Identification of Spatial-Temporal Switched ARX Systems

1. Identification of Spatial-Temporal Switched ARX Systems René Vidal Center for Imaging Science Johns Hopkins University

2. Identification of hybrid systems Given input/output data, identify Number of discrete states Model parameters of linear systems Hybrid state (continuous & discrete) Switching parameters (partition of state space) Challenging “chicken-and-egg” problem Given switching times, estimate model parameters Given the model parameters, estimate hybrid state Given all above, estimate switching parameters Possible solution: iterate Very sensitive to initialization Needs a minimum dwell time Does not use all data

3. Prior work on hybrid system identification • Piecewise ARX systems • Clustering approach: k-means clustering + regression + classification + iterative refinement: (Ferrari-Trecate et al. ‘03) • Bayesian approach: maximum likelihood via expectation maximization algorithm (Juloski et al. ‘05) • Mixed integer quadratic programming: (Bemporad et al. ‘01) • Greedy/iterative approach: (Bemporad et al. ‘03) • Switched ARX systems • Batch algebraic approach: (Vidal et al. ‘03 ‘04) • Recursive algebraic approach: (Vidal et al. ‘04 ‘05)

4. Segmentation problems in dynamic vision • Dynamical vision problems involve • Appearing and disappearing motions • Multiple rigid and non-rigid motions

5. Spatial-temporal switched ARX models • Model output with mixture of dynamical models exhibiting changes in • Space: multiple motions in a video • Time: appearing and disappearing motions in a video • Solve a very complex hybrid system identification problem SARX1 SARXnt SARX2

6. Spatial-temporal switched ARX models • The dynamics of each mode are in ARX form • input/output • discrete state • order of the ARX models • model parameters • Input/output data lives in a hyperplane • I/O data • Model parameters

7. Recursive ID of ARX models • True model parameters • Equation error identifier • Persistence of excitation:

8. Recursive ID of STSARX model parameters • In the SARX case input/output data satisfy the hybrid decoupling polynomial • For STARX model, the situation is more complicated due to dependency on spatial location

9. IDofSTSARXmodel = IDofliftedARXmodel • Identification of a STSARX model is equivalent to identification of a single lifted ARX model • Can apply equation error identifier and derive persistence of excitation condition in lifted space Embedding Lifting Embedding

10. Recursive ID of STSARX model parameters • Equations of the lifted ARX model (hybrid decoupling polynomial) • We wish to minimize the mean prediction error • Normalized gradient descent gives hybrid equation error identifier • When |X|=1 and n=1 we get standard equation error identifier

11. Recursive ID of STSARX - Convergence • Theorem : For a minimal STSARX If there exists such that where then exponentially

12. Recursive ID of ARX model parameters: 1/2

13. Recursive ID of ARX model parameters: 2/2 • Estimation of Spatial Regions • Minimize • Algorithm • Given estimated normals, calculate membership as • Given estimated membership, estimate parameters as • Iterate until membership converges K-means

14. Spatial temporal video segmentation Video Batch Recursive

15. Spatial-temporal video segmentation

16. Spatial-temporal video segmentation

17. Conclusions • Contributions • A recursive identification algorithm for spatial-temporal switched ARX models of unknown number of modes and order • A persistence of excitation condition on the input/output data that guarantees exponential convergence • Open issues • Persistence of excitation condition on the mode and input sequences only • Extend the model to more general, possibly non-linear hybrid systems