Renato Coppi

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 linear regression model for imprecise response

A linear regression model with imprecise response and p real explanatory variables is analyzed. The imprecision of the response variable is functionally described by means of certain kinds of fuzzy sets, the LR fuzzy sets. The LR fuzzy random variables are introduced to model usual random experiments when the characteristic observed on each result can be described with fuzzy numbers of a particular class, determined by 3 random values: the center, the left spread and the right spread. In fact, these constitute a natural generalization of the interval data. To deal with the estimation problem the space of the LR fuzzy numbers is proved to be isometric to a closed and convex cone of R^3 with respect to a generalization of the most used metric for LR fuzzy numbers. The expression of the estimators in terms of moments is established, their limit distribution and asymptotic properties are analyzed and applied to the determination of confidence regions and hypothesis testing procedures. The results are illustrated by means of some case-studies.

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.

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.

Fuzzy K-means clustering models for triangular fuzzy time trajectories

We focus our attention on the classification of fuzzy time trajectories with triangular membership function, described by a given set of individuals. To this purpose, we adopt a fullyinformational approach, explicitly recognizing the informational nature shared by the ingredients of the classification procedure: the observed data (Empirical Information) and the classification model (Theoretical Information). In particular, by supposing that the informational paradigm has a fuzzy nature, we suggest three fuzzy clustering models allowing the classification of the triangular fuzzy time trajectories, based on the analysis of the cross sectional and/or longitudinal characteristics of their components (centers and spreads). Two applicative examples are illustrated.

A Fuzzy Clustering Model for Multivariate Spatial Time Series

Clustering of multivariate spatial-time series should consider: 1) the spatial nature of the objects to be clustered; 2) the characteristics of the feature space, namely the space of multivariate time trajectories; 3) the uncertainty associated to the assignment of a spatial unit to a given cluster on the basis of the above complex features. The last aspect is dealt with by using the Fuzzy C-Means objective function, based on appropriate measures of dissimilarity between time trajectories, by distinguishing the cross-sectional and longitudinal aspects of the trajectories. In order to take into account the spatial nature of the statistical units, a spatial penalization term is added to the above function, depending on a suitable spatial proximity/ contiguity matrix. A tuning coefficient takes care of the balance between, on one side, discriminating according to the pattern of the time trajectories and, on the other side, ensuring an approximate spatial homogeneity of the clusters. A technique for determining an optimal value of this coefficient is proposed, based on an appropriate spatial autocorrelation measure. Finally, the proposed models are applied to the classification of the Italian provinces, on the basis of the observed dynamics of some socio-economical indicators.

The Geometric Approach to the Comparison of Multivariate Time Trajectories

To compare time trajectories different approaches might be envisaged. In this paper, considering the geometric approach, several dissimilarity measures between time trajectories are taken into account. An empirical comparison of the dissimilarity measures is also shown.

Wiktor stent for treatment of chronic total coronary artery occlusions: short- and long-term clinical and angiographic results from a large multicenter experience.

OBJECTIVES This study reports the first multicenter experience with the Wiktor coil stent for treatment of chronic total coronary artery occlusions (CTOs). BACKGROUND Percutaneous transluminal coronary angioplasty (PTCA) of CTO is associated with very high restenosis and reocclusion rates. Coronary stenting has been proposed as a means of improving outcome. However, the Wiktor device for CTOs has never been tested in a large patient sample. METHODS From January 1993 to December 1996, 89 patients with 91 CTOs underwent Wiktor stent implantation after successful PTCA. The post-stenting regimen consisted of warfarin (Coumadin) plus aspirin in the initial 49 patients (55%) and aspirin plus ticlopidine in 40 patients (45%). RESULTS Stenting was successful in 87 patients (98%). At 1 month, 6% of patients had subacute stent thrombosis, 3% had a major bleeding event, and 1% had access-site complications. Subacute stent thrombosis showed univariate association with warfarin therapy (p = 0.009). Angiographic follow-up was obtained in 76 (93%) of 82 eligible patients. The restenosis rate was 32%, including 4% reocclusions. By multiple logistic regression analysis, restenosis was independently associated with multiple stents (adjusted odds ratio [OR] 27.67, 95% confidence interval [CI] 4.25 to 79.95, p = 0.0008) and increasing values of occlusion length (adjusted OR 1.23, 95% CI 1.09 to 1.39, p = 0.001). Freedom from death, myocardial infarction or stented vessel revascularization was 87% and 72% at 1 and 3 years, respectively. CONCLUSIONS Short- and long-term clinical and angiographic outcomes are favorable in patients undergoing Wiktor stent implantation in CTO. Further technical improvement is needed to reduce the restenosis rate in patients with long lesions and multiple stents.

Fuzzy C-medoids clustering models for time-varying data

Abstract In this paper , by considering suitable dissimilarity measures for multivariate time trajectories (time series) [ 1 , 2 , 3 ], we suggest different fuzzy C–medoids clustering models , which are particular relational versions of the dynamic clustering models proposed in [ 2 , 3 ]. In particular , our models classify time trajectories and select , in the set of the observed trajectories , typical trajectories that synthetically representthe structural characteristics of the identified clusters (medoid time trajectories) . Asimulation study and an application are also discussed .

Fuzzy Time Arrays and Dissimilarity Measures For Fuzzy Time Trajectories

In this paper we define a fuzzy extension of a time array. The algebraic and geometric characteristics of the fuzzy time array are analyzed. Furthermore, considering the objects space ℜ J+1, where J is the number of variables and the remaining dimension is related to time, we suggest different dissimilarity measures for fuzzy time trajectories.