arXiv subjects

Marco Forti

Publications and source records attributed to Marco Forti.

10 recordsLinked to original sources

arXiv2025

Extending finite mixture models with skew-normal distributions and hidden Markov models for time series

We introduce an extension of finite mixture models by incorporating skew-normal distributions within a Hidden Markov Model framework. By assuming a constant transition probability matrix and allowing emission distributions to vary according to hidden states, the proposed model effectively captures dynamic dependencies between variables. Through the estimation of state-specific parameters, including location, scale, and skewness, the proposed model enables the detection of structural changes, such as shifts in the observed data distribution, while addressing challenges such as overfitting and computational inefficiencies inherent in Gaussian mixtures. Both simulation studies and real data analysis demonstrate the robustness and flexibility of the approach, highlighting its ability to accurately model asymmetric data and detect regime transitions. This methodological advancement broadens the applicability of a finite mixture of hidden Markov models across various fields, including demography, economics, finance, and environmental studies, offering a powerful tool for understanding complex temporal dynamics.

stat.ME↗

arXiv2022

A Euclidean comparison theory for the size of sets

We discuss two main ways in comparing and evaluating the size of sets: the "Cantorian" way, grounded on the so called Hume principle (two sets have equal size if they are equipotent), and the "Euclidean" way, maintaining Euclid's principle "the whole is greater than the part". The former being deeply investigated since the very birth of set theory, we concentrate here on the "Euclidean" notion of size (numerosity), that maintains the Cantorain defiitions of order, addition and multiplication, while preserving the natural idea that a set is (strictly) larger than its proper subsets. These numerosities satisfy the five Euclid's common notions, and constitute a semiring of nonstandarda natural numbers, thus enjoying the best arithmetic. Most relevant is the natural set theoretic definition} of the set-preordering: $$X\prec Y\ \ \Iff\ \ \exists Z\ X\simeq Z\subset Y$$ Extending this ``proper subset property" from countable to uncountable sets has been the main open question in this area from the beginning of the century.

math.LO↗

arXiv2022

Euclidean integers, Euclidean ultrafilters, and Euclidean numerosities

We introduce axiomatically the ring $\bf{Z}_\kappa$ of the Euclidean integers, that can be viewed as the ``integral part" of the field $\mathbb{E}$ of Euclidean numbers of [4], where the transfinite sum of ordinal indexed $\kappa$-sequences of integers is well defined. In particular any ordinal might be identified with the transfiite sum of its characteristic function, preserving the so called natural operations. The ordered ring $\bf{Z}_\kappa$ may be obtained as an ultrapower of $\mathbb{Z}$ modulo suitable ultrafilters, thus constituting a \it{ring of nonstandard integers.} Most relevant is the \it{algebraic} characterization of the ordering: a Euclidean integer is \it{positive} if and only if it is \it{the transfinite sum of natural numbers.} This property requires the use of special ultrafilters called Euclidean, here introduced to ths end. The ring $\bf{Z}_\kappa$ allows to assign a ``Euclidean" size (\it{numerosity}) to ``ordinal Punktmengen", i.e. sets of tuples of ordinals, as the transfinite sum of their characteristic functions: so every set becomes equinumerous to a set of ordinals, the Cantorian defiitions of \it{order, addition and multiplication} are maintained, while the Euclidean principle ``the whole is greater than the part" (\it{a set is (strictly) larger than its proper subsets}) is fulfilled.

math.LO↗

arXiv2020

The Euclidean numbers

We introduce axiomatically a Nonarchimedean field E, called the field of the Euclidean numbers, where a transfinite sum is defined that is indicized by ordinal numbers less than the first inaccessible Ω. Thanks to this sum, E becomes a saturated hyperreal field isomorphic to the so called Kiesler field of cardinality Ω, and suitable topologies can be put on E and on Ω \cup {Ω} so as to obtain the transfinite sums as limits of a suitable class of their finite subsums. Moreover there is a natural isomorphic embedding into E of the semiring Ω equipped by the natural sum and product. Finally a notion of numerosity satisfying all Euclidean common notions is given, whose values are nonnegative nonstandard integers of E. Then E can be charachterized as the hyperreal field generated by the real numbers and together with the semiring of numerosities (and this explains the name Euclidean numbers).

math.LO↗

arXiv2017

A topological interpretation of three Leibnizian principles within the functional extensions

Three philosophical principles are often quoted in connection with Leibniz: "objects sharing the same properties are the same object" (Identity of indiscernibles), "everything can possibly exist, unless it yields contradiction" (Possibility as consistency), and "the ideal elements correctly determine the real things" (Transfer). Here we give a precise logico-mathematical formulation of these principles within the framework of the Functional Extensions, mathematical structures that generalize at once compactifications, completions, and elementary extensions of models. In this context, the above Leibnizian principles appear as topological or algebraic properties, namely: a property of separation, a property of compactness, and a property of directeness, respectively. Abiding by this interpretation, we obtain the somehow surprising conclusion that these Leibnizian principles may be fulfilled in pairs, but not all three together.

math.LO↗

arXiv2012

Natural numerosities of sets of tuples

We consider a notion of "numerosity" for sets of tuples of natural numbers, that satisfies the five common notions of Euclid's Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a notion, we show that, contrasting to cardinal arithmetic, the natural "Cantorian" definitions of order relation and arithmetical operations provide a very good algebraic structure. In fact, numerosities can be taken as the non-negative part of a discretely ordered ring, namely the quotient of a formal power series ring modulo a suitable ("gauge") ideal. In particular, special numerosities, called "natural", can be identified with the semiring of hypernatural numbers of appropriate ultrapowers of N.

math.LO↗

arXiv2011

Quasi-selective ultrafilters and asymptotic numerosities

We isolate a new class of ultrafilters on N, called "quasi-selective" because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of quasi-selective ultrafilters is equivalent to the existence of "asymptotic numerosities" for all sets of tuples of natural numbers. Such numerosities are hypernatural numbers that generalize finite cardinalities to countable point sets. Most notably, they maintain the structure of ordered semiring, and, in a precise sense, they allow for a natural extension of asymptotic density to all sequences of tuples of natural numbers.

math.LO↗

arXiv2011

Quasi-selective and weakly Ramsey ultrafilters

Selective ultrafilters are characterized by many equivalent properties, in particular the Ramsey property that every finite colouring of unordered pairs of integers has a homogeneous set in U, and the equivalent property that every function is nondecreasing on some set in U. Natural weakenings of these properties led to the inequivalent notions of weakly Ramsey and of quasi-selective ultrafilter, introduced and studied in [1] and [4], respectively. U is weakly Ramsey if for every finite colouring of unordered pairs of integers there is a set in U whose pairs share only two colours, while U is f-quasi-selective if every function g < f is nondecreasing on some set in U. (So the quasi-selective ultrafilters of [4] are here id-quasi selective.) In this paper we consider the relations between various natural cuts of the ultrapowers of N modulo weakly Ramsey and f-quasi-selective ultrafilters. In particular we characterize those weakly Ramsey ultrafilters that are isomorphic to a quasi-selective ultrafilter.

math.LO↗

arXiv2011

A simple algebraic characterization of nonstandard extensions

We introduce the notion of "functional extension" of a set X, by means of two natural algebraic properties of the operator * on unary functions. We study the connections with ultrapowers of structures with universe X, and we give a simple characterization of those functional extensions that correspond to limit ultrapower extensions. In particular we obtain a purely algebraic proof of Keisler's characterization of nonstandard (= complete elementary) extensions.

math.LO↗

arXiv2010

Leibniz's Principles and Topological Extensions

Three philosophical principles are often quoted in connection with Leibniz: "objects sharing the same properties are the same object", "everything can possibly exist, unless it yields contradiction", "the ideal elements correctly determine the real things". Here we give a precise formulation of these principles within the framework of the Topological Extensions of [8], structures that generalize at once compactifications, completions, and nonstandard extensions. In this topological context, the above Leibniz's principles appear as a property of separation, a property of compactness, and a property of analyticity, respectively. Abiding by this interpretation, we obtain the somehow surprising conclusion that these Leibnz's principles can be fulfilled in pairs, but not all three together.

math.GN↗