on the choice of parameterization for identification the effect on an identification procedure of the choice of system parameterization is considered first a formal framework is presented in which to assess this effect and some previous results are reviewed in this framework then the class of methods that minimize an input output objective function are studied sufficient conditions on identifiable parameterizations of linear models are stated which render the associated model estimators equally accurate in asymptotic statistical terms these are shown to apply to parameterizations with maximal number of free parameters

from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

new developments for matrix fraction descriptions this article aims at giving a new answer for the challenging problem of the parametrisation of multi input multi output matrix fraction descriptions in order to reach this goal new parametrisations of matrix fraction descriptions called fully parametrised left matrix fraction descriptions f lmfd are first introduced their structural properties as well as their suitability for multi input multi output model description are more precisely analysed as any over parametrised model description the f lmfd cannot describe a transfer function uniquely the structure of the space of equivalent f lmfd is then investigated through the determination of its basis the study carried out in this article is the prelude to a computational improvement of the identification of matrix fraction descriptions with gradient based optimisation methods

the stochastic heat equation feynman kac formula and intermittence we study in one space dimension the heat equation with a random potential that is a white noise in space and time this equation is a linearized model for the evolution of a scalar field in a space time dependent random medium it has also been related to the distribution of two dimensional directed polymers in a random environment to the kpz model of growing interfaces and to the burgers equation with conservative noise we show how the solution can be expressed via a generalized feynman kac formula we then investigate the statistical properties the two point correlation function is explicitly computed and the intermittence of the solution is proven this analysis is carried out showing how the statistical moments can be expressed through local times of independent brownian motions

vector arma estimation a reliable subspace approach a parameter estimation method for finite dimensional multivariate linear stochastic systems which is guaranteed to produce valid models approximating the true underlying system in a computational time of a polynomial order in the system dimension is presented this is achieved by combining the main features of certain stochastic subspace identification techniques with sound matrix schur restabilizing procedures and multivariate covariance fitting both of which are formulated as linear matrix inequality problems all aspects of the identification method are discussed with an emphasis on the two issues mentioned above and examples of the overall performance are provided for two different systems

estimating the number of sinusoids in additive white noise the test proposed uses the eigenvector decomposition of the estimated autocorrelation matrix and is based on matrix perturbation analysis the estimator is shown to be able to resolve closely spaced sinusoids at lower signal to noise ratios than heuristic tests simulation results for two closely spaced sinusoids are detailed several unanswered questions are discussed

arma canonical forms obtained from constructibility invariants in recent approaches multivariable arma models have been derived from observable canonical mfds but it was realized that these models have a number of serious limitations and disadvantages this paper presents a new approach based on the idea of observing the state from past outputs which leads to the construction of monic arma models defined by the constructibility invariants the derivation and structural properties of these constructibility forms are investigated

consistency and asymptotic normality of some subspace algorithms for systems without observed inputs asymptotic normality for a class of subspace algorithms which estimate the state in a first step is derived expressions for the asymptotic variance are given linear systems with unobserved white noise inputs are considered a class of subspace estimates for the system matrices obtained by estimating the state in the first step is analyzed the main result presented here states asymptotic normality of subspace estimates in addition a consistency result for the system matrix estimates is given an algorithm to compute the asymptotic variances of the estimates is presented in a final section the implications of the result are discussed

the frisch scheme in dynamic system identification abstract the use of the frisch scheme in the identification of linear dynamic systems is investigated in order to describe the whole family of models that can explain given input output noisy sequences unlike the algebraic case it is shown that in general only a single model is compatible with the data these results are first proposed for single input single output systems and then generalized to the multivariable case

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

frequency domain subspace based identification of discrete time power spectra from nonuniformly spaced measurements in this paper we present a new subspace based algorithm for the identification of multi input multi output square discrete time linear time invariant systems from nonuniformly spaced power spectrum measurements the algorithm is strongly consistent and it is illustrated with one practical example that solves a stochastic road modeling problem

model structure selection for multivariable systems by cross validation methods using cross validation ideas two procedures are proposed for making a choice between different model structures used for approximate modelling of multivariable systems the procedures are derived under fairly general conditions the true system does not need to be contained in the model set model structures do not need to be nested and different criteria may be used for model estimation and validation the proposed structure selection rules are shown to be invariant to parameter scaling under certain conditions essentially requiring that the system belongs to the model set and that the maximum likelihood method is used for parameter estimation they are shown to be asymptotically equivalent to the generalized akaike structure selection criteria

structure and order estimation of multivariable stochastic processes an easy to implement numerically efficient algorithm which estimates the kronecker invariants is presented a procedure allowing estimation of the structure of a state space representation for a multivariable stationary stochastic process from measured output data is presented it is assumed that the observed vector time series is a realization of a process with rational spectrum or the output of a stable time invariant linear system driven by white noise an algorithm is proposed which selects a maximal set of linearly independent rows of the hankel matrix built upon the estimated covariance sequence and thus yields estimates of the kronecker invariants when applied to simulated examples it systematically yielded the good structure without any ambiguity i e with a surprising robustness with respect to the choice of the probability of false alarm the numerical efficiency of the procedure is remarkable and no exhaustive search over the set of all possible kronecker indexes has to be performed

brief minimal representations of mimo time varying systems and realization of cyclostationary models this paper describes a family of minimally parameterized state space models for completely n step observable multi input multi output time varying systems and their algebraical links with equivalent input output models these results are then used to develop an algorithm for the realization of generic input output sequences generated by a periodic system a complete numerical example is also developed to illustrate all steps of the algorithm described in the paper

strong consistency of ml estimators using partial fraction and elementary subsystem representation of multivariable systems the strong consistency of ml estimators for partial fraction matrix pfm and elementary subsystem ess representations of vector arma processes will be discussed for the case when the structure is overestimated the main results of the paper are the off line strong consistency theorems of ml estimators for pfm representation when the number of poles is overestimated and for the ess representations when the number of ess s associated with the poles is overestimated

estimation and identification for 2 d block kalman filtering the development of a recursive identification and estimation procedure for two dimensional block kalman filtering is discussed the recursive identification scheme can be used online to update the image model parameters at each iteration based on the local statistics within a block of the observed noisy image the covariance matrix of the driving noise can also be estimated at each iteration of this algorithm a recursive procedure for computing the parameters of the higher order models is given simulation results are also provided

spatial asymptotics for the parabolic anderson models with generalized time space gaussian noise partially motivated by the recent papers of conus joseph and khoshnevisan ann probab 41 2013 2225 2260 and conus et al probab theory related fields 156 2013 483 533 this work is concerned with the precise spatial asymptotic behavior for the parabolic anderson equation u t t x 12 u t x v t x u t x u 0 x u0 x where the homogeneous generalized gaussian noise v t x is among other forms white or fractional white in time and space associated with the cole hopf solution to the kpz equation in particular the precise asymptotic form limr logr 2 3logmax x ru t x 342t3 3a s is obtained for the parabolic anderson model tu 12 2xxu w u with the 1 1 white noise w t x in addition some links between time and space asymptotics for the parabolic anderson equation are also pursued

model structure selection by cross validation two criteria for choosing between different model structures are proposed their derivation is within a natural cross validatory assessment context and is fairly assumption free in particular the two criteria can be used for discriminating between non nested model structures and more importantly the true system is not required to belong to the considered set of models should the true system belong to the model set the two proposed criteria will asymptotically reduce to some well known structure selection criteria this is believed to be a desirable feature of our proposals on the other hand it provides a nice cross validation interpretation of some well known model structure selection rules also the cross validation interpretation helps one to choose which criteria to use in a given application the paper also has a second purpose somewhat decoupled from that mentioned above it contains an extensive survey of the literature that is useful in its own right

paper uniquely identifiable state space and arma parametrizations for multivariable linear systems multivariable systems can be represented in a uniquely identifiable way either by canonical forms or by so called overlapping forms the advantage of the latter is that they do not require the a priori estimation of a set of structural invariants e g kronecker invariants we show here how to define uniquely identifiable overlapping parametrizations for state space and arma models we show that these parametrizations are all related to a set of intrinsic invariants which are obtained from the markov parameters of the system different forms of overlapping arma parametrizations are derived and their properties discussed

the determination of optimum structures for the state space representation of multivariate stochastic processes when identifying a model for a multivariate stationary stochastic process an important problem is that of determining the structure of the state variable model several overlapping parameterizations can usually be fitted to a given process and the question arises as to which structure leads to the most accurate parameter estimates the accuracy of parameter estimates is often measured by the determinant of the fisher information matrix we show that all admissible structures will give asymptotically the same value to this criterion for finite data some structures may still be better than others and two heuristic structure estimation methods are analyzed some simulation results are also presented

representations of jointly stationary stochastic feedback processes abstract stable constant linear closed loop systems relating an input vector u to an output vector u and vice versa produce a jointly stationary y u procoss on the other hand it is often natural to split up a stationary vector random process z into component vectors yand u and to examine the closed loop relations between y and u this paper presents a number of new results on the spectral factorization and the closed loop representation of a jointly stationary vector y u proccss conditions are derived on the closed loop models for the joint process model to bo oe minimal degree stable and minimum phase relations between different joint process models producing the same spectrum yu z are established

brief identification of dynamic errors in variables models approaches based on two dimensional arma modeling of the data in this paper we propose a parametric and a non parametric identification algorithm for dynamic errors in variables model we show that the two dimensional process composed of the input output data admits a finite order arma representation the non parametric method uses the arma structure to compute a consistent estimate of the joint spectrum of the input and the output a frisch scheme is then employed to extract an estimate of the joint spectrum of the noise free input output data which in turn is used to estimate the transfer function of the system the parametric method exploits the arma structure to give estimates of the system parameters the performances of the algorithms are illustrated using the results obtained from a numerical simulation study

on harmonic renewal measures let be a probability and nu h sum limits n 1 infty frac 1 n mu n the corresponding harmonic renewal measure complementing earlier results where is concentrated on a halfline we investigate the behaviour ofvh x x 1 and the harmonic renewal functiong x vh x asx ifm1 x dx 0 we also consider the casem1 0

on diffeomorphisms between classes of linear systems abstract in this paper we show that the following classes of linear dynamical systems are diffeomorphic to each other fixed order asymptotically stable strictly bounded real strictly positive real stable and minimum phase and invertible these results are then used to prove the diffeomorphisms between the state bundles associated with these classes of systems and between their associated principal fibre bundles we shall conclude this paper by discussing how these results can be used to deduce geometric properties of other classes of linear systems and to construct overlapping parametrizations

data driven local coordinates for multivariable linear systems and their application to system identification in this paper we introduce a new parametrization for state space systems data driven local coordinates ddlc the parametrization is obtained by restricting the full state space parametrization where all matrix entries are considered to be free to an affine plane containing a given nominal state space realization this affine plane is chosen to be perpendicular to the tangent space to the manifold of observationally equivalent state space systems at the nominal realization the application of the parametrization to prediction error identification is exemplified simulations indicate that the proposed parametrization has numerical advantages as compared to e g the more commonly used observable canonical form

paper pseudo canonical forms identifiable parametrizations and simple parameter estimation for linear multivariable systems parameter estimation in this paper the estimation of parameters for the pseudo canonical input output models introduced in correa g o and glover k automatica 20 429 442 1984 is considered modifications of three equation error type estimators linear least squares compensated and instrumental variable estimator are derived which incorporate the constraint on parameter values associated with these models the consistency of the proposed estimators is analysed finally results are presented on the construction of parametrization independent instrumental variables

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

positive output state maps and quasi positive realization of mimo discrete systems the positive realization of externally positive systems described by their transfer matrix g z is a complex problem whose analysis has been completed only recently for the siso case on the basis of algebraic and geometric approaches that underline the many constraints that condition its solution these constraints concern the minimal order of the obtained models the minimality of their parameterizations and even their existence this paper considers the new category of quasi positive state space models introduced in guidorzi 2014 that limit the assumptions on the nonnegativeness of the state space trajectories to the only trajectories that can be actually generated by the system under nonnegative controls it is shown that all externally positive systems admit a quasi positive minimally parameterized state space realization whose existence is not conditioned by restrictions on the signs of the parameters of the polynomials in g z

software for weighted structured low rank approximation a software package is presented that computes locally optimal solutions to low rank approximation problems with the following features mosaic hankel structure constraint on the approximating matrix weighted 2 norm approximation criterion fixed elements in the approximating matrix missing elements in the data matrix and linear constraints on an approximating matrix s left kernel basis it implements a variable projection type algorithm and allows the user to choose standard local optimization methods for the solution of the parameter optimization problem for an mxn data matrix with n m the computational complexity of the cost function and derivative evaluation is o m 2n the package is suitable for applications with n m in statistical estimation and data modeling the main application areas of the package n m corresponds to modeling of large amount of data by a low complexity model performance results on benchmark system identification problems from the database daisy and approximate common divisor problems are presented

identifiability and persistent excitation in full matrix fraction parameter estimation this paper analyzes the identifiability of the full matrix fraction description of multivariable systems and provides a lower bound on the persistence of excitation of the input output pair in terms of the persistence of excitation of the input

on line structure selection for multivariable state space models an algorithm is described for the selection of model structure for identifying state space models of black box character the algorithm receives as input a given system in a given parametrization it is then tested whether this parametrization is suitable well conditioned for identification purposes if not a better one is selected and the transformation of the system to the new representation is performed this algorithm can be used as a block both in an iterative off line identification procedure and for recursive on line identification it can be called whenever there is some indication that the model structure is ill conditioned it is discussed how the model structure selection algorithm can be interfaced with an off line identification procedure a complete procedure is described and tested on real and simulated data

a macroscopic multifractal analysis of parabolic stochastic pdes it is generally argued that the solution to a stochastic pde with multiplicative noise such as dot u frac12 u u xi where xi denotes space time white noise routinely produces exceptionally large peaks that are macroscopically multifractal see for example gibbon and doering arch ration mech anal 177 115 150 2005 gibbon and titi proc r soc a 461 3089 3097 2005 and zimmermann et al phys rev lett 85 17 3612 3615 2000 a few years ago we proved that the spatial peaks of the solution to the mentioned stochastic pde indeed form a random multifractal in the macroscopic sense of barlow and taylor j phys a 22 13 2621 2626 1989 proc lond math soc 3 64 125 152 1992 the main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio temporal peaks of the solution form infinitely many different multifractals on infinitely many different scales which we sometimes refer to as stretch factors a simpler though still complex such structure is shown to also exist for the constant coefficient version of the said stochastic pde

paper pseudo canonical forms identifiable parametrizations and simple parameter estimation for linear multivariable systems input output models this paper examines the use of pseudo canonical forms as opposed to canonical forms in the identification of linear discrete time multivariable systems firstly a relationship is established between state space models in a pseudo canonical form and a set of corresponding matrix fraction descriptions or equivalently finite difference equations the causality of the systems implies that there will be dependencies amongst the parameters in the matrix fraction description in certain circumstances by introducing an ordering of the output variables and associated parameters this dependency can be handled sequentially in simple parameter estimation algorithms

a generalised instrumental variable estimator for multivariable errors in variables identification problems a new and very general estimation method the generalised instrumental variable estimator give has been introduced to deal with the errors in variables identification the give contains several well known estimation methods as special cases that can be obtained by various user choices in give these methods include the bias eliminating least squares method a number of different versions of the frisch scheme and the extended compensated least squares method this article shows how previous analysis for single input single output systems can be extended in a non trivial way to also cope with multi input multi output systems various computational aspects of give are presented and it is shown how the parameter estimates can be computed in an efficient way the asymptotic distribution of the parameter estimates is also derived and analysed

indirect adaptive pole placement control of mimo stochastic systems self tuning results in this paper we consider indirect adaptive pole placement control appc of linear multivariable stochastic systems instead of the canonical representation often used in the literature we propose using a non minimal but otherwise uniquely identifiable pseudo canonical parameterization that is more suitable for multivariable armax model identification to identify the plant we use the weighted extended least squares wels algorithm a least squares method with slowly decreasing weights which was introduced in bercu 1995 the pole placement controller parameters are then calculated by using a certain perturbation of the parameter estimates such that the linear models corresponding to the perturbed estimates are uniformly controllable and observable we prove that with a reasonable amount of prior information the resulting appc scheme is globally stabilizing and asymptotically self tuning regardless of the degree of persistency of external excitation these results represent the most complete study of stochastic multivariable appc systems to this date

new method of order estimation for arma armax processes let the observation y k be generated by the multivariate arma process a z y k c z w k with unknown coefficients theta a theta c and orders p r where w k is a sequence of independent and identically distributed i i d random vectors with zero mean and unknown covariance matrix r w 0 a new method for estimating the orders p r is introduced in contrast to most of the existing results the new method is not based on optimizing a certain criterion and the order estimates given in the paper are rather easy to update computationally in comparison with the criterion optimization based methods when new data arrive the method is then extended to determining the orders of armax processes under reasonable conditions the estimates are proved to converge to the true orders with probability one as time tends to infinity

a strong renewal theorem for generalized renewal functions in the infinite mean case letf x be a nonarithmetic c d f on 0 such that 1 f x x l x wherel x is slowly varying and 0 1 leta x be regularly varying with exponent 1 a strong renewal theorem of blackwell type for generalized renewal functions of the form g t equiv sum limits n 0 infty a n f n t is proved here thus extending the recent work of embrechts maejima and omey 1 and that of erickson 4

statistics of a passive scalar advected by a large scale two dimensional velocity field analytic solution steady statistics of a passive scalar advected by a random two dimensional flow of an incompressible fluid is described in the range of scales between the correlation length of the flow and the diffusion scale this corresponds to the so called batchelor regime where the velocity is replaced by its large scale gradient the probability distribution of the scalar in the locally comoving reference frame is expressed via the probability distribution of the line stretching rate the description of line stretching can be reduced to a classical problem of the product of many random matrices with a unit determinant we have found the change of variables that allows one to map the matrix problem onto a scalar one and to thereby prove the central limit theorem for the stretching rate statistics the proof is valid for any finite correlation time of the velocity field whatever the statistics of the velocity field the statistics of the passive scalar averaged over time locally in space is shown to approach gaussian statistics with increase in the p eclet number pe the pumping to diffusion scale ratio the first nsimultaneous correlation functions are expressed via the flux of the square of the scalar and only one factor depending on the velocity field the mean stretching rate which can be calculated analytically in limiting cases non gaussian tails of the probability distributions at finite pe are found to be exponential

on admissibility of pseudoobservability and pseudocontrollability indexes it has been shown recently that a given mimo system may be represented by several equivalent state space representations referred to as overlapping or pseudocanonical forms instead of the unique observability controllability canonical form the structure of a pseudocanonical form is determined by a set of pseudoobservability controllability indexes analogous to the observability controllability indexes determining the structure of the canonical form a mimo system can be represented by several different pseudocanonical forms specified by their structure indexes in this note we give an upper bound on the admissible structure indexes of a given system in terms of a new set of invariants of this system an illustrative example is included

the range error test in the structural identification of linear multivariable systems among the approaches that have been proposed in recent years for the identification of discrete time linear multivariable systems very efficient procedures are based on the introduction of a set of input output canonical models and on the separate determination of the process structure structural identification and parameters parametric identification this paper introduces a new simple test which allows us to perform the structural identification without the drawbacks usually associated with singularity tests the proposed test gives an a priori prediction of the model performance in order to avoid trial and error procedures in the determination of a suitable structure for the process to be identified

exact modelling and identifiability of linear systems in this paper we consider the question of identifiability of a system on the basis of observed data in particular we present a framework for exact modelling and identifiability of finite dimensional linear time invariant systems on the basis of time series of infinite or finite length systems are characterized in terms of their behaviour i e the set of time series which are allowed by the system in this behavioural approach no a priori distinction is made between input and output variables and also non controllable systems can be considered we describe identification procedures for exact modelling of multivariable time series of infinite or finite length these procedures are based on the concept of corroboration of models by data reflecting that the data should incorporate evidence for the acceptability of the identified models the assumed a priori information on the data generating system consists of the qualitative behavioural properties of linearity time invariance and completeness no knowledge is required concerning quantitative properties like the number of input and output variables the number of state variables or the observability indices a complete characterization is obtained of the minimal number of time series required for system identifiability the class of systems identifiable by one time series contains the controllable systems as a strict subclass

kpz line ensemble for each t ge 1 we construct an mathbb n indexed ensemble of random continuous curves with three properties 1 the lowest indexed curve is distributed as the time t hopf cole solution to the kardar parisi zhang kpz stochastic partial differential equation with narrow wedge initial data 2 the entire ensemble satisfies a resampling invariance which we call the mathbf h brownian gibbs property with mathbf h x e x 3 increments of the lowest indexed curve when centered by t 24 and scaled down vertically by t 1 3 and horizontally by t 2 3 remain uniformly absolutely continuous i e have tight radon nikodym derivatives with respect to brownian bridges as time t goes to infinity this construction uses as inputs the diffusion that o connell discovered ann probab 40 437 458 2012 in relation to the o connell yor semi discrete brownian polymer the convergence result of moreno flores et al in preparation of the lowest indexed curve of that diffusion to the solution of the kpz equation with narrow wedge initial data and the one point distribution formula proved by amir et al commun pure appl math 64 466 537 2011 for the solution of the kpz equation with narrow wedge initial data we provide four main applications of this construction 1 uniform as t goes to infinity brownian absolute continuity of the time t solution to the kpz equation with narrow wedge initial data even when scaled vertically by t 1 3 and horizontally by t 2 3 2 universality of the t 1 3 one point vertical fluctuation scale for the solution of the kpz equation with general initial data 3 concentration in the t 2 3 scale for the endpoint of the continuum directed random polymer 4 exponential upper and lower tail bounds for the solution at fixed time of the kpz equation with general initial data

invariants and canonical forms for systems structural and parametric identification the basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled with reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set then the set of polynomial input output models of the type p z y t q z u t and the equivalence relation induced by the premultiplication of p and q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced since the two sets of canonical forms share common sets of complete independent invariants very simple algebraical links between state space and input output canonical forms can be deduced the previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input output sequences generated by a linear discrete time invariant multivariable system the results obtained in the identification of a real process are then reported

mfd models and time delays some consequences for identification a general relationship is established between the coefficients of a general mfd model and its associated markov parameters this result is used to derive necessary and sufficient conditions for the appearance of time delays in general mfd models these conditions involve dependencies between some coefficients of the mfd models finally it is indicated how these dependencies can possibly be incorporated in equation error identification techniques retaining the favourable property of linearity in the parameters

intermittency for the wave and heat equations with fractional noise in time in this article we consider the stochastic wave and heat equations driven by a gaussian noise which is spatially homogeneous and behaves in time like a fractional brownian motion with hurst index h 1 2 the solutions of these equations are interpreted in the skorohod sense using malliavin calculus techniques we obtain an upper bound for the moments of order p 2 of the solution in the case of the wave equation we derive a feynman kac type formula for the second moment of the solution based on the points of a planar poisson process this is an extension of the formula given by dalang mueller and tribe trans amer math soc 360 2008 4681 4703 in the case h 1 2 and allows us to obtain a lower bound for the second moment of the solution these results suggest that the moments of the solution grow much faster in the case of the fractional noise in time than in the case of the white noise in time

model sets and parametrizations for identification of multivariable equation error models abstract equation error or linear regression models are known to inherently require the a priori choice for specific signal variables to be considered as regressand and or regressor this implies that a model set should be a priori restricted in some way in order to define an acceptable identification problem in the case of approximate identification i e the system to be modelled is not contained in the model set this restriction acts as a design variable with the identified models being dependent on its specific choice in this paper the necessity of this restriction is quantified by the property of discriminability i e the ability of an identification criterion to distinguish between all the different models in a model set employing a deterministic signal oriented framework several sets of sufficient conditions are derived for model sets to be discriminable by a least squares identification criterion to this end use is made of polynomial model representations in two shift operators although it is of a different nature the problem discussed is shown to be closely related to the problem of constructing identifiable parametrizations for sets of rational transfer functions it is shown that the pseudo canonical or overlapping parametrization of all transfer functions with fixed mcmillan degree constitutes a nonoverlapping set of equation error models that is discriminable by a least squares identification criterion

a complete classification of minimal realizations of nonsingular and strictly proper transfers under similarity transformations we consider minimal realizations of square nonsingular and strictly proper transfer matrices equivalence classes of all such realizations under similarity transformations are characterized via a canonical form

arma order estimation via matrix perturbation theory using matrix perturbation theory we obtain the statistical distribution of the smallest eigenvalue of the hankel matrix built upon the estimated covariances under the hypothesis that the corresponding exact hankel matrix possesses one single zero eigenvalue this allows us to develop and justify a new order determination scheme whose performance appears similar to those based on likelihood maximization but with a much lower computational complexity

subspace based methods for the identification of linear time invariant systems subspace based methods for system identification have attracted much attention during the past few years this interest is due to the ability of providing accurate state space models for multivariable linear systems directly from input output data the methods have their origin in classical state space realization theory as developed in the 1960s the main computational tools are the qr and the singular value decompositions here an overview of existing subspace based techniques for system identification is given the methods are grouped into the classes of realization based and direct techniques similarities between different algorithms are pointed out and their applicability is commented upon we also discuss some recent ideas for improving and extending the methods a simulation example is included for comparing different algorithms the subspace based approach is found to perform competitive with respect to prediction error methods provided the system is properly excited

identification of invariants of over parameterized models finite sample results this paper shows that a certain class of estimates of invariants of over parameterized models are with probability one independent of the particular over parameterization chosen the result is nonasymptotic and the estimators considered minimize a cost function that is invariant with respect to the same parameter transformation as the overparameterized model

parameter and structure identification of linear multivariable systems abstract in this paper a parameter and structure identification scheme which can be used for both continuous and discrete time systems is presented this scheme has an on line nature and it incorporates standard techniques such as state variable filters and adaptation laws developed through stability analysis the canonical form being used is based on the output injection which is constructed to assign all the poles of the system to an arbitrary chosen real number it is therefore named the canonical form

criterion based equivalence for equation error models the problem of modeling an input output data sequence is considered when applying equation error least squares identification methods to matrix fraction description mfd models in the case that a set of models has been parametrized using the notion of i o input output equivalence it is shown that the identified model will be essentially dependent on the specific parameterization chosen as an alternative a notion of model equivalence is introduced that is related to the identification criterion the pseudocanonical or overlapping parametrization of mfd models appears to constitute a set of canonical forms for this least squares based model equivalence

properties of the parametrization of monic arma systems abstract some properties of the parametrization of multi input multi output monic arma systems with prescribed column degrees are considered in particular we give a minimal cover for the set of transfer functions corresponding to all arma systems in terms of this parametrization we investigate the relation between the kronecker indices and the prescribed column degrees and we investigate the boundary of a certain parameter space and of the corresponding set of transfer functions

brief optimal errors in variables filtering this paper deals with optimal minimal variance filtering in an errors in variables framework differently from many other contexts errors in variables models treat all variables in a symmetric way no partition of the variables into inputs and outputs is required and assume additive noise on all the variables the filtering technique described in this paper can be easily implemented in a recursive way and does not require the use of a riccati equation at every update the results of monte carlo simulations have shown the effectiveness and consistency of the approach

on the chaotic character of the stochastic heat equation before the onset of intermitttency we consider a nonlinear stochastic heat equation tu 1 xxu u xtw where xtw denotes space time white noise and r r is lipschitz continuous we establish that at every xed time t 0 the global behavior of the solution depends in a critical manner on the structure of the initial function u0 under suitable technical conditions on u0 and supx2rut x is a s nite when u0 has compact support whereas with probability one lim supjxj 1ut x logjxj 1 6 0 when u0 is bounded uniformly away from zero the mentioned sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at xed times well before the onset of intermittency

system identifiability from finite time series in this paper we investigate the identification of systems from time series observed over a finite time interval the data generating system is supposed to be finite dimensional linear and time invariant but not necessarily controllable the minimal number of time series needed to identify a system is characterized by the identifiability index of a system which measures the rank drop of autoregressive representations we formulate a procedure for modelling finite time series which takes the corroboration of system restrictions into account this also gives a new solution for the partial realization problem

intermittency and multifractality a case study via parabolic stochastic pdes let denote space time white noise and consider the following stochastic partial dierential equations i u 1 u 00 u started identically at one and ii z 1 z 00 started identically at zero it is well known that the solution to i is intermittent whereas the solution to ii is not and the two equations are known to be in dierent universality classes we prove that the tall peaks of both systems are multifractals in a natural large scale sense some of this work is extended to also establish the multifractal behavior of the peaks of stochastic pdes on r r d with d 2 g lawler has asked us if intermittency is the same as multifractality the present work gives a negative answer to this question as a byproduct of our methods we prove also that the peaks of the brownian motion form a large scale monofractal whereas the peaks of the ornstein uhlenbeck process on r are multifractal throughout we make extensive use of the macroscopic fractal theory of m t barlow and s j taylor 3 4 we expand on aspects of the barlow taylor theory as well

a robust matched detector we address the matched detector problem in the case the signal to be detected is imperfectly known while in the standard detector the signal is known to lie along a particular direction we consider the case where this direction is known up to additive white gaussian noise this somehow amounts to assuming that the signal lies in a cone the aperture of which depends upon the level of uncertainty we build the associated generalized likelihood ratio glr analyze its statistical properties indicate how to set the threshold to achieve a given false alarm rate and how to predict the associated probability of detection the so obtained detector reduces to the conventional one when the uncertainty vanishes and we analyze its behavior when the level of uncertainty which has to be known a priori is mis evaluated it appears that the sensitivity of the detector is quite low with respect to this kind of errors more importantly several realistic examples are presented that indicate that the proposed detector remains quite efficient when the true signals are far from being of the assumed model and whatever the model of the uncertainty actually is it is this robustness that makes the detector valuable