1 Introduction Principally, behaviours focus our attention on the primary level of system dynamics, that is, on empirical observations. In a sense the

Transcription

1 Identiæcation of Factor Models by Behavioural and Subspace Methods Wolfgang Scherrer a Christiaan Heij bæ version a Institut fíur íokonometrie, Operations Research und Systemtheorie, Technische Universitíat Wien, Austria b Econometric Institute, Erasmus University Rotterdam, Netherlands Abstract The behavioural framework has several attractions to oæer for the identiæcation of multivariable systems. Some of the variables may be left unexplained without the need for a distinction between inputs and outputs; criteria for model quality are independent of the chosen parametrization; and behaviours allow for a global èi.e., non-localè approximation of the system dynamics. This is illustrated by the identiæcation of dynamic factor models. Behavioural least squares is a natural method for this problem, and a comparison is given with non-behavioural methods. Keywords: behaviour, linear system, system identiæcation, least squares, factor models, principal components, model reduction. *Corresponding author, heij@few.eur.nl 1

2 1 Introduction Principally, behaviours focus our attention on the primary level of system dynamics, that is, on empirical observations. In a sense the behaviour consists of the objective information on the system. Other issues, like input-output decomposition and parametric representation, are secondary as they depend on the subjective choices of the user. This explicit distinction between data information, model representation, and model use is one of the charms of the behavioural approach in system theory. Several behavioural methods for system identiæcation have been proposed since the introduction of this framework by Willems in ë22ë. One approach is based on state models, realization theory and model reduction as in ë23ë, see for example ë3, 4, 20ë. Subspace methods as in ë17ë can be seen as approximate versions of the realization method in ë23ë. Earlier roots of these ideas are in ë1,2ë for stochastic systems. Another approach is more equation oriented as in ë8, 24ë, and this related to prediction error methods ë11ë. A third approach, based on least squares approximation of behaviours, is developed in ë14ë, see also ë10, 16ë, and ë21ë for the case of exponential time series. In this paper we consider the behavioural least squares method èblsè, also called global total least squares. In a sense, this method adheres most strictly to the behavioural principle of focusing on the external system characteristics. Subspace methods and realization based methods for data modelling employ states as secondary objects in their approximation, and equation oriented procedures are faced with the choice of canonical parameters. In BLS, the model quality is expressed simply as the least squares distance between the empirical data and the system behaviour. This criterion is parameter independent, and one can freely choose parametrizations that are suitable from a computational point of view. This paper intends to communicate the following two messages : 1. behavioural methods can be applied for 'noisy' data; 2. behavioural identiæcation is the most appropriate approach for some problems of practical interest. As concerns è1è, the behavioural identiæcation methods in ë3, 4, 20ë seem not to be appropriate for the modelling of 'inexact' data. These methods require that the observed data can be modelled without error by a ènon-trivialè linear system. This is of interest for purposes of coding and realization, but in many circumstances the data are not of this type. In this paper we consider the identiæcation problem under assumptions that are common in statistics, that is, the data generating system is supposed to be a èfull rankè stationary process. In this case the data satisfy no exact relations, and the realization-based methods ë3, 4, 20ë can not be applied. However, the BLS method can be used to approximate this kind of data. As concerns è2è, it will come as no surprise that behavioural methods are also optimal for a class of identiæcation problems. This holds true for virtually 2

3 every method, by deæning the problem appropriately. However, in this paper we consider an identiæcation problem of independent interest, that is, the identiæcation of dynamic factor models. This problem has received broad attention, both in the statistical literature èsee for example ë5, 7ëè and in the system and control literature èsee for instance ë6, 10, 13, 19ëè. It turns out that BLS is a natural identiæcation method within this setting. In order to evaluate the real contribution of the behavioural approach, it is of interest to investigate also non-behavioural methods for the identiæcation of dynamic factor models. As we know of no existing methods for this problem, we develop for this purpose two subspace identiæcation methods, one within an input-output setting and the other based on equation errors. It is shown that these methods, though computationally somewhat less demanding, lead to clearly inferior results as compared to BLS. It remains an open question if there exist other input-output methods or equation error methods that perform better than the two proposed methods. However, in our opinion the behavioural approach is the most natural one in this case. The paper is organized as follows. In Section 2 we brieæy review the BLS method. This is applied in Section 3 for the identiæcation of dynamic factor models for stationary processes. Alternative, non-behavioural methods are presented that are based on subspace identiæcation of frequency data. Section 4 illustrates the methods by a simulation example, and Section 5 concludes. 2 Behavioural Least Squares We assume that the reader is familiar with the behavioural framework as developed in ë22, 23, 24, 25ë. The empirical data are denoted by w, aq-dimensional vector time series observed in discrete time over the interval T ç Z. A behaviour is a subset B ç èr q è Z of the set of all time series over the full time axis Z: We will only consider behaviours that correspond to linear systems, that is, B is a linear, shift invariant set that is closed in the topology of pointwise convergence. In general, unless the observed data are very structured, linear systems that model the observed data without error will be very complex. If T = Z, generically the only linear system with w 2 B is given by B = èr q è Z. If T is a ænite interval of length N, then it can be shown that there always exists a linear system B with p = 1 output and m = q, 1 inputs and state dimension n ç èn +1è=èq+ 1è with the property that w 2B T, the restriction of B to T. Such systems are not helpful in describing the data in a less complex way, and this is only possible by allowing for some kind of approximation. We measure the approximation error by the squared distance between the data and the linear system, that is d T èw; Bè = minfk w, ^w k; ^w 2B T g; è1è where k w, ^w k 2 = P t2t P q j=1 fw jètè, ^w j ètèg 2. Because of the foregoing result, we consider only approximating systems with pé0 and nén, in which case 3

4 the dimension is given by dimèb T è=mn + n: è2è The complexity of a system is deæned by the pair èm; nè, that is, by the number of inputs and the number of states of the system. The behavioural least squares èblsè problem is to determine the optimal system for given complexity, that is, d T èw; m; nè = minfd T èw; Bè; B has complexity èm; nèg: è3è For a structural analysis of this minimization problem we refer to ë9ë and for a Gauss-Newton algorithm to ë14, 15ë. In practice one can determine the errors d T èw; m; nè for a range of complixities èm; nè and choose the model that provides an acceptable trade-oæ between complexity and æt. This choice is subjective, no formal criteria have been developed until now. As compared with other system identiæcation methods, the above BLS approach has the following characteristic features. The model quality, in terms of æt and complexity, is deæned on the observational level, in terms of the data w and the behaviour B. Further, all variables are treated in a symmetric way, and the error criterion measures the global misæt of models in the sense that ^w is required to satisfy all the system restrictions. In comparison, more conventional identiæcation procedures like prediction error methods start from an input-output decomposition of the system variables and consider the onestep-ahead forecasting quality, a local criterion of æt. As compared with other behavioural approaches, the equation oriented methods in ë8, 24ë are symmetric, but with a local criterion. The realization based behavioural methods in ë3, 4ë determine an exactly ætting system, that is, with d T èw; Bè = 0, and then model reduction procedures could be employed to lower the complexity. The resulting model is very sensitive to data variations. Further, the behavioural method in ë20ë always gives autonomous systems, that is, systems with no inputs. Such systems are of not so much value for modelling noisy data. In our opinion, BLS is an attractive method if one is interested in a descriptive, approximate model of the data and, in the case of input-output systems, if both the inputs and the outputs are subject to noise. 3 Identiæcation of Factor Models We suppose that the data are generated by a full rank stationary process w with a spectral density S that is bounded on the unit circle. A factor model is a representation w = ^w + ~w è4è characterized by the condition that the factor process ^w has less degrees of freedom than w. The process ~w is the error resulting from the approximation of w by the reduced process ^w. We denote by ^S the spectrum of ^w and by Bè^wè the factor behaviour, that is, the smallest linear system with the property that ^w 2Balmost surely. 4

5 3.1 Principal components and BLS In the non-dynamic case, the spectrum S is the covariance matrix of w and factor models have the property that the covariance matrix ^S has reduced rank. If we æx the allowed rank m of this matrix, i.e., the dimension of B, then the model that minimizes the error E k ~wètè k 2 is given by principal P components. q If the covariance matrix has eigenvalue decomposition S = ç j=1 ju j u æ j, with ç 1 çæææç ç m éç m+1 çæææç ç q é0, then the principal component model of complexity m is given by ^w = mx qx u j u æ j w; ~w = j=1 j=m+1 u j u æ j w with behaviour Bè^wè = spanfu j ; j =1;æææ;mg. In the dynamic case we can apply this decomposition frequency-wise, as is proposed in ë5ë. Under certain regularity conditions, the functions u j can be chosen to be analytic so that P P m the factor process ^w PC = j=1 u jèçèu j èçè æ w q and the error process ~w PC = u j=m+1 jèçèu j èçè æ w are well-deæned, with ç the shift operator on Z. The resulting error Ek ~w PC ètèk 2 = P q j=m+1 R ç,ç ç jèe,iç èdç is minimal among all factor models with ^w of rank at most m. The disadvantage of this model is that the restrictions on the factor process ^w are in general nonrational. This means that it is hard to give an explicit description of the factors. More precisely, in general the factor behaviour Bè^w PC è=èr q è Z so that, in the sense of linear systems, the factor process ^w PC is not simpler than the original process w. Suppose that the allowed complexity èm; nè of the factor behaviour has been æxed. There are now several ways to ænd a model that satisæes this restriction, according to diæerent ways of approximation. An obvious method similar to principal components is to ænd a solution for where minfdèw; Bè; B has complexity èm; nèg è6è dèw; Bè :=minfèekwètè, ^wètèk 2 è 1=2 ; Bè^wè=Bg è7è This can be seen as the 'inænite sample' analogon of BLS in è1è and è3è, see ë9ë for further details. An example is given in Section Subspace identiæcation of frequency data In order to evaluate the contribution of the above behavioural approach for the identiæcation problem è6,7è, we will also consider what can be achieved by other approaches. A possible alternative, somewhat in the spirit of the realization approach, is to take the principal component model as a starting point. This is the optimal model with the given number of inputs, but in general it has inænitely many states. Therefore it remains to approximate the principal component model by a system with the given number of states. We è5è 5

6 ærst describe a subspace method for modelling frequency data, which is applied in the next section for the identiæcation of factor models. The factor process of the principal component model è5è satisæes the restrictions Uèçè^w=0 è8è where U is the èin general non-rationalè èq, mè æ q matrix function with rows u æ j ;j = m+1;æææ;q. The idea is to approximate U by a rational function ^U, based on the frequency data Uèe,iç j è, ç j = 2çj=n f, j = 0;:::;n f,1. For this purpose we extend frequency domain algorithms in ë12ë for causal transfer functions and in ë18ë for miniphase spectral factors to the non-causal case. To describe this in more detail, consider the following sæt rational transfer function with n 1 stable and n 2 unstable poles Gèe,iç è=c 1 èe iç I n1, A 1 è,1 B 1 + C 2 èe iç I n2, A 2 è,1 B 2 + D; è9è with A 1 an n 1 æ n 1 stable matrix and A 2 an n 2 æ n 2 antistable matrix. The parameters on the right hand side of è9è can be estimated from measured frequency data Gèe,iç j è, ç j =2çj=n f, j =0;:::;n f,1, by the following algorithm, that we call subspace identiæcation of frequency data èsifdè. We assume that n f é 2èn 1 + n 2 è and, only for notational simplicity, that n f is even and that s ç t. Algorithm SIFD 1. Input : n f equally spaced frequency measurements Gèe,iç j è, ç j =2çj=n f, j =0;:::;n f,1, and a given dimension n of the state space, with 2n én f. 2. Determine the inverse discrete Fourier transform of the data, denoted by çg j ;j =0;1;æææ;n f,1. 3. Determine a singular value decomposition of the following èsn f =2èæètn f =2è block Hankel matrix 0 çg 1 G2 ç æææ çg 2 G3 ç æææ Gnf ç =2 G1+nf ç =2 æ æ æææ æ æ æ æææ æ æ æ æææ æ çg nf =2 çg 1+nf =2 æææ Gnf ç 1 C A =V 1 æv T 2 è10è with V 1 and V 2 unitary and with æ = ëdiagèç 1 ;ç 2 ;æææ;ç snf =2èj0ë where ç 1 ç ç 2 ç æææ ç ç snf =2 ç 0 and the 0-matrix is of size èsn f =2è æ èèt, sèn f =2è. Let æ n be the èsn f =2èæn matrix consisting of the ærst n columns of æ, and deæne K = V 1 æ n. 4. Estimate A by regression in K u = K d A, where K u èk d è is obtained from K by deleting the ærst èlastè block row of K. Transform A to block diagonal form A = diagèa 1 ;A 2 è, with A 1 stable and A 2 antistable, and 6

7 transform and partition the columns of K = ëk 1 ;K 2 ë correspondingly. Deæne C 1 as the ærst block rowofk 1 and C 2 as the èn f =2è-th block row of K Estimate B 1 ;B 2 and D by regression in è9è, with A 1 ;A 2 ;C 1 and C 2 as determined in step 4. This algorithm identiæes reduced order systems of given state dimension n. If the data are noise-free and generated by the system è9è of dimension n 1 +n 2 = n, then the transformed data in step 2 are given by the following expressions èfor certain matrices M i è çg 0 = D + M 0 ; çg j = C 1 A j,1 1 M 1 + C 2 A,n f =2+1 2 A j,1 2 M 2 ; j =1;:::;n f,1: è11è In this case the block Hankel matrix è10è has rank n and it can be shown that SIFD gives an exact reconstruction of the state space parameters in è9è èup to a block diagonal similarity transformationè. 3.3 Identiæcation by IOM and IRM We now describe a non-behavioural approach for the identiæcation of factor models. As the observed process has full rank and the factor model has méq inputs and n states, the approximation should reduce both the number of inputs and the number of states. The reduction of the number of inputs is a nonconventional problem. Within this setting, the principal components model of complexity m in è5è seems a natural starting point, that is, the system with equations Uèçè^w= 0 in è8è. This system has m inputs, but in general it is not ænite dimensional so that it remains to reduce the state dimension of this system. Because the principal component model is most easily described in the frequency domain, it is convenient to employ frequency domain methods to reduce the state dimension. We consider two methods that both use the subspace method SIFD. One method follows an input-output approach, by decomposing the variables of the system è8è into inputs and outputs. The other method is equation oriented and approximates the coeæcients of the ènormalizedè non-rational equations è8è by rational equations. To describe the methods in more detail, we assume that the available information is given in terms of n f values of the spectrum S at the frequencies ç j ;j =0;:::;n f,1. If the data consists of an observed time series, then the spectrum is ærst estimated by smoothing the periodogram of the data. For each frequency the èq, mè æ q matrix Uèe,iç j è is determined from the eigenvalue decomposition of Sèe,iç j è. Note that these matrices are only deæned up to left multiplication with unitary matrices V èe,iç j è. The two methods described below identify systems that do not depend on the chosen eigenvalue decomposition. The ærst method, that we call the input-output method èiomè, is deæned as follows. Let the observed variables be decomposed in m inputs and p = 7

8 q, m outputs, and let the variables be ordered with the outputs ærst and the inputs last. If U = ëu 1 ;,U 2 ë is a corresponding partitioning of the columns of U in è8è, then the transfer function is given by G = U,1 1 U 2. This transfer function is in general non-rational and non-causal, and it does not depend on the chosen eigenvalue decomposition of S. For given frequency data Gèe,iç j è, j =0;æææ;n f,1, a rational approximation ^G = P,1 Q of McMillan degree n is obtained by SIFD. The corresponding behaviour B IOM is described by the polynomial relations ^Uèçè^w= 0 where ^U =ëp;,që. A disadvantage of IOM is the arbitrary selection of inputs and outputs. The second method, that we call the iterative relation method èirmè, does not require this selection. In IRM the relation U is approximated, taking into account that U is only deæned up to left multiplication with a p æ p unitary matrix function V. As a starting point we take the system B IOM. This system can be represented as B IOM =kerèu n è with U n a pæq isometric rational matrix function of McMillan degree n, see ë14ë. Now for each frequency ç j a unitary p æ p matrix V j is determined such that ku n èe,iç j è, V j Uèe,iç j èk is minimal. Then a rational approximation of the frequency data V j Uèe,iç j è is obtained by SIFD, with a corresponding isometric rational function ^Un. The foregoing steps are iterated with this new approximation ^Un,until convergence is reached. If ^U is the ænal approximation, then the identiæed factor behaviour is described by ^Uèçè^w=0. We do not claim that IOM and IRM would be optimal non-behavioural methods. However, IOM suggests itself quite naturally within an input-output framework. Further, IRM is an equation oriented method that does not require achoice of inputs and outputs. As presented here, by taking B IOM as starting point, IRM is meant to improve the approximation obtained by IOM. Perhaps another canonical parametrization of the relation U would give better results, but this is an open question. 4 Simulation Example We illustrate the identiæcation of dynamic factor models with a simple simulation example. We consider two situations, one where the process spectrum S is known and another where the empirical data consists of a ænite sample of length N of the process, in which case the spectrum is estimated by smoothing the periodogram. In both situations we apply the three methods described before, that is, BLS, IOM and IRM. The simulated process is a stationary ARMA process with q = 2 variables and with state dimension n = 3. Represented in state space form, the data generating process is given by çx = Ax + B"; w = Cx + D": è12è Here w denotes the bivariate observed process, " is an èunobservedè two-dimensional white noise process with unit covariance matrix, and x is an èunobservedè threedimensional Markov process. We restrict the attention to representations è12è 8

9 in innovation form, that is, with A and A,BC stable. In our simulation we take D = I and the parameters èa; B; Cè are chosen at random, under the above restrictions. The actual parameters are given by ç A C B D ç = 2 6 4,0:1965 0:0000,0:3143 0:8770,0:4949 0:5200,0:2538,0:6434,0:6621,0:0280 0:0952 0:7954 0:2210,0:2319,0:6759,0:4646,0:2678,0:1856 1:0000 0:0000 0:1833 0:3058 0:4734 0:0000 1: è13è The spectrum S of this process is a 2 æ 2 rational matrix with rank 2 and McMillan degree 6. For this process, behaviours are estimated with complexities m = 1 and n = 1; 2. This corresponds to factor processes ^w that can be generated as in è12è with a single driving process " ènot necessarily white noiseè and with state dimension n = 1; 2. Note that all models with m = 1 are approximations of the process, for all values of the state dimension n é 1. This is in contrast with stochastic model reduction, where m = 2 and hence n = 1; 2 are the only relevant cases in model reduction. An essential step in factor modelling is to reduce the number of inputs m, and then in principle every dimension n is of relevance. However, for simplicity of the presentation we report only the results for n =1;2. In Table 1 we present the results in terms of the error criterion è7è. The principal component model with m = 1 is non-rational and has error èek ~w PC ètèk 2 è 1=2 = 0:8209. In Table 1 the column with N = 1 relates to the case where the true spectral density is used in IOM and IRM and where for BLS è7è is minimized. The next columns show summary statistics on the errors dèw; Bè in è7è of factor models of complexity m = 1 and n =1;2 obtained in 500 simulation runs with samples of length N = For IOM and IRM, n f = 16 frequencies were used, the results are similar for larger values of n f. Further, in IOM the ærst variable is taken as output and the second one as input, the reverse selection gives similar results. Table 1 shows the quality of the approximation in terms of the mean squared error è7è, that is, of dèw; Bè = Z ç tracef Sèe ~,iç ègdç,ç è14è where S ~ is the spectrum of the error process ~w. Figures 1 and 2 show the contributions to this error per frequency, for a single simulation run of length N = 1024 and using n f = 16 frequencies. Similar results are obtained for the other simulation runs. Figure 1 corresponds to the estimated systems of complexity èm; nè =è1;1è, Figure 2 to èm; nè =è1;2è. The four curves in each plot correspond to four diæerent factor models, that is, PCA is the principal component model, OPT is the optimal model of required complexity, and BLS and IRM are the models estimated by these methods for the observed time series of length N. The results of IOM are not shown, as they are comparable with and somewhat inferior to those of IRM. In both ægures, the ærst plot shows 9

10 n N = 1 Mean StDev Median Min Max 1 BLS IOM IRM BLS IOM IRM Table 1: Summary statistics of the error è7è of systems with m = 1 and n =1;2, estimated by BLS, IOM and IRM, for data generated by the process è12,13è. The column with N = 1 corresponds to a known spectrum, the next columns summarize the results of 500 simulations of the process with sample length N = the frequency-wise squared approximation error, that is, tracef ~ Sèe,iç èg, which integrated over ë,ç; çë gives the error è7è. For each frequency, the PCA model givesalower bound for the error of all systems with m = 1 input and with an arbitrary number of states. The second plot is the Nyquist plot of the transfer functions of the estimated systems, taking the ærst variable as output and the second variable as input. That is, if the factor behaviour is described by the equations ëp;,që^w = 0 then the transfer function is given by G = P,1 Q. In the PCA model G is non-rational, whereas for the other three models G has McMillan degree n = 1 in Figure 1 and n = 2 in Figure 2. This leads to the following conclusions. BLS does not only give the smallest errors on average, but as it has also the smallest standard deviation the BLS models are also least aæected by random variations in the data. Further, Figures 1 and 2 show that BLS performs better than IRM for a wide range of frequencies. The same holds true when BLS is compared with IOM, as IOM gives worse results than IRM. If a particular frequency region would be of special interest, this could be taken into accountby applying a weighting ælter W = TT æ in è14è. This problem is simply solved by applying èunweightedè BLS to the transformed data Tw. Summarizing, the simulation results indicate that BLS can give reasonably good approximations of stationary processes by factor processes and that the methods IOM and IRM perform less well. This is of course partly because the approximation step of these algorithms, i.e., the rational approximation of U, is not directly related to the criterion dèw; Bè. Alternative methods might give better results, but in any case the required choice of inputs and outputs or the choice of canonical parameters remains arbitrary in this setting. A more complete analysis and development of alternative methods are topics for further investigation. 10

11 PCA d(w,b)= OPT d(w,b)= BLS d(w,b)= IRM d(w,b)= misfit frequency 3 2 PCA OPT BLS IRM 1 imag axis real axis Figure 1: Frequency domain characteristics of factor models of complexity èm; nè = è1; 1è, estimated for a time series of length N = 1024 generated by the process è12,13è. The ærst plot shows the squared approximation error tracef Sèe ~,iç èg for the factor models PCA èprincipal components with m = 1è, OPT èthe optimal system with èm; nè = è1; 1èè, BLS and IRM èthe systems with èm; nè =è1;1è estimated by BLS and IRMè. The second plot is the Nyquist plot of the corresponding transfer functions, taking the ærst variable as output and the second as input. 11

12 PCA d(w,b)= OPT d(w,b)= BLS d(w,b)= IRM d(w,b)= misfit frequency 3 2 PCA OPT BLS IRM 1 imag axis real axis Figure 2: Frequency domain characteristics of factor models of complexity èm; nè = è1; 2è, estimated for a time series of length N = 1024 generated by the process è12,13è. The ærst plot shows the squared approximation error tracef Sèe ~,iç èg for the factor models PCA èprincipal components with m = 1è, OPT èthe optimal system with èm; nè = è1; 2èè, BLS and IRM èthe systems with èm; nè =è1;2è estimated by BLS and IRMè. The second plot is the Nyquist plot for the corresponding transfer functions, taking the ærst variable as output and the second as input. 12

13 5 Conclusion In this paper we discussed a behavioural approach for the identiæcation of linear systems. The behavioural least squares èblsè method expresses model quality on the observational level, without the need to choose inputs and outputs or a parametric representation of the model. The criterion function evaluates the global æt of the model, that is, it not only considers the local restrictions but also all behavioural restrictions over time intervals of arbitrary length. We applied BLS for the identiæcation of dynamic factor models, with the advantages that all variables are treated in a similar way èno inputs and outputsè and that the obtained model for the factors is a linear system èas compared to the inænite dimensional system obtained by principal componentsè. For reasons of comparison we described two non-behavioural methods for factor modelling. Both methods are based on subspace techniques, one within an input-output setting and the other within an equation error setting. The methods are compared by a simulation example. The results suggest that the behavioural method can be applied with reasonable success in factor modelling. Further, it seems not easy to replace BLS by non-behavioural methods without signiæcantly increasing the approximation error and the sensitivity to random variations. The results in this paper are merely indicative, and a more thorough comparison with existing methods is needed before more general conclusions can be drawn. With respect to the application considered in this paper, this concerns the development of methods to approximate a given spectrum by one of reduced rank and McMillan degree. References ë1ë H. Akaike, Stochastic theory of minimal realization, IEEE Transactions on Automatic Control 19, 1974, pp ë2ë H. Akaike, Canonical correlation analysis of time series and the use of an information criterion, in R. Mehra and D. Lainiotis èeds.è, System Identiæcation: Advances and Case Studies, Academic Press, New York, 1976, pp ë3ë A.C. Antoulas, A new approach to modeling for control, Linear Algebra and its Applications , 1994, pp ë4ë A.C. Antoulas and J.C. Willems, A behavioural approach to linear exact modeling, IEEE Transactions on Automatic Control 38, 1993, pp ë5ë D.R. Brillinger, Time Series: Data Analysis and Theory, Holt, Rinehart and Winston, New York, ë6ë M. Deistler, Symmetric modeling in system identiæcation, in H. Nijmeijer and J.M. Schumacher èeds.è, Three Decades of Mathematical System Theory, Springer, 1989, pp

14 ë7ë J.F. Geweke, The dynamic factor analysis of economic time series models, in D.J. Aigner and A.S. Goldberger èeds.è, Latent Variables in Socio-economic Models, North Holland, 1977, pp ë8ë C. Heij, Deterministic Identiæcation of Dynamical Systems, Lecture notes in control and information sciences 127, Springer, Berlin, ë9ë C. Heij and W. Scherrer, Consistency of system identiæcation by global total least squares, Report 9635, Econometric Institute, Erasmus University Rotterdam, Submitted for publication. ë10ë C. Heij, W. Scherrer and M. Deistler, System identiæcation by dynamic factor models, SIAM Journal on Control and Optimization 35, 1997, to appear. ë11ë L. Ljung, System Identiæcation: Theory for the User, Prentice-Hall, Englewood Cliæs, New Jersey, ë12ë T. McKelvey, Identiæcation of State Space Models from Time and Frequency Data, Linkíoping Studies in Science and Technology, Dissertation no. 380, Linkíoping, ë13ë G. Picci and S. Pinzoni, Dynamic factor analysis models for stationary processes, IMA Journal of Mathematical Control and Information 3, 1986, pp ë14ë B. Roorda, Global Total Least Squares, Tinbergen Institute Research Series 88, Thesis Publishers, Amsterdam, Netherlands, ë15ë B. Roorda, Algorithms for global total least squares modelling of ænite multivariable time series, Automatica 31, 1995, pp ë16ë B. Roorda and C. Heij, Global total least squares modelling of multivariable time series, IEEE Transactions on Automatic Control 40, 1995, pp ë17ë P. Van Overschee and B. De Moor, Subspace algorithms for the stochastic identiæcation problem, Automatica 29, 1993, pp ë18ë P. Van Overschee, B. de Moor, W. Dehandschouter and J. Swevers, A subspace algorithm for the identiæcation of discrete time frequency domain power spectra, Report ESAT-SISTAèTR, Leuven, ë19ë J.H. Van Schuppen, Stochastic realization problems, in H. Nijmeijer and J.M. Schumacher èeds.è, Three Decades of Mathematical System Theory, Springer, 1989, pp ë20ë S. Weiland and A.A. Stoorvogel, Optimal Hankel norm identiæcation of dynamical systems, Automatica 33, 1997, pp

15 ë21ë E. Weyer, System Identiæcation in the Behavioural Framework, Thesis, Department of Engineering Cybernetics, The Norwegian Institute of Technology, Trondheim, Norway, ë22ë J.C. Willems, From time series to linear system, part I: Finite dimensional linear time invariant systems, Automatica 22, 1986, pp ë23ë J.C. Willems, From time series to linear system, part II: Exact modelling, Automatica 22, 1986, pp ë24ë J.C. Willems, From time series to linear system, part III: Approximate modelling, Automatica 23, 1987, pp ë25ë J.C. Willems, Paradigms and puzzles in the theory of dynamical systems, IEEE Transactions on Automatic Control 36, 1991, pp