Zsuzsanna Vágó

Optimization over discrete sets via SPSA

A fixed gain version of the SPSA (simultaneous perturbation stochastic approximation) method (J.C. Spall, 1992) for function minimization is developed and the error process is characterized. The new procedure is applicable to optimization problems over /spl Zscr//sup p/., the grid of points in /spl Rscr//sup p/ with integer components. Simulation results and a closely related application, a resource allocation problem, is also described.

Spatio-temporal analysis of monofractal and multifractal properties of the human sleep EEG.

Fractality is a common property in nature. It can also be observed in time series representing dynamics of complex processes. Therefore fractal analysis could be a useful tool to describe the dynamics of brain electrical activities in physiological and pathological conditions. In this study, we carried out a spatio-temporal analysis of monofractal and multifractal properties of whole-night sleep EEG recordings. We estimated the Hurst exponent (H) and the range of fractal spectra (dD) in 10 healthy subjects. We found higher H values during NREM4 compared to NREM2 and REM in all electrodes. Measure dD showed an opposite trend. Differences of H and dD between NREM2 and REM reached significancy at circumscribed regions only. Our results contribute to a deeper understanding of the fractal nature of brain electrical activities and may have implications for automatic classification of sleep stages.

The use of the SPSA method in ECG analysis

The classification, monitoring, and compression of electrocardiogram (ECG) signals recorded of a single patient over a relatively long period of time is considered. The particular application we have in mind is high-resolution ECG analysis, such as late potential analysis, morphology changes in QRS during arrythmias, T-wave alternants, or the study of drug effects on ventricular activation. We propose to apply a modification of a classical method of cluster analysis or vector quantization. The novelty of our approach is that we use a new distortion measure to quantify the distance of two ECG cycles, and the class-distortion measure is defined using a min-max criterion. The new class-distortion-measure is much more sensitive to outliers than the usual distortion measures using average-distance. The price of this practical advantage is that computational complexity is significantly increased. The resulting nonsmooth optimization problem is solved by an adapted version of the simultaneous perturbation stochastic approximation (SPSA) method of J. Spall (IEEE Trans. Automat. Contr., vol. 37, p. 332-41, Mar. 1992). The main idea is to generate a smooth approximation by a randomization procedure. The viability of the method is demonstrated on both simulated and real data. An experimental comparison with the widely used correlation method is given on real data.

New methods for the statistical analysis of Hidden Markov Models

The estimation of Hidden Markov Models has attracted a lot of attention recently. The purpose of this paper is to lay the foundation for a new approach for the analysis of the maximum-likelihood estimation of HMM-s, using representation of HMM-s due to Borkar (1993). A useful connection between the estimation theory of HMM-s and linear stochastic systems is established via the theory of L-mixing processes. The results are potentially useful for deriving strong approximation results, which are in turn applicable to analyze adaptive predictors and change detection methods.

Randomized iterative feedback tuning

Abstract In this contribution we present a controller tuning method which combines some of the advantages of Iterative Feedback Tuning (IFT) with some of the advantages of simultaneous perturbation stochastic approximation (SPSA). In particular this scheme can be shown to converge with geometric rate when a pure gradient search is used and the system is noise free. The number of experiments required to obtain an unbiased estimate of the gradient can be reduced significantly for multi-input multi-output systems. In particular we study the problem where the reference signal is periodic and when the noise is negligible.

SPSA in noise free optimization

The SPSA (simultaneous perturbation stochastic approximation) method for function minimization developed in Spall (1992) is analyzed for optimization problems without measurement noise. We prove the result that under appropriate technical conditions the estimator sequence converges to the optimum with geometric rate with probability 1. Numerical experiments support the conjecture that the top Lyapunov-exponent defined in terms of the SPSA method is smaller than the Lyapunov-exponent of its deterministic counterpart. We conclude that randomization improves convergence rate while dramatically reducing the number of function evaluations.

Randomization Methods in Optimization and Adaptive Control

We consider simultaneous perturbation stochastic approximation (SPSA) methods applied to noise-free problems in optimization and adaptive control. More generally, we consider discrete-time fixed gain stochastic approximation processes that are defined in terms of a random field that is identically zero at some point θ*. The boundedness of the estimator process is enforced by a resetting mechanism. Under appropriate technical conditions the estimator sequence converges to θ* with geometric rate almost surely. This result is in striking contrast to classical stochastic approximation theory where the typical convergence rate is n −1/2. For the proof a discrete-time version of the ODE-method is used and the techniques of [10] are extended. A simple variant of noise free-SPSA is applied to extend a direct controller tuning method named Iterative Feedback Tuning (IFT), see [16]. Using randomization, the number of experiments required to obtain an unbiased estimate of the gradient of the cost function can be reduced significantly for multi-input multi-output systems.

Fixed-gain estimation in continuous time

We prove that the parameter estimation error of continuous-time linear stochastic systems that is obtained in connection with a fixed-gain estimation method can be written as a stochastic integral plus a residual term, the moments of which are of orderλ+o(1) whereλ is the forgetting factor.

P371 Simulation and estimation of Hawkes processes for modeling EEG data to forecast epileptic seizures

Objectives Prediction of epileptic seizures has been in the forefront of the research for a long time but without the development of a satisfactory solution so far. Our aim is to develop a forecasting method – previously applied in seismology – that is based on fitting so-called Hawkes processes to EEG data, both exhibiting self-exiting dynamics. The choice of this approach is justified by the observation of Sornette and Osorio, according to which earthquakes and epileptic seizures show numerous similarities in their dynamics. Methods As an essential preliminary work for model-fitting we first implemented an algorithm for the simulation of self-exiting point processes. This was followed by the implementation of a maximum likelihood estimation (MLE) method based on the work of Ozaki. Results By visual inspection of projections of the three-dimensional parameter space we found a variety of behaviors of the cost function including some degenerate constellations of the parameters, where the behavior of the cost function relies almost solely on one parameter. Despite the simplicity of the local optimization method applied during the off-line estimation of the parameters, we succeeded to estimate the parameters with a fairly good approximation. Discussion and conclusion Based on our preliminary results, we conclude that the simulation of Hawkes-processes and fitting them to a point-process derived from EEG data is promising. As a next step, we plan to implement an on-line MLE method, by which real-time change detection in the EEG-derived parameters becomes possible. Significance As far as we know, self-exiting point processes have not been applied in seizure prediction before.

Discrete stochastic approximation via simultaneous difference approximations

A stochastic approximation method for optimizing a class of discrete functions is considered. The procedure is a version of the simultaneous perturbation stochastic approximation (SPSA) method that has been modified to obtain a stochastic optimization method for cost functions defined on a discrete set of points. We discuss the algorithm and examine its convergence and also the rate of convergence.