Fawad Rauf

Calculation of Lyapunov exponents through nonlinear adaptive filters

The authors present a novel approach, using nonlinear adaptive filter, to model, filter, and predict chaotic time series. Lyapunov exponents can be estimated from the prediction error growth rate of such filters. The technique is very effective for detecting and quantifying low-dimensional chaos. The technique performs accurately even under noisy environments. Measurement noise and catastrophic noise are both handled equally well in this technique and do not hinder the estimate of the Lyapunov exponent since they are adequately filtered. The authors have demonstrated their technique with Henon and logistic maps. They have also presented a more reliable and accurate algorithm for estimation of Lyapunov exponents from finite and noisy time series data for applications in econometrics and other fields where large numbers of data points are not available.<<ETX>>

Adaptive state dependent filters

An approach known as state dependent embedding for developing nonlinear adaptive filters is presented. Many types of nonlinear filters including Volterra, bilinear, and polynomial autoregressive (PAR) are unified under this method. By recognizing the functional relationships between the channels of an equivalent linearly embedded system, state dependent embedding creates much more efficient filters than previous approaches. A filter called the layered structure emerges from the embedding. Its virtues include low computation, modularity, and local adaptation, allowing nonlinear filters to be implemented with linear adaptive building blocks. The layered structure is very amenable to VLSI. The state dependent embedding can also be used to develop very efficient lattice filters.<<ETX>>

New Nonlinear Adaptive Filters with Applications to Chaos

We present a new approach to nonlinear adaptive filtering based on Successive Linearization. Our approach provides a simple, modular and unified implementation for a broad class of polynomial filters. We refer to this implementation as the layered structure and note that it offers substantial computational efficiency over previous methods. A new class of Polynomial Autoregressive filters is introduced which can model limit cycle and chaotic dynamics. Existing geometric methods for modeling and characterizing chaotic processes suffer from several drawbacks. They require a huge number of data points to reconstruct the attractor geometry and performance is severely limited by noisy experimental measurements. We present a new method for processing chaotic signals using nonlinear adaptive filters. We demonstrate the modeling, prediction and filtering of these signals. We also show how the prediction error growth rate can be used to estimate the effective Lyapunov exponent of the chaotic map. Our approach requires orders of magnitude fewer data points and is robust to noise in the experimental data. Although reconstruction of the attractor geometry is unnecessary, the adaptive filter contains most of the geometric information.

Adaptive signal processing techniques for chaotic systems

The issue of modeling chaotic systems is addressed. Present methods for treating chaotic dynamics are based on state space reconstruction through delay embedding. These approaches are computationally intensive and are adversely affected by noise in the experimental time series. The authors take a different approach and apply an adaptive layered structure for estimation of chaotic dynamics. They show that presently used spatial local approximations are not necessary and that their temporal adaptive local approximations perform better, are tolerant to noise factors, and save an order of magnitude in computations, and data requirements.<<ETX>>

Nadine-a feedforward neural network for arbitrary nonlinear time series

It is shown that Madaline networks applied to the modeling of time series realize a constrained Volterra series because of the fixed nature of the nonlinearity. The authors introduce a novel feedforward structure, named Nadine, that can model arbitrary Volterra series and hence arbitrary, analytic nonlinearities with memory. Nadine can be realized using layers of adaptive linear combiners in which the outputs of one layer are used as the weights rather than the activities of the next layer. This structure admits local adaptation of the linear combiners, making it possible to implement backpropagation-style learning without actually propagating adaptation information between the layers. Nadine is therefore very modular, easy to implement, and readily extendible. High-order neural networks and polynomial discriminant-based methods are the special cases of Nadine which can now be implemented modularly without involving preprocessing.<<ETX>>

Fast learning neural networks using transform domain LMS structure

Transform domain adaptive filtering, which uses orthogonal transforms to partially uncorrelate the colored input, thus reducing the eigenvalue spread, is extended to the nonlinear domain and used as the basis of a learning algorithm for neural networks. Computer simulations show that this neural structure learns much faster than least-mean-square-based structure.<<ETX>>

Adaptive bilinear inverse filtering

A parsimonious adaptive nonlinear filtering structure based on bilinear models is presented. A state-dependent embedding for developing nonlinear filters is presented. It is used to develop a computationally efficient bilinear adaptive filter which requires only local adaptation. Modularity and local connectivity make the structure amenable to VLSI implementation. In contrast to previous input-output pair (system identification) frameworks, an inverse filtering problem is considered where only output is observable. The adaptation is shown to be dependent on both past and present gradients.<<ETX>>

A universal structure for artificial neural networks

An approximation procedure, named successive linearization, is introduced for unified implementation of a large class of neural networks. A nonlinear neural model with dynamic sensitivity is presented. It is modular and has rapid learning schemes. Arbitrary nonlinear functions with memory which are commonly used for modeling dynamical systems, as well as static nonlinear classification boundaries, can both be implemented equally well. Fast learning algorithms for the universal structure are presented.<<ETX>>

Fast recursive adaptation for nonlinear filters

A new recursive procedure is presented for developing fast algorithms for nonlinear adaptive filters. Fast recursive adaptation is achieved at a computational cost comparable to simple stochastic gradient algorithms. Successive linearization is shown to be an adequate approximation procedure for developing a general class of nonlinear adaptive filters.<<ETX>>