Showing papers in "Kybernetika in 2012"

The gamma-uniform distribution and its applications

Abstract: Up to present for modelling and analyzing of random phenomenons, some statistical distributions are proposed. This paper considers a new general class of distributions, generated from the logit of the gamma random variable. A special case of this family is the Gamma-Uniform distribution. We derive expressions for the four moments, variance, skewness, kurtosis, Shannon and Renyi entropy of this distribution. We also discuss the asymptotic distribution of the extreme order statistics, simulation issues, estimation by method of maximum likelihood and the expected information matrix. We show that the Gamma-Uniform distribution provides great flexibility in modelling for negatively and positively skewed, convex-concave shape and reverse `J' shaped distributions. The usefulness of the new distribution is illustrated through two real data sets by showing that it is more flexible in analysing of the data than of the Beta Generalized-Exponential, Beta-Exponential, Beta-Pareto, Generalized Exponential, Exponential Poisson, Beta Generalized Half-Normal and Generalized Half-Normal distributions.

Several applications of divergence criteria in continuous families.

Abstract: This paper deals with four types of point estimators based on minimization of informationtheoretic divergences between hypothetical and empirical distributions. These were introduced (i) by Liese & Vajda (2006) and independently Broniatowski & Keziou (2006), called here power superdivergence estimators, (ii) by Broniatowski & Keziou (2009), called here power subdivergence estimators, (iii) by Basu et al. (1998), called here power pseudodistance estimators, and (iv) by Vajda (2008) called here Renyi pseudodistance estimators. The paper studies and compares general properties of these estimators such as consistency and influence curves, and illustrates these properties by detailed analysis of the applications to the estimation of normal location and scale.

Generalized synchronization and control for incommensurate fractional unified chaotic system and applications in secure communication

Abstract: A fractional differential controller for incommensurate fractional unified chaotic system is described and proved by using the Gershgorin circle theorem in this paper. Also, based on the idea of a nonlinear observer, a new method for generalized synchronization (GS) of this system is proposed. Furthermore, the GS technique is applied in secure communication (SC), and a chaotic masking system is designed. Finally, the proposed fractional differential controller, GS and chaotic masking scheme are showed by using numerical and experimental simulations.

Convexity inequalities for estimating generalized conditional entropies from below

Abstract: Generalized entropic functionals are in an active area of research. Hence lower and upper bounds on these functionals are of interest. Lower bounds for estimating Renyi conditional $\alpha$-entropy and two kinds of non-extensive conditional $\alpha$-entropy are obtained. These bounds are expressed in terms of error probability of the standard decision and extend the inequalities known for the regular conditional entropy. The presented inequalities are mainly based on the convexity of some functions. In a certain sense, they are complementary to generalized inequalities of Fano type.

On extremal dependence of block vectors

Abstract: Due to globalization and relaxed market regulation, we have assisted to an increasing of extremal dependence in international markets. As a consequence, several measures of tail dependence have been stated in literature in recent years, based on multivariate extreme-value theory. In this paper we present a tail dependence function and an extremal coefficient of dependence between two random vectors that extend existing ones. We shall see that in weakening the usual required dependence allows to assess the amount of dependence in $d$-variate random vectors based on bidimensional techniques. Simple estimators will be stated and can be applied to the well-known stable tail dependence function. Asymptotic normality and strong consistency will be derived too. An application to financial markets will be presented at the end.

Leader-following consensus of multiple linear systems under switching topologies: an averaging method

Abstract: The leader-following consensus of multiple linear time invariant (LTI) systems under switching topology is considered. The leader-following consensus problem consists of designing for each agent a distributed protocol to make all agents track a leader vehicle, which has the same LTI dynamics as the agents. The interaction topology describing the information exchange of these agents is time-varying. An averaging method is proposed. Unlike the existing results in the literatures which assume the LTI agents to be neutrally stable, we relax this condition, only making assumption that the LTI agents are stablizable and detectable. Observer-based leader-following consensus is also considered.

On the extremal behavior of a Pareto process: an alternative for ARMAX modeling

Abstract: In what concerns extreme values modeling, heavy tailed autoregressive processes defined with the minimum or maximum operator have proved to be good alternatives to classical linear ARMA with heavy tailed marginals (Davis and Resnick [8], Ferreira and Canto e Castro [13]). In this paper we present a complete characterization of the tail behavior of the autoregressive Pareto process known as Yeh-Arnold-Robertson Pareto(III) (Yeh et al. [32]). We shall see that it is quite similar to the first order max-autoregressive ARMAX, but has a more robust parameter estimation procedure, being therefore more attractive for modeling purposes. Consistency and asymptotic normality of the presented estimators will also be stated.

Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities

Abstract: Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are finite on the positive and infinite on the negative numbers, strictly convex but not necessarily differentiable. The minimization is viewed as a primal problem and studied together with a dual one in the framework of convex duality. The effective domain of the value function is described by a conic core, a modification of the earlier concept of convex core. Minimizers and generalized minimizers are explicitly constructed from solutions of modified dual problems, not assuming the primal constraint qualification. A generalized Pythagorean identity is presented using Bregman distance and a correction term for lack of essential smoothness in integrands. Results are applied to minimization of Bregman distances. Existence of a generalized dual solution is established whenever the dual value is finite, assuming the dual constraint qualification. Examples of ‘irregular’ situations are included, pointing to the limitations of generality of certain key results.

Moment estimation methods for stationary spatial Cox processes - A comparison

Abstract: In the present paper we consider the problem of fitting parametric spatial Cox point process models. We concentrate on the moment estimation methods based on the second order characteristics of the point process in question. These methods represent a simulation-free faster-to-compute alternative to the computationally intense maximum likelihood estimation. We give an overview of the available methods, discuss their properties and applicability. Further we present results of a simulation study in which performance of these estimating methods was compared for planar point processes with different types and strength of clustering and inter-point interactions.

A Lyapunov-based design tool of impedance controllers for robot manipulators

Abstract: This paper presents a design tool of impedance controllers for robot manipulators, based on the formulation of Lyapunov functions. The proposed control approach addresses two challenges: the regulation of the interaction forces, ensured by the impedance error converging to zero, while preserving a suitable path tracking despite constraints imposed by the environment. The asymptotic stability of an equilibrium point of the system, composed by full nonlinear robot dynamics and the impedance control, is demonstrated according to Lyapunov's direct method. The system's performance was tested through the real-time experimental implementation of an interaction task involving a two degree-of-freedom, direct-drive robot.

Arbology: Trees and pushdown automata

Abstract: Trees are (data) structures used in many areas of human activity. Tree as the formal notion has been introduced in the theory of graphs. Nevertheless, trees have been used a long time before the foundation of the graph theory. An example is the notion of a genealogical tree. The area of family relationships was an origin of some terminology in the area of the tree theory (parent, child, sibling, ...) in addition to the terms originating from the area of the dendrology (root, branch, leaf, ...).

Chance constrained problems: penalty reformulation and performance of sample approximation technique

Abstract: We explore reformulation of nonlinear stochastic programs with several joint chance constraints by stochastic programs with suitably chosen penalty-type objectives. We show that the two problems are asymptotically equivalent. Simpler cases with one chance constraint and particular penalty functions were studied in [6,11]. The obtained problems with penalties and with a fixed set of feasible solutions are simpler to solve and analyze then the chance constrained programs. We discuss solving both problems using Monte-Carlo simulation techniques for the cases when the set of feasible solution is finite or infinite bounded. The approach is applied to a financial optimization problem with Value at Risk constraint, transaction costs and integer allocations. We compare the ability to generate a feasible solution of the original chance constrained problem using the sample approximations of the chance constraints directly or via sample approximation of the penalty function objective.

Finite-time output feedback stabilization and control for a quadrotor mini-aircraft

Abstract: This paper focuses on the finite-time output feedback control problem for a quad-rotor mini-aircraft system. First, a finite-time state feedback controller is designed by utilizing the finite-time control theory. Then, considering the case that the velocity states are not measurable, a finite-time stable observer is developed to estimate the unmeasurable states. Thus a finite-time output feedback controller is obtained and the stability analysis is provided to ensure the finite-time stability of the closed loop system. The proposed control method improves the convergence and disturbance rejection properties with respect to the asymptotically control results. Simulation results show the effectiveness of the proposed method.

Generalized Thue-Morse words and palindromic richness

Abstract: We prove that the generalized Thue-Morse word $\mathbf{t}_{b,m}$ defined for $b \geq 2$ and $m \geq 1$ as $\mathbf{t}_{b,m} = (s_b(n) \mod m)_{n=0}^{+\infty}$, where $s_b(n)$ denotes the sum of digits in the base-$b$ representation of the integer $n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering simultaneously antimorphisms $\Theta \in D_m$, we show that $\mathbf{t}_{b,m}$ is saturated by $\Theta$-palindromes up to the highest possible level. Using the terminology generalizing the notion of palindromic richness for more antimorphisms recently introduced by the author and E. Pelantov\'a, we show that $\mathbf{t}_{b,m}$ is $D_m$-rich. We also calculate the factor complexity of $\mathbf{t}_{b,m}$.

An optimality system for finite average Markov decision chains under risk-aversion

Abstract: This work concerns controlled Markov chains with finite state space and compact action sets. The decision maker is risk-averse with constant risk-sensitivity, and the performance of a control policy is measured by the long-run average cost criterion. Under standard continuity-compactness conditions, it is shown that the (possibly non-constant) optimal value function is characterized by a system of optimality equations which allows to obtain an optimal stationary policy. Also, it is shown that the optimal superior and inferior limit average cost functions coincide.

Convergence model of interest rates of CKLS type

Abstract: This paper deals with convergence model of interest rates, which explains the evolution of interest rate in connection with the adoption of Euro currency. Its dynamics is described by two stochastic differential equations – the domestic and the European short rate. Bond prices are then solutions to partial differential equations. For the special case with constant volatilities closed form solutions for bond prices are known. Substituting its constant volatilities by instantaneous volatilities we obtain an approximation of the solution for a more general model. We compute the order of accuracy for this approximation, propose an algorithm for calibration of the model and we test it on the simulated and real market data.

Modified power divergence estimators in normal models - simulation and comparative study

Abstract: Point estimators based on minimization of information-theoretic divergences between empirical and hypothetical distribution induce a problem when working with continuous families which are measure-theoretically orthogonal with the family of empirical distributions. In this case, the -divergence is always equal to its upper bound, and the minimum -divergence estimates are trivial. Broniatowski and Vajda [3] proposed several modifications of the minimum divergence rule to provide a solution to the above mentioned problem. We examine these new estimation methods with respect to consistency, robustness and eciency through an extended simulation study. We focus on the well-known family of power divergences parametrized by 2 R in the Gaussian model, and we perform a comparative computer simulation for several randomly selected contaminated and uncontaminated data sets, dierent sample sizes and dierent -divergence parameters.

Holt-Winters method with general seasonality

Abstract: The paper suggests a generalization of widely used Holt-Winters smoothing and forecasting method for seasonal time series. The general concept of seasonality modeling is introduced both for the additive and multiplicative case. Several special cases are discussed, including a linear interpolation of seasonal indices and a usage of trigonometric functions. Both methods are fully applicable for time series with irregularly observed data (just the special case of missing observations was covered up to now). Moreover, they sometimes outperform the classical Holt-Winters method even for regular time series. A simulation study and real data examples compare the suggested methods with the classical one.

Semidefinite characterisation of invariant measures for one-dimensional discrete dynamical systems

Abstract: Using recent results on measure theory and algebraic geometry, we show how semidefinite programming can be used to construct invariant measures of onedimensional discrete dynamical systems (iterated maps on a real interval). In particular we show that both discrete measures (corresponding to finite cycles) and continuous measures (corresponding to chaotic behavior) can be recovered using standard software.

$\phi $PHI-divergences, sufficiency, Bayes sufficiency, and deficiency

Abstract: The paper studies the relations between $\phi$-divergences and fundamental concepts of decision theory such as sufficiency, Bayes sufficiency, and LeCam's deficiency. A new and considerably simplified approach is given to the spectral representation of $\phi $-divergences already established in Osterreicher and Feldman [28] under restrictive conditions and in Liese and Vajda [22], [23] in the general form. The simplification is achieved by a new integral representation of convex functions in terms of elementary convex functions which are strictly convex at one point only. Bayes sufficiency is characterized with the help of a binary model that consists of the joint distribution and the product of the marginal distributions of the observation and the parameter, respectively. LeCam's deficiency is expressed in terms of $\phi $-divergences where $\phi $ belongs to a class of convex functions whose curvature measures are finite and satisfy a normalization condition.

On an algorithm for testing T4 solvability of max-plus interval systems

Abstract: Max-plus algebra is the algebraic structure in which classical addition and multiplication are replaced by a⊕b = max{a, b} and a⊗b = a+b, respectively. Each system of linear equation in max-plus algebra we can write in the matrix form A⊗ x = b, where A and b are matrix and vector of suitable size. If we replace the matrix elements with matrix interval A = [A,A] and vector elements by vector interval b = [b, b], we get an interval system of linear equations. We can define several types of solvability of interval systems in max-plus algebra. In this paper, we shall deal with one of them, the so called T5 solvability. We give the algorithm which answers the question whether the given interval system is T5 solvable or not.

An unbounded Berge's minimum theorem with applications to discounted Markov decision processes

Abstract: This paper deals with a certain class of unbounded optimization problems. The optimization problems taken into account depend on a parameter. Firstly, there are established conditions which permit to guarantee the continuity with respect to the parameter of the minimum of the optimization problems under consideration, and the upper semicontinuity of the multifunction which applies each parameter into its set of minimizers. Besides, with the additional condition of uniqueness of the minimizer, its continuity is given. Some examples of nonconvex optimization problems that satisfy the conditions of the article are supplied. Secondly, the theory developed is applied to discounted Markov decision processes with unbounded cost functions and with possibly noncompact actions sets in order to obtain continuous optimal policies. This part of the paper is illustrated with two examples of the controlled Lindley's random walk. One of these examples has nonconstant action sets.

Probabilistic properties of the continuous double auction.

Abstract: In this paper we formulate a general model of the continuous double auction. We (recursively) describe the distribution of the model. As a useful by-product, we give a (recursive) analytic description of the distribution of the process of the best quotes (bid and ask).

A backward selection procedure for approximating a discrete probability distribution by decomposable models

Abstract: Decomposable (probabilistic) models are log-linear models generated by acyclic hypergraphs, and a number of nice properties enjoyed by them are known. In many applications the following selection problem naturally arises: given a probability distribution p over a finite set V of n discrete variables and a positive integer k, find a decomposable model with tree-width k that best fits p. If ℌ is the generating hypergraph of a decomposable model and p ℌ is the estimate of p under the model, we can measure the closeness of p ℌ to p by the information divergence D(p: p ℌ ), so that the problem above reads: given p and k, find an acyclic, connected hypergraph ℌ of tree-width k such that D(p: p ℌ ) is minimum. It is well-known that this problem is NP-hard. However, for k = 1 it was solved by Chow and Liu in a very efficient way; thus, starting from an optimal Chow―Liu solution, a few forward-selection procedures have been proposed with the aim at finding a 'good' solution for an arbitrary k. We propose a backward-selection procedure which starts from the (trivial) optimal solution for k = n ― 1, and we show that, in a study case taken from literature, our procedure succeeds in finding an optimal solution for every k.

Max-min interval systems of linear equations with bounded solution

Abstract: Max-min algebra is an algebraic structure in which classical addition and multiplication are replaced by and , where a b = max{a,b}, a b = min{a,b}. The notation A x = b represents an interval system of linear equations, where A = [A,A], b = [b,b] are given interval matrix and interval vector, respectively, and a solution is from a given interval vector x = [x,x]. We define six types of solvability of max-min interval systems with bounded solution and give necessary and sucient conditions for them.

Invariant subspaces for grasping internal forces and non-interacting force-motion control in robotic manipulation

Abstract: This paper presents a parametrization of a feed-forward control based on structures of subspaces for a non-interacting regulation. With advances in technological development, robotics is increasingly being used in many industrial sectors, including medical applications (e. g., micromanipulation of internal tissues or laparoscopy). Typical problems in robotics and general mechanisms may be mathematically formalized and analyzed, resulting in outcomes so general that it is possible to speak of structural properties in robotic manipulation and mechanisms. This work shows an explicit formula for the reachable internal contact forces of a general manipulation system. The main contribution of the paper consists of investigating the design of a feed-forward force-motion control which, together with a feedback structure, realizes a decoupling force-motion control. A generalized linear model is used to perform a careful analysis, resulting in the proposed general geometric structure for the study of mechanisms. In particular, a lemma and a theorem are presented which offer a parametrization of a feed-forward control for a task-oriented choice of input subspaces. The existence of these input subspaces is a necessary condition for the structural non-interaction property. A simulation example in which the subspaces and the control structure are explicitly calculated is shown and widely explicated.

Goffin's algorithm for zonotopes

Abstract: The Löwner-John ellipse of a full-dimensional bounded convex set is a circumscribed ellipse with the property that if we shrink it by a factor n (where n is dimension), we obtain an inscribed ellipse. Goffin’s algorithm constructs, in polynomial time, a tight approximation of the Löwner-John ellipse of a polyhedron given by facet description. In this text we adapt the algorithm for zonotopes given by generator descriptions. We show that the adapted version works in time polynomial in the size of the generator description (which may be superpolynomially shorter than the facet description). AMS Classification. 90C57, 52B12, 52B55, 68U05.