Krzysztof Jajuga

L 1 -norm based fuzzy clustering

Abstract The paper presents the L1 version of the well-known fuzzy clustering method, namely fuzzy ISODATA, proposed by Bezdek and Dunn. Due to their robustness, L1-norm based methods gained much attention in statistics. The presented fuzzy clustering problem uses the distance between observations and location parameter vectors, which is based on the L1-norm, instead of the inner product induced norm used in classical fuzzy ISODATA. Two alternative methods to solve the L1 fuzzy clustering problem are derived. In practice both membership grades and location parameter vectors are unknown. The paper presents two iterative algorithms, each the implementation of the derived method. Finally, numerical examples are presented. One of them refers to famous Iris data.

Fuzzy clustering with squared Minkowski distances

Abstract This paper presents a new fuzzy clustering model based on a root of the squared Minkowski distance which includes squared and unsquared Euclidean distances and the L 1 -distance. An algorithm is presented that is based on iterative majorization and yields a convergent series of monotone nonincreasing loss function values. This algorithm coincides under some condition with the ISODATA algorithm of Dunn (J. Cybernet. 3 (1974) 32–57) and the fuzzy c -means algorithm of Bezdek (Ph.D. Thesis, Cornell University, Ithaca, 1973) for squared Euclidean distance and with an algorithm of Jajuga (Fuzzy Sets and Systems 39 (1991) 43–50) for L 1 -distances. To find a global minimum we compare a special strategy called fuzzy steps with fuzzy Kohonen clustering networks (FKCN) (Pattern Recognition 27 (1994) 757–764) and multistart. Fuzzy steps and FKCN are based on finding updates for a decreasing weighting exponent, which seems to work particularly well for hard clustering. To assess the performance of the methods, two numerical experiments and a simulation study are performed.

Linear fuzzy regression

Abstract This paper presents an approach which is useful for regression analysis in the case of heterogeneity of a set of observations, for which regression is evaluated. The proposed procedure consists of two stages. First, for a set of observations, fuzzy classification is determined. Due to this, homogenous classes of observations which are of hyperellipsoidal shape, are obtained. Then for each fuzzy class, the so called linear fuzzy regression is evaluated. In the paper the method of calculating linear fuzzy regression coefficients is given. It is a generalized version of the least squares method. The formula for the values of coefficients is given. Some properties of linear fuzzy regression are analyzed. It is proved that in one- and two-dimensional cases, the formulae are analogous to those for usual regression. A measure of goodness-of-fit and the method of determination of the number of fuzzy classes are also given. Presented examples indicate the superiority of fuzzy regression in comparison to usual regression in the case of heterogenous observations.

On The General Distance Measure

In [1993], pp. 44–45 the distance measure was proposed, which can be used for the ordinal data. In the paper the proposal of the general distance measure is given. This measure can be used for data measured in ratio, interval and ordinal scale. The proposal is based on the idea of the generalised correlation coefficient.

Standardisation of Data Set under Different Measurement Scales

Standardisation of multivariate observations is the important stage that precedes the determination of distances (dissimilarities) in clustering and multidimensional scaling. Different studies (e.g. Milligan, Cooper (1988)) show the effect of standardisation on the cluster structure in various data configurations. In the paper a survey of standardisation formulas is given. Then we consider the problem of different scales of measurement and their impact on: — the selection of the standardisation formula; — the selections of the appropriate dissimilarity (or similarity) measure.

Copula Functions in Model Based Clustering

Model based clustering is common approach used in cluster analysis. Here each cluster is characterized by some kind of model, for example — multivariate distribution, regression, principal component etc. One of the most well known approaches in model based clustering is the one proposed by Banfield and Raftery (1993), where each class is described by multivariate normal distribution. Due to the eigenvalue decomposition, one gets flexibility in modeling size, shape and orientation of the clusters, still assuming general elliptical shape of the set of observations. In the paper we consider the other proposal based on the general stochastic approach in two versions: classification likelihood approach, where each observation comes from one of several populations; mixture approach, where observations are distributed as a mixture of several distributions.

Tail Dependence in Multivariate Data — Review of Some Problems

Tail dependence is understood as the dependence between the variables assuming that these variables take the values from the tails of univariate distributions. In the paper two approaches of tail dependence determination are discussed: conditional correlation coefficient and tail dependence coefficients. It can be argued that both approaches are the generalizations of the well-known univariate approach, based on conditional excess distribution. In the paper the proposal is also given to extend tail dependence coefficients to the general multivariate case and to represent these coefficients through copula function.

Model-Based Clustering — Discussion on Some Approaches

One of the most well grounded approaches in clustering is model-based clustering, where one assumes particular multivariate distribution for each class. Most results in model-based clustering were obtained under multivariate normal distribution. In the paper we propose to adopt other approach, namely copula analysis in model-based clustering. Two possible stochastic approaches, namely classification approach and mixture approach, are considered as the framework to apply copula analysis. In the paper iterative algorithms are proposed to find optimal solution of clustering problem.

Risk of Options — Impact of Volatility Parameter

A derivative instrument is a financial instrument whose value depends on the value on the underlying index, for example on the price of another financial instrument (e.g. currency, stock). Derivative instruments gained a lot of attention of financial theoreticians and practitioners in the last decade. The main reason is the increasing risk both on the financial markets and in the whole economy. On the one hand, the derivative instruments proved to be very useful instruments in risk management. This refers to the following types of risk: market risk (interest rate risk, exchange rate risk, stock price risk, commodity price risk); credit risk; enterprise-wide risk; weather risk; insurance (catastrophe) risk.

Statistics and Data Analysis in Market Risk Measurement

The paper contains some remarks on the application of statistical methods in market risk analysis and measurement. First of all, the historical sketch of the development in this area is given, as well as the systematization of the particular groups of applications, including modeling and forecasting financial prices and financial risk analysis. Then the review of three groups of market risk measures is given, namely: volatility measures, sensitivity measures and downside risk measures. Finally some possible directions of future research are discussed.