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

from time series to linear system part iii approximate modelling abstract in the third part of this paper the problem of finding a linear time invariant complete system which models an observed time series will be continued however in this part it will be assumed that the model is required to explain the observations only approximately this raises the question of what is meant by an approximate model and how one should judge a model against data which in a strict sense falsify the model the initial sections of this paper introduce the methodology either the maximal admissible complexity is fixed and the best fitting model in the model class is sought or the maximally tolerated misfit is fixed and the least complex model in the model class is sought this is illustrated by means of finite dimensional models where natural complexity and misfit functions suggest themselves subsequently what is meant by the complexity of a linear time invariant complete system is discussed a concrete complexity function is proposed which is defined in terms of the richness of the behaviour on finite time intervals and is in one to one correspondence with the number of inputs the number of states and the observability indices of the underlying minimal i s o system the next definition is that of the misfit between an observed time series and a system a concrete misfit function is proposed which takes into consideration the lag structure of the system and in how far the observations fail to corroborate the ar equations which define the system s behaviour how the misfit may be computed by means of the correlation function of the data is then shown these concepts are subsequently used in order to set up two algorithms for approximate modelling of an observed q dimensional time series the first algorithm fixes the maximal complexity in a sense this comes down to fixing the number and the lags of the required ar equations the second algorithm fixes the tolerated misfit it is adaptive in nature in the sense that the number and the structure of the resulting ar equations will depend on the observations these ideas and algorithms are illustrated by means of two simulated examples

state maps for linear systems modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential algebraic equations as such state space models are not a natural starting point for modeling while they have utmost importance in the simulation and control phase the paper addresses the problem of computing state variables for systems of linear differential algebraic equations of various forms the point of view from which the problem is considered is the behavioral one as put forward in j c willems automatica j ifac 22 1986 pp 561 580 dynamics reported 2 1989 pp 171 269 ieee trans automat control 36 1991 pp 259 294

prediction diagnostics and updating in multivariate calibration summary in multivariate calibration the relationship between a q variate response vector y and p explanatory variables x are estimated from training data in order to predict an unknown x denoted by e from further observed responses when q p both the profile likelihood for e and bayesian inference for e depend on a prediction inconsistency diagnostic which highlights those response vectors used for prediction which are internally inconsistent in the prediction of e when several further response vectors together display systematic anomalies one is led to questioning the estimated model the information in prediction data about changes in parameters is investigated under various assumptions the results indicate systematic anomalies in prediction data may be detected in a variety of ways but corrected only under strong assumptions so that recalibration might be needed

on numerical improvement of gauss lobatto quadrature rules it is well known that gauss lobatto quadrature email protected 11f x email protected i 1nw if x i pf 1 qf 1 is exact for polynomials of degree at most 2n 1 in this paper we are going to find a formula which is approximately exact for monomials x j j 0 1 2n 3 instead of being analytically exact for monomials x j j 0 1 2n 1 we also consider a class of functions for which the new formula produces better results

estimating fetal age by ultrasound measurements an example of multivariate calibration a problem in the estimation of gestational age by ultrasound measurement of fetal bone lengths the femur length f and the biparietal diameter bpd is described by means of multivariate calibration techniques f and bpd are combined to give an index of gestational age which is significantly more accurate than existing indices based on f or bpd alone a detailed description of the statistical analysis is given ranging from the initial steps of data analysis and model choice to the final step of determining the index and its accuracy a number of interesting aspects of multivariate calibration theory are illustrated

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

improving upon standard estimators in discrete exponential families with applications to poisson and negative binomial cases assume that xi xp are independent random observations having discrete exponential densities p1 0 ti x rj 1 i 1 p respectively a general technique of improving upon the uniform minimum variance unbiased estimator umvue of o1 op is developed under possibly weighted squared error loss functions it is shown that improved estimators can be constructed by solving a difference inequality typical difference inequalities of a fairly general type are presented and solved when specialized to poisson and negative binomial cases broad classes of estimators are given that dominate the umvue these results unify many known results in this rapidly diverging field and some of them are new especially those related to negative binomial distributions improved estimators are also obtained for the problems in which some of the observations are from poisson families and some from negative binomial families for sum of squared errors loss estimators which dominate the umvue in the discrete exponential families are also given explicitly

state representations of arma models a state representation of an arbitrary arma model is computed explicitly it is shown then that every arma model is homotopy equivalent to its state representation and that two state models are homotopy equivalent if and only if they are similar

exact scheffe type confidence intervals for output from groundwater flow models 2 combined use of hydrogeologic information and calibration data calibration data observed values corresponding to model computed values of dependent variables are incorporated into a general method of computing exact scheffe type confidence intervals analogous to the confidence intervals developed in part 1 cooley this issue for a function of parameters derived from a groundwater flow model parameter uncertainty is specified by a distribution of parameters conditioned on the calibration data this distribution was obtained as a posterior distribution by applying bayes theorem to the hydrogeologically derived prior distribution of parameters from part 1 and a distribution of differences between the calibration data and corresponding model computed dependent variables tests show that the new confidence intervals can be much smaller than the intervals of part 1 because the prior parameter variance covariance structure is altered so that combinations of parameters that give poor model fit to the data are unlikely the confidence intervals of part 1 and the new confidence intervals can be effectively employed in a sequential method of model construction whereby new information is used to reduce confidence interval widths at each stage

on the state of behaviors the theme of the present paper is the study of the concept of state and the corresponding state maps in the context of willems behavioral theory we concentrate on markovian system and their representation in terms of first order difference or differential systems we follow by a full analysis of the special case of state systems the embedding of a linear system in a state system via the use of state maps arriving at state representations or equivalently to a realization theory for behaviors minimality is defined and characterized and a state space isomorphism theorem is established realization procedures based on the shift realization are developed as well as a rigorous analysis of the construction of state maps the paper owes much to rapisarda and willems p rapisarda j c willems state maps for linear systems siam j contr optim 35 1997 1053 1091

on numerical improvement of closed newton cotes quadrature rules abstract in this paper we discuss about numerical improvement of the open newton cotes integration rules that are in forms of a x 1 b x n 1 x 1 n 2 h f x d x k 0 n b k n f x 1 k 1 h it is known that the precision degree of above formula is n 1 for even n s and is n for odd n s however if the integral bounds are considered as two additional variables i e a and h in fact we reach a nonlinear system that numerically improves the precision degree of the above integration formula up to degree n 2 in this way some numerical tests are given to show the numerical superiority of our idea with respect to the usual open newton cotes integration rules

combining independent studies in a calibration problem abstract the problem of calibration in which the response variable is measured by k different methods or using different instruments is considered it is well known that the usual classical estimator for the unknown explanatory variable has infinite mean and mean squared error when k 1 in this article a linear combination of the classical estimators is proposed it is shown that the combined estimator has finite mean provided that k 2 and finite mean squared error provided that k 3 expressions for asymptotic bias and mean squared error are given also two confidence sets for the unknown exploratory variable are developed sufficient conditions under which they will be finite intervals are given the results are illustrated by a practical example

arma models and their equivalences the classical theory of strict system equivalence of rosenbrock and fuhrmann is presented in a very general setting namely in the setting of arma models defined over an arbitrary commutative ring

the semi open newton cotes quadrature rule and its numerical improvement in this paper we discuss about numerical improvement of the semi open newton cotes integration rules that are in forms of a x 1b x n 1 x 1 n 1 hf x email protected k 0nb k n f x 1 k 1 h it is known that the precision degree of above formula is n 1 for even n s and is n for odd n s however if the integral bounds are considered as two additional variables i e a and h in fact we reach a nonlinear system that numerically improves the precision degree of the above integration formula up to degree n 2 in this way some computational tests are given to show the numerical superiority of our approach with respect to the usual semi open newton cotes integration rules

a note on equal coefficient quadrature rules abstract in this paper we introduce an integration method with equal coefficient in the following form a b w x f x d x c n i 1 n f x i then by using the formulas of newton s equations and degree of precision we introduce a method which express the nodes and coefficients in this integration formulas finally some examples are presented to illustrate the procedure

on behavior homomorphisms and system equivalence in this paper we introduce the concept of a behavior homomorphisms and isomorphisms and use it to present a uni ed approach to the study of equivalence of di erent behavior representations concentrating on a special representation we call normalized arma narma representation c 2001 elsevier science b v all rights reserved

the first kind chebyshev lobatto quadrature rule and its numerical improvement one of the integration methods is the first kind chebyshev lobatto quadrature rule denoted email protected 11f x 1 x email protected email protected k 1nfcos 2k 1 email protected n 1 f 1 email protected n 1 f 1 according to this rule the precision degree of above formula is the highest i e 2n 1 hence it is not possible to increase the precision degree of chebyshev lobatto integration formulas anymore we will present a matrix proof for this matter but on the other hand we claim that one can improve the above formula numerically to do this we consider the integral bounds as two new unknown variables this causes to numerically be extended the monomial space f x x j from j 0 1 2n 1 to j 0 1 2n 3 two monomials more than the first kind chebyshev lobatto integration method in other words we present an approximate formula email protected abf x 1 x email protected i 1nw if x i v 1f a v 2f b in which a b and w 1 w 2 w n and x 1 x 2 x n are all unknowns and the formula is almost exact for the monomial basis f x x j j 0 1 2n 3 several tests are finally given to show the excellent superiority of the proposed nodes and weights with respect to the usual first kind chebyshev lobatto nodes and weights let us add that in this part we have also some wonderful 5 point formulas that are comparable with 100002 point formulas of the first kind chebyshev lobatto quadrature rules in average

a study of behaviors abstract the paper presents the results of the study of behavior theory as developed by j c willems from the point of view of polynomial and rational models considering behaviors in the discrete time case to be generalizations of rational models a natural focal point becomes the concept of a behavior homomorphism we give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices these results which are of intrinsic interest are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations to a certain extent these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of rosenbrock a study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory

realization of autoregressive equations in pencil and descriptor form a linear system described by autoregressive equations with a given input output structure cannot be transformed to standard state space form if the implied input output relation is nonproper instead a realization in descriptor form must be used in this paper it is shown how to obtain minimal descriptor realizations from autoregressive equations without separating finite and infinite frequencies and without going through a reduction process external equivalence is used so that even situations in which there is no transfer matrix can be considered the approach is based on the so called pencil representation of linear systems and it is shown that there is a natural realization of autoregressive equations in pencil form in this way the link between the realization theories of willems and fuhrmann can also be clarified

exact confidence intervals for the common mean of several normal populations summary the problem of interval estimation of the common mean of several normal populations when the variances are unknown and unequal is considered some new confidence intervals are proposed one of them is centered at the well known graybill deal estimator of the common mean these new intervals and an existing exact confidence interval are numerically compared with respect to their expected lengths based on comparison studies some recommendations are made regarding the choice of the intervals to be used in applications the problem of combining the results of independent investigations for the purpose of a common objective has received considerable attention in the literature in particular it may be necessary to combine the data sets collected by independent agencies to estimate or to test the common parameter of interest this problem arises for example when two or more independent agencies are involved in measuring the effect of a new drug when several measuring instruments are used to measure the products produced by the same production process to assess the average quality or when different laboratories are used to measure the amount of toxic waste in a river meier 1953 gave an illustrative example in which four experiments are used to estimate the mean percentage of albumin in the plasma protein of normal human subjects other examples can be found in the recent papers by eberhardt reeve and spiegelman 1989 and skinner 1991 the former paper gives three examples each of which is concerned with estimating a chemical substance in nonfat milk powder by combining the results of alternative analytical methods the latter gives examples related to clinical trials and determining the density of nitrogen using different laboratory methods in these situations the variances are typically unequal due to the variation in different methods or measuring instruments if it is assumed that the samples collected by independent studies are from normal populations then the problem of interest may be to estimate or test the common mean a of these populations

the second kind chebyshev quadrature rules of semi open type and its numerical improvement abstract one of the less known integration methods is the weighted newton cotes quadrature rule of semi open type which is shown by a x 0 b x n 1 x 0 n 1 h f x w x d x k 0 n w k f x 0 kh where w x is a weight function on a b and h b a n 1 is the step size there are various cases for the weight function w x that one can use here we consider the weight function w x 1 x 2 which is important in numerical analysis therefore in this paper we would face with the following formula 1 1 f x 1 x 2 d x k 0 n w k f 1 2 k n 1 it is known that the precision degree of above formula is n 1 for even n s and is n for odd n s but we consider the upper and lower bounds as two additional variables to reach a nonlinear system that numerically improves the precision degree of above formula up to degree n 2 in this way we also give some examples to show the numerical superiority of our idea

the first kind chebyshev newton cotes quadrature rules closed type and its numerical improvement one of the less known integration methods is the weighted newton cotes of closed type quadrature rule which is denoted by a x 0b x n x 0 nhf x w x email protected k 0nw kf x 0 kh where w x is a positive function and h b an is a positive value there are various cases for the weight function w x that one can use because of special importance of the weight function of gauss chebyshev quadrature rules i e w x 11 x 2 in numerical analysis we consider this function as the main weight hence in this paper we face with the following formula in fact 1 1f x 1 x email protected k 0nw kf 1 2kn it is known that the precision degree of above formula is n 1 for even n s and is n for odd n s however if we consider its bounds as two additional variables we reach a nonlinear system that numerically improves the precision degree of above formula up to degree n 2 in this way we give several examples which show the numerical superiority of our approach

singular state models and singular ltid systems in this paper we offer a systematic development of state representation theory for singular linear systems in the behavioural setting of willems

on the parametrization of conditioned invariant subspaces and observer theory the present paper is an in depth analysis of the set of conditioned invariant subspaces of a given observable pair c a we do this analysis in two different ways one based on polynomial models starting with a characterization obtained in p a fuhrmann linear operators and systems in hilbert space 1981 ieee trans automat control ac 26 1981 284 the other being a state space approach toeplitz operators projections in polynomial and rational models wiener hopf factorizations and factorization indices all appear and are tools in the characterizations we single out an important subclass of conditioned invariant subspaces namely the tight ones which already made an appearance in p a fuhrmann u helmke systems control lett 30 1997 217 a precursor of the present paper of particular importance for the study of the parametrization of the set of conditioned invariant subspaces of an observable pair c a is the structural map that associates with any reachable pair with only the input dimension constrained a uniquely determined conditioned invariant subspace the construction of this map uses polynomial models and the shift realization new objects the partial observability and reachability matrices are introduced which are needed for the state space characterizations kernel and image representations for conditioned invariant subspaces are derived uniqueness of a kernel representation of a conditioned invariant subspace is shown to be equivalent to tightness we pass on to an analysis and derivation of the kronecker hermite canonical form for full column rank rectangular polynomial matrices this extends the work of a e eckberg a characterization of linear systems via polynomial matrices and module theory ph d thesis mit cambridge ma 1974 g d forney siam j control optim 13 1973 493 and d hinrichsen h f munzner and d pratzel wolters systems

confidence regions for random effects calibration curves with heteroscedastic errors we construct confidence bounds for a random effects calibration curve model an example application is analysis of analytical chemistry data in which the calibration curve contains measurements y for several values of known concentration x in each of q laboratories laboratory is considered a random effect in this design and the intercept and slope of the calibration curve are allowed to have laboratory specific values here we a develop an appropriate interlaboratory calibration curve for heteroscedastic data of the type commonly observed in analytical chemistry b compute a point estimate for an unknown true concentration x when corresponding measured concentrations y1 y2 yq are provided from q laboratories i e a subset of the original q laboratories used to calibrate the model where 1 q q c compute the asymptotic mean and variance of the estimate and d construct a confidence region for x we then illustrate the methods are using both simulated and typical interlaboratory calib

on numerical improvement of the first kind gauss chebyshev quadrature rules one of the integration methods of the equality type is gauss chebyshev quadrature rule which is in the following form 11f x 1 x 2dx pn k 1nfcos 2k 1 p2n 2 p2 2 n 2n f 2 n h 1 h 1 according to gauss quadrature rules the precision degree of above formula is the highest i e 2n 1 hence it is not possible to increase the precision degree of gauss chebyshev integration formulas anymore in this way we present a matrix proof for this matter but on the other hand we claim that we can improve the above formula numerically to do this we consider the integral bounds as two unknown variables this causes to numerically be extended the monomial space f x x j from j 0 1 2n 1 to j 0 1 2n 1 this means that we have two monomials more than gauss chebyshev integration method in other words we give an approximate formula as abf x 1 x 2dx i 1nw if x i in which a b and w 1 w 2 w n and x 1 x 2 x n are all unknowns and the formula is almost exact for the monomial basis f x x j j 0 1 2n 1 some important examples are finally given to show the excellent superiority of the proposed nodes and weights with respect to the usual gauss chebyshev nodes and weights let us add that in this part we have also some wonderful 2 point formulas that are comparable with 71 point formulas of gauss chebyshev quadrature rules in average