Michael J. Korenberg

The identification of nonlinear biological systems: Wiener and Hammerstein cascade models

Systems that can be represented by a cascade of a dynamic linear subsystem preceded (Hammerstein cascade model) or followed (Wiener cascade model) by a static nonlinearity are considered. Various identification schemes that have been proposed for the Hammerstein and Wiener systems are critically reviewed with reference to the special problems that arise in the identification of nonlinear biological systems. Examples of Wiener and Hammerstein systems are identified from limited duration input-output data using an iterative identification scheme.

The identification of nonlinear biological systems: LNL cascade models

Systems that can be represented by a cascade of a dynamic linear (L), a static nonlinear (N) and a dynamic linear (L) subsystem are considered. Various identification schemes that have been proposed for these LNL systems are critically reviewed with reference to the special problems that arise in the identification of nonlinear biological systems. A simulated LNL system is identified from limited duration input-output data using an iterative identification scheme.

Identifying nonlinear difference equation and functional expansion representations: The fast orthogonal algorithm

A method is presented for identifying functional expansion and difference equation representations for nonlinear systems. The method relies on an orthogonal approach which does not require explicit creation of orthogonal functions. This greatly reduces computing time, so that 15-fold increases in speed of estimating kernels or difference equation coefficients are readily obtainable, compared with a previous orthogonal technique. In addition, storage requirements are considerably diminished. A wide variety of input excitation, both random and deterministic, can be used, and the method is not limited to inputs which are Gaussian, white or lengthy. A model of the peripheral auditory system is simulated to show kernel measurement is free of artifacts using the present method, in contrast to the crosscorrelation approach.

Parallel cascade identification and kernel estimation for nonlinear systems

We consider the representation and identification of nonlinear systems through the use of parallel cascades of alternating dynamic linear and static nonlinear elements. Building on the work of Palm and others, we show that any discrete-time finite-memory nonlinear system having a finite-order Volterra series representation can be exactly represented by a finite number of parallel LN cascade paths. Each LN path consists of a dynamic linear system followed by a static nonlinearity (which can be a polynomial). In particular, we provide an upper bound for the number of parallel LN paths required to represent exactly a discrete-time finite-memory Volterra functional of a given order. Next, we show how to obtain a parallel cascade representation of a nonlinear system from a single input-output record. The input is not required to be Gaussian or white, nor to have special autocorrelation properties. Next, our parallel cascade identification is applied to measure accurately the kernels of nonlinear systems (even those with lengthy memory), and to discover the significant terms to include in a nonlinear difference equation model for a system. In addition, the kernel estimation is used as a means of studying individual signals to distinguish deterministic from random behaviour, in an alternative to the use of chaotic dynamics. Finally, an alternate kernel estimation scheme is presented.

A robust orthogonal algorithm for system identification and time-series analysis

We describe and illustrate methods for obtaining a parsimonious sinusoidal series representation or model of biological time-series data. The methods are also used to identify nonlinear systems with unknown structure. A key aspect is a rapid search for significant terms to include in the model for the system or the time-series. For example, the methods use fast and robust orthogonal searches for significant frequencies in the time-series, and differ from conventional Fourier series analysis in several important respects. In particular, the frequencies in our resulting sinusoidal series need not be commensurate, nor integral multiples of the fundamental frequency corresponding to the record length. Freed of these restrictions, the methods produce a more economical sinusoidal series representation (than a Fourier series), finding the most significant frequencies first, and automatically determine model order. The methods are also capable of higher resolution than a conventional Fourier series analysis. In addition, the methods can cope with unequally-spaced or missing data, and are applicable to time-series corrupted by noise. Fially, we compare one of our methods with a wellknown technique for resolving sinusoidal signals in noise using published data for the test time-series.

The identification of nonlinear biological systems : Volterra kernel approaches

Detection, representation, and identification of nonlinearities in biological systems are considered. We begin by briefly but critically examining a well-known test of system nonlinearity, and point out that this test cannot be used to prove that a system is linear. We then concentrate on the representation of nonlinear systems by Wieners orthogonal functional series, discussing its advantages, limitations, and biological applications. System identification through estimating the kernels in the functional series is considered in detail. An efficient time-domain method of correcting for coloring in inputs is examined and shown to result in significantly improved kernel estimates in a biologically realistic system.

Modeling the Stochastic Drift of a MEMS-Based Gyroscope in Gyro/Odometer/GPS Integrated Navigation

To have a continuous navigation solution that does not suffer from interruption, GPS is integrated with relative positioning techniques such as odometry and inertial navigation. Targeting a low-cost navigation solution for land vehicles, this paper uses a reduced multisensor system consisting of one microelectromechanical-system (MEMS)-based single-axis gyroscope used together with the vehicles odometer, and the whole system is integrated with GPS. This system provides a 2-D navigation solution, which is adequate for land vehicles. The traditional technique for this multisensor integration problem is Kalman filtering (KF). Due to the inherent errors of MEMS inertial sensors and their stochastic nature, which is difficult to model, the KF with its linearized models has limited capabilities in providing accurate positioning. Particle filtering (PF) has recently been suggested as a nonlinear filtering technique to accommodate arbitrary inertial sensor characteristics, motion dynamics, and noise distributions. An enhanced version of PF is utilized in this paper and is called the Mixture PF. Since PF can accommodate nonlinear models, this paper uses total-state nonlinear system and measurement models. In addition, sophisticated models are used to model the stochastic drift of the MEMS-based gyroscope. A nonlinear system identification technique based on parallel cascade identification (PCI) is used to model this stochastic gyroscope drift. In this paper, the performance of the PCI model is compared with that of higher order autoregressive (AR) stochastic models. Such higher order models are difficult to use with KF since the size of the dynamic matrix and the error-covariance matrix becomes very large and complicates the KF operation. The performance of the proposed 2-D navigation solution using Mixture PF with both PCI and higher order AR models is examined by road-test trajectories in a land vehicle. The two proposed combinations are compared with four other 2-D solutions: a Mixture PF with the Gauss-Markov (GM) model for the gyro drift, a Mixture PF with only white Gaussian noise (WGN) for stochastic gyro errors, and two different KF solutions with GM model for the gyro drift. The experimental results show that the two proposed solutions outperform all the compared counterparts.

Low-Cost Three-Dimensional Navigation Solution for RISS/GPS Integration Using Mixture Particle Filter

Recent technological advances in both GPS and low-cost microelectromechanical-system (MEMS)-based inertial sensors have enabled the monitoring of the location of moving platforms for numerous positioning and navigation (POS/NAV) applications. GPS is presently widely used in land vehicles. However, in some environments, the GPS signal may suffer from signal blockage and multipath effects that deteriorate the positioning accuracy. When miniaturized inside any moving platforms, the MEMS-based inertial navigation system (INS) can be integrated with GPS and enhance the performance in denied GPS environments (like in urban canyons). Targeting a low-cost navigation solution for land vehicles, this paper uses a reduced inertial sensor system (RISS) with MEMS-based inertial sensors. In this paper, the RISS consists of one single-axis gyroscope and a two-axis accelerometer used together with the vehicles odometer, and the whole system is integrated with GPS to obtain a 3-D navigation solution. The traditional technique for this integration problem is Kalman filtering (KF). Due to the inherent errors of MEMS inertial sensors and the relatively high noise levels associated with their measurements, KF has limited capabilities in providing accurate positioning. Particle filtering (PF) was recently suggested as a nonlinear filtering technique to accommodate arbitrary inertial sensor characteristics, motion dynamics, and noise distributions. An enhanced version of PF is utilized in this paper and is called Mixture PF. The performance of the proposed 3-D navigation solution using Mixture PF for RISS/GPS integration is examined by road-test trajectories in a land vehicle. The proposed method is compared with four other solutions: 1) 3-D solution using KF for full INS/GPS integration; 2) 2-D solution using KF for RISS/GPS integration; 3) 2-D solution using Mixture PF for RISS/GPS integration; and 4) 3-D solution using sampling/importance resampling (SIR) PF for RISS/GPS integration. The experimental results show that the proposed solution outperforms all the compared counterparts.

Exact orthogonal kernel estimation from finite data records: Extending Wiener's identification of nonlinear systems

A technique is described for exact estimation of kernels in functional expansions for nonlinear systems. The technique operates by orthogonalizing over the data record and in so doing permits a wide variety of input excitation. In particular, the excitation is not limited to inputs that are white, Gaussian, or lengthy. Diagonal kernel values can be estimated, without modification, as accurately as off-diagonal values. Simulations are provided to demonstrate that the technique is more accurate than the Lee-Schetzen method with a white Gaussian input of limited duration, retaining its superiority when the system output is corrupted by noise.

Dynamic Online-Calibrated Radio Maps for Indoor Positioning in Wireless Local Area Networks

Context-awareness and Location-Based-Services are of great importance in mobile computing environments. Although fingerprinting provides accurate indoor positioning in Wireless Local Area Networks (WLAN), difficulty of offline site surveys and the dynamic environment changes prevent it from being practically implemented and commercially adopted. This paper introduces a novel client/server-based system that dynamically estimates and continuously calibrates a fine radio map for indoor positioning without extra network hardware or prior knowledge about the area and without time-consuming offline surveys. A modified Bayesian regression algorithm is introduced to estimate a posterior signal strength probability distribution over all locations based on online observations from WLAN access points (AP) assuming Gaussian prior centered over a logarithmic pass loss mean. To continuously adapt to dynamic changes, Bayesian kernels parameters are continuously updated and optimized genetically based on recent APs observations. The radio map is further optimized by a fast features reduction algorithm to select the most informative APs. Additionally, the system provides reliable integrity monitor (accuracy measure). Two different experiments on IEEE 802.11 networks show that the dynamic radio map provides 2-3m accuracy, which is comparable to results of an up-to-date offline radio map. Also results show the consistency of estimated accuracy measure with actual positioning accuracy.