algebraic approach to system identification a variety of identification procedures exists for estimating the parameters of an autoregressive moving average arma process from noise free excitation and noise contaminated response data in this paper an identification procedure is proposed for the more realistic situation in which both the excitation and response are contaminated by white noises the method is based upon the null space characterization of an associated data matrix some of the more important algebraic properties possessed by this data matrix are first established in the ideal noise free data case in particular it is found that an overordering of the arma model will not impair the identification of the underlying system in the more realistic noise contaminated data case an approximation of the data matrix s null space is affected by using an eigenvalue eigenvector decomposition by incorporating this null space approximation the deleterious effects of the noise are significantly reduced thereby giving rise to improved modeling performance this improvement is demonstrated by means of a standard example in which the proposed identification method is shown to produce a better modeling behavior than does the classical least squares method the corresponding iterative generalized least squares method and a commonly employed instrumental variable method

an approach to time series analysis and arma spectral estimation this paper presents an approach to time series analysis and arma spectral estimation from only the output data corrupted by noise it is shown that the generalized not well known modified yule walker myw equations hold when the residual is some correlated noise to solve such equations a new version of the generalized least squares gls method is proposed yielding ar parameter estimates with higher accuracy this gls method can also be used to enhance the estimation accuracy of ar parameters for short and noisy data in case of the myw equation holding theoretically furthermore a simple procedure for improving ma parameter estimates is studied our approach as cadzow s singular value decomposition svd method has provided significantly higher performance spectral estimates for a low order arma model than those obtained via usual techniques which are based upon direct solution of the myw equations and which use a high order model without considering an additive noise

on continuous solutions of a class of systems of nonlinear functional difference equations with deviating argument abstract bounded continuous solutions on r of a class of systems of nonlinear functional difference equations with a complicated nonlinear deviation of the argument that depends on unknown functions are studied here

the rise algorithm for recursive eigenspace decomposition a recursive method for generating the eigenvalues and eigenvectors of hermitian matrices is presented this algorithm is closely related to recursive iterative toeplitz eigenspace decomposition rite and thus may appropriately be named recursive iterative self adjoint eigenspace decomposition rise since it is formulated for the more general hermitian matrix rise may be performed at a cost of o n sup 3 multiplies per matrix order while rite requires o 4n sup 3 it is demonstrated that rise as is also true with rite is numerically stable and thus does not appreciably accumulate errors as the recursion progresses additionally a variation of rite is presented which allows the computation of the eigenvalues of successively smaller hermitian toeplitz matrices the initial presentation of rite only provided for the determination of the eigenvalues of successively larger matrices

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

survival probabilities in time dependent random walks we analyze the dynamics of random walks in which the jumping probabilities are periodic time dependent functions in particular we determine the survival probability of biased walkers who are drifted towards an absorbing boundary the typical lifetime of the walkers is found to decrease with an increment in the oscillation amplitude of the jumping probabilities we discuss the applicability of the results in the context of complex adaptive systems

arma parameter estimation using a novel recursive estimation algorithm with selective updating the authors investigate an extension of a recursive estimation algorithm the so called optimal bounding ellipsoid or obe algorithm which features a discerning update strategy in particular an extension of the algorithm to arma autoregressive moving average parameter estimation is presented along with a convergence analysis the extension is similar to the extended least squares els algorithm however the convergence analysis is complicated due to the discerning update strategy which incorporates an information dependent updating factor the virtues of such an update strategy are more efficient use of the input data in terms of information processing and a modular adaptive filter structure which would facilitate the development of a parallel pipelined signal processing architecture it is shown that if the input noise is bounded and the moving average parameters satisfy a certain magnitude bound then the a posteriori prediction errors are uniformly bounded with an additional persistence of excitation condition the parameter estimates are shown to converge to a neighborhood of the true parameters and the a priori prediction errors are shown to be asymptotically bounded simulation results show that the parameter estimation error for the eobe algorithm is comparable to that for the els algorithm

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

two dimensional autoregressive 2 d ar model order estimation much research has been devoted to the area of one dimensional autoregressive 1 d ar and autoregressive moving average arma model order selection the most well known solutions for this problem are the akaike information criterion aic mdl and the minimum eigenvalue mev criteria on the other hand all works in the 2 d case have focused on the problem of parameter estimation in this correspondence we extend the previous criteria to the 2 d ar model order determination the model is assumed causal stable and spatially invariant with p sub 1 spl times p sub 2 quarter plane qp support numerical examples are given to illustrate the effectiveness of each method

a genetic algorithm approach to the spectral estimation of time series with noise and missed observations this study considers the problem of estimating the autoregressive moving average arma power spectral density when measurements are corrupted by noise and by missed observations the missed observations model is based on a probabilistic structure unlike conventional cases of missed observation in parameter estimation problems the variance of noise is unavailable that is the time points of missed observations are unknown and the probability of missing data needs to be estimated in this situation spectral estimation is more difficult to solve and becomes a highly nonlinear optimization problem with many local minima in this paper we use the genetic algorithm ga method to achieve a global optimal solution with a fast convergence rate for this spectral estimation problem from the simulation results we have determined that the performance is significantly improved if the probability of data loss is considered in the spectral estimation problem

on periodic and solutions converging to zero of some systems of differential difference equations sufficient conditions for the existence of a periodic c 1 solution of a system of differential difference equations are given we also give some sufficient conditions for the existence of an n parameter family of solutions of a system of differential difference equations converging to zero along with their first derivatives

simultaneous identification and adaptive control of unknown systems over finite parameter sets the problem considered is one of simultaneously identifying an unknown system while adequately controlling it the system can be any fairly general discrete time system and the cost criterion can be either of a discounted type or of a long term average type the chief restriction being that the unknown parameter lies in a finite parameter set for a previously introduced scheme of identification and control based on biased maximum likelihood estimates it is shown that 1 every cesaro limit point of the parameter estimates is closed loop equivalent to the unknown parameter 2 for both the discounted and long term average cost criteria the adaptive control law cesaro converges to the set of optimal control laws and 3 in the case of the long term average cost criterion the actual cost incurred by the use of the adaptive controller is optimal and cannot be bettered even if one knew the value of the unknown parameter at the start

brief papers linear identification of arma processes a new method for estimating the parameters of an arma process is presented the method consists of three linear least squares estimations in the first an autoregressive model is fitted to the observation sequence yielding an estimate of the values of the driving white noise sequence linear least squares is then used to fit an arma model to the observation and estimated white noise sequences this model is used to filter the observation and estimated white noise sequences finally an arma model is fitted to the filtered sequences it is shown that the resultant estimator is p consistent the asymptotic bias tends to zero as the degree p of the autoregressive model tends to infinity and is p efficient the asymptotic efficiency approaches the theoretical maximum as p tends to infinity

a universal prior for integers and estimation by minimum description length of the number of bits required to write down the observed data has been reformulated to extend the classical maximum likelihood principle the principle permits estimation of the number of the parameters in statistical models in addition to their values and even of the way the parameters appear in the models i e of the model structures the principle rests on a new way to interpret and construct a universal prior distribution for the integers which makes sense even when the parameter is an individual object truncated realvalued parameters are converted to integers by dividing them by their precision and their prior is determined from the universal prior for the integers by optimizing the precision 1 introduction in this paper we study estimation based upon the principle of minimizing the total number of binary digits required to rewrite the observed data when each observation is given with some precision instead of attempting at an absolutely shortest description which would be futile we look for the optimum relative to a class of parametrically given distributions this minimum description length mdl principle which we introduced in a less comprehensive form in 25 turns out to degenerate to the more familiar maximum likelihood ml principle in case the number of parameters in the models is fixed so that the description length of the parameters themselves can be ignored in another extreme case where the parameters determine the data it similarly degenerates to jaynes s principle of maximum entropy 14 but the main power of the new criterion is that it permits estimates of the entire model its parameters their number and even the way the parameters appear in the model i e the model structure hence there will be no need to supplement the estimated parameters with a separate hypothesis test to decide whether a model is adequately parameterized or perhaps over parameterized

asymptotically convergent solutions of a system of nonlinear functional differential equations of neutral type with iterated deviating arguments some conditions that guarantee the existence of a family of solutions converging to zero of a system of nonlinear functional differential equations with iterated deviating arguments are given

retract principle for neutral functional differential equations abstract the investigation of asymptotic behaviour of solutions of ordinary differential equations is often based on the application of the retract principle initially developed for ordinary differential equations this technique was extended to other classes of equations not answered remains a problem concerning the possibility of extending this principle to neutral differential equations the goal of the present paper is to partially fill this gap and develop a corresponding technique for the application of this principle the applicability of the main result is illustrated on a nonlinear equation and sufficient conditions for existence of a positive solution are derived

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

characterization and identification of matrix fraction descriptions for lti systems this paper shows that the matrix fraction description given by a pair a z b z of matrix polynomials of z for a linear time invariant system may not be unique even if a z is monic a z and b z have no common left factor and the matrix coefficients corresponding to the highest order terms of a z and b z are full row rank the orders of all possible matrix fraction descriptions mfds of a given system are completely characterized testing criteria for determining whether a matrix pair is an mfd of the system are derived which involve rank tests of certain toeplitz matrices derived from either the impulse response or output correlation functions of the system a decision procedure is devised that generates sequentially all mfds for a given system identification algorithms are introduced that estimate all mfds of a given system from its input output data or output data only the results are then extended to cover armax systems

a new method for ar order determination of an arma process using a combination of the minimum description length mdl criterion and overdetermined instrumental variable method a computationally efficient method for ar order determination of an arma process is proposed it is shown that the well known singular value decomposition svd method for ar order determination is a special case of the proposed method numerical simulations are presented to show the effectiveness of this method

unique existence of bounded continuous solutions on the real line of a class of nonlinear functional equations with complicated deviations abstract the existence and uniqueness of bounded continuous solutions on the real line of a general class of nonlinear functional equations with quite complicated deviations dependent of unknown functions are established in this paper

on solutions of a class of systems of nonlinear functional equations in a neighborhood of zero satisfying lipschitz condition unique existence of solutions of a system of nonlinear functional equations with iterated deviations in a neighborhood of zero and satisfying the lipschitz condition is studied here

model selection through a statistical analysis of the global minimum of a weighted nonlinear least squares cost function this paper presents a model selection algorithm for the identification of parametric models that are linear in the measurements it is based on the mean and variance expressions of the global minimum of a weighted nonlinear least squares cost function the method requires the knowledge of the noise covariance matrix but does not assume that the true model belongs to the model set unlike the traditional order estimation methods available in literature the presented technique allows to detect undermodeling the theory is illustrated by simulations on signal modeling and system identification problems and by one real measurement example

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

order and disorder in the local evolutionary minority game abstract we study a modification of the evolutionary minority game emg in which agents are placed in the nodes of a regular or a random graph a neighborhood for each agent can thus be defined and a modification of the usual relaxation dynamics can be made in which each agent updates her decision scheme depending upon the options made in her immediate neighborhood we name this model the local evolutionary minority game lemg we report numerical results for the topologies of a ring a torus and a random graph changing the size of the neighborhood we focus our discussion in a one dimensional system and perform a detailed comparison of the results obtained from the random relaxation dynamics of the lemg and from a linear chain of interacting spin like variables with temperature we provide a physical interpretation of the surprising result that in the lemg a better coordination a lower frustration is achieved if agents base their actions on local information we show how the lemg can be regarded as a model that gradually interpolates between a fully ordered antiferromagnetic like system and a fully disordered system that can be assimilated to a spin glass

an algorithm for pole zero system model order estimation in system identification estimates of the unknown system model orders are often required an algorithm for estimating model orders is described that looks at input output data covariance matrix eigenvectors when model orders are overestimated zeros appear in the noise subspace eigenvectors the number of zeros present can be used to estimate model orders

time dependent random walks and the theory of complex adaptive systems motivated by novel results in the theory of complex adaptive systems we analyze the dynamics of random walks in which the jumping probabilities are time dependent we determine the survival probability in the presence of an absorbing boundary for an unbiased walk the survival probability is maximized in the case of large temporal oscillations in the jumping probabilities on the other hand a random walker who is drifted towards the absorbing boundary performs best with a constant jumping probability we use the results to reveal the underlying dynamics responsible for the phenomenon of self segregation and clustering observed in the evolutionary minority game

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

robust and accurate arx and arma model order estimation of non gaussian processes the field of higher order statistics is emerging rapidly for analyzing non gaussian processes there are several motivations behind the use of higher order statistics the emphasis of this paper is based on the property that higher order cumulants are blind to any kind of gaussian process hence when the processed signal is non gaussian stationary and the additive noise is stationary gaussian the noise will vanish in the cumulant domain this paper presents an extension to the results of liang et al 1993 this extension is a straightforward generalization of liang s approach to third order cumulants tocs the new cumulant based algorithm provides a higher level of accuracy in the presence of noise than the original second order algorithm in addition the original results of liang have been extended to the case of colored gaussian noise examples are given to demonstrate the performance of this algorithm even when the observed signal is heavily corrupted by gaussian noise

sequential estimation of hidden arma processes by particle filtering part ii this is part ii of a series of two papers where we address sequential estimation of wide sense stationary autoregressive moving average arma state processes by particle filtering in part i we considered a state space model where the state was an arma process of known order and where the parameters of the process could be known or unknown in this paper we extend our work from part i by considering the same type of models with the added complexity that the arma processes are now of unknown order instead of working on a scheme that first tracks the state by operating with different assumed models and then selects the best model by using a predefined criterion we present a method that directly estimates the state without the need of knowing the model order we derive the transition density of the state for unknown arma model order and propose a particle filter based on that density and the empirical bayesian methodology we demonstrate the performance of the proposed method with computer simulations and compare it with the methods from part i

an optimal instrumental variable method for arma spectral estimation a multistep iterative and fast recursive algorithm for autoregressive moving average arma spectral estimation is presented the ar parameters of an arma process are estimated using the extended instrumental variable eiv method the optimal choice of instruments prefilter and weighting matrix is investigated a bootstrapping procedure that has computational convenience is proposed for the algorithm the statistical analysis and experiments show that the optimal iv estimate is unbiased consistent efficient asymptotically normal and equivalent to the maximum likelihood ml estimate and the prediction error pe estimate and the proposed algorithm has the advantages of sharper resolution less frequency bias and better efficiency of convergence

adaptive stochastic filters with no strict positive real condition a new adaptive stochastic filter structure is introduced which avoids the strict passivity test used as a sufficient condition for convergence required by existing adaptive schemes the proposed algorithm consists of three stages in the first stage an autoregressive model is fitted and the residue obtained is used as an estimate of the noise in the second stage an autoregressive recursive moving average model is fitted using the residual of the first stage a modified residual is then filtered using a parameter and the model obtained from the second stage to generate an improved estimate of the noise in the third stage this improved estimate of the noise is used to obtain a better autoregressive moving average model it is shown that the proposed algorithm will also reduce the bias in the estimated parameters the simulation results given show that the proposed filter compares favorably to the algorithm introduced by mayne and clark and also landau this filter is then applied to the adaptive line enhancement sinusoidal detection and adaptive spectral estimation problems to illustrate its usefulness

a covariance extension approach to identification of time series in this paper we consider a three step procedure for identification of time series based on covariance extension and model reduction and we present a complete analysis of its statistical convergence properties a partial covariance sequence is estimated from statistical data then a high order maximum entropy model is determined which is finally approximated by a lower order model by stochastically balanced model reduction such procedures have been studied before in various combinations but an overall convergence analysis comprising all three steps has been lacking supposing the data is generated from a true finite dimensional system which is minimum phase it is shown that the transfer function of the estimated system tends in h to the true transfer function as the data length tends to infinity if the covariance extension and the model reduction is done properly the proposed identification procedure and some variations of it are evaluated by simulations

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

arma spectral estimation of narrow band processes via model reduction the problem of estimating autoregressive moving average arma models for narrowband processes is considered the following approach is proposed estimate a high order autoregressive ar approximation of the process by model reduction based on a truncated internally balanced realization or optimal hankel norm model reduction reduce the order of this high order ar estimate to find a lower order arma model this algorithm gives arma spectral estimates with excellent resolution properties without using iterative numerical minimization methods as for the maximum likelihood method how to take the narrowband assumption into account in the model reduction step is discussed in detail

3 d ar model order selection via rank test procedure this paper deals with the problem of three dimensional autoregressive 3 d ar model order estimation we show that the information for the 3 d ar model order is implicitly contained in an appropriate matrix rank built from the autocorrelation function acf of the underlying 3 d gaussian process exploiting this property we develop an algorithm to estimate the order p1 p2 p3 corresponding to the quarter space qs region of support the proposed method is based upon a rank test procedure rtp using singular value decomposition svd and solving nonlinear system equations numerical simulations are presented to illustrate the performances of the proposed algorithm

note on the existence of periodic solutions of a class of systems of differential difference equations abstract we prove a result on the existence of periodic solutions of a class of systems of differential difference equations partially solved with respect to the first order derivative by constructing it in a form of a series

a reverse formulation of the rise algorithm the recently developed recursive iterative self adjoint eigenspace rise decomposition algorithm recursively computes the complete eigenspace decomposition of successively larger hermitian matrices however some practical applications also require that the decomposition of successively smaller matrices be computed a modification to the rise algorithm is presented that makes it possible to run this algorithm backward on successively smaller hermitian matrices this important modification increases the number of practical applications of this algorithm

a method for autoregressive moving average estimation summary the problem considered is that of estimating an autoregressive moving average system including estimating the degrees of the autoregressive and moving average lag operators the basic method is that introduced by hannan rissanen 1982 however that method may sometimes overestimate the degrees and modifications are here introduced to correct this the problem is itself due to the use of a long autoregression of order clog t when t is large in the first stage of the process the effect of this is investigated and in particular its effect on the speed of convergence of the estimates

solutions converging to zero of some systems of nonlinear functional differential equations with iterated deviating argument abstract some sufficient conditions for the existence of a p parameter family of solutions converging to zero of a system of nonlinear functional differential equations with an iterated deviating argument dependent of unknown function are given

statistical analysis of the evolutionary minority game with different capacities abstract in this paper we consider the evolutionary minority game with different capacities with the increase of capacity level 1 the probability n of room1 with n agents changes from one normal distribution to two normal distributions and back to one normal distribution again when room2 is the basic strategy the probability of room1 with n agents is a small normal distribution and when room1 is the basic strategy the probability of room1 with n agents is a large normal distribution we calculate the integral of the normal distributions and explain the characters in the figures for instance the room with more capacity represents the interest of the majority and the room has more probability to win

existence of bounded solutions of some systems of nonlinear functional differential equations with complicated deviating argument abstract sufficient conditions for the existence of bounded solutions on the real line satisfying lipschitz s condition of a class of systems of nonlinear functional differential equations with complicated deviating argument dependent of unknown functions are given in this paper

arma model order estimation based on the eigenvalues of the covariance matrix an approach to model order determination based on the minimum description length mdl criterion is proposed and shown to depend on the minimum eigenvalues of a covariance matrix derived from the observed data a selection procedure for estimating the model order by means of the mdl method is proposed examples are given to illustrate the significantly improved accuracy of the technique

temporal oscillations and phase transitions in the evolutionary minority game the study of societies of adaptive agents seeking minority status is an active area of research recently it has been demonstrated that such systems display an intriguing phase transition agents tend to self segregate or to cluster according to the value of the prize to fine ratio r we show that such systems do not establish a true stationary distribution the winning probabilities of the agents display temporal oscillations the amplitude and frequency of the oscillations depend on the value of r the temporal oscillations that characterize the system explain the transition in the global behavior from self segregation to clustering in the r 1 case

model reductions of high order estimated models the asymptotic ml approach abstract the reduction of order of high order models obtained from an identification experiment is discussed from a statistical point of view the asymptotic maximum likelihood ml approach is defined to reduce the order of an estimated model this approach considers the maximum likelihood criterion given the asymptotic statistics both in the number of data and the order of the estimated model and corresponds to frequency weighted l 2 norm model reduction by using the insight from the asymptotic ml approach an identification algorithm is proposed based on a high order arx estimate and model reduction via a frequency weighted balanced realization the advantage of this algorithm is that iterative minimization methods are not required to find the estimate

parametric identification using weighted null space fitting in identification of dynamical systems the prediction error method with a quadratic cost function provides asymptotically efficient estimates under gaussian noise but in general it requires solving a nonconvex optimization problem which may imply convergence to nonglobal minima an alternative class of methods uses a nonparametric model as intermediate step to obtain the model of interest weighted null space fitting wnsf belongs to this class starting with the estimate of a nonparametric arx model with least squares then the reduction to a parametric model is a multistep procedure where each step consists of the solution of a quadratic optimization problem which can be obtained with weighted least squares the method is suitable for both open and closed loop data and can be applied to many common parametric model structures including output error armax and box jenkins the price to pay is the increase of dimensionality in the nonparametric model which needs to tend to infinity as function of the sample size for certain asymptotic statistical properties to hold in this paper we conduct a rigorous analysis of these properties namely consistency and asymptotic efficiency also we perform a simulation study illustrating the performance of wnsf and identify scenarios where it can be particularly advantageous compared with state of the art methods

noisy data and impulse response estimation this paper investigates the impulse response estimation of linear time invariant lti systems when only noisy finite length input output data of the system is available the competing parametric candidates are the least square impulse response estimates of possibly different lengths it is known that the presence of noise prohibits using model sets with large number of parameters as the resulting parameter estimation error can be quite large model selection methods acknowledge this problem hence they provide metrics to compare estimates in different model classes such metrics typically involve a combination of the available least square output error which decreases as the number of parameters increases and a function that penalizes the size of the model in this paper we approach the model class selection problem from a different perspective that is closely related to the involved denoising problem the method primarily focuses on estimating the parameter error in a given model class of finite order using the available least square output error we show that such an estimate which is provided in terms of upper and lower bounds with certain level of confidence contains the appropriate tradeoffs between the bias and variance of the estimation error consequently these measures can be used as the basis for model comparison and model selection furthermore we demonstrate how this approach reduces to the celebrated aic method for a specific confidence level the performance of the method as the noise variance and or the data length varies is explored and consistency of the approach as the data length grows is analyzed

technical communique linear estimation of arma processes the method of estimating arma parameters first discussed by mayne and firoozan is investigated it is shown that if at the first stage the order of the fitted autoregression is allowed to depend on the number of time points in a reasonable manner then it will still be true that the final estimate of the parameter vector will converge almost surely to the true value this is to be compared to the result in the original paper where the order is fixed and it is shown that as the sample size increases the estimate converges to a value which if the order of the autoregression is high enough will be arbitrarily near to the true value some comments are made on other extensions on the law of the iterated logarithm on the central limit theorem and on the choice of the order of the fitted autoregression the innovation sequence need not be gaussian and for the convergence result only a natural condition relating to prediction needs be imposed

on a nonlinear generalized max type difference equation abstract the boundedness character of positive solutions of the following max type difference equation x n max a x n 1 p x n k r n n 0 where k n 1 the parameters a and r are positive and p is a nonnegative real number is studied in this paper our main results considerably improve results appearing in the literature

paper a new algorithm for recursive estimation of parameters in controlled arma processes a recursive algorithm for estimating the constant but unknown parameters of a controlled arma process is presented the algorithm is a recursive version of an off line algorithm using three stages of standard least squares in the first stage the parameters of a controlled ar model of degree p are estimated the residuals used in this stage are employed in the second stage to estimate the parameters of a controlled arma process the first two stages constitute a recursive version of durbin s algorithm the model obtained in the second stage is used to filter the input output and residuals and these filtered variables are used in the final stage to obtain improved estimates of the controlled arma process it is shown that the estimate is globally p consistent i e that the estimate converges a s as the number of data tends to infinity to a vector which in turn converges to the true parameter vector as the degree p of the ar model tends to infinity

periodicity of a class of nonautonomous max type difference equations abstract in this note we prove several general and easily applicable results regarding the periodicity of positive solutions of the following difference equations x n max f 1 x n k 1 n f m x n k m n x n s n n 0 and x n min f 1 x n k 1 n f m x n k m n x n s n n 0 where 1 k 1 k m m s n and where the functions f i 0 n 0 0 1 i m satisfy some additional conditions these results considerably extend and unify numerous results on max type difference equations existing in the literature

on a system of difference equations abstract we show that the system of difference equations x n 1 ax n 1 by n x n 1 c y n 1 y n 1 x n y n 1 n n 0 where the parameters a b c and initial values x 1 x 0 y 1 y 0 are real numbers can be solved considerably improving the results in the literature

a fast estimation method for arma processes an improvement of a batch algorithm for parameter estimation of an arma process is presented a fast and iterative algorithm is presented and provides estimates with the same asymptotic distribution as the maximum likelihood estimates this algorithm is a modified version of the method suggested by mayne and firoozan 1982 linear identification of arma processes automatica 18 461 466 and requires one less regression than the previous method if only one pass is executed