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58 lines
2.1 KiB
Python
58 lines
2.1 KiB
Python
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# -*- coding: utf-8 -*-
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"""
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Classes in this module enhance Matern covariance functions with the
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Stochastic Differential Equation (SDE) functionality.
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"""
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from .stationary import Matern32
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import numpy as np
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class sde_Matern32(Matern32):
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"""
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Class provide extra functionality to transfer this covariance function into
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SDE forrm.
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Matern 3/2 kernel:
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.. math::
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k(r) = \\sigma^2 (1 + \\sqrt{3} r) \exp(- \sqrt{3} r) \\ \\ \\ \\ \\text{ where } r = \sqrt{\sum_{i=1}^input_dim \\frac{(x_i-y_i)^2}{\ell_i^2} }
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"""
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def sde(self):
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"""
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Return the state space representation of the covariance.
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"""
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variance = float(self.variance.values)
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lengthscale = float(self.lengthscale.values)
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foo = np.sqrt(3.)/lengthscale
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F = np.array([[0, 1], [-foo**2, -2*foo]])
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L = np.array([[0], [1]])
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Qc = np.array([[12.*np.sqrt(3) / lengthscale**3 * variance]])
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H = np.array([[1, 0]])
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Pinf = np.array([[variance, 0],
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[0, 3.*variance/(lengthscale**2)]])
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# Allocate space for the derivatives
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dF = np.empty([F.shape[0],F.shape[1],2])
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dQc = np.empty([Qc.shape[0],Qc.shape[1],2])
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dPinf = np.empty([Pinf.shape[0],Pinf.shape[1],2])
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# The partial derivatives
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dFvariance = np.zeros([2,2])
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dFlengthscale = np.array([[0,0],
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[6./lengthscale**3,2*np.sqrt(3)/lengthscale**2]])
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dQcvariance = np.array([12.*np.sqrt(3)/lengthscale**3])
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dQclengthscale = np.array([-3*12*np.sqrt(3)/lengthscale**4*variance])
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dPinfvariance = np.array([[1,0],[0,3./lengthscale**2]])
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dPinflengthscale = np.array([[0,0],
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[0,-6*variance/lengthscale**3]])
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# Combine the derivatives
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dF[:,:,0] = dFvariance
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dF[:,:,1] = dFlengthscale
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dQc[:,:,0] = dQcvariance
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dQc[:,:,1] = dQclengthscale
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dPinf[:,:,0] = dPinfvariance
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dPinf[:,:,1] = dPinflengthscale
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return (F, L, Qc, H, Pinf, dF, dQc, dPinf)
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