ENH: Added SDE for all basic kernels except Rationale Quadratic.

Some necessary modifications for the previous code are performed.

GPy/kern/_src/sde_brownian.py

@@ -0,0 +1,57 @@
+# -*- coding: utf-8 -*-
+"""
+Classes in this module enhance Brownian motion covariance function with the
+Stochastic Differential Equation (SDE) functionality.
+"""
+
+from .brownian import Brownian
+
+import numpy as np
+
+class sde_Brownian(Brownian):
+    """
+    
+    Class provide extra functionality to transfer this covariance function into
+    SDE form.
+    
+    Linear kernel:
+
+    .. math::
+
+       k(x,y) = \sigma^2 min(x,y)
+
+    """
+    
+    def sde_update_gradient_full(self, gradients):
+        """
+        Update gradient in the order in which parameters are represented in the
+        kernel
+        """
+    
+        self.variance.gradient = gradients[0]
+        
+    def sde(self): 
+        """ 
+        Return the state space representation of the covariance. 
+        """ 
+        
+        variance = float(self.variance.values) # this is initial variancve in Bayesian linear regression
+        
+        F = np.array( ((0,1.0),(0,0) ))
+        L = np.array( ((1.0,),(0,)) )
+        Qc = np.array( ((variance,),) )
+        H = np.array( ((1.0,0),) )
+        
+        Pinf   = np.array( ( (0, -0.5*variance ), (-0.5*variance, 0) ) )
+        #P0 = Pinf.copy() 
+        P0 = np.zeros((2,2))   
+        #Pinf   = np.array( ( (t0, 1.0), (1.0, 1.0/t0) ) ) * variance
+        dF = np.zeros((2,2,1))
+        dQc    = np.ones( (1,1,1) )
+        
+        dPinf = np.zeros((2,2,1))
+        dPinf[:,:,0] = np.array( ( (0, -0.5), (-0.5, 0) ) )
+        #dP0 = dPinf.copy() 
+        dP0 = np.zeros((2,2,1))
+  
+        return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)

GPy/kern/_src/sde_linear.py

@@ -1,6 +1,6 @@
 # -*- coding: utf-8 -*-
 """
-Classes in this module enhance Matern covariance functions with the
+Classes in this module enhance Linear covariance function with the
 Stochastic Differential Equation (SDE) functionality.
 """
 from .linear import Linear
@@ -20,16 +20,45 @@ class sde_Linear(Linear):
        k(x,y) = \sum_{i=1}^{input dim} \sigma^2_i x_iy_i
 
     """
-
+    def __init__(self, input_dim, X, variances=None, ARD=False, active_dims=None, name='linear'):
+        """
+        Modify the init method, because one extra parameter is required. X - points
+        on the X axis.
+        """
+        
+        super(sde_Linear, self).__init__(input_dim, variances, ARD, active_dims, name)
+        
+        self.t0 = np.min(X)
+        
+    
+    def sde_update_gradient_full(self, gradients):
+        """
+        Update gradient in the order in which parameters are represented in the
+        kernel
+        """
+    
+        self.variances.gradient = gradients[0]
+        
     def sde(self): 
         """ 
         Return the state space representation of the covariance. 
         """ 
         
-        # Arno, insert your code here
+        variance = float(self.variances.values) # this is initial variancve in Bayesian linear regression
-
+        t0 = float(self.t0)
-        # Params to use:
+        
-
+        F = np.array( ((0,1.0),(0,0) ))
-        # self.variances
+        L = np.array( ((0,),(1.0,)) )
-
+        Qc = np.zeros((1,1))
-        #return (F, L, Qc, H, Pinf, dF, dQc, dPinf)
+        H = np.array( ((1.0,0),) )
+        
+        Pinf   = np.zeros((2,2))
+        P0 = np.array( ( (t0**2, t0), (t0, 1) ) ) * variance        
+        dF = np.zeros((2,2,1))
+        dQc    = np.zeros( (1,1,1) )
+        
+        dPinf = np.zeros((2,2,1))
+        dP0 = np.zeros((2,2,1))
+        dP0[:,:,0]  = P0 / variance
+  
+        return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)

GPy/kern/_src/sde_matern.py

@@ -38,25 +38,24 @@ class sde_Matern32(Matern32):
         lengthscale = float(self.lengthscale.values)
         
         foo  = np.sqrt(3.)/lengthscale 
-        F    = np.array([[0, 1], [-foo**2, -2*foo]]) 
+        F    = np.array(((0, 1), (-foo**2, -2*foo))) 
-        L    = np.array([[0], [1]]) 
+        L    = np.array(( (0,), (1,) ))
-        Qc   = np.array([[12.*np.sqrt(3) / lengthscale**3 * variance]]) 
+        Qc   = np.array(((12.*np.sqrt(3) / lengthscale**3 * variance,),)) 
-        H    = np.array([[1, 0]]) 
+        H    = np.array(((1, 0),)) 
-        Pinf = np.array([[variance, 0],  
+        Pinf = np.array(((variance, 0), (0, 3.*variance/(lengthscale**2))))
-        [0,              3.*variance/(lengthscale**2)]]) 
+        P0 = Pinf.copy()
+        
         # 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]) 
+        dFvariance       = np.zeros((2,2)) 
-        dFlengthscale    = np.array([[0,0], 
+        dFlengthscale    = np.array(((0,0), (6./lengthscale**3,2*np.sqrt(3)/lengthscale**2))) 
-        [6./lengthscale**3,2*np.sqrt(3)/lengthscale**2]]) 
+        dQcvariance      = np.array((12.*np.sqrt(3)/lengthscale**3)) 
-        dQcvariance      = np.array([12.*np.sqrt(3)/lengthscale**3]) 
+        dQclengthscale   = np.array((-3*12*np.sqrt(3)/lengthscale**4*variance)) 
-        dQclengthscale   = np.array([-3*12*np.sqrt(3)/lengthscale**4*variance]) 
+        dPinfvariance    = np.array(((1,0),(0,3./lengthscale**2))) 
-        dPinfvariance    = np.array([[1,0],[0,3./lengthscale**2]]) 
+        dPinflengthscale = np.array(((0,0), (0,-6*variance/lengthscale**3))) 
-        dPinflengthscale = np.array([[0,0], 
-        [0,-6*variance/lengthscale**3]]) 
         # Combine the derivatives 
         dF[:,:,0]    = dFvariance 
         dF[:,:,1]    = dFlengthscale 
@@ -64,8 +63,9 @@ class sde_Matern32(Matern32):
         dQc[:,:,1]   = dQclengthscale 
         dPinf[:,:,0] = dPinfvariance 
         dPinf[:,:,1] = dPinflengthscale 
-
+        dP0 = dPinf.copy()
-        return (F, L, Qc, H, Pinf, dF, dQc, dPinf)
+        
+        return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)
 
 class sde_Matern52(Matern52):
     """
@@ -106,7 +106,7 @@ class sde_Matern52(Matern52):
         H = np.array(((1,0,0),))        
         
         Pinf = np.array(((variance,0,-kappa), (0, kappa, 0), (-kappa, 0, 25.0*variance/lengthscale**4)))
-        
+        P0 = Pinf.copy()
         # Allocate space for the derivatives         
         dF = np.empty((3,3,2))        
         dQc = np.empty((1,1,2))        
@@ -130,75 +130,6 @@ class sde_Matern52(Matern52):
         dQc[:,:,1] = dQclengthscale        
         dPinf[:,:,0] = dPinf_variance
         dPinf[:,:,1] = dPinf_lengthscale
+        dP0 = dPinf.copy()
         
-#        % Derivative of F w.r.t. parameter magnSigma2
+        return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)  
-#    dFmagnSigma2    =  [0,  0,  0;
-#                        0,  0,  0;
-#                        0,  0,  0];
-#    
-#    % Derivative of F w.r.t parameter lengthScale
-#    dFlengthScale   =  [0,                          0,                  0;
-#                        0,                          0,                  0;
-#                        15*sqrt(5)/lengthScale^4,    30/lengthScale^3,   3*sqrt(5)/lengthScale^2];
-#    
-#    % Derivative of Qc w.r.t. parameter magnSigma2
-#    dQcmagnSigma2   =   400*sqrt(5)/3/lengthScale^5;
-#    
-#    % Derivative of Qc w.r.t. parameter lengthScale
-#    dQclengthScale  =   -magnSigma2*2000*sqrt(5)/3/lengthScale^6;
-#    
-#    % Derivative of Pinf w.r.t. parameter magnSigma2    
-#    dPinfmagnSigma2 = Pinf/magnSigma2;
-#    
-#    % Derivative of Pinf w.r.t. parameter lengthScale
-#    kappa2 = -2*kappa/lengthScale;
-#    dPinflengthScale = [0,          0,       -kappa2;
-#                        0,          kappa2,  0;
-#                        -kappa2,    0,       -100*magnSigma2/lengthScale^5];
-#  
-#    % Stack all derivatives
-#    dF = zeros(3,3,2);  
-#    dQc = zeros(1,1,2); 
-#    dPinf = zeros(3,3,2);
-#  
-#    dF(:,:,1) = dFmagnSigma2;
-#    dF(:,:,2) = dFlengthScale;
-#    dQc(:,:,1) = dQcmagnSigma2;
-#    dQc(:,:,2) = dQclengthScale;
-#    dPinf(:,:,1) = dPinfmagnSigma2;
-#    dPinf(:,:,2) = dPinflengthScale; 
-  
-#        % Derived constants
-#          lambda = sqrt(5)/lengthScale;
-#        
-#          % Feedback matrix
-#          F = [ 0,          1,          0;
-#                0,          0,          1;
-#               -lambda^3, -3*lambda^2, -3*lambda];
-#        
-#          % Noise effect matrix
-#          L = [0; 0; 1];
-#        
-#          % Spectral density
-#          Qc = magnSigma2*400*sqrt(5)/3/lengthScale^5;
-#        
-#          % Observation model
-#          H = [1, 0, 0];
-  
-  
-#        %% Stationary covariance
-#          
-#          % Calculate Pinf only if requested
-#          if nargout > 4,
-#              
-#            % Derived constant
-#            kappa = 5/3*magnSigma2/lengthScale^2;
-#            
-#            % Stationary covariance
-#            Pinf = [magnSigma2, 0,      -kappa;
-#                    0,          kappa,  0;
-#                    -kappa,     0,      25*magnSigma2/lengthScale^4];
-#                
-#          end
-        
-        return (F, L, Qc, H, Pinf, dF, dQc, dPinf)  

GPy/kern/_src/sde_standard_periodic.py

@@ -75,7 +75,7 @@ class sde_StdPeriodic(StdPeriodic):
         Qc   = np.zeros((2*(N+1), 2*(N+1)))
         P_inf = np.kron(np.diag(q2),np.eye(2))
         H    = np.kron(np.ones((1,N+1)),np.array((1,0)) )
-        
+        P0 = P_inf.copy()
         
         # Derivatives
         dF = np.empty((F.shape[0], F.shape[1], 3))
@@ -96,9 +96,9 @@ class sde_StdPeriodic(StdPeriodic):
         dF[:,:,2] = np.zeros(F.shape)
         dQc[:,:,2] = np.zeros(Qc.shape)
         dP_inf[:,:,2] = np.kron(np.diag(dq2l),np.eye(2))
-        
+        dP0 = dP_inf.copy()
 
-        return (F, L, Qc, H, P_inf, dF, dQc, dP_inf)
+        return (F, L, Qc, H, P_inf, P0, dF, dQc, dP_inf, dP0)
         
         
         

GPy/kern/_src/sde_static.py

@@ -1,6 +1,6 @@
 # -*- coding: utf-8 -*-
 """
-Classes in this module enhance Matern covariance functions with the
+Classes in this module enhance Static covariance functions with the
 Stochastic Differential Equation (SDE) functionality.
 """
 from .static import White
@@ -14,25 +14,47 @@ class sde_White(White):
     Class provide extra functionality to transfer this covariance function into
     SDE forrm.
     
-    Linear kernel:
+    White kernel:
 
     .. math::
 
-       k(x,y) = \alpha
+       k(x,y) = \alpha*\delta(x-y)
 
     """
-
+    
+    def sde_update_gradient_full(self, gradients):
+        """
+        Update gradient in the order in which parameters are represented in the
+        kernel
+        """
+    
+        self.variance.gradient = gradients[0]
+        
     def sde(self): 
         """ 
         Return the state space representation of the covariance. 
         """ 
         
-        # Arno, insert your code here
+        variance = float(self.variance.values) 
+        
+        F = np.array( ((-np.inf,),) )
+        L = np.array( ((1.0,),)  )
+        Qc = np.array( ((variance,),)  )
+        H = np.array( ((1.0,),) )
+        
+        Pinf   = np.array( ((variance,),)  )
+        P0 = Pinf.copy()     
+        
+        dF = np.zeros((1,1,1))
+        dQc = np.zeros((1,1,1))
+        dQc[:,:,0]    = np.array( ((1.0,),) )
+        
+        dPinf = np.zeros((1,1,1))
+        dPinf[:,:,0] = np.array( ((1.0,),) )
+        dP0 = dPinf.copy()
+        
+        return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)
 
-        # Params to use:
-        # self.variance
-
-        #return (F, L, Qc, H, Pinf, dF, dQc, dPinf)
 
 class sde_Bias(Bias):
     """
@@ -40,22 +62,40 @@ class sde_Bias(Bias):
     Class provide extra functionality to transfer this covariance function into
     SDE forrm.
     
-    Linear kernel:
+    Bias kernel:
 
     .. math::
 
-       k(x,y) = \alpha*\delta(x-y)
+       k(x,y) = \alpha
 
     """
-
+    def sde_update_gradient_full(self, gradients):
+        """
+        Update gradient in the order in which parameters are represented in the
+        kernel
+        """
+    
+        self.variance.gradient = gradients[0]
+        
     def sde(self): 
         """ 
         Return the state space representation of the covariance. 
         """ 
+        variance = float(self.variance.values) 
         
-        # Arno, insert your code here
+        F = np.array( ((0.0,),))
+        L = np.array( ((1.0,),))
+        Qc = np.zeros((1,1))
+        H = np.array( ((1.0,),))
         
-        # Params to use:
+        Pinf   = np.zeros((1,1))
-        # self.variance
+        P0 = np.array( ((variance,),) )      
         
-        #return (F, L, Qc, H, Pinf, dF, dQc, dPinf)
+        dF = np.zeros((1,1,1))
+        dQc    = np.zeros((1,1,1))
+        
+        dPinf = np.zeros((1,1,1))
+        dP0 = np.zeros((1,1,1))
+        dP0[:,:,0] = np.array( ((1.0,),) )
+        
+        return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)

GPy/kern/_src/sde_stationary.py

@@ -8,6 +8,7 @@ from .stationary import Exponential
 from .stationary import RatQuad
 
 import numpy as np
+import scipy as sp
 
 class sde_RBF(RBF):
     """
@@ -22,20 +23,87 @@ class sde_RBF(RBF):
         k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r^2 \\bigg) \\ \\ \\ \\  \text{ where  } r = \sqrt{\sum_{i=1}^{input dim} \frac{(x_i-y_i)^2}{\ell_i^2} }
 
     """
+    def sde_update_gradient_full(self, gradients):
+        """
+        Update gradient in the order in which parameters are represented in the
+        kernel
+        """
+    
+        self.variance.gradient = gradients[0]
+        self.lengthscale.gradient = gradients[1]
 
     def sde(self): 
         """ 
         Return the state space representation of the covariance. 
         """ 
         
-        # Arno, insert your code here
+        N = 10# approximation order ( number of terms in exponent series expansion)
-
+        roots_rounding_decimals = 6
-        # Params to use:
-
-        # self.lengthscale
-        # self.variance
         
-        #return (F, L, Qc, H, Pinf, dF, dQc, dPinf)
+        fn = np.math.factorial(N)        
+        
+        kappa = 1.0/2.0/self.lengthscale**2
+       
+        Qc = np.array((self.variance*np.sqrt(np.pi/kappa)*fn*(4*kappa)**N,),)
+       
+        pp = np.zeros((2*N+1,)) # array of polynomial coefficients from higher power to lower
+        
+        for n in range(0, N+1): # (2N+1) - number of polynomial coefficients
+            pp[2*(N-n)] = fn*(4.0*kappa)**(N-n)/np.math.factorial(n)*(-1)**n 
+        
+        pp = sp.poly1d(pp)
+        roots = sp.roots(pp)        
+        
+        neg_real_part_roots = roots[np.round(np.real(roots) ,roots_rounding_decimals) < 0]
+        aa = sp.poly1d(neg_real_part_roots, r=True).coeffs        
+        
+        F = np.diag(np.ones((N-1,)),1)
+        F[-1,:] = -aa[-1:0:-1]
+        
+        L= np.zeros((N,1))
+        L[N-1,0] = 1
+        
+        H = np.zeros((1,N))
+        H[0,0] = 1
+        
+        # Infinite covariance:
+        Pinf = sp.linalg.solve_lyapunov(F, -np.dot(L,np.dot( Qc[0,0],L.T)))
+        
+        # Allocating space for 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]) 
+        
+        # Derivatives:
+        dFvariance = np.zeros(F.shape)
+        dFlengthscale = np.zeros(F.shape)
+        dFlengthscale[-1,:] = -aa[-1:0:-1]/self.lengthscale * np.arange(-N,0,1)
+
+        dQcvariance = Qc/self.variance
+        dQclengthscale = np.array(((self.variance*np.sqrt(2*np.pi)*fn*2**N*self.lengthscale**(-2*N)*(1-2*N,),)))         
+        
+        dPinf_variance = Pinf/self.variance
+        
+        lp = Pinf.shape[0]
+        coeff = np.arange(1,lp+1).reshape(lp,1) + np.arange(1,lp+1).reshape(1,lp) - 2
+        coeff[np.mod(coeff,2) != 0] = 0
+        dPinf_lengthscale = -1/self.lengthscale*Pinf*coeff
+        
+        dF[:,:,0]    = dFvariance 
+        dF[:,:,1]    = dFlengthscale 
+        dQc[:,:,0]   = dQcvariance 
+        dQc[:,:,1]   = dQclengthscale 
+        dPinf[:,:,0] = dPinf_variance 
+        dPinf[:,:,1] = dPinf_lengthscale
+        
+        # Benefits of this are unjustified
+        #import GPy.models.state_space_main as ssm
+        #(F, L, Qc, H, Pinf, dF, dQc, dPinf,T) = ssm.balance_ss_model(F, L, Qc, H, Pinf, dF, dQc, dPinf)
+        
+        P0 = Pinf.copy()
+        dP0 = dPinf.copy()
+        
+        return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)
 
 class sde_Exponential(Exponential):
     """
@@ -50,29 +118,47 @@ class sde_Exponential(Exponential):
        k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r \\bigg) \\ \\ \\ \\  \text{ where  } r = \sqrt{\sum_{i=1}^{input dim} \frac{(x_i-y_i)^2}{\ell_i^2} }
 
     """
-
+    
+    def sde_update_gradient_full(self, gradients):
+        """
+        Update gradient in the order in which parameters are represented in the
+        kernel
+        """
+    
+        self.variance.gradient = gradients[0]
+        self.lengthscale.gradient = gradients[1]
+        
     def sde(self): 
         """ 
         Return the state space representation of the covariance. 
         """ 
-        F  = np.array([[-1/self.lengthscale]]) 
+        variance = float(self.variance.values)
-        L  = np.array([[1]]) 
+        lengthscale = float(self.lengthscale)        
-        Qc = np.array([[2*self.variance/self.lengthscale]]) 
-        H = np.array([[1]]) 
-        Pinf = np.array([[self.variance]]) 
-        # TODO: return the derivatives as well 
         
-        return (F, L, Qc, H, Pinf)
+        F  = np.array(((-1.0/lengthscale,),))
+        L  = np.array(((1.0,),)) 
+        Qc = np.array( ((2.0*variance/lengthscale,),) ) 
+        H = np.array(((1,),)) 
+        Pinf = np.array(((variance,),)) 
+        P0 = Pinf.copy()        
         
-        # Arno, insert your code here
+        dF = np.zeros((1,1,2));  
+        dQc = np.zeros((1,1,2)); 
+        dPinf = np.zeros((1,1,2));
         
-        # Params to use:
+        dF[:,:,0] = 0.0        
-
+        dF[:,:,1] = 1.0/lengthscale**2
-        # self.lengthscale
+        
-        # self.variance
+        dQc[:,:,0] = 2.0/lengthscale       
-
+        dQc[:,:,1] = -2.0*variance/lengthscale**2
-        #return (F, L, Qc, H, Pinf, dF, dQc, dPinf) 
+        
+        dPinf[:,:,0] = 1.0
+        dPinf[:,:,1] = 0.0
+        
+        dP0 = dPinf.copy()        
 
+        return (F, L, Qc, H, Pinf, P0, dF, dQc, dPinf, dP0)
+        
 class sde_RatQuad(RatQuad):
     """
     
@@ -92,7 +178,7 @@ class sde_RatQuad(RatQuad):
         Return the state space representation of the covariance. 
         """ 
         
-        # Arno, insert your code here
+        assert False, 'Not Implemented'
         
         # Params to use:
 
@@ -100,4 +186,4 @@ class sde_RatQuad(RatQuad):
         # self.variance
         #self.power
         
-        #return (F, L, Qc, H, Pinf, dF, dQc, dPinf)  
+        #return (F, L, Qc, H, Pinf, dF, dQc, dPinf)  

GPy/kern/src/add.py

@@ -290,23 +290,26 @@ class Add(CombinationKernel):
         Qc    = None
         H     = None
         Pinf  = None
+        P0    = None
         dF    = None
         dQc   = None
-        dPinf = None    
+        dPinf = None
+        dP0   = None
         n = 0
         nq = 0
         nd = 0
 
          # Assign models
         for p in self.parts:
-            (Ft,Lt,Qct,Ht,Pinft,dFt,dQct,dPinft) = p.sde()
+            (Ft,Lt,Qct,Ht,Pinft,P0t,dFt,dQct,dPinft,dP0t) = p.sde()
             F = la.block_diag(F,Ft) if (F is not None) else Ft
             L = la.block_diag(L,Lt) if (L is not None) else Lt
             Qc = la.block_diag(Qc,Qct) if (Qc is not None) else Qct
             H = np.hstack((H,Ht)) if (H is not None) else Ht
              
             Pinf = la.block_diag(Pinf,Pinft) if (Pinf is not None) else Pinft
-             
+            P0 = la.block_diag(P0,P0t) if (P0 is not None) else P0t
+            
             if dF is not None:
                 dF = np.pad(dF,((0,dFt.shape[0]),(0,dFt.shape[1]),(0,dFt.shape[2])),
                         'constant', constant_values=0)
@@ -327,7 +330,14 @@ class Add(CombinationKernel):
                 dPinf[-dPinft.shape[0]:,-dPinft.shape[1]:,-dPinft.shape[2]:] = dPinft
             else:
                 dPinf = dPinft
-             
+                
+            if dP0 is not None:
+                dP0 = np.pad(dP0,((0,dP0t.shape[0]),(0,dP0t.shape[1]),(0,dP0t.shape[2])),
+                        'constant', constant_values=0)
+                dP0[-dP0t.shape[0]:,-dP0t.shape[1]:,-dP0t.shape[2]:] = dP0t
+            else:
+                dP0 = dP0t
+                
             n += Ft.shape[0]
             nq += Qct.shape[0]
             nd += dFt.shape[2]
@@ -336,9 +346,11 @@ class Add(CombinationKernel):
         assert (L.shape[0] == n and L.shape[1]==nq), "SDE add: Check of L Dimensions failed"
         assert (Qc.shape[0] == nq and Qc.shape[1]==nq), "SDE add: Check of Qc Dimensions failed"
         assert (H.shape[0] == 1 and H.shape[1]==n), "SDE add: Check of H Dimensions failed"
-        assert (Pinf.shape[0] == n and Pinf.shape[1]==n), "SDE add: Check of Pinf Dimensions failed"        
+        assert (Pinf.shape[0] == n and Pinf.shape[1]==n), "SDE add: Check of Pinf Dimensions failed"
+        assert (P0.shape[0] == n and P0.shape[1]==n), "SDE add: Check of P0 Dimensions failed"        
         assert (dF.shape[0] == n and dF.shape[1]==n and dF.shape[2]==nd), "SDE add: Check of dF Dimensions failed"
         assert (dQc.shape[0] == nq and dQc.shape[1]==nq and dQc.shape[2]==nd), "SDE add: Check of dQc Dimensions failed"
         assert (dPinf.shape[0] == n and dPinf.shape[1]==n and dPinf.shape[2]==nd), "SDE add: Check of dPinf Dimensions failed"
+        assert (dP0.shape[0] == n and dP0.shape[1]==n and dP0.shape[2]==nd), "SDE add: Check of dP0 Dimensions failed"
         
-        return (F,L,Qc,H,Pinf,dF,dQc,dPinf)
+        return (F,L,Qc,H,Pinf,P0,dF,dQc,dPinf,dP0)

GPy/kern/src/prod.py

@@ -126,13 +126,15 @@ class Prod(CombinationKernel):
         Qc     = np.array((1,), ndmin=2)
         H      = np.array((1,), ndmin=2)
         Pinf   = np.array((1,), ndmin=2)
+        P0   = np.array((1,), ndmin=2)
         dF     = None
         dQc    = None
         dPinf  = None
-          
+        dP0  = None
+        
          # Assign models
         for p in self.parts:
-            (Ft,Lt,Qct,Ht,P_inft,dFt,dQct,dP_inft) = p.sde()
+            (Ft,Lt,Qct,Ht,P_inft, P0t, dFt,dQct,dP_inft,dP0t) = p.sde()
             
             # check derivative dimensions ->
             number_of_parameters = len(p.param_array)            
@@ -149,14 +151,16 @@ class Prod(CombinationKernel):
             dF    = dkron(F,dF,Ft,dFt,'sum')
             dQc   = dkron(Qc,dQc,Qct,dQct,'prod')
             dPinf = dkron(Pinf,dPinf,P_inft,dP_inft,'prod')
-             
+            dP0 = dkron(P0,dP0,P0t,dP0t,'prod')
+            
             F    = np.kron(F,np.eye(Ft.shape[0])) + np.kron(np.eye(F.shape[0]),Ft)
             L    = np.kron(L,Lt)
             Qc   = np.kron(Qc,Qct)
             Pinf = np.kron(Pinf,P_inft)
+            P0 = np.kron(P0,P_inft)
             H    = np.kron(H,Ht)
             
-        return (F,L,Qc,H,Pinf,dF,dQc,dPinf)
+        return (F,L,Qc,H,Pinf,P0,dF,dQc,dPinf,dP0)
 
 def dkron(A,dA,B,dB, operation='prod'):
     """

GPy/models/state_space_new.py

@@ -16,6 +16,7 @@
 
 import numpy as np
 from scipy import linalg
+from scipy import stats
 from ..core import Model
 from .. import kern
 from GPy.plotting.matplot_dep.models_plots import gpplot
@@ -26,17 +27,18 @@ from GPy.core.parameterization.param import Param
 
 import GPy
 from .. import likelihoods
-import GPy.models.state_space_main as ssm
+
-#import state_space_main as ssm
+from . import state_space_main as ssm
-reload(ssm)
-print ssm.__file__
 
 class StateSpace(Model):
     def __init__(self, X, Y, kernel=None, sigma2=1.0, name='StateSpace'):
         super(StateSpace, self).__init__(name=name)
         self.num_data, input_dim = X.shape
         assert input_dim==1, "State space methods for time only"
-        num_data_Y, self.output_dim = Y.shape
+        if len(Y.shape) ==2: # TODO make this nice
+            num_data_Y, self.output_dim = Y.shape
+        elif len(Y.shape) ==3:
+            num_data_Y, self.output_dim, ts_number = Y.shape
         assert num_data_Y == self.num_data, "X and Y data don't match"
         assert self.output_dim == 1, "State space methods for single outputs only"
 
@@ -68,8 +70,8 @@ class StateSpace(Model):
         """
         
         # Get the model matrices from the kernel
-        (F,L,Qc,H,P_inf,dFt,dQct,dP_inft) = self.kern.sde()
+        (F,L,Qc,H,P_inf, P0, dFt,dQct,dP_inft, dP0t) = self.kern.sde()
-
+        
         # necessary parameters
         measurement_dim = self.output_dim
         grad_params_no = dFt.shape[2]+1 # we also add measurement noise as a parameter
@@ -78,17 +80,19 @@ class StateSpace(Model):
         dF    = np.zeros([dFt.shape[0],dFt.shape[1],grad_params_no])
         dQc   = np.zeros([dQct.shape[0],dQct.shape[1],grad_params_no])
         dP_inf = np.zeros([dP_inft.shape[0],dP_inft.shape[1],grad_params_no])
-
+        dP0 = np.zeros([dP0t.shape[0],dP0t.shape[1],grad_params_no])
+        
         # Assign the values for the kernel function
         dF[:,:,:-1] = dFt
         dQc[:,:,:-1] = dQct
         dP_inf[:,:,:-1] = dP_inft
-
+        dP0[:,:,:-1] = dP0t
+        
         # The sigma2 derivative
         dR = np.zeros([measurement_dim,measurement_dim,grad_params_no])
         dR[:,:,-1] = np.eye(measurement_dim)
 
-
+        #(F,L,Qc,H,P_inf,dF,dQc,dP_inf) = ssm.balance_ss_model(F,L,Qc,H,P_inf,dF,dQc,dP_inf)
         # Use the Kalman filter to evaluate the likelihood
         
         grad_calc_params = {}
@@ -96,26 +100,53 @@ class StateSpace(Model):
         grad_calc_params['dF'] = dF
         grad_calc_params['dQc'] = dQc
         grad_calc_params['dR'] = dR
+        grad_calc_params['dP_init'] = dP0
         
         (filter_means, filter_covs, log_likelihood, 
-         grad_log_likelihood,SmootherMatrObject) = ssm.ContDescrStateSpace.cont_discr_kalman_filter(F,L,Qc,H,self.Gaussian_noise.variance,P_inf,self.X,self.Y,m_init=None,
+         grad_log_likelihood,SmootherMatrObject) = ssm.ContDescrStateSpace.cont_discr_kalman_filter(F,L,Qc,H,
-                                      P_init=None, calc_log_likelihood=True, 
+                                      float(self.Gaussian_noise.variance),P_inf,self.X,self.Y,m_init=None,
+                                      P_init=P0, calc_log_likelihood=True, 
                                       calc_grad_log_likelihood=True, 
                                       grad_params_no=grad_params_no, 
                                       grad_calc_params=grad_calc_params)
-                      
-        self._log_marginal_likelihood = log_likelihood
-        #gradients  = self.compute_gradients()
-        self.likelihood.update_gradients(grad_log_likelihood[-1,0])
         
-        self.kern.sde_update_gradient_full(grad_log_likelihood[:-1,0])
+        grad_log_likelihood_sum = np.sum(grad_log_likelihood,axis=1)
+        grad_log_likelihood_sum.shape = (grad_log_likelihood_sum.shape[0],1)
+        self._log_marginal_likelihood = np.sum( log_likelihood,axis=1 )
+        self.likelihood.update_gradients(grad_log_likelihood_sum[-1,0])
+        
+        self.kern.sde_update_gradient_full(grad_log_likelihood_sum[:-1,0])
         
     def log_likelihood(self):
         return self._log_marginal_likelihood
 
-    def _predict_raw(self, Xnew, Ynew=None, filteronly=False):
+    def _raw_predict(self, Xnew, Ynew=None, filteronly=False):
         """
-        Inner function. It is called only from inside this class
+        Performs the actual prediction for new X points.
+        Inner function. It is called only from inside this class.
+        
+        Input:
+        ---------------------
+        
+        Xnews: vector or (n_points,1) matrix
+            New time points where to evaluate predictions.
+            
+        Ynews: (n_train_points, ts_no) matrix
+            This matrix can substitude the original training points (in order 
+            to use only the parameters of the model).
+            
+        filteronly: bool
+            Use only Kalman Filter for prediction. In this case the output does
+            not coincide with corresponding Gaussian process.
+            
+        Output:
+        --------------------
+        
+        m: vector
+            Mean prediction
+        
+        V: vector
+            Variance in every point
         """
         
         # Set defaults
@@ -128,41 +159,44 @@ class StateSpace(Model):
 
         # Sort the matrix (save the order)
         _, return_index, return_inverse = np.unique(X,True,True)
-        X = X[return_index]
+        X = X[return_index] # TODO they are not used
         Y = Y[return_index]
 
         # Get the model matrices from the kernel
-        (F,L,Qc,H,P_inf,dF,dQc,dP_inf) = self.kern.sde()
+        (F,L,Qc,H,P_inf, P0, dF,dQc,dP_inf,dP0) = self.kern.sde()
         state_dim = F.shape[0]        
         
+        #import pdb; pdb.set_trace()
+        #Y = self.Y[:, 0,0]
         # Run the Kalman filter
-        (M, P, tmp_log_likelihood, 
+        #import pdb; pdb.set_trace()
-         tmp_grad_log_likelihood,SmootherMatrObject) = ssm.ContDescrStateSpace.cont_discr_kalman_filter(F,L,Qc,H,self.sigma2,P_inf,self.X,self.Y,m_init=None,
+        (M, P, log_likelihood, 
-                                      P_init=None, calc_log_likelihood=False, 
+         grad_log_likelihood,SmootherMatrObject) = ssm.ContDescrStateSpace.cont_discr_kalman_filter(
-                                      calc_grad_log_likelihood=False) 
+                                      F,L,Qc,H,float(self.Gaussian_noise.variance),P_inf,self.X,Y,m_init=None,
-                                      
+                                      P_init=P0, calc_log_likelihood=False, 
+                                      calc_grad_log_likelihood=False)                              
         # Run the Rauch-Tung-Striebel smoother
         if not filteronly:
             (M, P) = ssm.ContDescrStateSpace.cont_discr_rts_smoother(state_dim, M, P, 
                                 AQcomp=SmootherMatrObject, X=X, F=F,L=L,Qc=Qc)
         
         # remove initial values        
-        M = M[:,1:]
+        M = M[1:,:]
-        P = P[:,:,1:]        
+        P = P[1:,:,:]        
         
         # Put the data back in the original order
-        M = M[:,return_inverse]
+        M = M[return_inverse,:]
-        P = P[:,:,return_inverse]
+        P = P[return_inverse,:,:]
 
         # Only return the values for Xnew
-        M = M[:,self.num_data:]
+        M = M[self.num_data:,:]
-        P = P[:,:,self.num_data:]
+        P = P[self.num_data:,:,:]
 
         # Calculate the mean and variance
-        m = H.dot(M).T
+        m = np.dot(M,H.T)
-        V = np.tensordot(H[0],P,(0,0))
+        V = np.einsum('ij,ajk,kl', H, P, H.T)
-        V = np.tensordot(V,H[0],(0,0))
+        
-        V = V[:,None]
+        V.shape = (V.shape[0], V.shape[1]) # remove the third dimension
 
         # Return the posterior of the state
         return (m, V)
@@ -170,10 +204,10 @@ class StateSpace(Model):
     def predict(self, Xnew, filteronly=False):
 
         # Run the Kalman filter to get the state
-        (m, V) = self._predict_raw(Xnew,filteronly=filteronly)
+        (m, V) = self._raw_predict(Xnew,filteronly=filteronly)
 
         # Add the noise variance to the state variance
-        V += self.sigma2
+        V += float(self.Gaussian_noise.variance)
 
         # Lower and upper bounds
         lower = m - 2*np.sqrt(V)
@@ -181,143 +215,149 @@ class StateSpace(Model):
 
         # Return mean and variance
         return (m, V, lower, upper)
-
+        
-    def plot(self, plot_limits=None, levels=20, samples=0, fignum=None,
+    def predict_quantiles(self, Xnew, quantiles=(2.5, 97.5)):
-            ax=None, resolution=None, plot_raw=False, plot_filter=False,
+        mu, var = self._raw_predict(Xnew)
-            linecol=Tango.colorsHex['darkBlue'],fillcol=Tango.colorsHex['lightBlue']):
+        #import pdb; pdb.set_trace()
-
+        return  [stats.norm.ppf(q/100.)*np.sqrt(var + float(self.Gaussian_noise.variance)) + mu for q in quantiles]
-        # Deal with optional parameters
-        if ax is None:
-            fig = pb.figure(num=fignum)
-            ax = fig.add_subplot(111)
-
-        # Define the frame on which to plot
-        resolution = resolution or 200
-        Xgrid, xmin, xmax = x_frame1D(self.X, plot_limits=plot_limits)
-
-        # Make a prediction on the frame and plot it
-        if plot_raw:
-            m, v = self.predict_raw(Xgrid,filteronly=plot_filter)
-            lower = m - 2*np.sqrt(v)
-            upper = m + 2*np.sqrt(v)
-            Y = self.Y
-        else:
-            m, v, lower, upper = self.predict(Xgrid,filteronly=plot_filter)
-            Y = self.Y
-
-        # Plot the values
-        gpplot(Xgrid, m, lower, upper, axes=ax, edgecol=linecol, fillcol=fillcol)
-        ax.plot(self.X, self.Y, 'kx', mew=1.5)
-
-        # Optionally plot some samples
-        if samples:
-            if plot_raw:
-                Ysim = self.posterior_samples_f(Xgrid, samples)
-            else:
-                Ysim = self.posterior_samples(Xgrid, samples)
-            for yi in Ysim.T:
-                ax.plot(Xgrid, yi, Tango.colorsHex['darkBlue'], linewidth=0.25)
-
-        # Set the limits of the plot to some sensible values
-        ymin, ymax = min(np.append(Y.flatten(), lower.flatten())), max(np.append(Y.flatten(), upper.flatten()))
-        ymin, ymax = ymin - 0.1 * (ymax - ymin), ymax + 0.1 * (ymax - ymin)
-        ax.set_xlim(xmin, xmax)
-        ax.set_ylim(ymin, ymax)
-
-    def prior_samples_f(self,X,size=10):
-
-        # Sort the matrix (save the order)
-        (_, return_index, return_inverse) = np.unique(X,True,True)
-        X = X[return_index]
-
-        # Get the model matrices from the kernel
-        (F,L,Qc,H,Pinf,dF,dQc,dPinf) = self.kern.sde()
-
-        # Allocate space for results
-        Y = np.empty((size,X.shape[0]))
-
-        # Simulate random draws
-        #for j in range(0,size):
-        #    Y[j,:] = H.dot(self.simulate(F,L,Qc,Pinf,X.T))
-        Y = self.simulate(F,L,Qc,Pinf,X.T,size)
-
-        # Only observations
-        Y = np.tensordot(H[0],Y,(0,0))
-
-        # Reorder simulated values
-        Y = Y[:,return_inverse]
-
-        # Return trajectory
-        return Y.T
-
-    def posterior_samples_f(self,X,size=10):
-
-        # Sort the matrix (save the order)
-        (_, return_index, return_inverse) = np.unique(X,True,True)
-        X = X[return_index]
-
-        # Get the model matrices from the kernel
-        (F,L,Qc,H,Pinf,dF,dQc,dPinf) = self.kern.sde()
-
-        # Run smoother on original data
-        (m,V) = self.predict_raw(X)
-
-        # Simulate random draws from the GP prior
-        y = self.prior_samples_f(np.vstack((self.X, X)),size)
-
-        # Allocate space for sample trajectories
-        Y = np.empty((size,X.shape[0]))
-
-        # Run the RTS smoother on each of these values
-        for j in range(0,size):
-            yobs =  y[0:self.num_data,j:j+1] + np.sqrt(self.sigma2)*np.random.randn(self.num_data,1)
-            (m2,V2) = self.predict_raw(X,Ynew=yobs)
-            Y[j,:] = m.T + y[self.num_data:,j].T - m2.T
-
-        # Reorder simulated values
-        Y = Y[:,return_inverse]
-
-        # Return posterior sample trajectories
-        return Y.T
-
-    def posterior_samples(self, X, size=10):
-
-        # Make samples of f
-        Y = self.posterior_samples_f(X,size)
-
-        # Add noise
-        Y += np.sqrt(self.sigma2)*np.random.randn(Y.shape[0],Y.shape[1])
-
-        # Return trajectory
-        return Y
         
         
-    def simulate(self,F,L,Qc,Pinf,X,size=1):
+#    def plot(self, plot_limits=None, levels=20, samples=0, fignum=None,
-        # Simulate a trajectory using the state space model
+#            ax=None, resolution=None, plot_raw=False, plot_filter=False,
-
+#            linecol=Tango.colorsHex['darkBlue'],fillcol=Tango.colorsHex['lightBlue']):
-        # Allocate space for results
+#
-        f = np.zeros((F.shape[0],size,X.shape[1]))
+#        # Deal with optional parameters
-
+#        if ax is None:
-        # Initial state
+#            fig = pb.figure(num=fignum)
-        f[:,:,1] = np.linalg.cholesky(Pinf).dot(np.random.randn(F.shape[0],size))
+#            ax = fig.add_subplot(111)
-
+#
-        # Time step lengths
+#        # Define the frame on which to plot
-        dt = np.empty(X.shape)
+#        resolution = resolution or 200
-        dt[:,0] = X[:,1]-X[:,0]
+#        Xgrid, xmin, xmax = x_frame1D(self.X, plot_limits=plot_limits)
-        dt[:,1:] = np.diff(X)
+#
-
+#        # Make a prediction on the frame and plot it
-        # Solve the LTI SDE for these time steps
+#        if plot_raw:
-        As, Qs, index = ssm.ContDescrStateSpace.lti_sde_to_descrete(F,L,Qc,dt)
+#            m, v = self.predict_raw(Xgrid,filteronly=plot_filter)
-
+#            lower = m - 2*np.sqrt(v)
-        # Sweep through remaining time points
+#            upper = m + 2*np.sqrt(v)
-        for k in range(1,X.shape[1]):
+#            Y = self.Y
-
+#        else:
-            # Form discrete-time model
+#            m, v, lower, upper = self.predict(Xgrid,filteronly=plot_filter)
-            A = As[:,:,index[1-k]]
+#            Y = self.Y
-            Q = Qs[:,:,index[1-k]]
+#
-
+#        # Plot the values
-            # Draw the state
+#        gpplot(Xgrid, m, lower, upper, axes=ax, edgecol=linecol, fillcol=fillcol)
-            f[:,:,k] = A.dot(f[:,:,k-1]) + np.dot(np.linalg.cholesky(Q),np.random.randn(A.shape[0],size))
+#        ax.plot(self.X, self.Y, 'kx', mew=1.5)
-
+#
-        # Return values
+#        # Optionally plot some samples
-        return f
+#        if samples:
+#            if plot_raw:
+#                Ysim = self.posterior_samples_f(Xgrid, samples)
+#            else:
+#                Ysim = self.posterior_samples(Xgrid, samples)
+#            for yi in Ysim.T:
+#                ax.plot(Xgrid, yi, Tango.colorsHex['darkBlue'], linewidth=0.25)
+#
+#        # Set the limits of the plot to some sensible values
+#        ymin, ymax = min(np.append(Y.flatten(), lower.flatten())), max(np.append(Y.flatten(), upper.flatten()))
+#        ymin, ymax = ymin - 0.1 * (ymax - ymin), ymax + 0.1 * (ymax - ymin)
+#        ax.set_xlim(xmin, xmax)
+#        ax.set_ylim(ymin, ymax)
+#
+#    def prior_samples_f(self,X,size=10):
+#
+#        # Sort the matrix (save the order)
+#        (_, return_index, return_inverse) = np.unique(X,True,True)
+#        X = X[return_index]
+#
+#        # Get the model matrices from the kernel
+#        (F,L,Qc,H,Pinf,dF,dQc,dPinf) = self.kern.sde()
+#
+#        # Allocate space for results
+#        Y = np.empty((size,X.shape[0]))
+#
+#        # Simulate random draws
+#        #for j in range(0,size):
+#        #    Y[j,:] = H.dot(self.simulate(F,L,Qc,Pinf,X.T))
+#        Y = self.simulate(F,L,Qc,Pinf,X.T,size)
+#
+#        # Only observations
+#        Y = np.tensordot(H[0],Y,(0,0))
+#
+#        # Reorder simulated values
+#        Y = Y[:,return_inverse]
+#
+#        # Return trajectory
+#        return Y.T
+#
+#    def posterior_samples_f(self,X,size=10):
+#
+#        # Sort the matrix (save the order)
+#        (_, return_index, return_inverse) = np.unique(X,True,True)
+#        X = X[return_index]
+#
+#        # Get the model matrices from the kernel
+#        (F,L,Qc,H,Pinf,dF,dQc,dPinf) = self.kern.sde()
+#
+#        # Run smoother on original data
+#        (m,V) = self.predict_raw(X)
+#
+#        # Simulate random draws from the GP prior
+#        y = self.prior_samples_f(np.vstack((self.X, X)),size)
+#
+#        # Allocate space for sample trajectories
+#        Y = np.empty((size,X.shape[0]))
+#
+#        # Run the RTS smoother on each of these values
+#        for j in range(0,size):
+#            yobs =  y[0:self.num_data,j:j+1] + np.sqrt(self.sigma2)*np.random.randn(self.num_data,1)
+#            (m2,V2) = self.predict_raw(X,Ynew=yobs)
+#            Y[j,:] = m.T + y[self.num_data:,j].T - m2.T
+#
+#        # Reorder simulated values
+#        Y = Y[:,return_inverse]
+#
+#        # Return posterior sample trajectories
+#        return Y.T
+#
+#    def posterior_samples(self, X, size=10):
+#
+#        # Make samples of f
+#        Y = self.posterior_samples_f(X,size)
+#
+#        # Add noise
+#        Y += np.sqrt(self.sigma2)*np.random.randn(Y.shape[0],Y.shape[1])
+#
+#        # Return trajectory
+#        return Y
+#        
+#        
+#    def simulate(self,F,L,Qc,Pinf,X,size=1):
+#        # Simulate a trajectory using the state space model
+#
+#        # Allocate space for results
+#        f = np.zeros((F.shape[0],size,X.shape[1]))
+#
+#        # Initial state
+#        f[:,:,1] = np.linalg.cholesky(Pinf).dot(np.random.randn(F.shape[0],size))
+#
+#        # Time step lengths
+#        dt = np.empty(X.shape)
+#        dt[:,0] = X[:,1]-X[:,0]
+#        dt[:,1:] = np.diff(X)
+#
+#        # Solve the LTI SDE for these time steps
+#        As, Qs, index = ssm.ContDescrStateSpace.lti_sde_to_descrete(F,L,Qc,dt)
+#
+#        # Sweep through remaining time points
+#        for k in range(1,X.shape[1]):
+#
+#            # Form discrete-time model
+#            A = As[:,:,index[1-k]]
+#            Q = Qs[:,:,index[1-k]]
+#
+#            # Draw the state
+#            f[:,:,k] = A.dot(f[:,:,k-1]) + np.dot(np.linalg.cholesky(Q),np.random.randn(A.shape[0],size))
+#
+#        # Return values
+#        return f