# Abstracts

### Invited lectures

Milan Bašta: Time series analysis with wavelets

Piotr Jaworski: Invariant multivariate dependence structure under univariate truncation

Jozef Komorník, Magda Komorníková: Regime switching copula models for relations between returns of stock indexes

Radko Mesiar: Copulas and integrals

Oľga Nánásiová: Modeling of non-compatible random events via multidimensional states

Pál Rakonczai: Bivariate generalized Pareto distribution in practice: models and estimation

### Lectures

Tomáš Bacigál: Recent tools for modelling dependence with copulas and R

Mária Bohdalová, Michal Greguš: Monte Carlo simulation Value at Risk and PCA

Dana Hliněná, Martin Kalina, Pavol Kráľ: Implicators and I-partitions

Vladimír Jágr: Generalization and construction of Archimax copulas for higher dimensions

Jana Kalická, Tomáš Kulla: Optimal bandwith in nonparametric regression

Anna Kolesárová, Andrea Stupňanová: On the structure of associative n-dimensional copulas

Anna Kolesárová, Andrea Stupňanová, Juliana Beganová: Aggregation-based extensions of utility functions

Darina Kyselová: Aggregation functions-based building of transitive preference structures

Jana Lencuchova: Comparing the power properties of the proposed test and some other nonlinearity tests for markov-switching time series models

Oľga Nánásiová, Miroslav Sabo: Clustering by two methods simultaneously

Monika Pekárová: On some insurance risk applications of copulas

Anna Petričková: Modelling of the ARMA models residuals using autocopulas

G.Beliakov, S.James, J.Beganová, T.Rückschlossová and R.R.Yager: Bonferroni mean operators in multi-criteria aggregation

Danuša Szőkeová, Silvia Kohnová: SETAR models in the streamflow modeling

Lucia Vavríková: Application of aggregation operators on the assessment of public universities and their faculties

Petra Zacharovská: Comparison of descriptive and predictive properties of MSW models with different probability distribution of residuals

#### Milan Bašta: Time series analysis with wavelets

Wavelets are a relatively new tool of the data analysis. They have been successfully applied in meteorology, geophysics, hydrology, astronomy, signal processing, statistics, medicine etc. The results are often very interesting and bring new insights into the phenomenon which is being studied. Recently wavelets have also found their way to economics and finance. I give a short introduction to the time series analysis with wavelets and present a few applications of wavelets in the analysis of economic and financial time series.

#### Piotr Jaworski: Invariant multivariate dependence structure under univariate truncation

The interest in the construction of multivariate stochastic models describing the dependence among several variables has grown in the last years. In particular, the recent financial crisis has underlined the necessity of considering models that can serve to estimate better the occurrence of extremal events. Indeed, the crucial interest in modelling the real world phenomena often lies in the proper description of the evolution of dependency between different factors when one of them is achieving the extreme values. Copulas are the most general measures of dependence. Hence, it is convenient to state the results in copula based language.

Following the approach proposed in [2], the first step is to study copulas invariant under univariate conditioning i.e. copulas C, such that if C is the copula of the random vector X = (X_{1},...,X_{n+1}), then C is also the copula of X supposing that X_{1} is smaller than its α quantile. The complete characterization of bivariate invariant copulas is given in [1]. In my talk I am going to extend these results for multivariate copulas. Specifically, I will characterize the (n+1)-dimensional copulas invariant under the conditioning of the first variable X_{1} in terms of the conditional n-dimensional copula of (X_{2},...,X_{n+1}) under the condition X = x and bivariate marginal copulas of pairs (X_{1},X_{i}), i = 2,... , n + 1. (full abstract)

#### Jozef Komorník, Magda Komorníková: Regime switching copula models for relations between returns of stock indexes

We have investigated relations between the returns of some major global stock indexes (New York, Tokyo, London) as well as their influence on the returns of regional indexes in South-East Asia. The existence of the time lag between New York and East Asia enabled us to identify a prevailing global leading influence of the N.Y. stock index (expressed by greatly increased values of the Kendall correlation coefficient of its lagged time series with those of the Asian stocks). However, the intensity of the relations between all studied indexes has manifested a strong time variability (with major changes in the time periods of global recessions, as well as when major local economic disturbances with global consequences occurred). In order to obtain more realistic models for such changes, we applied regime–switching modeling procedures utilizing Archimedean class copula models and their convex combinations. (full abstract)

#### Radko Mesiar: Copulas and integrals

The stochastic dependence structure of bivariate random variables captured by copulas can be applied in the integration approaches. Indeed, a copula C : [0, 1]^{2} → [0, 1] can express the connection between function values and measure values through an integral I_{C,m} by I_{C,m} (a, 1_{A} ) = C(a, m(A)), where m is the (monotone) measure and a ∈ [0, 1] is a constant. Looking on integrals I_{C,·} as functionals, one can introduce an axiomatic approach, too. For example, comonotone additivity is a genuine property of the Choquet integral, while the comonotone maxitivity and the min–homogeneity characterize the Sugeno integral. We give a general axiomatic characterization of discrete copula–based universal integrals. (full abstract)

#### Oľga Nánásiová: Modeling of non-compatible random events via multidimensional states

The theory of quantum logics began in the beginning of the 20th century because the classical theory of probability theory did not explain events that occurred in quantum physics. In these days, many scientific schools are interested in the study of non-compatibility and in the study of uncertainty of random events.

Algebraic approach is based on the study of more general structures such as Boolean algebra. We will use an orthomodular lattice with at least one state. This structure is called a quantum logic. States on a quantum logic represent probability measures and observables on a quantum logic represent random variables. We will focus on multivariable states that represent measures of intersection, union and symmetric difference in the case of compatibility. Multidimensional states are possible to use for modeling of non-compatible observables, for example, for modeling of joint distribution. (full abstract)

#### Pál Rakonczai: Bivariate generalized Pareto distribution in practice: models and estimation

Extreme values are of substantial interest in fields of environmental science, engineering or finance, because they are associated with rare but hazardous events (such as flooding, mechanical failure or severe financial loss). There is often interest in understanding how the extremes of two different processes are related to each other. One possible way to tackle this problem is an asymptotic approach which involves fitting multivariate generalised Pareto distribution (MGPD) to data that exceed a suitably high threshold. As exceedances can be defined different ways, there are a few non-equivalent definitions of MGPD in use. A rather classical way is the first type definition which is based on exceedances being over the threshold in all components and a second type definition considers those exceedances, which are over a threshold in at least one of the components regardless of the rest. The first type definition is widely investigated in the recent literature but the second type definition attracted less attention. One aim of this paper is to investigate the applicability of classical parametric dependence models within the second type definition of MGPD. Due to continuity problems the set of available dependence models narrows, especially if asymmetry property is also required. As an alternative solution, a general transformation is proposed for creating asymmetric models from the well-known symmetric ones. We apply the proposed approach to the exceedances of wind speed data and outline methods for calculating prediction regions as well as evaluating the goodness-of-fit. (full abstract)

#### Tomáš Bacigál: Recent tools for modelling dependence with copulas and R

R is a powerful software for statistical computation and visualization with open-source license and support of wide community of users and developers. We give an overview of built-in tools and extensions for analysis of dependence structure in random vector with copulas, from data preprocessing and visualization, through parameters estimation and goodness-of-fit tests to prediction. Purpose and state of development of several function libraries (including that of ours) are discussed and experience with some graphical user interfaces, editors and integrated development environments are shared.

#### Mária Bohdalová, Michal Greguš: Monte Carlo simulation Value at Risk and PCA

Financial portfolios include many of risk factors, such as asset prices, interest rates, foreign exchange rates, that affect the portfolios profit/loss. Risk factors are often very highly correlated, therefore it is convenient to use principal component analysis to reduce the dimension of the risk factor space. Moreover, the construction of the principal components guarantees that they are uncorrelated, because they are generated by orthogonal eigenvectors. The ability of PCA to reduce dimensions, combined with the use of orthogonal variables for risk factors, makes this technique an extremely attractive option for Monte Carlo simulation. In highly correlated term structures of risk factors the replacement of the original risk factors by just a few orthogonal risk factors introduces very little error into the simulations, and increases the efficiency of the simulations enormously. In this paper the PCA in Monte Carlo simulation with multivariate normal and Student t VaR will be used. (full abstract)

#### Dana Hliněná, Martin Kalina, Pavol Kráľ: Implicators and I-partitions

Jayaram and Mesiar (2009) introduced an I-fuzzy equivalence relation. We say that the concept of I-equivalence relation admits a value a from [0,1] iff there exists an I-equivalence relation E and some elements x,y such that E(x,y) = a. In this contribution we will investigate properties of the induces I-partitions for some special types of implicators. Particularly, we will be interested I-implicators which do not admit all values form [0,1].

#### Vladimír Jágr: Generalization and construction of Archimax copulas for higher dimensions

Extreme-value copulas of higher dimensions are discussed and we propose some new construction methods for tail and Pickands dependence functions. In short we recall an overview of some known results for d-dimensional Archimedean copulas. Finally, we focus to the main aim of this paper, which are higher dimensions of Archimax copulas. Some new classes of d-dimensional Archimax copulas are also introduced. (full abstract)

#### Jana Kalická, Tomáš Kulla: Optimal bandwith in nonparametric regression

Nonparametric regression can be used when parametric regression insufficiently describes the real data. One of nonparametric methods is kernel smoothing. Very important is the choise of optimal bandwith. We compare some methods for choosing this smoothing parameter.

#### Anna Kolesárová and Andrea Stupňanová: On the structure of associative n-dimensional copulas

The associativity of n-dimensional copulas in the sense of Post is studied. The structure of associative n-dimensional copulas is clarified. It is shown that associative n-dimensional copulas are n-ary extensions of associative 2-dimensional copulas with special constraints. The main result solves an open problem formulated by R. Mesiar. (full abstract)

#### Anna Kolesárová, Andrea Stupňanová, Juliana Beganová: Aggregation-based extensions of utility functions

The aim of the paper is to present a method extending fuzzy measures (utility functions) on N = {1,...,n} to n-ary aggregation functions (fuzzy utility functions) by means of a suitable n-ary aggregation function and the Möbius transform of the considered fuzzy measure. The method generalizes the well-known Lovász and Owen extensions of fuzzy measures.

#### Darina Kyselová: Aggregation functions-based building of transitive preference structures

Aggregation functions are often needed when building preference relations in multicriteria decision making problems. We bring an overview of methods that enable aggregation-based rankings to be refined. In our contribution we deal with orderings on finite scales and aggregation functions with range in the given finite scale. We discuss the orders based on a system of k-ary aggregation functions and also reversibility and redundancy of order induced by this system.

#### Jana Lencuchova: Comparing the power properties of the proposed test and some other nonlinearity tests for markov-switching time series models

The comparison of the power properties of some already known nonlinearity tests and a new proposed test is provided. We are testing the validity of Markov assumptions by the proposed test as an alternative to the classical test for linearity against the Markov-switching type of nonlinearity, which is very time-consuming. The RESET-type tests, McLeod-Li test and Tsay test are used to investigate their ability to reveal the Markov-switching type of nonlinearity since they test only some departures from linearity in general without a specific nonlinear parametric alternative. (full abstract)

#### Oľga Nánásiová, Miroslav Sabo: Clustering by two methods simultaneously

Clustering is a method of unsupervised learning, that finds clusters (objects with similar properties) in data. Since fifties, there were proposed many different clustering algorithms. Our method combine two arbitrary clustering algorithms to obtain more accurate results. Method is based on concept of similarity coefficients. Also practical application on hydrological data in R language will be shown. (full abstract)

#### Monika Pekárová: On some insurance risk applications of copulas

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. The goal of this paper is to provide simple applications for the practical use of copulas for risk management from an insurance point of view. In this paper we focus on special class of copulas and goodness-of-fit testing. (full abstract)

#### Anna Petričková: Modelling of the ARMA models residuals using autocopulas

The ’k-lag auto-copula’ is a 2-dimensional joint distribution function of the bivariate random vector (Y_{t} , Y_{t−k} ) of time lagged values of random variables that generate time series. In the contribution we extend the idea to the use of auto-copulas (originally used by Rakonczai for tests of the residual independence of time series models) also for investigation of the residuals dependence structures of the linear ARMA (Autoregressive Moving-Average) time series models. We model the residual dependence of ARMA time series models with auto-copulas (Archimedean, Extreme Value and their convex combinations). The fitting quality (both in-the-sample and out-of-the-sample) of the resulting models was considerable improved for a large class of economic time series. (full abstract)

#### G.Beliakov, S.James, J.Beganová, T.Rückschlossová and R.R.Yager: Bonferroni mean operators in multi-criteria aggregation

In this paper we provide a systematic investigation of a family of composed aggregation functions which generalize the Bonferroni mean.Such extenxions of the Bonferroni mean are capable of modeling the concepts of hard and soft partial conjunction and disjunction as well as that of k-tolerance and k-intolerance. There are several interesting special cases with quite an intuitive interpretation for application.

#### Danuša Szőkeová, Silvia Kohnová: SETAR models in the streamflow modeling

However linear models dominate empirical time series modeling for simplicity of estimation and forecasting there are many processes out of realy linear structure. The self exciting treshold autoregressive (SETAR) model is linear within a regime but switches between regimes if the ddelay value crosses a treshold. In the presentation we provide briefly the description of SETAR model structure, methods of parameter estimation, the determination of regimes number and the threshold values, the method of testing for treshold nonlinearity and the point forecast evaluation. We illustrate two regimes SETAR(p1,p2,d) models and three regimes SETAR(p1,p2,p3,d) models in the streamflow research of several Slovak rivers for the time period 1961-2000. For SETAR modeling we consider the residuals time series of observed data after subtraction systematic components (linear trend, seasonal and cyclical components) and the observed time series. In conclusion we focuse on the usefulness of SETAR models for in-of-sample modeling and out-of-sample forecasting relative to linear AR models.

#### Lucia Vavríková: Application of aggregation operators on the assessment of public universities and their faculties

In order to solve decision making problem we have to compare and rank a finite set of alternatives according to finite number of criteria. In this paper we want to show some approaches how to create the preference structure (ranking) of alternatives. These approaches lead us to use chosen multicriteria decision methods and aggregation operators. ARRA (Academic ranking and rating agency) uses one fixed way to create a ranking of public universities and their faculties. Here many more approaches how to create such rankings (or preference structures) of alternatives are discussed. (full abstract)

#### Petra Zacharovská: Comparison of descriptive and predictive properties of MSW models with different probability distribution of residuals

We modeled the series of annual counts of major earthquakes (i.e. magnitude seven and above) for the years 1900–2006 by MSW models with different probability distribution of residuals. We considered three probability distributions of residuals – two continuous (normal distribution and Student t-distribution) and one discrete (Poisson) probability distribution. We also considered an extension to continuous distributions, including models with different continuous distribution types (mentioned above) in different states. We compared descriptive and predictive properties of these models. The parameters in the MSW model can be estimated using maximum likelihood techniques. The aim of the estimation procedure is not only to obtain estimates of the parameters in the autoregressive models in different regimes and the probabilities of transition from one regime to the other, but also to obtain an estimate of the probabilities of occurrence of each state at each point in time.