Comparison of simulation-based algorithms for parameter estimation and state reconstruction in nonlinear state-space models

Figure 1.  Impact of parameter values on smoothing distributions for the Lorenz-63 model (20). The true state (black curve) and observations (black points) have been simulated with $ \theta^* = ({\bf{Q}},{\bf{R}}) = (0.01{\bf{I}}_3,2{\bf{I}}_3) $. The mean of the smoothing distributions (red curve) are computed using a standard particle smoother [16] with $ 100 $ particles. Results obtained with the true parameter value $ \theta^* = (0.01{\bf{I}}_3,2{\bf{I}}_3) $ (left panel) and a wrong parameter value $ \tilde{\theta} = ({\bf{I}}_3,{\bf{I}}_3) $ (right panel) are plotted

Figure 2.  Comparison of one iteration of PF and CPF algorithms using $ N_f = 5 $ particles (light grey points). The differences are highlighted in black : CPF replaces the particle $ {\bf{x}}_t^{(N_f)} $ of the PF with the conditioning particle $ {\bf{x}}_t^* $ (dark grey point)

Figure 3.  Comparisons of PF and CPF algorithms with 10 particles on the Kitagawa model defined in Section 3.2. Grey lines show the ancestors of the particles

Figure 4.  Example of ancestor tracking based on ancestral links of filtering particles. Particles (grey balls) are obtained using a filtering algorithm with $ N_f = 3 $ particles

Figure 5.  Comparison of CPF (left), CPFAS (middle), and CPFBS (right). The state (black line) and the observations (black points) have been simulated using the Kitagawa model (19) with $ Q = 1 $ and $ R = 10 $. $ N_f = 10 $ particles (grey points with grey lines showing the genealogy) are used in the three algorithms. The red curves show $ N_s = 10 $ realizations simulated with the algorithms

Figure 6.  Four iterations of the CPFBS smoother (Algorithm 3). The state (black line) and the observations (black points) have been simulated using the Kitagawa model (19) with $ Q = 1 $ and $ R = 10 $. At the first iteration, the conditioning trajectory (grey dotted line) is initialized with the constant sequence equal to $ 0 $. CPFBS is run with $ N_f = 10 $ particles (grey points) and $ N_s = 10 $ trajectories (red curves)

Figure 7.  Sequence simulated with the linear Gaussian SSM model (18) with $ \theta^* = (0.9, 1, 1) $. The mean of the smoothing distribution (red curve) and $ 95\% $ prediction interval (light red area) are computed based on the smoothing trajectories simulated in the last 10 iterations of CPFBS-SEM algorithm with $ N_f = N_s = 10 $ particles

Figure 8.  Distribution of the estimates obtained with CPFBS-SEM and CPFAS-SEM algorithms as a function of the number of EM iterations for the linear Gaussian SSM model (18) with $ \theta^* = (0.9,1,1) $, $ T = 100 $, $ N_f = N_s = 10 $. The empirical distributions are computed using $ 100 $ simulated samples. The median (grey dotted line) and $ 95\% $ confidence interval (grey shaded area) are computed using $ 10^3 $ iterations of the KS-EM algorithm

Figure 9.  Distribution of the estimates obtained with CPFBS-SEM, CPFAS-SEM, PFBS-EM, and EnKS-EM algorithms as a function of the number of particles for the linear Gaussian SSM model (18) with $ \theta^* = (0.9,1,1) $ and $ T = 100 $. Results obtained by running $ 100 $ iterations of the algorithms. The empirical distributions are computed using $ 100 $ simulated samples

Figure 10.  Sequence simulated with the Kitagawa model (19) with $ \theta^* = (1, 10) $. The mean of the smoothing distribution (red curve) and $ 95\% $ prediction interval (light red area) are computed based on the smoothing trajectories simulated in the last 10 iterations of CPFBS-SEM algorithm with $ N_f = N_s = 10 $ particles

Figure 11.  Distribution of the estimates obtained with CPFBS-SEM and CPFAS-SEM algorithms as a function of the number of SEM iterations for the Kitagawa model (19) with $ \theta^* = (1,10) $, $ T = 100 $, $ N_f = N_s = 10 $. The empirical distributions are computed using $ 100 $ simulated samples

Figure 12.  Distribution of the estimates obtained with CPFBS-SEM and CPFAS-SEM algorithms for the Kitagawa model (19) with $ \theta^* = (1,R^*) $, $ R^* \in \{0.1, 1, 5, 10\} $, $ T = 100 $, $ N_f = N_s = 10 $. Results obtained by running $ 100 $ iterations of the SEM algorithms. The empirical distributions are computed using $ 100 $ simulated samples

Figure 13.  Sequence simulated with the Lorenz-63 model (20) with $ \theta^* = (0.01, 2) $ and time step $ \triangle = 0.15 $. The mean of the smoothing distribution (red curve) and $ 95\% $ prediction interval (light red area) are computed based on the smoothing trajectories simulated in the last 10 iterations of CPFBS-SEM algorithm with $ N_f = N_s = 10 $ particles

Figure 14.  Distribution of the estimates obtained with CPFBS-SEM and CPFAS-SEM algorithms as a function of the number of EM iterations for the Lorenz-63 models (20) with $ \theta^* = (0.01, 2) $, $ \triangle = 0.15 $ $ T = 100 $, $ N_f = N_s = 20 $. The empirical distributions are computed using $ 100 $ simulated samples

Figure 15.  Distribution of the estimates obtained with CPFBS-SEM, CPFAS-SEM, and EnKS-EM algorithms as a function of the time step $ \triangle $ for the Lorenz-63 models (20) with $ \theta^* = (0.01, 2) $, $ T = 100 $, $ N_f = N_s = 20 $ and $ 20 $ members for the EnKS algorithm. The empirical distributions are computed using $ 100 $ simulated samples