Pierpaolo D'Urso

Linear regression analysis for fuzzy/crisp input and fuzzy/crisp output data

In order to estimate fuzzy regression models, possibilistic and least-squares procedures can be considered. By taking into account a least-squares approach, regression models with crisp or fuzzy inputs and crisp or fuzzy output are suggested. In particular, for these fuzzy regression models, unconstrained and constrained (with inequality restrictions) least-squares estimation procedures are developed. Furthermore, for the various models presented, explanatory examples are shown and some concluding remarks are also included.

Least squares estimation of a linear regression model with LR fuzzy response

The problem of regression analysis in a fuzzy setting is discussed. A general linear regression model for studying the dependence of a LR fuzzy response variable on a set of crisp explanatory variables, along with a suitable iterative least squares estimation procedure, is introduced. This model is then framed within a wider strategy of analysis, capable to manage various types of uncertainty. These include the imprecision of the regression coefficients and the choice of a specific parametric model within a given class of models. The first source of uncertainty is dealt with by exploiting the implicit fuzzy arithmetic relationships between the spreads of the regression coefficients and the spreads of the response variable. Concerning the second kind of uncertainty, a suitable selection procedure is illustrated. This consists in maximizing an appropriately introduced goodness of fit index, within the given class of parametric models. The above strategy is illustrated in detail, with reference to an application to real data collected in the framework of an environmental study. In the final remarks, some critical points are underlined, along with a few indications for future research in this field.

A least-squares approach to fuzzy linear regression analysis

This paper deals with a new approach to fuzzy linear regression analysis. A doubly linear adaptive fuzzy regression model is proposed, based on two linear models: a core regression model and a spread regression model. The first one “explains” the centers of the fuzzy observations, while the second one is for their spreads. As dependence between centers and spreads is often encountered in real world applications, our model is defined in such a way as to take into account a possible linear relationship among centers and spreads. Illustrative examples are also discussed, and a computer program which implements our procedure is enclosed.

Autocorrelation-based fuzzy clustering of time series

The traditional approaches to clustering a set of time series are generally applicable if there is a fixed underlying structure to the time series so that each will belong to one cluster or the other. However, time series often display dynamic behaviour in their evolution over time. This dynamic behaviour should be taken into account when attempting to cluster time series. For instance, during a certain period, a time series might belong to a certain cluster; afterwards its dynamics might be closer to that of another cluster. In this case, the traditional clustering approaches are unlikely to find and represent the underlying structure in the given time series. This switch from one time state to another, which is typically vague, can be naturally treated following a fuzzy approach. This paper proposes a fuzzy clustering approach based on the autocorrelation functions of time series, in which each time series is not assigned exclusively to only one cluster, but it is allowed to belong to different clusters with various membership degrees.

Fuzzy clustering of time series in the frequency domain

Traditional and fuzzy cluster analyses are applicable to variables whose values are uncorrelated. Hence, in order to cluster time series data which are usually serially correlated, one needs to extract features from the time series, the values of which are uncorrelated. The periodogram which is an estimator of the spectral density function of a time series is a feature that can be used in the cluster analysis of time series because its ordinates are uncorrelated. Additionally, the normalized periodogram and the logarithm of the normalized periodogram are also features that can be used. In this paper, we consider a fuzzy clustering approach for time series based on the estimated cepstrum. The cepstrum is the spectrum of the logarithm of the spectral density function. We show in our simulation studies for the typical generating processes that have been considered, fuzzy clustering based on the cepstral coefficients performs very well compared to when it is based on other features.

Fuzzy and possibilistic clustering for fuzzy data

The Fuzzy k-Means clustering model (FkM) is a powerful tool for classifying objects into a set of k homogeneous clusters by means of the membership degrees of an object in a cluster. In FkM, for each object, the sum of the membership degrees in the clusters must be equal to one. Such a constraint may cause meaningless results, especially when noise is present. To avoid this drawback, it is possible to relax the constraint, leading to the so-called Possibilistic k-Means clustering model (PkM). In particular, attention is paid to the case in which the empirical information is affected by imprecision or vagueness. This is handled by means of LR fuzzy numbers. An FkM model for LR fuzzy data is firstly developed and a PkM model for the same type of data is then proposed. The results of a simulation experiment and of two applications to real world fuzzy data confirm the validity of both models, while providing indications as to some advantages connected with the use of the possibilistic approach.

Fuzzy Clustering for Data Time Arrays With Inlier and Outlier Time Trajectories

In many knowledge discovery and data mining tasks, fuzzy clustering is one of the most common tools for data partitioning. In this paper dynamic fuzzy clustering models for classifying a set of multivariate time trajectories (time series, sequences) are developed. In particular, by adopting an exploratory approach, based on a geometric-algebraic formulation of the data time array, different kinds of dynamic fuzzy clustering models, based on cross sectional and longitudinal aspects, are suggested. Furthermore, a modified version of the previous clustering models, that can be seen as a generalization of these models, is proposed. By utilizing these models we can obtain beneficial effects in the clustering process when anomalous trajectories (trajectories with anomalous positions and slopes) are present in the dataset; in fact the models are suitable for detecting structures of time trajectories with anomalous patterns that are not uniformly distributed over the structures domains and are characterized by strange slopes. In these models, the disruptive effect of the anomalous trajectories is neutralized and smoothed and the information on the influence of individual time trajectories on the detected groups is given. Furthermore, some remarks on dynamic three-way extensions of a few robust fuzzy clustering models for two-way data are suggested. Demonstrative examples are shown and a comparison assessment based on artificial multivariate time-varying data is carried out

Robust fuzzy regression analysis

In this paper we propose a robust fuzzy linear regression model based on the Least Median Squares-Weighted Least Squares (LMS-WLS) estimation procedure. The proposed model is general enough to deal with data contaminated by outliers due to measurement errors or extracted from highly skewed or heavy tailed distributions. We also define suitable goodness of fit indices useful to evaluate the performances of the proposed model. The effectiveness of our model in reducing the outliers influence is shown by using applicative examples, based both on simulated and real data, and by a simulation study.

Three-way fuzzy clustering models for LR fuzzy time trajectories

Fuzzy multivariate time trajectories are defined. For a suitable class, called LR time trajectories, three types of dissimilarity measures are introduced: the instantaneous, the velocity and the simultaneous measures, respectively. Correspondingly, three different kinds of dynamic fuzzy clustering models are suggested, based on a generalization of the Bezdek and Yang and Ko objective functions for fuzzy clustering. The solutions and characteristics of the three models are then illustrated. A comparative appraisal of their practical meaning is proposed by means of an application to the time pattern of the subjective judgments expressed by a sample of web navigators on different types of banners. Some indications for future research in this methodological domain are finally provided.

An orderwise polynomial regression procedure for fuzzy data

A procedure for polynomial fit with fuzzy data is presented. As in real case studies, there is often dependence between modes and spreads, we propose a regression polynomial model capable to take into account possible interactions between the estimated spreads and modes. This method finds applications in several fields such as reliability, quality control, psychometrics, marketing, image processing, etc. An application to a software reliability problem is also presented.