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GPy/models/state_space_ode.py

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+# CURRENTLY UNDER PROGRESS
+# Copyright (c) 2013, Mu Niu,Arno Solin, Simo Sarkka.
+# Licensed under the BSD 3-clause license (see LICENSE.txt)
+#
+# This implementation of converting GPs to state space models is based on the article:
+#
+#  @article{Sarkka+Solin+Hartikainen:2013,
+#     author = {Simo S\"arkk\"a and Arno Solin and Jouni Hartikainen},
+#       year = {2013},
+#      title = {Spatiotemporal learning via infinite-dimensional {B}ayesian filtering and smoothing},
+#    journal = {IEEE Signal Processing Magazine},
+#     volume = {30},
+#     number = {4},
+#      pages = {51--61}
+#  }
+#
+
+import numpy as np
+from scipy import linalg
+from ..core import Model
+from .. import kern
+from GPy.util.plot import gpplot, Tango, x_frame1D
+import pylab as pb
+
+class StateSpace_ODE(Model):
+    def __init__(self, X, Y, kernel=None):
+        super(StateSpace_ODE, self).__init__()
+        self.num_data, input_dim = X.shape
+        assert input_dim==1, "State space methods for time only but for two outputs"
+        num_data_Y, self.output_dim = Y.shape
+        assert num_data_Y == self.num_data, "X and Y data don't match"
+        assert self.output_dim == 2, "State space methods for single outputs only"
+
+        # Make sure the observations are ordered in time
+        sort_index = np.argsort(X[:,0])
+        self.X = X[sort_index]
+        self.Y = Y[sort_index]
+
+        self.a = 1.
+        self.b = 1.
+
+        # Noise variance
+        self.sigma2 = .1
+
+        # Default kernel
+        if kernel is None:
+            self.kern = kern.Matern32(1)
+        else:
+            self.kern = kernel
+
+        # Make sure all parameters are positive
+        self.ensure_default_constraints()
+
+        # Assert that the kernel is supported
+        #assert self.kern.sde() not False, "This kernel is not supported for state space estimation"
+
+    def _set_params(self, x):
+        self.kern._set_params(x[:self.kern.num_params_transformed()])
+        self.sigma2 = x[-3]
+        self.a = x[-2]
+        self.b = x[-1]
+
+    def _get_params(self):
+        #return np.append(self.kern._get_params_transformed(), self.sigma2, self.a, self.b)
+        return np.hstack([ self.kern._get_params_transformed(), self.sigma2, self.a, self.b ])
+
+    def _get_param_names(self):
+        return self.kern._get_param_names_transformed() + ['noise_variance','a','b']
+
+    def log_likelihood(self):
+
+        # Get the model matrices from the kernel
+        #(F,L,Qc,H,Pinf) = self.kern.sde()
+        (F,L,Qc,H,Pinf,use1,use2,use3) = self.kern.sde()
+
+        Fm = np.zeros((3,3))
+        Fm[1:,1:] = F
+        Fm[0,0] = -self.a
+        Fm[0,1] = self.b
+
+        Lm = np.zeros((3,1))
+        Lm[1:,0] = L.flatten()
+
+        Hm = np.zeros((2,3))
+        Hm[0,0] = 1
+        Hm[1,1:] = H
+
+        Pinfm = linalg.solve_lyapunov(Fm,-Lm.dot(Qc).dot(Lm.T))
+        # Use the Kalman filter to evaluate the likelihood
+        #return self.kf_likelihood(F,L,Qc,H,self.sigma2,Pinf,self.X.T,self.Y.T)
+        return self.kf_likelihood(Fm,Lm,Qc,Hm,self.sigma2,Pinfm,self.X.T,self.Y.T)
+
+    def _log_likelihood_gradients(self):
+
+        # Get the model matrices from the kernel
+        #(F,L,Qc,H,Pinf,dF,dQc,dPinf) = self.kern.sde()
+        (F,L,Qc,H,Pinf,use1,use2,use3) = self.kern.sde()
+
+        # Calculate the likelihood gradients TODO
+        #return self.kf_likelihood_g(F,L,Qc,self.sigma2,H,Pinf,dF,dQc,dPinf,self.X,self.Y) 
+        return False
+
+    def predict_raw(self, Xnew, filteronly=False):
+
+        # Make a single matrix containing training and testing points
+        X = np.vstack((self.X, Xnew))
+        Y = np.vstack((self.Y, np.nan*np.zeros((Xnew.shape[0],2))))
+
+        # Sort the matrix (save the order)
+        _, return_index, return_inverse = np.unique(X,True,True)
+        X = X[return_index]
+        Y = Y[return_index]
+
+        # Get the model matrices from the kernel
+        #(F,L,Qc,H,Pinf) = self.kern.sde()
+        (F,L,Qc,H,Pinf,use1,use2,use3) = self.kern.sde()
+        
+        Fm = np.zeros((3,3))
+        Fm[1:,1:] = F
+        Fm[0,0] = -self.a
+        Fm[0,1] = self.b
+
+        Lm = np.zeros((3,1))
+        Lm[1:,0] = L.flatten()
+
+        Hm = np.zeros((2,3))
+        Hm[0,0] = 1
+        Hm[1,1:] = H
+
+        Pinfm = linalg.solve_lyapunov(Fm,-Lm.dot(Qc).dot(Lm.T))
+        # Run the Kalman filter
+        #stop
+        (M, P) = self.kalman_filter(Fm,Lm,Qc,Hm,self.sigma2,Pinfm,X.T,Y.T)
+        
+        # Run the Rauch-Tung-Striebel smoother
+        #if not filter:
+        (M, P) = self.rts_smoother(Fm,Lm,Qc,X.T,M,P)
+
+        # Put the data back in the original order
+        M = M[:,return_inverse]
+        P = P[:,:,return_inverse]
+
+        # Only return the values for Xnew
+        M = M[:,self.num_data:]
+        P = P[:,:,self.num_data:]
+        
+        # Calculate the mean and variance
+        m = Hm.dot(M).T
+        V=P[0:2,0:2,:]
+        #V = np.tensordot(H[0],P,(0,0))
+        #V = np.tensordot(V,H[0],(0,0))
+        #V = V[:,None]
+        #stop
+        # Return the posterior of the state
+        return (m, V)
+
+    def predict(self, Xnew, filteronly=False):
+
+        # Run the Kalman filter to get the state
+        (m, V) = self.predict_raw(Xnew,filteronly=filteronly)
+
+        # Add the noise variance to the state variance
+        V[0,0,:] += self.sigma2
+        V[1,1,:] += self.sigma2
+        
+        # Lower and upper bounds
+        lower = m[:,0] - 2*np.sqrt(V[0,0,:])
+        upper = m[:,0] + 2*np.sqrt(V[0,0,:])
+        #stop
+        # Return mean and variance
+        return (m[:,0], V[0,0,:], lower, upper)
+
+    def plot(self, plot_limits=None, levels=20, samples=0, fignum=None,
+            ax=None, resolution=None, plot_raw=False, plot_filter=False,
+            linecol=Tango.colorsHex['darkBlue'],fillcol=Tango.colorsHex['lightBlue']):
+
+        # 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
+        Xnew, xmin, xmax = x_frame1D(self.X, plot_limits=plot_limits)
+        Xgrid = np.empty((Xnew.shape[0],2))
+        Xgrid[:,0] = Xnew.flatten()
+        Xgrid[:,1] = 0
+        #stop
+        # 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)
+            m, v, lower, upper = self.predict(Xnew,filteronly=plot_filter)
+            Y = self.Y
+            #stop
+        # Plot the values
+        gpplot(Xnew, m, lower, upper, axes=ax, edgecol=linecol, fillcol=fillcol)
+        #ax.plot(self.X, self.Y, 'kx', mew=1.5)
+        ax.plot(self.X, self.Y[:,0], 'kx', mew=1.5)
+
+        # Optionally plot some samples
+        if samples:
+            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 posterior_samples_f(self,X,size=10):
+
+        # Reorder X values
+        sort_index = np.argsort(X[:,0])
+        X = X[sort_index]
+
+        # Get the model matrices from the kernel
+        #(F,L,Qc,H,Pinf) = self.kern.sde()
+        (F,L,Qc,H,Pinf,use1,use2,use3) = 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))
+
+        # Reorder simulated values
+        Y[:,sort_index] = Y[:,:]
+
+        # Return trajectory
+        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 kalman_filter(self,F,L,Qc,H,R,Pinf,X,Y):
+        # KALMAN_FILTER - Run the Kalman filter for a given model and data
+
+        # Allocate space for results
+        MF = np.empty((F.shape[0],Y.shape[1]))
+        PF = np.empty((F.shape[0],F.shape[0],Y.shape[1]))
+
+        # Initialize
+        MF[:,-1] = np.zeros(F.shape[0])
+        PF[:,:,-1] = Pinf.copy()
+
+        # 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 = self.lti_disc(F,L,Qc,dt)
+        #stop
+        # Kalman filter
+        for k in range(0,Y.shape[1]):
+
+            # Form discrete-time model
+            #(A, Q) = self.lti_disc(F,L,Qc,dt[:,k])
+            A = As[:,:,index[k]];
+            Q = Qs[:,:,index[k]];
+
+            # Prediction step
+            MF[:,k] = A.dot(MF[:,k-1])
+            PF[:,:,k] = A.dot(PF[:,:,k-1]).dot(A.T) + Q
+
+            # Update step (only if there is data)
+            #if not np.isnan(Y[:,k]):
+            if not np.isnan(Y[0,k]):
+                 if Y.shape[0]==1:
+                     K = PF[:,:,k].dot(H.T)/(H.dot(PF[:,:,k]).dot(H.T) + R)
+                 else:
+                     #stop
+                     LL = linalg.cho_factor(H.dot(PF[:,:,k]).dot(H.T) + R)
+                     K = linalg.cho_solve(LL, H.dot(PF[:,:,k].T)).T
+                 MF[:,k] += K.dot(Y[:,k]-H.dot(MF[:,k]))
+                 PF[:,:,k] -= K.dot(H).dot(PF[:,:,k])
+
+        # Return values
+        return (MF, PF)
+
+    def rts_smoother(self,F,L,Qc,X,MS,PS):
+        # RTS_SMOOTHER - Run the RTS smoother for a given model and data
+
+        # 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 = self.lti_disc(F,L,Qc,dt)
+
+        # Sequentially smooth states starting from the end
+        for k in range(2,X.shape[1]+1):
+
+            # Form discrete-time model
+            #(A, Q) = self.lti_disc(F,L,Qc,dt[:,1-k])
+            A = As[:,:,index[1-k]];
+            Q = Qs[:,:,index[1-k]];
+
+            # Smoothing step
+            LL = linalg.cho_factor(A.dot(PS[:,:,-k]).dot(A.T)+Q)
+            G = linalg.cho_solve(LL,A.dot(PS[:,:,-k])).T
+            MS[:,-k] += G.dot(MS[:,1-k]-A.dot(MS[:,-k]))
+            PS[:,:,-k] += G.dot(PS[:,:,1-k]-A.dot(PS[:,:,-k]).dot(A.T)-Q).dot(G.T)
+
+        # Return
+        return (MS, PS)
+
+    def kf_likelihood(self,F,L,Qc,H,R,Pinf,X,Y):
+        # Evaluate marginal likelihood
+
+        # Initialize
+        lik = 0
+        m = np.zeros((F.shape[0],1))
+        P = Pinf.copy()
+
+        # 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 = self.lti_disc(F,L,Qc,dt)
+
+        # Kalman filter for likelihood evaluation
+        for k in range(0,Y.shape[1]):
+
+            # Form discrete-time model
+            #(A,Q) = self.lti_disc(F,L,Qc,dt[:,k])
+            A = As[:,:,index[k]];
+            Q = Qs[:,:,index[k]];
+
+            # Prediction step
+            m = A.dot(m)
+            P = A.dot(P).dot(A.T) + Q
+            #stop
+            # Update step only if there is data
+            if not np.isnan(Y[0,k]):
+                 v = Y[:,k][:,None]-H.dot(m)
+                 if Y.shape[0]==1:
+                     S = H.dot(P).dot(H.T) + R
+                     K = P.dot(H.T)/S
+                     lik -= 0.5*np.log(S)
+                     lik -= 0.5*v.shape[0]*np.log(2*np.pi)
+                     lik -= 0.5*v*v/S
+                 else:
+                     LL, isupper = linalg.cho_factor(H.dot(P).dot(H.T) + R*np.eye(Y.shape[0]))
+                     lik -= np.sum(np.log(np.diag(LL)))
+                     lik -= 0.5*v.shape[0]*np.log(2*np.pi)
+                     lik -= 0.5*linalg.cho_solve((LL, isupper),v).T.dot(v)
+                     K = linalg.cho_solve((LL, isupper), H.dot(P.T)).T
+                 m += K.dot(v)
+                 P -= K.dot(H).dot(P)
+        #stop
+        # Return likelihood
+        return lik[0,0]
+        #return lik
+
+    def simulate(self,F,L,Qc,Pinf,X):
+        # Simulate a trajectory using the state space model
+
+        # Allocate space for results
+        f = np.zeros((F.shape[0],X.shape[1]))
+
+        # Initial state
+        f[:,0:1] = np.linalg.cholesky(Pinf).dot(np.random.randn(F.shape[0],1))
+
+        # Sweep through remaining time points
+        for k in range(1,X.shape[1]):
+
+            # Form discrete-time model
+            (A,Q) = self.lti_disc(F,L,Qc,X[:,k]-X[:,k-1])
+
+            # Draw the state
+            f[:,k] = A.dot(f[:,k-1]).T + np.dot(np.linalg.cholesky(Q),np.random.randn(A.shape[0],1)).T
+
+        # Return values
+        return f
+
+    def lti_disc(self,F,L,Qc,dt):
+        # Discrete-time solution to the LTI SDE
+
+        # Dimensionality
+        n = F.shape[0]
+        index = 0
+
+        # Check for numbers of time steps
+        if dt.flatten().shape[0]==1:
+
+            # The covariance matrix by matrix fraction decomposition
+            Phi = np.zeros((2*n,2*n))
+            Phi[:n,:n] = F
+            Phi[:n,n:] = L.dot(Qc).dot(L.T)
+            Phi[n:,n:] = -F.T
+            AB = linalg.expm(Phi*dt).dot(np.vstack((np.zeros((n,n)),np.eye(n))))
+            Q = linalg.solve(AB[n:,:].T,AB[:n,:].T)
+
+            # The dynamical model
+            A  = linalg.expm(F*dt)
+
+            # Return
+            return A, Q
+
+        # Optimize for cases where time steps occur repeatedly
+        else:
+
+            # Time discretizations (round to 14 decimals to avoid problems)
+            dt, _, index = np.unique(np.round(dt,14),True,True)
+
+            # Allocate space for A and Q
+            A = np.empty((n,n,dt.shape[0]))
+            Q = np.empty((n,n,dt.shape[0]))
+
+            # Call this function for each dt
+            for j in range(0,dt.shape[0]):
+                A[:,:,j], Q[:,:,j] = self.lti_disc(F,L,Qc,dt[j])
+
+            # Return
+            return A, Q, index
+