Classification methods for Hilbert data based on surrogate density

An unsupervised and a supervised classification approaches for Hilbert random curves are studied. Both rest on the use of a surrogate of the probability density which is defined, in a distribution-free mixture context, from an asymptotic factorization of the small-ball probability. That surrogate density is estimated by a kernel approach from the principal components of the data. The focus is on the illustration of the classification algorithms and the computational implications, with particular attention to the tuning of the parameters involved. Some asymptotic results are sketched. Applications on simulated and real datasets show how the proposed methods work.Comment: 33 pages, 11 figures, 6 table

A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation

Let $X$ be a fuzzy set--valued random variable (\frv{}), and \huku{X} the family of all fuzzy sets $B$ for which the Hukuhara difference X\HukuDiff B exists $\mathbb{P}$--almost surely. In this paper, we prove that $X$ can be decomposed as X(\omega)=C\Mink Y(\omega) for $\mathbb{P}$--almost every $\omega\in\Omega$, $C$ is the unique deterministic fuzzy set that minimizes $\mathbb{E}[d_2(X,B)^2]$ as $B$ is varying in \huku{X}, and $Y$ is a centered \frv{} (i.e. its generalized Steiner point is the origin). This decomposition allows us to characterize all \frv{} translation (i.e. X(\omega) = M \Mink \indicator{\xi(\omega)} for some deterministic fuzzy convex set $M$ and some random element in \Banach). In particular, $X$ is an \frv{} translation if and only if the Aumann expectation $\mathbb{E}X$ is equal to $C$ up to a translation. Examples, such as the Gaussian case, are provided.Comment: 12 pages, 1 figure. v2: minor revision. v3: minor revision; references, affiliation and acknowledgments added. Submitted versio

Describing the Concentration of Income Populations by Functional Principal Component Analysis on Lorenz Curves

Lorenz curves are widely used in economic studies (inequality, poverty, differentiation, etc.). From a model point of view, such curves can be seen as constrained functional data for which functional principal component analysis (FPCA) could be defined. Although statistically consistent, performing FPCA using the original data can lead to a suboptimal analysis from a mathematical and interpretation point of view. In fact, the family of Lorenz curves lacks very basic (e.g., vectorial) structures and, hence, must be treated with ad hoc methods. This work aims to provide a rigorous mathematical framework via an embedding approach to define a coherent FPCA for Lorenz curves. This approach is used to explore a functional dataset from the Bank of Italy income survey

Modeling functional data: a test procedure

The paper deals with a test procedure able to state the compatibility of observed data with a reference model, by using an estimate of the volumetric part in the small-ball probability factorization which plays the role of a real complexity index. As a preliminary by-product we state some asymptotics for a new estimator of the complexity index. A suitable test statistic is derived and, referring to the U-statistics theory, its asymptotic null distribution is obtained. A study of level and power of the test for finite sample sizes and a comparison with a competitor are carried out by Monte Carlo simulations. The test procedure is performed over a financial time series

A Note on Fuzzy Set--Valued Brownian Motion

In this paper, we prove that a fuzzy set--valued Brownian motion $B_t$, as defined in [1], can be handle by an $R^d$--valued Wiener process $b_t$, in the sense that B_t =\indicator{b_t}; i.e. it is actually the indicator function of a Wiener process

A set-valued framework for birth-and-growth process

We propose a set-valued framework for the well-posedness of birth-and-growth process. Our birth-and-growth model is rigorously defined as a suitable combination, involving Minkowski sum and Aumann integral, of two very general set-valued processes representing nucleation and growth respectively. The simplicity of the used geometrical approach leads us to avoid problems arising by an analytical definition of the front growth such as boundary regularities. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, it is not local, i.e. for a fixed time instant, growth is the same at each space point

School Bike Sharing Program: Will it Succeed?

Encouraging active and sustainable transport modes in order to limit the excessive use of cars, as well as reducing pollutant emissions and creating livable urban environments, has become one of the priorities for policymakers in recent years. The introduction of innovative systems increasingly being introduced in modern cities, such as bike sharing, can certainly contribute to the spread of cycling and thus allow a radical change in the mobility habits of their citizens. This can be especially true for high-school students who are often otherwise accompanied by their parents with private cars. This article aims to assess the influence that a bike sharing program for students has on modal share and on city mobility. As a case study, the city of Palermo was chosen, where the use of the car for home-school trips is prevalent. The "Go2School" project, which involves the creation of a bike sharing program for four schools, with the construction of cycle docks and cycle paths in the nearby areas, will soon become a reality. Thanks to appropriate surveys and questionnaires, a multinomial logit model was calibrated to estimate the modal share towards bike sharing for the students and evaluate the demand for this transport mode

Statistical aspects of birth--and--growth stochastic processes

The paper considers a particular family of set--valued stochastic processes modeling birth--and--growth processes. The proposed setting allows us to investigate the nucleation and the growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set--valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.Comment: simpler notations typo