A. Gramacki

Stability and controllability of a class of 2-D linear systems with dynamic boundary conditions

Discrete linear repetitive processes are a distinct class of two-dimensional (2-D) linear systems with applications in areas ranging from long-wall coal cutting through to iterative learning control schemes. The feature which makes them distinct from other classes of 2-D linear systems is that information propagation in one of the two independent directions only occurs over a finite duration. This, in turn, means that a distinct systems theory must be developed for them. In this paper a complete characterization of stability and so-called pass controllability (and several resulting features), essential building blocks for a rigorous systems theory, under a general set of initial, or boundary, conditions is developed. Finally, some significant new results on the problem of stabilization by choice of the pass state initial vector sequence are developed.

Graphics processing units in acceleration of bandwidth selection for kernel density estimation

Abstract The Probability Density Function (PDF) is a key concept in statistics. Constructing the most adequate PDF from the observed data is still an important and interesting scientific problem, especially for large datasets. PDFs are often estimated using nonparametric data-driven methods. One of the most popular nonparametric method is the Kernel Density Estimator (KDE). However, a very serious drawback of using KDEs is the large number of calculations required to compute them, especially to find the optimal bandwidth parameter. In this paper we investigate the possibility of utilizing Graphics Processing Units (GPUs) to accelerate the finding of the bandwidth. The contribution of this paper is threefold: (a) we propose algorithmic optimization to one of bandwidth finding algorithms, (b) we propose efficient GPU versions of three bandwidth finding algorithms and (c) we experimentally compare three of our GPU implementations with the ones which utilize only CPUs. Our experiments show orders of magnitude improvements over CPU implementations of classical algorithms.

Strong practical stability for a class of 2D linear systems

Linear repetitive processes are a distinct class of 2D linear systems of both theoretical and practical interest. The stability theory for these processes currently consists of two distinct concepts termed asymptotic stability and stability along the pass respectively where the former is a necessary condition for the latter. Recently applications have arisen where asymptotic stability is too weak and stability along the pass is too strong for meaningful progress to be made. This paper develops the concept of strong practical stability for such cases.

Differences in exterior conformation between primitive, Half-bred, and Thoroughbred horses: Anatomic-breeding approach

The study included 249 horses belonging to 3 horse breeds. Konik horses, comprising the first group, is an example of a breed similar to the extinct Tarpan. In our study, these horses were taken to be a primitive anatomical model of the horse body. The other groups comprised the Polish Half-bred horse and Thoroughbred horse. The biometric characteristics of the horses were compared based on 24 indices. The aim of the paper was to find a reduced set of indices that can be used to determine group membership of the horses. To do this, we used statistical methods to find the most important indices that best discriminate breeds from each other. Chi-squared statistics, linear discriminant analysis, logistic regression, and 1-way ANOVA showed that the discrimination among groups of horses is connected with these 5 indices: scapula, smaller trunk (distance between tubercle of humerus and coxal tuber), greater trunk (distance between tubercle of humerus and ischial tuberosity), metacarpus circumference, and hind autopodium-smaller trunk. Thoroughbred and Half-bred horses are clearly different in exterior conformation from Konik horses. The differences between Thoroughbred and Half-bred horses are more subtle. The conformation of Thoroughbreds is jointly determined by relatively small differences in a range of features.

FFT-Based Fast Computation of Multivariate Kernel Density Estimators With Unconstrained Bandwidth Matrices

ABSTRACT The problem of fast computation of multivariate kernel density estimation (KDE) is still an open research problem. In our view, the existing solutions do not resolve this matter in a satisfactory way. One of the most elegant and efficient approach uses the fast Fourier transform. Unfortunately, the existing FFT-based solution suffers from a serious limitation, as it can accurately operate only with the constrained (i.e., diagonal) multivariate bandwidth matrices. In this article, we describe the problem and give a satisfactory solution. The proposed solution may be successfully used also in other research problems, for example, for the fast computation of the optimal bandwidth for KDE. Supplementary materials for this article are available online.

FFT-based fast bandwidth selector for multivariate kernel density estimation

The performance of multivariate kernel density estimation (KDE) depends strongly on the choice of bandwidth matrix. The high computational cost required for its estimation provides a big motivation to develop fast and accurate methods. One of such methods is based on the Fast Fourier Transform. However, the currently available implementation works very well only for the univariate KDE and its multivariate extension suffers from a very serious limitation as it can accurately operate only with diagonal bandwidth matrices. A more general solution is presented where the above mentioned limitation is relaxed. Moreover, the presented solution can be easily adopted also for the task of efficient computation of integrated density derivative functionals involving an arbitrary derivative order. Consequently, bandwidth selection for kernel density derivative estimation is also supported. The practical usability of the new solution is demonstrated by comprehensive numerical simulations.

MATLAB based tools for 2D linear systems with application to iterative learning control schemes

Repetitive processes are a distinct class of 2D systems of both theoretic and practical interest. For example, they arise in the study of industrial processes such as long-wall coal cutting operations and also in the modeling of classes of iterative learning control schemes. This paper describes the development of MATLAB based tools for control related analysis/controller design in the case of so-called discrete linear repetitive processes with particular emphasis on the iterative learning control application. Some areas for short to medium term further development are also briefly noted.

Multitime Scale Systems - The ND Approach

Abstract In this paper so-called multitime scale systems are described in terms of discrete repetitive processes which belong to the wide family of so-called 2D systems.

Bandwidth Selectors for Kernel Density Estimation

This chapter describes the most popular bandwidth selection methods (also known as bandwidth selectors). It starts with a description of the constrained and unconstrained bandwidth selectors and moves on to an overview of the three major types of selectors (that is: rule-of-thumb (ROT), cross-validation (CV) and plug-in (PI) selectors). The next part of the chapter is devoted to describing these selectors in more detail, both for the uni- and multivariate cases. Finally, a few numerical examples are given. The chapter is rounded off with a short section on the computational issues related to bandwidth selectors.

Selected Applications Related to Kernel Density Estimation

This presents a number of applications related to KDE. The first one is discriminant analysis, a well-known data exploration method. The second is cluster analysis (this is also a well-researched field), where the so-called mean-shift algorithm is used. We illustrate the two areas with some simple numerical examples confirming the practical usability of these KDE-based variants of the algorithms used. Next, the nonparametric kernel regression is presented. It can be viewed as an interesting alternative to the classical parametric regression techniques. The fourth application is multidimensional statistical process control. Here, a kernel-based approach is a worth considering option if the underlying d-variate process is not multivariate normal. The final part is devoted to presenting a complete framework for the so-called gating procedure widely used in analyzing flow cytometry datasets. The framework is based on a smart adaptation of the so-called feature significance technique. To show that it can be used in practical terms, we provide a numerical example based on a real flow cytometry dataset. The described results show that the proposed method can be considered an alternative to classical gating methods.