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We present here a second step in solving the Algebraic Identification Problem for the causal analytic functionals in the sense of Fliess. These functionals are symbolically represented by noncommutative formal power series G=∑ w∈ Z ★ 〈G | w〉w , where w is a word on a finite-encoding alphabet Z . The problem consists in computing the coefficients 〈G | w〉 from the choice of a finite set of informations on the input/output behaviour of the functional. In a previous work, we already presented a first step: we showed that one can compute the contributions of G relative to a family of noncommutative polynomials gμ with integer coefficients, indexed by the set of partitions. Hence it remains to inverse these relations by computing the words w as linear combinations of the gμ. An answer could be found in two ways: firstly by providing an identification computation tool, secondly by solving the ‘Identifiability Problem’: is the previous identification effectively computable at any order? A computational tool is here presented, in the form of a concise Maple package IDENTALG that computes the polynomials gμ by a block recursive matrix implementation, and allows then to test the identification (when possible) at any order by matrix inversion. It requires a combinatorial study of the differential monomials on the inputs. The computation of a test set covering the identification of 2048 words is presented. This package is given in the widely significant case of functionals depending on ‘a single input with drift part’. It can be used without change in case of ‘two inputs without drift’. It could be extended very easily to the case of ‘several inputs with drift part’. Finally, we discuss the Identifiability Problem: we summarize the current state of our results, and we conclude with a conjecture in a weak form and in a strong form.

This paper presents the first step towards solving the problem of the exact algebraic identification of causal functionals. This problem consists in computing the coefficients of a noncommutative generating series when only the Taylor expansion of some inputs (at t =0), and the Taylor expansion (at t =0) of associated outputs are known.

International audience; A lot of computing tools are often used for analyzing DNA se- quences like trees, automata, dictionaries, every one being re- served for a particular problem. A. Blumer and al. have proposed a more general computing tool : the smaller automaton recogniz- ing the subwords of a text (DAWG). In this paper we propose the concept of "weighted factors au- tomaton" producing every occurrence of any factor. Its transi- tions are labeled by the read letter and weighted by the set of the indices of the factors beginnings. The factors are obtained by concatenating the read letters and the indices of the factors begin- nings are obtained by computing the intersection of the weight- ing sets, when advancing from the initial state to a final state. We think that this automaton can be more easily processed than DAWG and we present a comparison between DAWG and our automaton: the set of the factors beginnings indices and the fac- tors frequency are more easily obtained by our automaton and the restriction of our automaton to the factors of length k maintains the automaton structure, when DAWG cannot be eas- ily restricted. The applications are numerous: By selecting factors of length 1 , we obtain the coding regions, factors of length 3 , we obtain the expression level of some gene. The "weighted factors au- tomaton" allows us to find matches of pattern, to study homol- ogy, FASTA and BLAST algorithms being significantly simplified.

The Faa di Bruno formula enables us to compute the derivatives of a function of several variables. We show here, in spite of the noncommutativity of causal computations, that the Faa di Bruno grammar makes it possible to compute the derivatives of any causal functional. Using this property and some computations on derivatives of causal functionals, this paper presents the first step towards solving the problem of “Exact Algebraic Identification”. This problem consists in computing the coefficients of a noncommutative generating series when only the Taylor expansion of some inputs (at t = 0), and the Taylor expansion (at t = 0) of associated outputs are known.

We provide the drawing of the output of dynamical system (Σ), particularly when the output is rough or near instability points. (Σ) being analytical in a neighborhood of the initial state q(0) and described by its state equations, its output y(t) in a neighborhood of t=0 is obtained by "evaluating" its generating series. Our algorithm consists in juxtaposing local approximating outputs on successive time intervals [ti,ti+1]0≤i≤n−1, to draw y(t) everywhere as far as possible. At every point ti+1 we calculate at order k an approximated value of each component qr of the state; on every interval [ti,ti+1]0≤i≤n−1 we calculate an approximated output. These computings are obtained from the symbolic expressions of the generating series of qr and y, truncated at order k, specified for t=ti and "evaluated". A Maple package is built, providing a suitable result for oscillating outputs or near instability points when a Runge-Kutta method is wrong.

A dynamical system being described by its state equations and its initial state, we develop a method for drawing its output: It is based on the juxtaposition of local approximating outputs on successive time intervals $[t_i, t_(i+1)]0

National audience; Dans de précédents travaux, nous avons fourni un modèle du comportement "débit d'insuline/glycémie" du patient diabétique et une régulation de sa glycémie. Ce modèle comportemental est un modèle bilinéaire. Son acquisition consiste à identifier de 3 à 7 paramètres grâce aux données corrélées "débit de perfusion insulinique/glycémie" dont on dispose. Sur les tests initiaux, ce modèle de précision quadratique, présente en moyenne une erreur de 15% sur un intervalle de 15 minutes. Le problème posé est de savoir si ce modèle permet non seulement la prédiction sur les 15 minutes suivantes mais permet en plus de prévoir que le patient entre dans une période d'équilibre stable ou instable de sa glycémie. Il serait alors possible, à l'arrivée d'une perturbation (repas, activité physique, stress) de piloter au plus près les variations de distribution d'insuline, par une adaptation automatique du modèle de correction à la situation nouvelle détectée. En particulier, si l'équilibre est stable, la prédiction sera en principe valable sur un intervalle de temps plus long alors que si l'équilibre est instable, il y aura lieu de diminuer l'intervalle de temps. Dans ce papier, nous présentons un modèle flou de Tagaki-Sugeno (TS). Ce modèle consiste en une famille de modèles linéaires fusionnés grâce à des fonctions d'appartenance non-linéaires. Nous appliquons cette méthode au problème de traitement des diabétiques. Nous prenons le débit d'insuline comme une entrée et le taux de sucre dans le sang comme la sortie et nous considérons le patient comme une boîte noire [11],[12] et nous construisons le modèle à partir de mesures disponibles.

Recently, many papers have proposed automatic techniques for • extraction of knowledge from numerical data [18], [19], [20], [1]. • minimization of the number of rules [2], [3], [4]. But few works have been developed for design of experiments and datum plane covering. Most of optimization methods make the assumption that datum plane is sufficiently covered. If this assumption no longer holds, we will see that these methods may not work, since it implies that, before optimization, the fuzzy system gives acceptable results. We present in this paper, an algorithm for selection of fuzzy rules based on datum plane covering. We apply this method to two applications : • the problem of treatment of diabetics. Taking the insulin infusion rate as the input and the blood glucose rate as the output, we consider the patient as a black box [23], [22], whose model has to be obtained from the available measures of inputs-outputs. We dispose of a glycaemia file automatically produced for every person, and an insulin file shared by several persons. • the problem of detecting fraud in electronic payment systems. Along with the development of credit cards, fraudulent activities become a major problem in electronic payment systems [5], [6], [7]. Our model is based on specific and usual customer behavior, and deviation from such patterns is suspect.

International audience; The purpose of this paper is to apply combinatorial techniques for computing coefficients of rational formal series (Gk) in two noncommuting variables and their differences at orders k and k-1. This in turn may help one to study the reliability and the quality of a model for non-linear black-box identification. We investigate the quality of the model throughout the criteria of a measure of convergence. We provide, by a symbolic computation, a valuation relating to the convergence of the family (Bk). This computation is a sum of differ- ential monomials in the input functions and behavior system. We identify each differential monomial with its colored multiplicity and analyse our computation in the light of the free differential calcu- lus. We propose also a combinatorial interpretation of coefficients of (Gk). These coefficients are powers of an operator Θ which is in the monoid generated by two linear differential operators ∆ and Γ. More than a symbolic validation, these computing tools are param- eterized by the input and the system's behavior.

In previous papers, we provided a modeling of the behavior "insulin delivery/glycaemia" of the diabetic patient under continuous insulin infusion, continuous glucose monitoring and we provided a method of regulation of his glycaemia. This behavioral model is bilinear, predicting the behavior on an interval of 15 minutes, with an average error of 15%. And consequently, the model is adjusted for every 15 minutes. The aim of this paper is to study the Bounded-Input-Bounded-Output (BIBO) stability of the bilinear model in order to point out that the patient is entering in a period of stable/unstable equilibrium. In case of stable equilibrium, the prediction will be valid for a longer time interval, when in case of unstable equilibrium, it will leads one to reduce the time intervals and to pilot closely the variations of insulin delivery. The BIBO stability is studied by computing the generating series G of the model. This series, generalization of the transfer function, presents an useful tool for analyzing the stability of bilinear systems. It is a rational power series in noncommutative variables and by evaluating it, a formal expression of the output in form of iterated integrals is provided. Three cases arise: firstly, the output can be explicitly computed; secondly, the output can be bounded/unbounded if the input is bounded; thirdly, no conclusion seems available about the BIBO stability by using G. In this case, we propose a stabilizing constant input eta by studying the univariate series Geta.

Etant donnee une serie formelle en variables non commutatives sur un alphabet fini X, a coefficients dans un corps K, on propose une construction de serie rationnelle approximante au sens de la valuation X-adique, et qui soit de rang minimal

Given an analytic system, we compute a bilinear system of minimal dimension which approximates it up to order k (i.e. the outputs of these two systems have the same Taylor expansion up to order k).