Binary hidden Markov models and varieties

The technological applications of hidden Markov models have been extremely diverse and successful, including natural language processing, gesture recognition, gene sequencing, and Kalman filtering of physical measurements. HMMs are highly non-linear statistical models, and just as linear models are amenable to linear algebraic techniques, non-linear models are amenable to commutative algebra and algebraic geometry. This paper closely examines HMMs in which all the hidden random variables are binary. Its main contributions are (1) a birational parametrization for every such HMM, with an explicit inverse for recovering the hidden parameters in terms of observables, (2) a semialgebraic model membership test for every such HMM, and (3) minimal defining equations for the 4-node fully binary model, comprising 21 quadrics and 29 cubics, which were computed using Grobner bases in the cumulant coordinates of Sturmfels and Zwiernik. The new model parameters in (1) are rationally identifiable in the sense of Sullivant, Garcia-Puente, and Spielvogel, and each model's Zariski closure is therefore a rational projective variety of dimension 5. Grobner basis computations for the model and its graph are found to be considerably faster using these parameters. In the case of two hidden states, item (2) supersedes a previous algorithm of Schonhuth which is only generically defined, and the defining equations (3) yield new invariants for HMMs of all lengths $\geq 4$. Such invariants have been used successfully in model selection problems in phylogenetics, and one can hope for similar applications in the case of HMMs

Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence

We provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which we recently found evidence that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1+1 dimensions [Phys. Rev. Lett. 104, 230601 (2010); Sci. Rep. 1, 34 (2011)]. Here we investigate both circular and flat interfaces and report their statistics in detail. First we demonstrate that their fluctuations show not only the KPZ scaling exponents but beyond: they asymptotically share even the precise forms of the distribution function and the spatial correlation function in common with solvable models of the KPZ class, demonstrating also an intimate relation to random matrix theory. We then determine other statistical properties for which no exact theoretical predictions were made, in particular the temporal correlation function and the persistence probabilities. Experimental results on finite-time effects and extreme-value statistics are also presented. Throughout the paper, emphasis is put on how the universal statistical properties depend on the global geometry of the interfaces, i.e., whether the interfaces are circular or flat. We thereby corroborate the powerful yet geometry-dependent universality of the KPZ class, which governs growing interfaces driven out of equilibrium.Comment: 31 pages, 21 figures, 1 table; references updated (v2,v3); Fig.19 updated & minor changes in text (v3); final version (v4); J. Stat. Phys. Online First (2012

Blind identification of bilinear systems

Journal ArticleAbstract-This paper is concerned with the blind identification of a class of bilinear systems excited by non-Gaussian higher order white noise. The matrix of coefficients of mixed input-output terms of the bilinear system model is assumed to be triangular in this work. Under the additional assumption that the system output is corrupted by Gaussian measurement noise, we derive an exact parameter estimation procedure based on the output cumulants of orders up to four. Results of the simulation experiments presented in the paper demonstrate the validity and usefulness of our approach

New Trends in Statistical Physics of Complex Systems

A topical research activity in statistical physics concerns the study of complex and disordered systems. Generally, these systems are characterized by an elevated level of interconnection and interaction between the parts so that they give rise to a rich structure in the phase space that self-organizes under the control of internal non-linear dynamics. These emergent collective dynamics confer new behaviours to the whole system that are no longer the direct consequence of the properties of the single parts, but rather characterize the whole system as a new entity with its own features, giving rise to the birth of new phenomenologies. As is highlighted in this collection of papers, the methodologies of statistical physics have become very promising in understanding these new phenomena. This volume groups together 12 research works showing the use of typical tools developed within the framework of statistical mechanics, in non-linear kinetic and information geometry, to investigate emerging features in complex physical and physical-like systems. A topical research activity in statistical physics concerns the study of complex and disordered systems. Generally, these systems are characterized by an elevated level of interconnection and interaction between the parts so that they give rise to a rich structure in the phase space that self-organizes under the control of internal non-linear dynamics. These emergent collective dynamics confer new behaviours to the whole system that are no longer the direct consequence of the properties of the single parts, but rather characterize the whole system as a new entity with its own features, giving rise to the birth of new phenomenologies. As is highlighted in this collection of papers, the methodologies of statistical physics have become very promising in understanding these new phenomena. This volume groups together 12 research works showing the use of typical tools developed within the framework of statistical mechanics, in non-linear kinetic and information geometry, to investigate emerging features in complex physical and physical-like systems

VI Workshop on Computational Data Analysis and Numerical Methods: Book of Abstracts

The VI Workshop on Computational Data Analysis and Numerical Methods (WCDANM) is going to be held on June 27-29, 2019, in the Department of Mathematics of the University of Beira Interior (UBI), Covilhã, Portugal and it is a unique opportunity to disseminate scientific research related to the areas of Mathematics in general, with particular relevance to the areas of Computational Data Analysis and Numerical Methods in theoretical and/or practical field, using new techniques, giving especial emphasis to applications in Medicine, Biology, Biotechnology, Engineering, Industry, Environmental Sciences, Finance, Insurance, Management and Administration. The meeting will provide a forum for discussion and debate of ideas with interest to the scientific community in general. With this meeting new scientific collaborations among colleagues, namely new collaborations in Masters and PhD projects are expected. The event is open to the entire scientific community (with or without communication/poster)

Thermodynamic Bethe ansatz for non-equilibrium steady states: exact energy current and fluctuations in integrable QFT

We evaluate the exact energy current and scaled cumulant generating function (related to the large-deviation function) in non-equilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of Bernard and Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures Tl, Tr, and waiting for a long time. We evaluate the current J(Tl, Tr) using the exact QFT density matrix describing these non-equilibrium steady states and using Zamolodchikov's method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the high-temperature limit. We analyze numerically our non-equilibrium steady-state TBA equations for three models: the sinh-Gordon model, the roaming trajectories model, and the sine-Gordon model at a particular reflectionless point. Based on the numerics, we conjecture that an infinite family of non-equilibrium c-functions, associated with the scaled cumulants, can be defined, which we interpret physically. We study the full scaled distribution function and find that it can be described by a set of independent Poisson processes. Finally, we show that the 'additivity' property of the current, which is known to hold in CFT and was proposed to hold more generally, does not hold in general IQFT—that is, J(Tl, Tr) is not of the form f(Tl) − f(Tr)

Independent component analysis for non-standard data structures

Independent component analysis is a classical multivariate tool used for estimating independent sources among collections of mixed signals. However, modern forms of data are typically too complex for the basic theory to adequately handle. In this thesis extensions of independent component analysis to three cases of non-standard data structures are developed: noisy multivariate data, tensor-valued data and multivariate functional data. In each case we define the corresponding independent component model along with the related assumptions and implications. The proposed estimators are mostly based on the use of kurtosis and its analogues for the considered structures, resulting into functionals of rather unified form, regardless of the type of the data. We prove the Fisher consistencies of the estimators and particular weight is given to their limiting distributions, using which comparisons between the methods are also made.Riippumattomien komponenttien analyysi on moniulotteisen tilastotieteen työkalu,jota käytetään estimoimaan riippumattomia lähdesignaaleja sekoitettujen signaalien joukosta. Modernit havaintoaineistot ovat kuitenkin tyypillisesti rakenteeltaan liian monimutkaisia, jotta niitä voitaisiin lähestyä alan perinteisillä menetelmillä. Tässä väitöskirjatyössä esitellään laajennukset riippumattomien komponenttien analyysin teoriasta kolmelle epästandardille aineiston muodolle: kohinaiselle moniulotteiselle datalle, tensoriarvoiselle datalle ja moniulotteiselle funktionaaliselle datalle. Kaikissa tapauksissa määriteläään vastaava riippumattomien komponenttien malli oletuksineen ja seurauksineen. Esitellyt estimaattorit pohjautuvat enimmäkseen huipukkuuden ja sen laajennuksien käyttöönottoon ja saatavat funktionaalit ovat analyyttisesti varsin yhtenäisen muotoisia riippumatta aineiston tyypistä. Kaikille estimaattoreille näytetään niiden Fisher-konsistenttisuus ja painotettuna on erityisesti estimaattoreiden rajajakaumat, jotka mahdollistavat teoreettiset vertailut eri menetelmien välillä