Angela De Sanctis

FUNCTIONAL ANALYSIS FOR PARAMETRIC FAMILIES OF FUNCTIONAL DATA

Assuming a Parametric Family of Functional Data, the problem of computing summary statistics of the same functional form is investigated. The central idea is to compile the statistics on the parameters instead of on the functions themselves. With the hypothesis of a monotonic dependence from parameters, we highlight the special features of this statistics.

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.

Dealing with FDA Estimation Methods

In many different research fields, such as medicine, physics, economics, etc., the evaluation of real phenomena observed at each statistical unit is described by a curve or an assigned function. In this framework, a suitable statistical approach is Functional Data Analysis based on the use of basis functions. An alternative method, using Functional Analysis tools, is considered in order to estimate functional statistics. Assuming a parametric family of functional data, the problem of computing summary statistics of the same parametric form when the set of all functions having that parametric form does not constitute a linear space is investigated. The central idea is to make statistics on the parameters instead of on the functions themselves.

On the Geodesic Distance in Shapes K-means Clustering

In this paper, the problem of clustering rotationally invariant shapes is studied and a solution using Information Geometry tools is provided. Landmarks of a complex shape are defined as probability densities in a statistical manifold. Then, in the setting of shapes clustering through a K-means algorithm, the discriminative power of two different shapes distances are evaluated. The first, derived from Fisher–Rao metric, is related with the minimization of information in the Fisher sense and the other is derived from the Wasserstein distance which measures the minimal transportation cost. A modification of the K-means algorithm is also proposed which allows the variances to vary not only among the landmarks but also among the clusters.

Fisher Information Geometry for Shape Analysis

The aim of this study is to model shapes from complex systems using Information Geometry tools. It is well-known that the Fisher information endows the statistical manifold, defined by a family of probability distributions, with a Riemannian metric, called the Fisher-Rao metric. With respect to this, geodesic paths are determined, minimizing information in Fisher sense. Under the hypothesis that it is possible to extract from the shape a finite number of representing points, called landmarks, we propose to model each of them with a probability distribution, as for example a multivariate Gaussian distribution. Then using the geodesic distance, induced by the Fisher-Rao metric, we can define a shape metric which enables us to quantify differences between shapes. The discriminative power of the proposed shape metric is tested performing a cluster analysis on the shapes of three different groups of specimens corresponding to three species of flatfish. Results show a better ability in recovering the true cluster structure with respect to other existing shape distances.

Geodesic submanifolds of a family of statistical models

Given the Riemannian manifold, determined by the Fisher information metric in the statistical model with location parameters induced by the family of probability density functions on : P[Omega] = {exp( - c([omega])x2 + [phi]([omega])x - [psi]([omega])): [omega] [epsilon] [Omega]} with c([omega]) > 0, for every [omega] [set membership, variant] [Omega], conditions are found for the level submanifolds to be geodesic.

Non-Walrasian Equilibria in a Labour-Managed Economy

This paper studies the existence and stability of non-Walrasian equilibria in a labour-managed economy. The main finding is that non-Walrasian equilibria exist and constitute a one-dimensional smooth manifold; they are stable but not asymptotically stable.