Smoothing and mean-covariance estimation of functional data with a Bayesian hierarchical model

Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the issue of smoothing all functional observations simultaneously is less studied. Existing methods often focus on smoothing each individual function separately, at the risk of removing important systematic patterns common across functions. We propose a nonparametric Bayesian approach to smooth all functional observations simultaneously and nonparametrically. In the proposed approach, we assume that the functional observations are independent Gaussian processes subject to a common level of measurement errors, enabling the borrowing of strength across all observations. Unlike most Gaussian process regression models that rely on pre-specified structures for the covariance kernel, we adopt a hierarchical framework by assuming a Gaussian process prior for the mean function and an Inverse-Wishart process prior for the covariance function. These prior assumptions induce an automatic mean-covariance estimation in the posterior inference in addition to the simultaneous smoothing of all observations. Such a hierarchical framework is flexible enough to incorporate functional data with different characteristics, including data measured on either common or uncommon grids, and data with either stationary or nonstationary covariance structures. Simulations and real data analysis demonstrate that, in comparison with alternative methods, the proposed Bayesian approach achieves better smoothing accuracy and comparable mean-covariance estimation results. Furthermore, it can successfully retain the systematic patterns in the functional observations that are usually neglected by the existing functional data analyses based on individual-curve smoothing.Comment: Submitted to Bayesian Analysi

Visual Prediction of Rover Slip: Learning Algorithms and Field Experiments

Perception of the surrounding environment is an essential tool for intelligent navigation in any autonomous vehicle. In the context of Mars exploration, there is a strong motivation to enhance the perception of the rovers beyond geometry-based obstacle avoidance, so as to be able to predict potential interactions with the terrain. In this thesis we propose to remotely predict the amount of slip, which reflects the mobility of the vehicle on future terrain. The method is based on learning from experience and uses visual information from stereo imagery as input. We test the algorithm on several robot platforms and in different terrains. We also demonstrate its usefulness in an integrated system, onboard a Mars prototype rover in the JPL Mars Yard. Another desirable capability for an autonomous robot is to be able to learn about its interactions with the environment in a fully automatic fashion. We propose an algorithm which uses the robot's sensors as supervision for vision-based learning of different terrain types. This algorithm can work with noisy and ambiguous signals provided from onboard sensors. To be able to cope with rich, high-dimensional visual representations we propose a novel, nonlinear dimensionality reduction technique which exploits automatic supervision. The method is the first to consider supervised nonlinear dimensionality reduction in a probabilistic framework using supervision which can be noisy or ambiguous. Finally, we consider the problem of learning to recognize different terrains, which addresses the time constraints of an onboard autonomous system. We propose a method which automatically learns a variable-length feature representation depending on the complexity of the classification task. The proposed approach achieves a good trade-off between decrease in computational time and recognition performance.</p

A System Centric View of Modern Structured and Sparse Inference Tasks

University of Minnesota Ph.D. dissertation.June 2017. Major: Electrical/Computer Engineering. Advisor: Jarvis Haupt. 1 computer file (PDF); xii, 140 pages.We are living in the era of data deluge wherein we are collecting unprecedented amount of data from variety of sources. Modern inference tasks are centered around exploiting structure and sparsity in the data to extract relevant information. This thesis takes an end-to-end system centric view of these inference tasks which mainly consist of two sub-parts (i) data acquisition and (ii) data processing. In context of the data acquisition part of the system, we address issues pertaining to noise, clutter (the unwanted extraneous signals which accompany the desired signal), quantization, and missing observations. In the data processing part of the system we investigate the problems that arise in resource-constrained scenarios such as limited computational power and limited battery life. The first part of this thesis is centered around computationally-efficient approximations of a given linear dimensionality reduction (LDR) operator. In particular, we explore the partial circulant matrix (a matrix whose rows are related by circular shifts) based approximations as they allow for computationally-efficient implementations. We present several theoretical results that provide insight into existence of such approximations. We also propose a data-driven approach to numerically obtain such approximations and demonstrate the utility on real-life data. The second part of this thesis is focused around the issues of noise, missing observations, and quantization arising in matrix and tensor data. In particular, we propose a sparsity regularized maximum likelihood approach to completion of matrices following sparse factor models (matrices which can be expressed as a product of two matrices one of which is sparse). We provide general theoretical error bounds for the proposed approach which can be instantiated for variety of noise distributions. We also consider the problem of tensor completion and extend the results of matrix completion to the tensor setting. The problem of matrix completion from quantized and noisy observations is also investigated in as general terms as possible. We propose a constrained maximum likelihood approach to quantized matrix completion, provide probabilistic error bounds for this approach, and numerical algorithms which are used to provide numerical evidence for the proposed error bounds. The final part of this thesis is focused on issues related to clutter and limited battery life in signal acquisition. Specifically, we investigate the problem of compressive measurement design under a given sensing energy budget for estimating structured signals in structured clutter. We propose a novel approach that leverages the prior information about signal and clutter to judiciously allocate sensing energy to the compressive measurements. We also investigate the problem of processing Electrodermal Activity (EDA) signals recorded as the conductance over a user's skin. EDA signals contain information about the user's neuron ring and psychological state. These signals contain the desired information carrying signal superimposed with unwanted components which may be considered as clutter. We propose a novel compressed sensing based approach with provable error guarantees for processing EDA signals to extract relevant information, and demonstrate its efficacy, as compared to existing techniques, via numerical experiments

On evolutionary system identification with applications to nonlinear benchmarks

This paper presents a record of the participation of the authors in a workshop on nonlinear system identification held in 2016. It provides a summary of a keynote lecture by one of the authors and also gives an account of how the authors developed identification strategies and methods for a number of benchmark nonlinear systems presented as challenges, before and during the workshop. It is argued here that more general frameworks are now emerging for nonlinear system identification, which are capable of addressing substantial ranges of problems. One of these frameworks is based on evolutionary optimisation (EO); it is a framework developed by the authors in previous papers and extended here. As one might expect from the ‘no-free-lunch’ theorem for optimisation, the methodology is not particularly sensitive to the particular (EO) algorithm used, and a number of different variants are presented in this paper, some used for the first time in system identification problems, which show equal capability. In fact, the EO approach advocated in this paper succeeded in finding the best solutions to two of the three benchmark problems which motivated the workshop. The paper provides considerable discussion on the approaches used and makes a number of suggestions regarding best practice; one of the major new opportunities identified here concerns the application of grey-box models which combine the insight of any prior physical-law based models (white box) with the power of machine learners with universal approximation properties (black box)