James Ting-Ho Lo

Finite-dimensional sensor orbits and optimal nonlinear filtering

The filtering problem of a system with linear dynamics and non-Gaussian a priori distribution is investigated. A closed-form exact solution to the problem is presented along with an approximation scheme. The approximation is made in the construction of a mathematical model. It reduces optimal estimation to a combination of linear estimations. The asymptotic behavior of the filter is examined. The limiting distributions of the conditional mean and the conditional-error covariance exist as the time interval of observation becomes infinite. In the autonomous case, the estimate for the Wiener problem satisfies a linear stochastic differential equation. A large class of nonlinear problems with more nonlinear features than the one discussed above can be reduced to it through the idea of finite-dimensional sensor orbits. The general idea and a number of examples are discussed briefly.

Adaptive multilayer perceptrons with long- and short-term memories

Multilayer perceptrons (MLPs) with long- and short-term memories (LASTMs) are proposed for adaptive processing. The activation functions of the output neurons of such a network are linear, and thus the weights in the last layer affect the outputs of the network linearly and are called linear weights. These linear weights constitute the short-term memory and other weights the long-term memory. It is proven that virtually any function f(x, theta) with an environmental parameter theta can be approximated to any accuracy by an MLP with LASTMs whose long-term memory is independent of theta. This independency of theta allows the long-term memory to be determined in an a priori training and allows the online adjustment of only the short-term memory for adapting to the environmental parameter theta. The benefits of using an MLP with LASTMs include less online computation, no poor local extrema to fall into, and much more timely and better adaptation. Numerical examples illustrate that these benefits are realized satisfactorily.

Optimal estimation for the satellite attitude using star tracker measurements

An optimal estimation scheme is presented, which determines the satellite attitude using the gyro readings and the star tracker measurements of a commonly used satellite attitude measuring unit. The scheme is mainly based on the exponential Fourier densities that have the desirable closure property under conditioning. By updating a finite and fixed number of parameters, the conditional probability density, which is an exponential Fourier density, is recursively determined. Simulation results indicate that the scheme is more accurate and robust than extended Kalman filtering. It is believed that this approach is applicable to many other attitude measuring units. As no linearization and approximation are necessary in the approach, it is ideal for systems involving high levels of randomness and/or low levels of observability and systems for which accuracy is of overriding importance.

Optimal filters for bilinear systems with nilpotent Lie algebras

We consider a bilinear signal process driven by a Gauss-Markov process which is observed in additive, white, Gaussian noise. An exact stochastic differential equation for the least squares filter is derived when the Lie algebra associated with the signal process is nilpotent. It is shown that the filter is also bilinear and moreover that it satisfies an analogous nilpotency condition. Finally, some special cases and an example are discussed, indicating ways of reducing the filter dimensionality.

Exponential Fourier Densities on

A new representation of a probability density function on the three dimensional rotation group,

SO( 3 )

SO( 3 )

and Optimal Estimation and Detection for Rotational Processes

is presented, which generalizes the exponential Fourier densities on the circle. As in the circle case, this class of densities on

Exponential Fourier densities on S2 and optimal estimation and detection for directional processes

SO( 3 )

Exponential Fourier densities and optimal estimation and detection on the circle

is closed under the operation of taking conditional distributions. Several simple multistage estimation and detection models are considered. The closure property enables us to determine the sequential conditional densities by recursively updating a finite and fixed number of coefficients. It also enables us to express the likelihood ratio for signal detection explicitly in terms of the conditional densities.An error criterion, which is compatible with a Riemannian metric, is introduced and discussed. The optimal orientation estimates with respect to this error criterion are derived for a given probability distribution, illustrating how the updated conditional densities can be used to recursively determine the optimal estimates on

Unsupervised Hebbian learning by recurrent multilayer neural networks for temporal hierarchical pattern recognition

SO( 3 )