Francesca Fortuna

Bell-Shaped Fuzzy Numbers Associated with the Normal Curve

Statisticians often focus on fuzzy numbers with triangular or trapezoidal membership functions because they are very easy to apply. Although they offer a good approximation of a fuzzy variable, several doubts arise about the appropriateness of these kind of shapes. As known, fuzzy sets are useful for interval data when the “degree of truth” of the values varies within this range. In particular, they are desirable for translating human language into numbers. In this paper, we propose an alternative membership function that appears more appropriate to deal with linguistic variables. We refer to this function as “bell-shaped fuzzy number associated with the normal curve”. In particular, we highlight the specific properties of the proposed fuzzy number and illustrate the utility of linking this function with the normal distribution.

Functional confidence bands for lichen biodiversity profiles: A case study in Tuscany region central Italy

Biomonitoring techniques are widely used to assess environmental damages through the changes occurring in the composition of species communities. Among the living organisms used as bioindicators, epiphytic lichens, are recognized as reliable indicators of air pollution. However, lichen biodiversity studies are generally based on the analysis of a scalar measure that omits the species composition. For this reason, we propose to analyze lichen data through diversity profiles and the functional data analysis approach. Indeed, diversity profiles may be naturally considered as functional data because they are expressed as functions of the species abundance vector in a fixed domain. The peculiarity of these data is that the functional space is constituted by a set of curves belonging to the same family. In this context, simultaneous confidence bands are obtained for the mean diversity profile through the Karhunen-Love KL decomposition. The novelty of our method lies in exploiting the known form of the function underlying the data. This allows us to work directly on the functional space by avoiding smoothing techniques. The confidence band procedure is applied to a real data set concerning lichen data in Tuscany region central Italy. Confidence bands functional data analysis intrinsic diversity profile lichen data mean function KL expansion.

Clustering Functional Data on Convex Function Spaces

The curves in a functional data set often present a variety of distinctive patterns corresponding to different shapes that can be identified by clustering the functions. However, clustering functional data is a difficult task because the function space is, generally, of infinite dimension. Thus, the distance among functions may have infinity solutions and can be approximated in different ways leading to different clustering results. The paper deals with this problem and focuses on cases in which the functional form of the observations is known in advance. In this setting, the approximation of the function underlying the data is not required and the functional distance may be computed directly in the explicit form of the functions. Moreover, we restrict the space of the functions to a closed and convex subset in an Hilbert space to achieve desirable properties. In the proposed framework, an \(L^2\) metric is applied combined clustering algorithms for finite dimensional data. The method is applied to a real data set concerning lichen biodiversity in the province of Genoa, North Western Italy.

Cluster Analysis as a Decision-Making Tool: A Methodological Review

Cluster analysis has long played an important role in a broad variety of areas, such as psychology, biology, computer sciences. It has established as a precious tool for marketing and business areas, thanks to its capability to help in decision-making processes. Traditionally, clustering approaches concentrate on purely numerical or categorical data only. An important area of cluster analysis deals with mixed data, composed by both numerical and categorical attributes. Clustering mixed data is not simple, because there is a strong gap between the similarity metrics for these two kind of data. In this review we provide some technical details about the kind of distances that could be used with mixed-data types. Finally, we emphasize as in most applications of cluster analysis practitioners focus either on numeric or categorical variables, lessening the effectiveness of the method as a tool of decision-making.

Adaptive cluster double sampling with post stratification with application to an epiphytic lichen community

The implementation of an adaptive cluster sampling design often becomes logistically challenging because variation in the final sampling effort introduces uncertainty in survey planning. To overcome this drawback, an inexpensive and easy to measure auxiliary variable could be used in a two-phase survey strategy, called adaptive cluster double sampling (Félix-Medina and Thompson in Biometrika 91:877–891, 2004). In this paper, a two-phase sampling strategy is proposed which combines the idea of adaptive cluster double sampling with the principle of post-stratification. In the first-phase an adaptive cluster sample is selected by means of an inexpensive auxiliary variable. Networks from the first phase sampling are then post-stratified according to their size. In the second-phase, the network structure is used to select a subsample of units by means of stratified random sampling. The proposed sampling strategy employs stratification without requiring an a priori delineation of the strata. Indeed, the strata sizes are estimated in the course of the two-phase sampling process. Therefore, it is suitable for situations where stratification is suspected to be efficient but strata cannot be easily delineated in advance. In this framework, a new type of estimator for the population mean which mimics the stratified sampling mean estimator and an estimator of the sampling variance are proposed. The results of a simulation study confirm, as expected, that the use of post-stratification leads to gain in precision for the estimator. The proposed sampling strategy is applied for targeting an epiphytic lichen community Lobarion pulmonariae in a forest area of the Northern Apennines (N-Italy), characterized by several species of conservation concern.