Example of sde_Matern covarince function is added, along with other small changes.

State-space example is slightly modified.
Imports are corrected accordingly.

GPy/kern/_src/sde_Matern.py

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