case we're returning K_ff!! + $K_{ff}^{post} = K_{ff} - K_{fx} K_{xx}^{-1} K_{xf}$""" + K_ff = np.zeros(X.shape[0]) + for i,x in enumerate(X): + K_ff[i] = self.k_ff(x[0],x[0],self.lengthscale[0]) + return K_ff * self.variances[0] diff --git a/GPy/kern/src/multidimensional_integral_limits.py b/GPy/kern/src/multidimensional_integral_limits.py new file mode 100644 index 00000000..0f473742 --- /dev/null +++ b/GPy/kern/src/multidimensional_integral_limits.py @@ -0,0 +1,120 @@ +# Written by Mike Smith michaeltsmith.org.uk + +import numpy as np +from .kern import Kern +from ...core.parameterization import Param +from paramz.transformations import Logexp +import math + +class Multidimensional_Integral_Limits(Kern): #todo do I need to inherit from Stationary + """ + Integral kernel, can include limits on each integral value. + """ + + def __init__(self, input_dim, variances=None, lengthscale=None, ARD=False, active_dims=None, name='integral'): + super(Multidimensional_Integral_Limits, self).__init__(input_dim, active_dims, name) + + if lengthscale is None: + lengthscale = np.ones(1) + else: + lengthscale = np.asarray(lengthscale) + + self.lengthscale = Param('lengthscale', lengthscale, Logexp()) #Logexp - transforms to allow positive only values... + self.variances = Param('variances', variances, Logexp()) #and here. + self.link_parameters(self.variances, self.lengthscale) #this just takes a list of parameters we need to optimise. + + def h(self, z): + return 0.5 * z * np.sqrt(math.pi) * math.erf(z) + np.exp(-(z**2)) + + def dk_dl(self, t, tprime, s, sprime, l): #derivative of the kernel wrt lengthscale + return l * ( self.h((t-sprime)/l) - self.h((t - tprime)/l) + self.h((tprime-s)/l) - self.h((s-sprime)/l)) + + def update_gradients_full(self, dL_dK, X, X2=None): + #print self.variances + if X2 is None: #we're finding dK_xx/dTheta + dK_dl_term = np.zeros([X.shape[0],X.shape[0],self.lengthscale.shape[0]]) + k_term = np.zeros([X.shape[0],X.shape[0],self.lengthscale.shape[0]]) + dK_dl = np.zeros([X.shape[0],X.shape[0],self.lengthscale.shape[0]]) + dK_dv = np.zeros([X.shape[0],X.shape[0]]) + for il,l in enumerate(self.lengthscale): + idx = il*2 + for i,x in enumerate(X): + for j,x2 in enumerate(X): + dK_dl_term[i,j,il] = self.dk_dl(x[idx],x2[idx],x[idx+1],x2[idx+1],l) + k_term[i,j,il] = self.k_xx(x[idx],x2[idx],x[idx+1],x2[idx+1],l) + for il,l in enumerate(self.lengthscale): + dK_dl = self.variances[0] * dK_dl_term[:,:,il] + for jl, l in enumerate(self.lengthscale): + if jl!=il: + dK_dl *= k_term[:,:,jl] + #dK_dl = np.dot(dK_dl,k_term[:,:,il]) + #print k_term[:,:,il] + self.lengthscale.gradient[il] = np.sum(dK_dl * dL_dK) + dK_dv = self.calc_K_xx_wo_variance(X) #the gradient wrt the variance is k_xx. + self.variances.gradient = np.sum(dK_dv * dL_dK) + else: #we're finding dK_xf/Dtheta + print("NEED TO HANDLE TODO!") + #print self.variances[0],self.lengthscale[0],self.lengthscale[1] #np.sum(dK_dv*dL_dK) + + + #useful little function to help calculate the covariances. + def g(self,z): + return 1.0 * z * np.sqrt(math.pi) * math.erf(z) + np.exp(-(z**2)) + + def k_xx(self,t,tprime,s,sprime,l): + """Covariance between observed values. + + s and t are one domain of the integral (i.e. the integral between s and t) + sprime and tprime are another domain of the integral (i.e. the integral between sprime and tprime) + + We're interested in how correlated these two integrals are. + + Note: We've not multiplied by the variance, this is done in K.""" + return 0.5 * (l**2) * ( self.g((t-sprime)/l) + self.g((tprime-s)/l) - self.g((t - tprime)/l) - self.g((s-sprime)/l)) + + def k_ff(self,t,tprime,l): + """Doesn't need s or sprime as we're looking at the 'derivatives', so no domains over which to integrate are required""" + return np.exp(-((t-tprime)**2)/(l**2)) #rbf + + def k_xf(self,t,tprime,s,l): + """Covariance between the gradient (latent value) and the actual (observed) value. + + Note that sprime isn't actually used in this expression, presumably because the 'primes' are the gradient (latent) values which don't + involve an integration, and thus there is no domain over which they're integrated, just a single value that we want.""" + return 0.5 * np.sqrt(math.pi) * l * (math.erf((t-tprime)/l) + math.erf((tprime-s)/l)) + + def calc_K_xx_wo_variance(self,X): + """Calculates K_xx without the variance term""" + K_xx = np.ones([X.shape[0],X.shape[0]]) #ones now as a product occurs over each dimension + for i,x in enumerate(X): + for j,x2 in enumerate(X): + for il,l in enumerate(self.lengthscale): + idx = il*2 #each pair of input dimensions describe the limits on one actual dimension in the data + K_xx[i,j] *= self.k_xx(x[idx],x2[idx],x[idx+1],x2[idx+1],l) + return K_xx + + def K(self, X, X2=None): + if X2 is None: + #print "X x X" + K_xx = self.calc_K_xx_wo_variance(X) + return K_xx * self.variances[0] + else: + #print "X x X2" + K_xf = np.ones([X.shape[0],X2.shape[0]]) + for i,x in enumerate(X): + for j,x2 in enumerate(X2): + for il,l in enumerate(self.lengthscale): + idx = il*2 + K_xf[i,j] *= self.k_xf(x[idx],x2[idx],x[idx+1],l) + return K_xf * self.variances[0] + + def Kdiag(self, X): + """I've used the fact that we call this method for K_ff when finding the covariance as a hack so + I know if I should return K_ff or K_xx. In this case we're returning K_ff!! + $K_{ff}^{post} = K_{ff} - K_{fx} K_{xx}^{-1} K_{xf}$""" + K_ff = np.ones(X.shape[0]) + for i,x in enumerate(X): + for il,l in enumerate(self.lengthscale): + idx = il*2 + K_ff[i] *= self.k_ff(x[idx],x[idx],l) + return K_ff * self.variances[0]