C. P. Unsworth

Nonlinear prediction of chaotic signals using a normalised radial basis function network

In this paper, a simple and robust combination of architecture and training strategy is proposed for a radial basis function network (RBFN). The proposed network uses a normalised Gaussian kernel architecture with kernel centres randomly selected from a training data set. The output layer weights are adapted using the numerically robust Householder transform. The application of this normalised radial basis function network (NRBFN) to the prediction of chaotic signals is reported. NRBFNs are shown to perform better than un-normalised equivalent networks for the task of chaotic signal prediction. Chaotic signal prediction is also used to demonstrate that a NRBFN is less sensitive to basis function parameter selection than an equivalent un-normalised network. A novel structure and training Strategy are proposed for a forward-backward RBFN (FB-RBFN). FB-NRBFN chaotic signal prediction results are compared with those for a NRBFN. Normalisation is found to be a simple alternative to regularisation for the task of using a RBFN to recursively predict, and thus to capture the dynamics of, a chaotic signal corrupted by additive white Gaussian noise.

False detection of chaotic behaviour in the stochastic compound k-distribution model of radar sea clutter

There is current debate in the radar community whether sea clutter is stochastic or chaotic. In this paper, a stochastic k-distributed surrogate is generated for a typical sea clutter data set. The k-distributed set was then analysed using the methods recently applied to sea clutter by Haykin et al. (Haykin and Li, Proc. IEEE, vol.83, pp.95-122, 1995; Haykin and Puthusserypady, Proc. IEE Radar, pp.75-9, 1997). The k-distributed set is shown to have DML (maximum likelihood estimation of the correlation dimension) and FNN (false nearest neighbours) values in the same range as reported by Haykin et al. (1995; 1997) and with positive and negative Lyapunov exponents. In addition, various white and correlated noise distributed sets are analysed in the same way and found to produce a similar artefact. It is concluded that these chaotic invariants cannot be used to distinguish between chaotic and stochastic time series and are redundant in an application, such as radar sea clutter, where the time series is unknown and could be of a stochastic nature.

A new method to detect nonlinearity in a time-series: synthesizing surrogate data using a Kolmogorov-Smirnoff tested, hidden Markov model

Abstract A way of statistically testing for nonlinearity in a time-series is to employ the method of surrogate data. This method often makes use of the Fourier transform (FT) in order to generate the surrogate. As various authors have shown, this can lead to artefacts in the surrogates and spurious detection of nonlinearity can result. This paper documents a new method to synthesize surrogate data using a 1st order hidden Markov model (HMM) combined with a Kolmogorov–Smirnoff test (KS-test) to determine the required resolution of the HMM. Significance test results for a sinewave, Henon map and Gaussian noise time-series are presented. It is demonstrated that KS-tested HMM surrogates can be successfully used to distinguish between a deterministic and stochastic time-series. Then by applying a simple test for linearity, using linear and nonlinear predictors, it is possible to determine the nature of the deterministic class and hence conclude whether the system is linear deterministic or nonlinear deterministic. Furthermore, it is demonstrated that the method works for periodic functions too, where FT surrogates break down.

Determining the importance of learning the underlying dynamics of sea clutter for radar target detection

Existing evidence for and against sea clutter being chaotic and nonlinearly predictable is briefly discussed. Despite the uncertainty surrounding the chaotic nature of sea clutter, and its nonlinear predictability, the purpose of this paper is to examine what the best design criterion is for a nonlinear predictor which is to be used to detect targets against clutter which is known to be chaotic: mean square error performance or capturing the chaotic clutters underlying dynamics. Single pulse detection analysis using a Swerling I target and chaotic clutter is carried out using predictor-based detectors in an attempt to determine which criterion is most suitable. The predictor detectors are compared with standard detection strategies.

Detection of nonlinearity in a time-series: by the synthesis of surrogate data using a Kolmogorov-Smirnoff tested, hidden Markov model

Conventional methods of hypothesis testing for nonlinearity in a time-series employ the method of surrogate data which makes use of the Fourier transform (FT). As various authors have shown, this can lead to artifacts in the surrogates and spurious detection of nonlinearity can result. This paper documents a new method to synthesize surrogate data using a 1st order hidden Markov model (HMM) combined with a Kolmogorov-Smirnoff test (KS-test), to determine the required resolution of the HMM. The method provides a way to retain the dynamics of a time-series and impart the null hypothesis (H/sub 0/) onto the synthesized surrogate which avoids the FT and its associated artifact. Significance test results for a sinewave, Henon map and Gaussian noise time-series are presented. It is demonstrated through significance testing that KS-tested, HMM surrogates can be successfully used to distinguish between a deterministic and stochastic time-series. Then by applying a simple test for linearity, using linear and nonlinear predictors, it is possible to determine the nature of the deterministic class and hence, conclude whether the system is linear deterministic or nonlinear deterministic. Furthermore, it is demonstrated that the method works for periodic functions too, where FT surrogates break down.