Variable Weight Kernel Density Estimation

Nonparametric density estimation is a common and important task in many problems in machine learning. It consists in estimating a density function from available observations without making parametric assumptions on the generating distribution. Kernel means are nonparametric estimators composed of the average of simple functions, called kernels, centered at each data point. This work studies some relatives of these kernel means with structural similarity but which assign different weights to each kernel unit in order to attain certain desired characteristics. In particular, we present a sparse kernel mean estimator and a consistent kernel density estimator with fixed bandwidth parameter. First, regarding kernel means, we study the kernel density estimator (KDE) and the kernel mean embedding. These are frequently used to represent probability distributions, unfortunately, they face scalability issues. A single point evaluation of the kernel density estimator, for example, requires a computation time linear in the training sample size. To address this challenge, we present a method to efficiently construct a sparse approximation of a kernel mean. We do so by first establishing an incoherence-based bound on the approximation error. We then observe that, for any kernel with constant norm (which includes all translation invariant kernels), the bound can be efficiently minimized by solving the k-center problem. The outcome is a linear time construction of a sparse kernel mean, which also lends itself naturally to an automatic sparsity selection scheme. We demonstrate the computational gains of our method by looking at several benchmark data sets, as well as three applications involving kernel means: Euclidean embedding of distributions, class proportion estimation, and clustering using the mean-shift algorithm. Second we address the bandwidth selection problem in kernel density estimation. Consistency of the KDE requires that the kernel bandwidth tends to zero as the sample size grows. In this work, we investigate the question of whether consistency is still possible when the bandwidth is fixed, if we consider a more general class of weighted KDEs. To answer this question in the affirmative, we introduce the fixed-bandwidth KDE (fbKDE), obtained by solving a quadratic program, that consistently estimates any continuous square-integrable density. Rates of convergence are also established for the fbKDE for radial kernels and the box kernel under appropriate smoothness assumptions. Furthermore, in a simulation study we demonstrate that the fbKDE compares favorably to the standard KDE and the previously proposed variable bandwidth KDE.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138533/1/encc_1.pd

The computational magic of the ventral stream: sketch of a theory (and why some deep architectures work).

This paper explores the theoretical consequences of a simple assumption: the computational goal of the feedforward path in the ventral stream -- from V1, V2, V4 and to IT -- is to discount image transformations, after learning them during development

Risk management of energy derivatives: hedging and margin requirements

The recent growth of exchanges has generated large trading platforms for investors. The largest of these institutions, the Intercontinental Exchange and the Chicago Mercantile Exchange group are now responsible for clearing trades for the majority of investors worldwide and are perhaps, as large commercial banks are, too big to fail. This has attracted attention from international regulating bodies to impose strict risk management standards on the exchanges to ensure financial stability. In this thesis, we identify first, that an investor in the market is strongly affected by margins set by the exchanges in determining the transaction costs of a trade. We discuss the possibility that a volatile margin movement would introduce further risks for such an investor causing them to raise more capital to cover possible margin calls which can perhaps lead to procyclicality. We follow this work by addressing how margins can be determined in adherence to the new laws. Exchanges are now required to set margins based on the Value-at-Risk, hence we search for the best Value-at-Risk method for margining use. Here, we find that the simple Orthogonal Exponentially Weighted Moving Average method is sufficient in forecasting the Value-at-Risk, which contradicts a fair body of the literature who suggests that complex developments of GARCH are superior. We then offer methods for setting and evaluating margin requirements upon the Value-at-Risk estimates, concentrating on producing stable margin requirements. The automated methods produced in our work outperform all other methods available in the literature. Furthermore, we are the first to provide methods for assessing margin stability. Our work is timely in addressing the current affairs of the world economy and is among the first to tackle the margin stability issue in detail

Abstracts of manuscripts submitted in 1990 for publication

This volume contans the abstracts of manuscripts submitted for publication during calendar year 1990 by the staff and students of the Woods Hole Oceanographic Institution. We identify the journal of those manuscripts which are in press or have been published. The volume is intended to be informative, but not a bibliography. The abstracts are listed by title in the Table of Contents and are grouped into one of our five deparments, Marine Policy Center, Coastal Research Center, or the student category. An author index is presented in the back to facilitate locating specific papers

Sensors and Systems for Indoor Positioning

This reprint is a reprint of the articles that appeared in Sensors' (MDPI) Special Issue on “Sensors and Systems for Indoor Positioning". The published original contributions focused on systems and technologies to enable indoor applications

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal