Technical Transactions iss. 16. Fundamental Sciences iss. 2-NP

Technical Transactions iss. 16. Fundamental Sciences iss. 2-NP

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Collection items (13)


Data-driven score test of fit for class of GARCH models

Stawiarski, Bartosz
Article
2014
A data-driven score test of fit for testing the conditional distribution within the class of stationary GARCH(p,q) models is presented. In this paper extension of the complete results obtained by Inglot and Stawiarski in [7], as well as in Stawiarski [15] for the parsimonious GARCH(1,1) case is proposed. The null (composite) hypothesis subject to testing asserts that the innovations distribution, determining the GARCH conditional distribution, belongs to the specified parametric family. Generalized Error Distribution (called also Exponential Power) seems of special practical value. Applying the pioneer idea of Neyman [13] dating back to 1937, in combination with dimension selection device proposed by Ledwina [10] in 1994, lead to derivation uf the efficient score statistic and its data-driven version for this testing problem. In the case of GARCH(1,1) model both the asymptotic null distribution of the score statistic has been already established in [7] and [15], together with the asymptotics of the data-driven test statistic with appropriately regular estimators plugged in place of nuisance parameters. Main results are only stated herewith, while for detailed proofs inspection and power simulations, ample reference to these papers is provided. We show that the test derivation and asymptotic results carry over to stationary ARCH(q) models for any q ϵ N. Moreover, thanks to ARCH(1) representation of the GARCH(p,q) model, the test can asymptotically encompass the full GARCH family, which as a final result provides the flexible testing tool in the GARCH(p, q) framework.

Max-Plus linear system in control of data processing

Raszka, Jerzy, Jamroż, Lech
Article
2014
The increasing complexity of information processing in distributed computer systems and microprocessors requires the use of time-saving devices and extended capacities of transmission channels. Processes in computers systems need effective processing time. This article describes an application of the theory of the Max Plus Linear System (MPLS) to controlling digital information processing and transmission time in information systems. System processes are described by an MPLS state equation and an MPLS output equation. The MPLS model makes use of formal mathematical methods of max-plus algebra which include maximization and addition operations in the domain of non-negative real numbers with the addition of minus infinity. The input data and the structure of the processes under consideration are represented by the Timed Event Graph (TEG) formalism constituting a special case of Timed Petri Nets. The suggested MPLS methods are useful for investigating selected properties of network models. They may be applied, among others, to evaluate performance criteria, cycle time, predictive control etc. This article presents the theoretical considerations used to determine the input signals controlling discrete processes, which are then illustrated with examples of numerical computations.

Teaching, modeling and visualisation of ordinary differential equations

Mrozek, Zbigniew
Article
2014
Advances in computer technology and increased interest in dynamical systems influence the way of teaching ordinary differential equations. The paper presents inquiry oriented teaching, usage of modeling, visualisation and interactive web services. Last chapter describes the ways of using MATLAB or public domain software (e.g. Octave) to solve ordinary differential equations.

Application of the 3D objects surface shape analysis algorithms in biomedical engineering

Matan, Mateusz
Article
2014
The paper describes a concept of measuring possible error estimation and later the decomposition of the predefined model object into convex areas ECD (Exact Convex Decomposition) in order to find a solution to the problem of cavities location with the use of three-dimensional µ-tomography image of the tooth. Such an approach will enable the improvement of automatic cavities detection methods in the future. The paper is also concerned with the problem of a precise object acquisition and estimation of the error value during execution of automatic detection methods.

Adaptive unstructured solution to the problem of elastic-plastic hardening twist of prismatic bars

Kucwaj, Jan
Article
2014
This paper presents the appliction of a remeshing algorithm to solution of elastic-plastic torsion of bars with isotropic strain hardening. The remeshing algorithm uses a grid generator with mesh size function [7]. The method of grid generation is based on a coupling of the advancing front method and the Delaunay triangulation. The optimal mesh size for the posed problem is obtained iteratively. For the consequtive steps of the adaptation algorithm error indicators at nodes and in elements are used for mesh size modification. The discretized system of nonlinear algebraic equations is solved by the application of the Newton-Raphson method.

The existence of a weak solution of the semilinear first-order differential equation in a Banach space

Jużyniec, Mariusz
Article
2014
This paper is devoted to the investigation of the existence and uniqueness of a suitably defined weak solution of the abstract semilinear value problem ů(t) = Au(t) + f(t, u(t)), u(0) = x with x ϵ X, where X is a Banach space. We are concerned with two types of solutions: weak and mild. Under the assumption that A is the generator of a strongly continuous semigroup of linear, bounded operators, we also establish sufficient conditions such that if u is a weak (mild) solution of the initial value problem, then u is a mild (weak) solution of that problem.

Graphs with every path of length k in a hamiltonian cycle

Gancarzewicz, Grzegorz
Article
2014
In this paper we prove that if G is a (k+2)-connected graph on n≥3 vertices satisfying P(n+k) : dG(x, y) = 2⇒max {d(x), d(y)g}≥ n+k⁄2 for each pair of vertices x and y in G, then any path S⊂G of length k is contained in a hamiltonian cycle of G.

Cycles containing specified edges in a graph

Gancarzewicz, Grzegorz
Article
2014
The aim of this paper is to prove that if s≥1 and G is a graph of order n≥4s+6 satisfying σ2 ≥ 4n-4s-3⁄3, then every matching of G lies on a cycle of length at least n s and hence, in a path of length at least n-s+1.

Missing data analysis in cyclostationary models

Drake, Christiana, Knapik, Oskar, Leśkow, Jacek
Article
2014
In recent years, there has been a growing interest in modeling cyclostationary time series. The survey of Gardner and others [5] is quoting over 1500 different recently published papers that are dedicated to this topic. Data that can be reasonable modeled with such time series is often incomplete. To our knowledge, no systematic research has been conducted on that problem. This paper attempts to fill this gap. In this paper we propose to use EM algorithms to extend estimation for situation when some observations are missing.

On nonlocal evolution functional-differential problem in a Banach space

Byszewski, Ludwik, Winiarska, Teresa
Article
2014
The aim of this paper is to prove two theorems on the existence and uniqueness of mild and classical solutions of a nonlocal semilinear functional-differential evolution Cauchy problem in a Banach space. The method of semigroups, the Banach fixed-point theorem and the Bochenek theorem (see [3]) about the existence and uniqueness of the classical solution of the first order differential evolution problem in a not necessarily reflexive Banach space are used to prove the existence and uniqueness of the solutions of the considered problem. The results are based on publications [1 — 8].