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# CURRENTLY UNDER PROGRESS
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# Copyright (c) 2013, Arno Solin. Edited by Mu
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# Licensed under the BSD 3-clause license (see LICENSE.txt)
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#
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# This implementation of converting GPs to state space models is based on the article:
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#
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# @article{Sarkka+Solin+Hartikainen:2013,
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# author = {Simo S\"arkk\"a and Arno Solin and Jouni Hartikainen},
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# year = {2013},
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# title = {Spatiotemporal learning via infinite-dimensional {B}ayesian filtering and smoothing},
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# journal = {IEEE Signal Processing Magazine},
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# volume = {30},
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# number = {4},
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# pages = {51--61}
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# }
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#
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import numpy as np
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from scipy import linalg
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from ..core import Model
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from .. import kern
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from GPy.util.plot import gpplot, Tango, x_frame1D
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import pylab as pb
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class StateSpace_Old(Model):
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def __init__(self, X, Y, kernel=None):
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super(StateSpace_Old, self).__init__()
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self.num_data, input_dim = X.shape
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assert input_dim==1, "State space methods for time only but for two outputs"
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num_data_Y, self.output_dim = Y.shape
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assert num_data_Y == self.num_data, "X and Y data don't match"
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assert self.output_dim == 2, "State space methods for single outputs only"
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# Make sure the observations are ordered in time
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sort_index = np.argsort(X[:,0])
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self.X = X[sort_index]
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self.Y = Y[sort_index]
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self.a = 1.
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self.b = 1.
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# Noise variance
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self.sigma2 = .1
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# Default kernel
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if kernel is None:
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self.kern = kern.Matern32(1)
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else:
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self.kern = kernel
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# Make sure all parameters are positive
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self.ensure_default_constraints()
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# Assert that the kernel is supported
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#assert self.kern.sde() not False, "This kernel is not supported for state space estimation"
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def _set_params(self, x):
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self.kern._set_params(x[:self.kern.num_params_transformed()])
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self.sigma2 = x[-3]
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self.a = x[-2]
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self.b = x[-1]
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def _get_params(self):
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#return np.append(self.kern._get_params_transformed(), self.sigma2, self.a, self.b)
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return np.hstack([ self.kern._get_params_transformed(), self.sigma2, self.a, self.b ])
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def _get_param_names(self):
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return self.kern._get_param_names_transformed() + ['noise_variance','a','b']
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def log_likelihood(self):
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# Get the model matrices from the kernel
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#(F,L,Qc,H,Pinf) = self.kern.sde()
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(F,L,Qc,H,Pinf,use1,use2,use3) = self.kern.sde()
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Fm = np.zeros((3,3))
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Fm[1:,1:] = F
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Fm[0,0] = -self.a
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Fm[0,1] = self.b
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Lm = np.zeros((3,1))
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Lm[1:,0] = L.flatten()
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Hm = np.zeros((2,3))
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Hm[0,0] = 1
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Hm[1,1:] = H
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Pinfm = linalg.solve_lyapunov(Fm,-Lm.dot(Qc).dot(Lm.T))
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# Use the Kalman filter to evaluate the likelihood
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#return self.kf_likelihood(F,L,Qc,H,self.sigma2,Pinf,self.X.T,self.Y.T)
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return self.kf_likelihood(Fm,Lm,Qc,Hm,self.sigma2,Pinfm,self.X.T,self.Y.T)
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def _log_likelihood_gradients(self):
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# Get the model matrices from the kernel
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#(F,L,Qc,H,Pinf,dF,dQc,dPinf) = self.kern.sde()
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(F,L,Qc,H,Pinf,use1,use2,use3) = self.kern.sde()
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# Calculate the likelihood gradients TODO
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#return self.kf_likelihood_g(F,L,Qc,self.sigma2,H,Pinf,dF,dQc,dPinf,self.X,self.Y)
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return False
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def predict_raw(self, Xnew, filteronly=False):
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# Make a single matrix containing training and testing points
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X = np.vstack((self.X, Xnew))
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Y = np.vstack((self.Y, np.nan*np.zeros((Xnew.shape[0],2))))
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# Sort the matrix (save the order)
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_, return_index, return_inverse = np.unique(X,True,True)
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X = X[return_index]
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Y = Y[return_index]
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# Get the model matrices from the kernel
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#(F,L,Qc,H,Pinf) = self.kern.sde()
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(F,L,Qc,H,Pinf,use1,use2,use3) = self.kern.sde()
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Fm = np.zeros((3,3))
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Fm[1:,1:] = F
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Fm[0,0] = -self.a
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Fm[0,1] = self.b
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Lm = np.zeros((3,1))
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Lm[1:,0] = L.flatten()
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Hm = np.zeros((2,3))
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Hm[0,0] = 1
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Hm[1,1:] = H
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Pinfm = linalg.solve_lyapunov(Fm,-Lm.dot(Qc).dot(Lm.T))
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# Run the Kalman filter
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#stop
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(M, P) = self.kalman_filter(Fm,Lm,Qc,Hm,self.sigma2,Pinfm,X.T,Y.T)
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# Run the Rauch-Tung-Striebel smoother
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#if not filter:
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(M, P) = self.rts_smoother(Fm,Lm,Qc,X.T,M,P)
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# Put the data back in the original order
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M = M[:,return_inverse]
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P = P[:,:,return_inverse]
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# Only return the values for Xnew
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M = M[:,self.num_data:]
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P = P[:,:,self.num_data:]
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# Calculate the mean and variance
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m = Hm.dot(M).T
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V=P[0:2,0:2,:]
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#V = np.tensordot(H[0],P,(0,0))
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#V = np.tensordot(V,H[0],(0,0))
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#V = V[:,None]
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#stop
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# Return the posterior of the state
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return (m, V)
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def predict(self, Xnew, filteronly=False):
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# Run the Kalman filter to get the state
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(m, V) = self.predict_raw(Xnew,filteronly=filteronly)
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# Add the noise variance to the state variance
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V[0,0,:] += self.sigma2
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V[1,1,:] += self.sigma2
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# Lower and upper bounds
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lower = m[:,0] - 2*np.sqrt(V[0,0,:])
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upper = m[:,0] + 2*np.sqrt(V[0,0,:])
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#stop
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# Return mean and variance
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return (m[:,0], V[0,0,:], lower, upper)
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def plot(self, plot_limits=None, levels=20, samples=0, fignum=None,
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ax=None, resolution=None, plot_raw=False, plot_filter=False,
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linecol=Tango.colorsHex['darkBlue'],fillcol=Tango.colorsHex['lightBlue']):
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# Deal with optional parameters
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if ax is None:
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fig = pb.figure(num=fignum)
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ax = fig.add_subplot(111)
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# Define the frame on which to plot
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resolution = resolution or 200
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Xnew, xmin, xmax = x_frame1D(self.X, plot_limits=plot_limits)
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Xgrid = np.empty((Xnew.shape[0],2))
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Xgrid[:,0] = Xnew.flatten()
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Xgrid[:,1] = 0
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#stop
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# Make a prediction on the frame and plot it
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if plot_raw:
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m, v = self.predict_raw(Xgrid,filteronly=plot_filter)
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lower = m - 2*np.sqrt(v)
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upper = m + 2*np.sqrt(v)
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Y = self.Y
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else:
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#m, v, lower, upper = self.predict(Xgrid,filteronly=plot_filter)
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m, v, lower, upper = self.predict(Xnew,filteronly=plot_filter)
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Y = self.Y
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#stop
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# Plot the values
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gpplot(Xnew, m, lower, upper, axes=ax, edgecol=linecol, fillcol=fillcol)
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#ax.plot(self.X, self.Y, 'kx', mew=1.5)
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ax.plot(self.X, self.Y[:,0], 'kx', mew=1.5)
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# Optionally plot some samples
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if samples:
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Ysim = self.posterior_samples(Xgrid, samples)
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for yi in Ysim.T:
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ax.plot(Xgrid, yi, Tango.colorsHex['darkBlue'], linewidth=0.25)
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# Set the limits of the plot to some sensible values
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ymin, ymax = min(np.append(Y.flatten(), lower.flatten())), max(np.append(Y.flatten(), upper.flatten()))
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ymin, ymax = ymin - 0.1 * (ymax - ymin), ymax + 0.1 * (ymax - ymin)
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ax.set_xlim(xmin, xmax)
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ax.set_ylim(ymin, ymax)
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def posterior_samples_f(self,X,size=10):
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# Reorder X values
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sort_index = np.argsort(X[:,0])
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X = X[sort_index]
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# Get the model matrices from the kernel
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#(F,L,Qc,H,Pinf) = self.kern.sde()
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(F,L,Qc,H,Pinf,use1,use2,use3) = self.kern.sde()
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# Allocate space for results
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Y = np.empty((size,X.shape[0]))
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# Simulate random draws
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for j in range(0,size):
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Y[j,:] = H.dot(self.simulate(F,L,Qc,Pinf,X.T))
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# Reorder simulated values
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Y[:,sort_index] = Y[:,:]
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# Return trajectory
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return Y.T
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def posterior_samples(self, X, size=10):
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# Make samples of f
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Y = self.posterior_samples_f(X,size)
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# Add noise
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Y += np.sqrt(self.sigma2)*np.random.randn(Y.shape[0],Y.shape[1])
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# Return trajectory
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return Y
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def kalman_filter(self,F,L,Qc,H,R,Pinf,X,Y):
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# KALMAN_FILTER - Run the Kalman filter for a given model and data
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# Allocate space for results
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MF = np.empty((F.shape[0],Y.shape[1]))
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PF = np.empty((F.shape[0],F.shape[0],Y.shape[1]))
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# Initialize
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MF[:,-1] = np.zeros(F.shape[0])
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PF[:,:,-1] = Pinf.copy()
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# Time step lengths
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dt = np.empty(X.shape)
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dt[:,0] = X[:,1]-X[:,0]
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dt[:,1:] = np.diff(X)
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# Solve the LTI SDE for these time steps
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As, Qs, index = self.lti_disc(F,L,Qc,dt)
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#stop
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# Kalman filter
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for k in range(0,Y.shape[1]):
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# Form discrete-time model
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#(A, Q) = self.lti_disc(F,L,Qc,dt[:,k])
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A = As[:,:,index[k]];
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Q = Qs[:,:,index[k]];
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# Prediction step
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MF[:,k] = A.dot(MF[:,k-1])
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PF[:,:,k] = A.dot(PF[:,:,k-1]).dot(A.T) + Q
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# Update step (only if there is data)
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#if not np.isnan(Y[:,k]):
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if not np.isnan(Y[0,k]):
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if Y.shape[0]==1:
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K = PF[:,:,k].dot(H.T)/(H.dot(PF[:,:,k]).dot(H.T) + R)
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else:
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#stop
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LL = linalg.cho_factor(H.dot(PF[:,:,k]).dot(H.T) + R)
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K = linalg.cho_solve(LL, H.dot(PF[:,:,k].T)).T
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MF[:,k] += K.dot(Y[:,k]-H.dot(MF[:,k]))
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PF[:,:,k] -= K.dot(H).dot(PF[:,:,k])
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# Return values
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return (MF, PF)
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def rts_smoother(self,F,L,Qc,X,MS,PS):
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# RTS_SMOOTHER - Run the RTS smoother for a given model and data
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# Time step lengths
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dt = np.empty(X.shape)
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dt[:,0] = X[:,1]-X[:,0]
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dt[:,1:] = np.diff(X)
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# Solve the LTI SDE for these time steps
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As, Qs, index = self.lti_disc(F,L,Qc,dt)
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# Sequentially smooth states starting from the end
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for k in range(2,X.shape[1]+1):
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# Form discrete-time model
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#(A, Q) = self.lti_disc(F,L,Qc,dt[:,1-k])
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A = As[:,:,index[1-k]];
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Q = Qs[:,:,index[1-k]];
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# Smoothing step
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LL = linalg.cho_factor(A.dot(PS[:,:,-k]).dot(A.T)+Q)
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G = linalg.cho_solve(LL,A.dot(PS[:,:,-k])).T
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MS[:,-k] += G.dot(MS[:,1-k]-A.dot(MS[:,-k]))
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PS[:,:,-k] += G.dot(PS[:,:,1-k]-A.dot(PS[:,:,-k]).dot(A.T)-Q).dot(G.T)
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# Return
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return (MS, PS)
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def kf_likelihood(self,F,L,Qc,H,R,Pinf,X,Y):
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# Evaluate marginal likelihood
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# Initialize
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lik = 0
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m = np.zeros((F.shape[0],1))
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P = Pinf.copy()
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# Time step lengths
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dt = np.empty(X.shape)
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dt[:,0] = X[:,1]-X[:,0]
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dt[:,1:] = np.diff(X)
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# Solve the LTI SDE for these time steps
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As, Qs, index = self.lti_disc(F,L,Qc,dt)
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# Kalman filter for likelihood evaluation
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for k in range(0,Y.shape[1]):
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# Form discrete-time model
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#(A,Q) = self.lti_disc(F,L,Qc,dt[:,k])
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A = As[:,:,index[k]];
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Q = Qs[:,:,index[k]];
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# Prediction step
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m = A.dot(m)
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P = A.dot(P).dot(A.T) + Q
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#stop
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# Update step only if there is data
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if not np.isnan(Y[0,k]):
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v = Y[:,k][:,None]-H.dot(m)
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if Y.shape[0]==1:
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S = H.dot(P).dot(H.T) + R
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K = P.dot(H.T)/S
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lik -= 0.5*np.log(S)
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lik -= 0.5*v.shape[0]*np.log(2*np.pi)
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lik -= 0.5*v*v/S
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else:
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LL, isupper = linalg.cho_factor(H.dot(P).dot(H.T) + R*np.eye(Y.shape[0]))
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lik -= np.sum(np.log(np.diag(LL)))
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lik -= 0.5*v.shape[0]*np.log(2*np.pi)
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lik -= 0.5*linalg.cho_solve((LL, isupper),v).T.dot(v)
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K = linalg.cho_solve((LL, isupper), H.dot(P.T)).T
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m += K.dot(v)
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P -= K.dot(H).dot(P)
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#stop
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# Return likelihood
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return lik[0,0]
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#return lik
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def simulate(self,F,L,Qc,Pinf,X):
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# Simulate a trajectory using the state space model
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# Allocate space for results
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f = np.zeros((F.shape[0],X.shape[1]))
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# Initial state
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f[:,0:1] = np.linalg.cholesky(Pinf).dot(np.random.randn(F.shape[0],1))
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# Sweep through remaining time points
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for k in range(1,X.shape[1]):
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# Form discrete-time model
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(A,Q) = self.lti_disc(F,L,Qc,X[:,k]-X[:,k-1])
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# Draw the state
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f[:,k] = A.dot(f[:,k-1]).T + np.dot(np.linalg.cholesky(Q),np.random.randn(A.shape[0],1)).T
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# Return values
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return f
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def lti_disc(self,F,L,Qc,dt):
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# Discrete-time solution to the LTI SDE
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# Dimensionality
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n = F.shape[0]
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index = 0
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# Check for numbers of time steps
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if dt.flatten().shape[0]==1:
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# The covariance matrix by matrix fraction decomposition
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Phi = np.zeros((2*n,2*n))
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Phi[:n,:n] = F
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Phi[:n,n:] = L.dot(Qc).dot(L.T)
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Phi[n:,n:] = -F.T
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AB = linalg.expm(Phi*dt).dot(np.vstack((np.zeros((n,n)),np.eye(n))))
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Q = linalg.solve(AB[n:,:].T,AB[:n,:].T)
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# The dynamical model
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A = linalg.expm(F*dt)
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# Return
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return A, Q
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# Optimize for cases where time steps occur repeatedly
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else:
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# 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
|
||||
|
||||
Loading…
Add table
Add a link
Reference in a new issue