In this paper, we develop a Bayesian framework for the empirical estimation of the parameters of one of the best known
nonlinear models of the business cycle: The Marx-inspired model of a growth cycle introduced by R. M. Goodwin. The
model predicts a series of closed cycles representing the dynamics of labor's share and the employment rate in the
capitalist economy. The Bayesian framework is used to empirically estimate a modified Goodwin model. The original
model is extended in two ways. First, we allow for exogenous periodic variations of the otherwise steady growth rates of
the labor force and productivity per worker. Second, we allow for stochastic variations of those parameters. The resultant
modified Goodwin model is a stochastic predator-prey model with periodic forcing. The model is then estimated using a
newly developed Bayesian estimation method on data sets representing growth cycles in France and Italy during the
years 1960-2005. Results show that inference of the parameters of the stochastic Goodwin model can be achieved. The
comparison of the dynamics of the Goodwin model with the inferred values of parameters demonstrates quantitative
agreement with the growth cycle empirical data.
A Bayesian framework for parameter inference in non-stationary, nonlinear, stochastic, dynamical systems is
introduced. It is applied to decode time variation of control parameters from time-series data modelling physiological
signals. In this context a system of FitzHugh-Nagumo (FHN) oscillators is considered, for which synthetically
generated signals are mixed via a measurement matrix. For each oscillator only one of the dynamical
variables is assumed to be measured, while another variable remains hidden (unobservable). The control parameter
for each FHN oscillator is varying in time. It is shown that the proposed approach allows one: (i) to
reconstruct both unmeasured (hidden) variables of the FHN oscillators and the model parameters, (ii) to detect
stepwise changes of control parameters for each oscillator, and (iii) to follow a continuous evolution of the control
parameters in the quasi-adiabatic limit.
We consider the following general problem of applied stochastic
nonlinear dynamics. We observe a time series of signals y(t) = y(t0+hn) corrupted by noise. The actual state and the nonlinear vector field of the dynamical system is not known. The question is how and with what accuracy can we determine x(t) and functional form of f(x). In this talk we discuss a novel approach to the solution of this problem based on the application of the path-integral approach to the full Bayesian inference. We demonstrate a reconstruction of a dynamical state of a system from corrupted by noise measurements. Next we reconstruct the corresponding nonlinear vector field. The emphasis are on the theoretical analysis. The results are compared with the results of earlier research.
An application of the path-integral approach to an analysis of the
fluctuations in complex dynamical systems is discussed. It is
shown that essentially the same ideas underly recent progress in
the solutions of a number of long-standing problems in complex
dynamics. In particular, we consider the problems of prediction,
control and inference of chaotic dynamics perturbed by noise in
the framework of path-integral approach.
A new method of inferencing of coupled stochastic nonlinear
oscillators is described. The technique does not require extensive
global optimization, provides optimal compensation for noise-induced errors and is robust in a broad range of dynamical models. We illustrate the main ideas of the technique by inferencing a model of five globally and locally coupled noisy oscillators. Specific modifications of the technique for inferencing hidden degrees of freedom of coupled nonlinear oscillators is discussed in the context of physiological applications.
The vertical-cavity surface-emitting laser (VCSEL) is a novel type of semiconductor laser that is having a dramatic influence on many optical applications, including computing, communication, and sensing. Unfortunately, VCSELs suffer from a randomly fluctuating polarization whose dynamics are not fully understood. The model of stochastic polarization dynamics developed by San Miguel is rather complicated, and comparisons between experiment and simulations are quite difficult. One of the approaches to solve this problem was suggested in, where a simplified spin-eliminated linearized model (i.e., with dynamics reduced to that of a class-A laser) is used to analyze experimentally measured fluctuations and fluctuational switchings. In this talk, we present a recently developed technique that estimates the parameters of a nonlinear stochastic dynamical model by Bayesian inference, and demonstrate its application to the characterization of VCSELs. We start by considering the problem of diffusion of polarization in a potential well, onto which the dynamics of a class-A laser are usually mapped. We demonstrate the ability to infer laser parameters in numerical and analogue simulations, with the emphasis being placed on the role of large deviations. We specifically show that, contrary to one's intuition, the quality of inference can be improved by neglecting those data points in experimental time series that correspond to the rising part of large deviations. We then extend our technique to the full set of equations describing the polarization dynamics of a VCSEL in terms of
the motion of its Stokes vector on the Poincare sphere. Application of this technique to other standard problems encountered in characterizing semiconductor lasers, such as the identification of laser parameters from measurements of relaxational oscillations, is also discussed.
We suggest a fresh approach to the modeling of the human cardiovascular system. Taking advantage of a new Bayesian inference technique, able to deal with stochastic nonlinear systems, we show that one can estimate parameters for models of the cardiovascular system directly from measured time series. We present preliminary results of inference of parameters of a model of coupled oscillators from measured cardiovascular data addressing cardiorespiratory interaction. We argue that the inference technique offers a very promising tool for the modeling, able to contribute significantly towards the solution of a long standing challenge -- development of new diagnostic techniques based on noninvasive measurements.