Transformation invariant stochastic catastrophe theory

Eric Jan Wagenmakers, Peter Molenaar, Raoul P.P.P. Grasman, Pascal A.I. Hartelman, Han L.J. Van Der Maas

Research output: Contribution to journalArticle

41 Citations (Scopus)

Abstract

Catastrophe theory is a mathematical formalism for modeling nonlinear systems whose discontinuous behavior is determined by smooth changes in a small number of driving parameters. Fitting a catastrophe model to noisy data constitutes a serious challenge, however, because catastrophe theory was formulated specifically for deterministic systems. Loren Cobb addressed this challenge by developing a stochastic counterpart of catastrophe theory (SCT) based on Itô stochastic differential equations. In SCT, the stable and unstable equilibrium states of the system correspond to the modes and the antimodes of the empirical probability density function, respectively. Unfortunately, SCT is not invariant under smooth and invertible transformations of variables - this is an important limitation, since invariance to diffeomorphic transformations is essential in deterministic catastrophe theory. From the Itô transformation rules we derive a generalized version of SCT that does remain invariant under transformation and can include Cobb's SCT as a special case. We show that an invariant function is obtained by multiplying the probability density function with the diffusion function of the stochastic process. This invariant function can be estimated by a straightforward time series analysis based on level crossings. We illustrate the invariance problem and its solution with two applications.

Original languageEnglish (US)
Pages (from-to)263-276
Number of pages14
JournalPhysica D: Nonlinear Phenomena
Volume211
Issue number3-4
DOIs
StatePublished - Nov 15 2005

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catastrophe theory
probability density functions
invariance
time series analysis
stochastic processes
nonlinear systems
differential equations
formalism

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

Cite this

Wagenmakers, E. J., Molenaar, P., Grasman, R. P. P. P., Hartelman, P. A. I., & Van Der Maas, H. L. J. (2005). Transformation invariant stochastic catastrophe theory. Physica D: Nonlinear Phenomena, 211(3-4), 263-276. https://doi.org/10.1016/j.physd.2005.08.014
Wagenmakers, Eric Jan ; Molenaar, Peter ; Grasman, Raoul P.P.P. ; Hartelman, Pascal A.I. ; Van Der Maas, Han L.J. / Transformation invariant stochastic catastrophe theory. In: Physica D: Nonlinear Phenomena. 2005 ; Vol. 211, No. 3-4. pp. 263-276.
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Wagenmakers, EJ, Molenaar, P, Grasman, RPPP, Hartelman, PAI & Van Der Maas, HLJ 2005, 'Transformation invariant stochastic catastrophe theory', Physica D: Nonlinear Phenomena, vol. 211, no. 3-4, pp. 263-276. https://doi.org/10.1016/j.physd.2005.08.014

Transformation invariant stochastic catastrophe theory. / Wagenmakers, Eric Jan; Molenaar, Peter; Grasman, Raoul P.P.P.; Hartelman, Pascal A.I.; Van Der Maas, Han L.J.

In: Physica D: Nonlinear Phenomena, Vol. 211, No. 3-4, 15.11.2005, p. 263-276.

Research output: Contribution to journalArticle

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Wagenmakers EJ, Molenaar P, Grasman RPPP, Hartelman PAI, Van Der Maas HLJ. Transformation invariant stochastic catastrophe theory. Physica D: Nonlinear Phenomena. 2005 Nov 15;211(3-4):263-276. https://doi.org/10.1016/j.physd.2005.08.014