Formal Definitions of Conservative PDFs

Under ideal conditions, the probability density function (PDF) of a random variable, such as a sensor measurement, would be well known and amenable to computation and communication tasks. However, this is often not the case, so the user looks for some other PDF that approximates the true but intractable PDF. Conservativeness is a commonly sought property of this approximating PDF, especially in distributed or unstructured data systems where the data being fused may contain un-known correlations. Roughly, a conservative approximation is one that overestimates the uncertainty of a system. While prior work has introduced some definitions of conservativeness, these definitions either apply only to normal distributions or violate some of the intuitive appeal of (Gaussian) conservative definitions. This work provides a general and intuitive definition of conservativeness that is applicable to any probability distribution, including multi-modal and uniform distributions. Unfortunately, we show that this \emph{strong} definition of conservative cannot be used to evaluate data fusion techniques. Therefore, we also describe a weaker definition of conservative and show it is preserved through common data fusion methods such as the linear and log-linear opinion pool, and homogeneous functionals. In addition, we show that after fusion, weak conservativeness is preserved by Bayesian updates. These strong and weak definitions of conservativeness can help design and evaluate potential correlation-agnostic data fusion techniques

Coordinate Transformation and Polynomial Chaos for the Bayesian Inference of a Gaussian Process with Parametrized Prior Covariance Function

This paper addresses model dimensionality reduction for Bayesian inference based on prior Gaussian fields with uncertainty in the covariance function hyper-parameters. The dimensionality reduction is traditionally achieved using the Karhunen-\Loeve expansion of a prior Gaussian process assuming covariance function with fixed hyper-parameters, despite the fact that these are uncertain in nature. The posterior distribution of the Karhunen-Lo\`{e}ve coordinates is then inferred using available observations. The resulting inferred field is therefore dependent on the assumed hyper-parameters. Here, we seek to efficiently estimate both the field and covariance hyper-parameters using Bayesian inference. To this end, a generalized Karhunen-Lo\`{e}ve expansion is derived using a coordinate transformation to account for the dependence with respect to the covariance hyper-parameters. Polynomial Chaos expansions are employed for the acceleration of the Bayesian inference using similar coordinate transformations, enabling us to avoid expanding explicitly the solution dependence on the uncertain hyper-parameters. We demonstrate the feasibility of the proposed method on a transient diffusion equation by inferring spatially-varying log-diffusivity fields from noisy data. The inferred profiles were found closer to the true profiles when including the hyper-parameters' uncertainty in the inference formulation.Comment: 34 pages, 17 figure

Predicting Multiple Target Tracking Performance for Applications on Video Sequences

This dissertation presents a framework to predict the performance of multiple target tracking (MTT) techniques. The framework is based on the mathematical descriptors of point processes, the probability generating functional (p.g.fl). It is shown that conceptually the p.g.fls of MTT techniques can be interpreted as a transform that can be marginalized to an expression that encodes all the information regarding the likelihood model as well as the underlying assumptions present in a given tracking technique. In order to use this approach for tracker performance prediction in video sequences, a framework that combines video quality assessment concepts and the marginalized transform is introduced. The multiple hypothesis tracker (MHT), Joint Probabilistic Data Association (JPDA), Markov Chain Monte Carlo (MCMC) data association, and the Probability Hypothesis Density filter (PHD) are used as a test cases. We introduce their transforms and perform a numerical comparison to predict their performance under identical conditions. We also introduce the concepts that present the base for estimation in general and for applications in computer vision

Second-order statistics analysis and comparison between arithmetic and geometric average fusion: Application to multi-sensor target tracking

Two fundamental approaches to information averaging are based on linear and logarithmic combination, yielding the arithmetic average (AA) and geometric average (GA) of the fusing data, respectively. In the context of multisensor target tracking, the two most common formats of data to be fused are random variables and probability density functions, namely v-fusion and f-fusion, respectively. In this work, we analyze and compare the second-order statistics (including variance and mean square error) of AA and GA in terms of both v-fusion and f-fusion. The case of weighted Gaussian mixtures representing multitarget densities in the presence of false alarms and missed detections (whose weight sums are not necessarily unit) is also considered, the result of which turns out to be significantly different from that of a single target. In addition to exact derivation, exemplifying analyses and illustrations are also provided.This work was supported in part by the Marie Skłodowska-Curie Individual Fellowship under Grant 709267, in part by Shaanxi Natural Fund under Grant 2018MJ6048, in part by the Northwestern Polytechnical University, and in part by Junta Castilla y León, Consejería de Educación and FEDER funds under project SA267P18

Multiscale statistical methods for the segmentation of signals and images

Includes bibliographical references (p. 29-30).Supported by a National Science Foundation Graduate Fellowship, and by ONR. N00014-91-J-1004 Supported by AFOSR. F49620-95-1-0083 Supported by Boston University. GC123919NGN Supported by NIH. NINDS 1 R01 NS34189M.K. Schneider ... [et al.]

Predicting Multiple Target Tracking Performance for Applications on Video Sequences

This dissertation presents a framework to predict the performance of multiple target tracking (MTT) techniques. The framework is based on the mathematical descriptors of point processes, the probability generating functional (p.g.fl). It is shown that conceptually the p.g.fls of MTT techniques can be interpreted as a transform that can be marginalized to an expression that encodes all the information regarding the likelihood model as well as the underlying assumptions present in a given tracking technique. In order to use this approach for tracker performance prediction in video sequences, a framework that combines video quality assessment concepts and the marginalized transform is introduced. The multiple hypothesis tracker (MHT), Joint Probabilistic Data Association (JPDA), Markov Chain Monte Carlo (MCMC) data association, and the Probability Hypothesis Density filter (PHD) are used as a test cases. We introduce their transforms and perform a numerical comparison to predict their performance under identical conditions. We also introduce the concepts that present the base for estimation in general and for applications in computer vision