377 research outputs found
A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation
Let be a fuzzy set--valued random variable (\frv{}), and \huku{X} the
family of all fuzzy sets for which the Hukuhara difference X\HukuDiff B
exists --almost surely. In this paper, we prove that can be
decomposed as X(\omega)=C\Mink Y(\omega) for --almost every
, is the unique deterministic fuzzy set that minimizes
as is varying in \huku{X}, and 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 and some
random element in \Banach). In particular, is an \frv{} translation if
and only if the Aumann expectation is equal to 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
Twitter as an innovation process with damping effect
Understanding the innovation process, that is the underlying mechanisms
through which novelties emerge, diffuse and trigger further novelties is
undoubtedly of fundamental importance in many areas (biology, linguistics,
social science and others). The models introduced so far satisfy the Heaps'
law, regarding the rate at which novelties appear, and the Zipf's law, that
states a power law behavior for the frequency distribution of the elements.
However, there are empirical cases far from showing a pure power law behavior
and such a deviation is present for elements with high frequencies. We explain
this phenomenon by means of a suitable "damping" effect in the probability of a
repetition of an old element. While the proposed model is extremely general and
may be also employed in other contexts, it has been tested on some Twitter data
sets and demonstrated great performances with respect to Heaps' law and, above
all, with respect to the fitting of the frequency-rank plots for low and high
frequencies
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
Nonparametric covariate-adjusted response-adaptive design based on a functional urn model
In this paper we propose a general class of covariate-adjusted response-adaptive (CARA) designs based on a new functional urn model. We prove strong consistency concerning the functional urn proportion and the proportion of subjects assigned to the treatment groups, in the whole study and for each covariate profile, allowing the distribution of the responses conditioned on covariates to be estimated nonparametrically. In addition, we establish joint central limit theorems for the above quantities and the sufficient statistics of features of interest, which allow to construct procedures to make inference on the conditional response distributions. These results are then applied to typical situations concerning Gaussian and binary responses
Analytical confidence intervals for the number of different objects in data streams
This paper develops a new mathematical-statistical approach to analyze a class of Flajolet-Martin algorithms (FMa), and provides analytical confidence intervals for the number F0 of distinct elements in a stream, based on Chernoff bounds. The class of FMa has reached a significant popularity in bigdata stream learning, and the attention of the literature has mainly been based on algorithmic aspects, basically complexity optimality, while the statistical analysis of these class of algorithms has been often faced heuristically. The analysis provided here shows deep connections with mathematical special functions and with extreme value theory. The latter connection may help in explaining heuristic considerations, while the first opens many numerical issues, faced at the end of the present paper. Finally, the algorithms are tested on an anonymized real data stream and MonteCarlo simulations are provided to support our analytical choice in this context
Generation of discrete random variables in scalable framework
In this paper, we face the problem of simulating discrete random variables with general and varying distribution in a scalable framework, where fully parallelizable operations should be preferred. Compared with classical algorithms, we add randomness, that will be analyzed with a fully parallelizable operation, and we leave the final simulation of the random variable to a single associative operator. We characterize the set of algorithms that work in this way, and some classes of them related to an additive or multiplicative local noise. As a consequence, we could define a natural way to solve some popular simulation problems
An urn model with local reinforcement: a theoretical framework for a chi-squared goodness of fit test with a big sample
Motivated by recent studies of big samples, this work aims at constructing a parametric model which is characterized by the following features: (i) a "local" reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random fluctuation of the conditional probabilities, and (iii) a long-term convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness of fit result. This triple purpose has been achieved by the introduction of a new variant of the Eggenberger-Polya urn, that we call the "Rescaled" Polya urn. We provide a complete asymptotic characterization of this model and we underline that, for a certain choice of the parameters, it has properties different from the ones typically exhibited from the other urn models in the literature. As a byproduct, we also provide a Central Limit Theorem for a class of linear functionals of non-Harris Markov chains, where the asymptotic covariance matrix is explicitly given in linear form, and not in the usual form of a series
Taylor's law in innovation processes
Taylor's law quantifies the scaling properties of the fluctuations of the
number of innovations occurring in open systems.
Urn based modelling schemes have already proven to be effective in modelling
this complex behaviour.
Here, we present analytical estimations of Taylor's law exponents in such
models, by leveraging on their representation in terms of triangular urn
models.
We also highlight the correspondence of these models with Poisson-Dirichlet
processes and demonstrate how a non-trivial Taylor's law exponent is a kind of
universal feature in systems related to human activities.
We base this result on the analysis of four collections of data generated by
human activity: (i) written language (from a Gutenberg corpus); (ii) a n online
music website (Last.fm); (iii) Twitter hashtags; (iv) a on-line collaborative
tagging system (Del.icio.us).
While Taylor's law observed in the last two datasets agrees with the plain
model predictions, we need to introduce a generalization to fully characterize
the behaviour of the first two datasets, where temporal correlations are
possibly more relevant.
We suggest that Taylor's law is a fundamental complement to Zipf's and Heaps'
laws in unveiling the complex dynamical processes underlying the evolution of
systems featuring innovation.Comment: 17 page
A decomposition theorem for fuzzy set–valued random variables
In this paper, a decomposition theorem for a (square integrable) fuzzy random variable FRV is proposed. The paper is mainly divided in two part. In the first part, for any FRV X, we define the Hukuhara set as the family of (deterministic) fuzzy sets C for which the Hukuhara difference X 96HC exists almost surely; in particular, we prove that such a family is a closed (with respect to different well known metrics) convex subset of the family of all fuzzy sets. In the second part, we prove that any square integrable FRV can be decomposed, up to a random translation, as the sum of a FRV Y and an element C\u2032 chosen uniquely (thanks to a minimization argument) in the Hukuhara set. This decomposition allows us to characterize all fuzzy random translation; in particular, a FRV is a fuzzy random translation if and only if its Aumann expectation equals C\u2032 (given by the above decomposition) up to a deterministic translation. Examples and open problems are also presented
Interacting Generalized Pólya Urn Systems
We consider a system of interacting Generalized P\'olya Urns (GPUs) having irreducible mean replacement matrices. The interaction is modeled through the probability to sample the colors from each urn, that is defined as convex combination of the urn proportions in the system. From the weights of these combinations we individuate subsystems of urns evolving with different behaviors. We provide a complete description of the asymptotic properties of urn proportions in each subsystem by establishing limiting proportions, convergence rates and Central Limit Theorems. The main proofs are based on a detailed eigenanalysis and stochastic approximation techniques
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