Identification of SpatialTemporal Switched ARX Systems Ren Vidal

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

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

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)

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

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 SARX 1 SARXnt SARX 2 • Solve a very complex hybrid system identification problem

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

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

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

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

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

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

Recursive ID of ARX model parameters: 1/2

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

Spatial temporal video segmentation Video Batch Recursive

Spatial-temporal video segmentation

Spatial-temporal video segmentation

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