Optimal Identical Binary Quantizer Design for Distributed Estimation

We consider the design of identical one-bit probabilistic quantizers for distributed estimation in sensor networks. We assume the parameter-range to be finite and known and use the maximum Cram\'er-Rao Lower Bound (CRB) over the parameter-range as our performance metric. We restrict our theoretical analysis to the class of antisymmetric quantizers and determine a set of conditions for which the probabilistic quantizer function is greatly simplified. We identify a broad class of noise distributions, which includes Gaussian noise in the low-SNR regime, for which the often used threshold-quantizer is found to be minimax-optimal. Aided with theoretical results, we formulate an optimization problem to obtain the optimum minimax-CRB quantizer. For a wide range of noise distributions, we demonstrate the superior performance of the new quantizer - particularly in the moderate to high-SNR regime.Comment: 6 pages, 3 figures, This paper has been accepted for publication in IEEE Transactions in Signal Processin

Methods for Shape-Constrained Kernel Density Estimation

Nonparametric density estimators are used to estimate an unknown probability density while making minimal assumptions about its functional form. Although the low reliance of nonparametric estimators on modelling assumptions is a benefit, their performance will be improved if auxiliary information about the density\u27s shape is incorporated into the estimate. Auxiliary information can take the form of shape constraints, such as unimodality or symmetry, that the estimate must satisfy. Finding the constrained estimate is usually a difficult optimization problem, however, and a consistent framework for finding estimates across a variety of problems is lacking. It is proposed to find shape-constrained density estimates by starting with a pilot estimate obtained by standard methods, and subsequently adjusting its shape until the constraints are satisfied. This strategy is part of a general approach, in which a constrained estimation problem is defined by an estimator, a method of shape adjustment, a constraint, and an objective function. Optimization methods are developed to suit this approach, with a focus on kernel density estimation under a variety of constraints. Two methods of shape adjustment are examined in detail. The first is data sharpening, for which two optimization algorithms are proposed: a greedy algorithm that runs quickly but can handle a limited set of constraints, and a particle swarm algorithm that is suitable for a wider range of problems. The second is the method of adjustment curves, for which it is often possible to use quadratic programming to find optimal estimates. The methods presented here can be used for univariate or higher-dimensional kernel density estimation with shape constraints. They can also be extended to other estimators, in both the density estimation and regression settings. As such they constitute a step toward a truly general optimizer, that can be used on arbitrary combinations of estimator and constraint

Distributionally Robust Optimal Power Flow with Strengthened Ambiguity Sets

Uncertainties that result from renewable generation and load consumption can complicate the optimal power flow problem. These uncertainties normally influence the physical constraints stochastically and require special methodologies to solve. Hence, a variety of stochastic optimal power flow formulations using chance constraints have been proposed to reduce the risk of physical constraint violations and ensure a reliable dispatch solution under uncertainty. The true uncertainty distribution is required to exactly reformulate the problem, but it is generally difficult to obtain. Conventional approaches include randomized techniques (such as scenario-based methods) that provide a priori guarantees of the probability of constraint violations but generally require many scenarios and produce high-cost solutions. Another approach is to use an analytical reformulation, which assumes that the uncertainties follow specific distributions such as Gaussian distributions. However, if the actual uncertainty distributions do not follow the assumed distributions, the results often suffer from case-dependent reliability. Recently, researchers have also explored distributionally robust optimization, which requires probabilistic constraints to be satisfied at chosen probability levels for any uncertainty distributions within a pre-defined ambiguity set. The set is constructed based on the statistical information that is extracted from historical data. Existing literature applying distributionally robust optimization to the optimal power flow problem indicates that the approach has promising performance with low objective costs as well as high reliability compared with the randomized techniques and analytical reformulation. In this dissertation, we aim to analyze the conventional approaches and further improve the current distributionally robust methods. In Chapter II, we derive the analytical reformulation of a multi-period optimal power flow problem with uncertain renewable generation and load-based reserve. It is assumed that the capacities of the load-based reserves are affected by outdoor temperatures through non-linear relationships. Case studies compare the analytical reformulation with the scenario-based method and demonstrate that the scenario-based method generates overly-conservative results and the analytical reformulation results in lower cost solutions but it suffers from reliability issues. In Chapters III, IV, and V, we develop new methodologies in distributionally robust optimization by strengthening the moment-based ambiguity set by including a combination of the moment, support, and structural property information. Specifically, we consider unimodality and log-concavity as most practical uncertainties exhibit these properties. The strengthened ambiguity sets are used to develop tractable reformulations, approximations, and efficient algorithms for the optimal power flow problem. Case studies indicate that these strengthened ambiguity sets reduce the conservativeness of the solutions and result in sufficiently reliable solutions. In Chapter VI, we compare the performance of the conventional approaches and distributionally robust approaches including moment and unimodality information on large-scale systems with high uncertainty dimensions. Through case studies, we evaluate each approach's performance by exploring its objective cost, computational scalability, and reliability. Simulation results suggest that distributionally robust optimal power flow including unimodality information produces solutions with better trade-offs between objective cost and reliability as compared to the conventional approaches or the distributionally robust approaches that do not include unimodality assumptions. However, considering unimodality also leads to longer computational times.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/150051/1/libowen_1.pd

Metabolomics : a tool for studying plant biology

In recent years new technologies have allowed gene expression, protein and metabolite profiles in different tissues and developmental stages to be monitored. This is an emerging field in plant science and is applied to diverse plant systems in order to elucidate the regulation of growth and development. The goal in plant metabolomics is to analyze, identify and quantify all low molecular weight molecules of plant organisms. The plant metabolites are extracted and analyzed using various sensitive analytical techniques, usually mass spectrometry (MS) in combination with chromatography. In order to compare the metabolome of different plants in a high through-put manner, a number of biological, analytical and data processing steps have to be performed. In the work underlying this thesis we developed a fast and robust method for routine analysis of plant metabolite patterns using Gas Chromatography-Mass Spectrometry (GC/MS). The method was performed according to Design of Experiment (DOE) to investigate factors affecting the extraction and derivatization of the metabolites from leaves of the plant Arabidopsis thaliana. The outcome of metabolic analysis by GC/MS is a complex mixture of approximately 400 overlapping peaks. Resolving (deconvoluting) overlapping peaks is time-consuming, difficult to automate and additional processing is needed in order to compare samples. To avoid deconvolution being a major bottleneck in high through-put analyses we developed a new semi-automated strategy using hierarchical methods for processing GC/MS data that can be applied to all samples simultaneously. The two methods include base-line correction of the non-processed MS-data files, alignment, time-window determinations, Alternating Regression and multivariate analysis in order to detect metabolites that differ in relative concentrations between samples. The developed methodology was applied to study the effects of the plant hormone GA on the metabolome, with specific emphasis on auxin levels in Arabidopsis thaliana mutants defective in GA biosynthesis and signalling. A large series of plant samples was analysed and the resulting data were processed in less than one week with minimal labour; similar to the time required for the GC/MS analyses of the samples

Connected Attribute Filtering Based on Contour Smoothness

A new attribute measuring the contour smoothness of 2-D objects is presented in the context of morphological attribute filtering. The attribute is based on the ratio of the circularity and non-compactness, and has a maximum of 1 for a perfect circle. It decreases as the object boundary becomes irregular. Computation on hierarchical image representation structures relies on five auxiliary data members and is rapid. Contour smoothness is a suitable descriptor for detecting and discriminating man-made structures from other image features. An example is demonstrated on a very-high-resolution satellite image using connected pattern spectra and the switchboard platform

Essays in Risk Management and Asset Pricing with High Frequency Option Panels

The thesis investigates the information gains from high frequency equity option data with applications in risk management and empirical asset pricing. Chapter 1 provides the background and motivation of the thesis and outlines the key contributions. Chapter 2 describes the high frequency equity option data in detail. Chapter 3 reviews the theoretical treatments for Recovery Theorem. I derive the formulas for extracting risk neutral central moments from option prices in Chapter 4. In Chapter 5, I specify a perturbation theory on the recovered discount factor, pricing kernel, and the physical probability density. In Chapter 6, a fast and fully-identified sequential programming algorithm is built to apply the Recovery Theorem in practice with noisy market data. I document new empirical evidence on the recovered physical probability distributions and empirical pricing kernels extracted from both index and single-name equity options. Finally, I build a left tail index from the recovered physical probability densities for the S&P 500 index options and show that the left tail index can be used as an indicator of market downside risk. In Chapter 7, I uniquely introduce the higher dimensional option-implied average correlations and provide the procedures for estimating the higher dimensional option-implied average correlations from high frequency option data. In Chapter 8, I construct a market average correlation factor by sorting stocks according to their risk exposures to the option-implied average correlations. I find that (a) the market average correlation factor largely enhances the model-fitting of existing risk-adjusted asset pricing models. (b) the market average correlation factor yields persistent positive risk premiums in cross-sectional stock returns that cannot be explained by other existing risk factors and firm characteristic variables. Chapter 9 concludes the thesis

Análisis de sensibilidad bayesiana a través de clases de distribuciones a priori: teoría y aplicaciones

This Ph.D. dissertation provides contributions in the study of robustness in decision-making problems from a Bayesian point of view. We bring interesting results related with robust Bayesian analysis which make this studies easier. Then, these results are applied in the study of real problems in different contexts: actuarial or financial risk, metrology and reliability theory. Consequently, the thesis is divided into three chapters, where each of them are involved in these different backgrounds. Roughly speaking, Bayesian Statistics obtain the posterior distribution of an underlying univariate or multivariate parameter as a consequence of the likelihood function from an initial sample and a prior information of the parameter according to the Bayes' rule. So, the main interest of the Bayesian point of view is the contribution not only the initial sample that we have but also introducing more information stemming from some experts. That prior information may come in different forms, although it is usually based on the experts' knowledge, which will give enough information to make decisions. In Bayesian inference is fundamental a high precision in the decision maker's judgement, specially regarding his beliefs and preferences. In the Bayesian decision framework, the prior distribution is determined in the set of states given by the experts and it is used to obtain a posterior distribution and a posterior quantity of interest depending on the problem. Usually, that quantity is such that minimizes the expected loss, which is known as the Bayes action, especially in the univariate case. It is common in the Bayesian decision framework starting from a unique prior distribution. However, there are plenty of criticism on that issue: has been well selected the prior distribution from the prior knowledge? Has been introduced any biased information, i.e., there exist any subjectivity in the specific prior distribution? How difficult is to express mathematically the experts' prior knowledge? So, the problem gets more complicated when there exist inaccuracies in the choice of the prior distribution. Therefore, a Bayesian robust analysis seems to be essential. The main goal of Bayesian robustness is to quantify and interpret the uncertainty induced by partial knowledge of one (or more) of the three elements in the analysis. Those three elements are the prior distribution, the loss function and the likelihood function. However, thorough this work, we will mainly focus on prior uncertainty for two major reasons. First, use of priors have been criticized by detractors of the Bayesian approach and Bayesian robustness provides a way to address such issue. Second, there is a practical difficulty in specifying exactly a prior corresponding to the experts' knowledge. Bayesian analysis in complex problems typically entails messy computations, and most times one cannot afford the additional computational burden that would be imposed by a formal robustness analysis. Particularly when one try to compute the range of a posterior quantity of interest. So, we can find many papers in the literature where authors try to simplify that procedures. In particular, this work has focused on it. On the other hand, the Markov Chain - Monte Carlo (MCMC) algorithms appears as an essential tool in the Bayesian decision problem. We refer to the enormous impact that MCMC methods have had on Bayesian analysis, and taking into account that Bayesian robustness methodology will need to be compatible with MCMC methods to become widely used. Though much additional work needs to obtain good results of the robust Bayesian techniques by using MCMC, it is the best way to obtain quality samples and values for the quantity of interest. Then, this work is focused on replacing a single prior distribution by a class of priors but developing classes of prior distribution which make easier the computation of the classes of posterior distribution and, therefore, the computation of the set of the quantity of interest. In order to carry out the Bayesian sensitivity analysis, it will be important to define some useful tools: the univariate and multivariate stochastic orders and the distortion functions. First, stochastic orderings are specially important, providing information about how two distributions can be compared depending on what we are looking for. For example, the simplest one is by considering classical characteristics of the distributions, as the mean value or the standard deviation. However, these comparisons are not good enough because we summarize all information in just a single measure. In this way, stochastic orders represent a powerful tool which allows us to compare two random variables in terms of different criteria. Here we list some of the stochastic orders which we will use along this dissertation: the univariate and multivariate usual stochastic order, the increasing convex (concave) order, the univariate and multivariate likelihood ratio order and the uniform conditional variability order. We will see their formal definitions and some interesting properties, including the chain of implication among all of them. In addition, distortion functions play an important role not only in this dissertation but also in different fields that appear on it, as in actuarial theory. A distortion function h is a non-decreasing continuous function such that h(0)=0 and h(1)=1. For each distortion, we can find a distorted distribution function. This idea will be explained better in the introduction, besides more interesting properties of it. It is worth mentioning that in the multivariate case there exist different ways to define the distortion functions. So, rather than choice the more suitable option every time, we will use a natural extension given by the concept of weight functions and weighted densities in the multivariate case. We provide its definition and the main properties. To summarize, this PhD dissertation will focus on develop new ways to study Bayesian robustness in the prior distribution using different tools as the classical ones. Among all these tools, we will use stochastic orders, distortion functions and weight functions. It will be considered as the starting point of this work a new class of prior distribution that they show: the Distorted Band. We will find more information about this class along the dissertation. Also, this work gives several examples and ideas to indicate the importance and uses of robustness in a Bayesian setting. The main idea is to develop new results that allow us to make sensitivity analysis in different fields of application: actuarial risk, metrology and reliability theory. Finally, the key idea is to obtain a new multivariate class of prior distribution likewise the Distorted Band