Posteriori Density Function Research Articles

On Filtering Methods for State-Space Systems having Binary Output Measurements

Abstract In this paper we develop two filtering algorithms for state-space systems with binary outputs. We approximate the conditional probability mass function of the output signal given the state by using a Gaussian quadrature rule. This approximation naturally leads to a Gaussian Sum structure for the a posteriori density function. Our first algorithm is based on Gaussian Mixture models, and the second algorithm is based on Particle Filtering. Finally, we present numerical examples to illustrate the effectiveness of our proposal.

Spacecraft Attitude Estimation Using Adaptive Gaussian Sum Filter

This paper is concerned with improving the attitude estimation accuracy by implementing an adaptive Gaussian sum filter where the a posteriori density function is approximated by a sum of Gaussian density functions. Compared to the traditional Gaussian sum filter, this adaptive approach utilizes the Fokker-Planck-Kolmogorov residual minimization to update the weights associated with different components of the Gaussian mixture model. Updating the weights provides an accurate approximation of the a posteriori density function and thus superior estimates. Simulation results show that updating the weights during the propagation stage not only provides better estimates between the observations but also provides superior estimator performance where the measurements are ambiguous.

Maximum a posteriori resampling of noisy, spatially correlated data

In any geologic application, noisy data are sources of consternation for researchers, inhibiting interpretability and marring images with unsightly and unrealistic artifacts. Filtering is the typical solution to dealing with noisy data. However, filtering commonly suffers from ad hoc (i.e., uncalibrated, ungoverned) application. We present here an alternative to filtering: a newly developed method for correcting noise in data by finding the “best” value given available information. The motivating rationale is that data points that are close to each other in space cannot differ by “too much,” where “too much” is governed by the field covariance. Data with large uncertainties will frequently violate this condition and therefore ought to be corrected, or “resampled.” Our solution for resampling is determined by the maximum of the a posteriori density function defined by the intersection of (1) the data error probability density function (pdf) and (2) the conditional pdf, determined by the geostatistical kriging algorithm applied to proximal data values. A maximum a posteriori solution can be computed sequentially going through all the data, but the solution depends on the order in which the data are examined. We approximate the global a posteriori solution by randomizing this order and taking the average. A test with a synthetic data set sampled from a known field demonstrates quantitatively and qualitatively the improvement provided by the maximum a posteriori resampling algorithm. The method is also applied to three marine geology/geophysics data examples, demonstrating the viability of the method for diverse applications: (1) three generations of bathymetric data on the New Jersey shelf with disparate data uncertainties; (2) mean grain size data from the Adriatic Sea, which is a combination of both analytic (low uncertainty) and word‐based (higher uncertainty) sources; and (3) side‐scan backscatter data from the Martha's Vineyard Coastal Observatory which are, as is typical for such data, affected by speckle noise. Compared to filtering, maximum a posteriori resampling provides an objective and optimal method for reducing noise, and better preservation of the statistical properties of the sampled field. The primary disadvantage is that maximum a posteriori resampling is a computationally expensive procedure.

Fusion of range and intensity images on a connection machine (CM-2)

A new algorithm for fusion of range and intensity images in a Bayesian framework is presented. This framework allows the fusion problem to be posed in terms of maximization of a posteriori density function or, equibalently, minimization of an energy function. Experimental results of the fusion algorithm on registered range and intensity images indicate that the detected and labeled edges match the physical edges in the scene. To alleviate the heavy computational demands of our fusion algorithm, two energy a minimization algorithms have been implemented on a CM-2 Connection Machine, an SIMD type parallel hardware. It is demonstrated that a straightforward conversion of sequential code to parallel code results in poor computational performance. Experimental results show that computations in the fusion energy minimization operation can be carried out significantly faster (speedup > 50) on the CM-2 than on a Sun Sparcstation-2.

Estimation of BAYESian Confidence Intervals for the Product of Conditional Survival Probabilities

AbstractA model is derived to estimate the survival probability of a time interval when censorings occur. The time interval is divided into partial intervals in order to obtain the conditional survival probabilities, each of which is a parameter of a Binomial distributed random variable. To allow for the dependence between the events in the different intervals these parameters are transformed. Corresponding a priori density functions are formulated regarding both the Bayesian uniform distribution and the special model. The a posteriori density function is derived for the product of the conditional survival probabilities, and formulae for the BAYEsian confidence interval and the expectation are given. Lower and upper bounds for the confidence interval and the expectation are derived. Some examples are given to compare the results with other methods.

A Discrete Receiver Structure for Bit Detection Without Synchronization for Signals Corrupted by Additive and Multiplicative Noise

A discrete receiver structure is introduced that consolidates the problems of bit detection and bit synchronization to provide optimal estimates of bit values without explicit knowledge of the transition epoch. Equations are developed for the propagation of the discrete a posteriori density function on an interval-by-interval basis. The general model is developed for a signal corrupted by additive and multiplicative noise, and the receiver structure is shown to be equivalent to a synchronous discrete receiver. The model is applied to on-off-keyed (OOK) modulation in the presence of multipath.

Position Finding with Empirical Prior Knowledge

Information on the position of a source of radiation is often obtained from bearing observations made from two or more known locations. Some criticisms of existing procedures are enumerated and a new approach proposed. This approach allows for the incorporation of knowledge existing before the observations are made and provides simple computational procedures for obtaining both the a posteriori density function of the source position and an estimate of this position from the observed data.

Non-linear filtering by approximation of the a posteriori density

The problem of estimating from noisy measurement data the state of a dynamical system described by non-linear difference equations is considered. The measurement data have a non-linear relation with the state and are assumed to be available at discrete instants of time. A Bayesian approach to the problem is suggested in which the density function for the state conditioned upon the available measurement data is computed recursively. The evolution of the a posteriori density function cannot be described in a closed form for most systems; the class of linear systems with additive, white gaussian noise provides the major exception. Thus, the problem of non-linear filtering can be viewed as essentially a problem of approximating this density function. For linear systems with additive, white gaussian noise, the a posteriori density is gaussian. The results for linear systems are frequently applied to non-linear systems by introducing linear perturbation theory. Then, the linear equations and gaussian a posterio...

Optimal control and filtering of linear stochastic systems

This paper is in two parts. The first part treats the optimal control of a linear stochastic system with a. quadratic performance criterion. An explicit solution for the cost functional is given. The second port deals with the optimal filtering of the linear system considered in the first part. Modifications in the filtering equation have been made for the case in which the observation noise is of the truncated white noise type. An explicit solution for the a posteriori density function, defined by the filtering equation, is given. The paper concludes with a numerical solution to both the optimal control and the filtering problem associated with a six-plate chemical absorption tower.