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Research Papers

Causation Entropy Identifies Sparsity Structure for Parameter Estimation of Dynamic Systems

[+] Author and Article Information
Pileun Kim

George W. Woodruff School
of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332

Jonathan Rogers

Assistant Professor
George W. Woodruff School of
Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332

Jie Sun

Assistant Professor
Department of Mathematics,
Clarkson University,
Potsdam, NY 13699

Erik Bollt

Professor
Department of Mathematics,
Clarkson University,
Potsdam, NY 13699

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 19, 2016; final manuscript received June 29, 2016; published online September 1, 2016. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 12(1), 011008 (Sep 01, 2016) (14 pages) Paper No: CND-16-1026; doi: 10.1115/1.4034126 History: Received January 19, 2016; Revised June 29, 2016

Parameter estimation is an important topic in the field of system identification. This paper explores the role of a new information theory measure of data dependency in parameter estimation problems. Causation entropy is a recently proposed information-theoretic measure of influence between components of multivariate time series data. Because causation entropy measures the influence of one dataset upon another, it is naturally related to the parameters of a dynamical system. In this paper, it is shown that by numerically estimating causation entropy from the outputs of a dynamic system, it is possible to uncover the internal parametric structure of the system and thus establish the relative magnitude of system parameters. In the simple case of linear systems subject to Gaussian uncertainty, it is first shown that causation entropy can be represented in closed form as the logarithm of a rational function of system parameters. For more general systems, a causation entropy estimator is proposed, which allows causation entropy to be numerically estimated from measurement data. Results are provided for discrete linear and nonlinear systems, thus showing that numerical estimates of causation entropy can be used to identify the dependencies between system states directly from output data. Causation entropy estimates can therefore be used to inform parameter estimation by reducing the size of the parameter set or to generate a more accurate initial guess for subsequent parameter optimization.

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References

Figures

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Fig. 1

CEM table values versus a11 for example 2 × 2 linear system

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Fig. 2

CEM table values versus a12 for example 2 × 2 linear system—(o) case: σu=0.1, σv=0.15 and (+) case: σu=0.1, σv=0.7

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Fig. 3

CEM table values versus a21 for example 2 × 2 linear system (analytical and estimated with KDE)

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Fig. 4

Magnitude plot of A matrix components for example 5 × 5 linear system

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Fig. 5

Magnitude plot of theoretical CEM components for example 5 × 5 linear system

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Fig. 6

Magnitude plot of estimated CEM components for example 5 × 5 linear system

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Fig. 7

Example coupled linear harmonic oscillators

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Fig. 8

Magnitude plot of A matrix components for 10 × 10 coupled linear oscillator example

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Fig. 9

Magnitude plot of theoretical CEM components for coupled linear oscillator example

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Fig. 10

Magnitude plot of estimated CEM components for coupled linear oscillator example

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Fig. 11

State time history for Hénon map example

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Fig. 12

State time history for Duffing map example (traces for the two states look similar but are not the same)

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Fig. 13

State time history for 5 × 5 nonlinear system example

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Fig. 14

State time history for 6 × 6 nonlinear system example

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