Simon, H. and Ando, J. (1961). Aggregation of variables in dynamic systems. Econometrica, 29:111--138.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of which is iterative aggregation disaggregation (IAD) have been developed. IAD algorithms do not suffer from these limitations ( 8, 19, 1, 18, 17, 3] The idea in IAD methods is to observe the system in isolation in each of the diagonal blocks as if the system is completely decomposable (see [16]) and to compute the stationary probability distribution of each diagonal block. However, there are two problems with this approach. First of all, since the diagonal blocks are substochastic, the off diagonal probability mass must somehow be incorporated into the diagonal blocks. Secondly, the ....

H. Simon and A. Ando, Aggregation of variables in dynamic systems, Econometrica 29, (1961), 111--138.

....space generation and numerical analysis of very large monolithic CTMCs is often not feasible in practice due to memory and CPU time limitations, which is referred to as the notorious state space explosion problem. A large state space may become tractable if it is decomposed into smaller parts [95, 33]. Instead of analysing one large system, the decomposition approach relies on analysing several small subsystems, analysing an aggregated overall system, and afterwards combining the subsystems solutions accordingly. In general, this approach works well for nearly completely decomposable (NCD) ....

H.A. Simon and A. Ando. Aggregation of Variables in Dynamic Systems. Econometrica, 29:111--138, 1961.

....and very few hosts suffer from the GeoCities effect. 4 BlockRank Algorithm We now present the BlockRank algorithm that exploits the empirical findings of the previous section to speed up the computation of PageRank. This work is motivated by and builds on aggregation disaggregation techniques [5, 17] and domain decomposition techniques [6] in numerical linear algebra. Steps 2 and 3 of the BlockRank algorithm are similar to the Rayleigh Ritz refinement technique [13] We begin with a review of PageRank in Section 4.1. 4.1 Preliminaries In this section we summarize the definition of PageRank ....

H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29:111--138, 1961.

....and very few hosts suffer from the GeoCities effect. 4 BlockRank Algorithm We now present the BlockRank algorithm that exploits the empirical findings of the previous section to speed up the computation of PageRank. This work is motivated by and builds on aggregation disaggregation techniques [5, 17] and domain decomposition techniques [6] in numerical linear algebra. Steps 2 and 3 of the BlockRank algorithm are similar to the Rayleigh Ritz refinement technique [13] We begin with a review of PageRank in Section 4.1. 4.1 Preliminaries In this section we summarize the definition of PageRank ....

H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29:111--138, 1961.

....at the same level and lower interactions with other components. Near decomposability has been observed in other domains than computing: in economics, in biology, genetics, social sciences. The pioneers in this domain are Simon and Ando who studied several study cases in economics and in physics [10]. What they stated is that aggregation of variables in a nearly decomposable system must separate the analysis of the short term and long term dynamics. They proved two major theorems. The first says that a nearly decomposable system can be analysed by a completely decomposable system if the ....

.... projector (i.e. p ij = 0 8i; j 6= l, p ll = 1) So P 1 (1)P P I (1)P P 1 (1)P can be replaced by Z(i I ) The properties of Z(i I ) are given in [9] Similarly for A , we have = P P 1 (1)P P I (1)P Here we will give the first theorem of Simon and Ando [10] without demonstration: Theorem 1 For an arbitrary positive number , there exists a number ffl such that for ffl ffl , max k;l jz kl (i I ) Gamma z kl (i I )j (12) with 2 i n(I) 1 I N; 1 k; l n Let us now focus our attention on the implication of this theorem. The ....

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H. A. Simon and A. Ando, "Aggregation of variables in dynamic systems," Econometrica, no. 29, 1961. 13

....at the same level and lower interactions with other components. Near decomposability has been observed in other domains than computing: in economics, in biology, genetics, social sciences. The pioneers in this domain are Simon and Ando who studied several study cases in economics and in physics [16, 17, 18]. What they stated is that aggregation of variables in a nearly decomposable system must separate the analysis of the short term and long term dynamics. They proved two major theorems. The first says that a nearly decomposable system can be analyzed by a completely decomposable system if the ....

.... pi3 pi4 pi5 pi1 (decomposable matrix) pi2 pi3 pi4 pi5 state probabilities time (slots) Figure 2: Behaviors comparison of t and t , a = 6:7, b = 0:576, n = 5 Similarly for A , we have = P I=2 (1 I )P (14) Here we will give the first theorem of Simon and Ando [16] without demonstration: Theorem 1 For an arbitrary positive number , there exists a number ffl such that for ffl ffl , max k;l jz kl (i I ) Gamma z kl (i I )j (15) with 2 i n(I) 1 I M; 1 k; l n Let us now focus our attention on the implication of this theorem. The ....

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H. Simon and A. Ando, "Aggregation of variables in dynamic systems," Econometrica, no. 29, 1961.

....to mean first passage times is also presented. MARKOV CHAINS; FUNDAMENTAL MATRIX; SINGULAR PERTURBATION AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J10;47A55 SECONDARY 60J35;15A51 1. Introduction Singularly perturbed Markov chains have been studied in the pioneering works of Simon and Ando [31], Schweitzer [27] Courtois [6] Pervozvanskii et al. [10, 22, 23] Korolyuk and Turbin [17] Delebecque and Quadrat [7, 8] Phillips and Kokotovic [24] and later by Coderch et al. [5] Rohlicek and Willsky [25] Filar, Bielecki and Abbad [1, 3] Latouche and Louchard [20, 21] Hassin and Haviv ....

Simon, H.A., and Ando, A. (1961) Aggregation of variables in dynamic systems. Econometrica, 29, 111--138.

....And hence, as follows from formula (3.7) the ergodic projection corresponding to the unperturbed chain has a larger rank than the one corresponding to the perturbed Markov chain. Probably, the rst motivation to study the singular perturbed Markov chains was given in the paper by Simon and Ando [137]. They demonstrated that several problems in econometrics lead to the mathematical model based on singularly perturbed Markov chains. The rst rigorous theoretical developments of the singularly perturbed Markov chains have been carried out by Pervozvanski and Smirnov [117] and Gaitsgori and ....

H.A. Simon and A. Ando, \Aggregation of variables in dynamic systems", Econometrica, v.29, pp.111-138, 1961.

....technique with the hierarchical decomposition of the state space into its metastable sets. The key idea is to regard metastable sets as almost invariant sets w.r.t. some propagation operator corresponding to the Markov chain. UCMC combines aspects from aggregation disaggregation techniques [30], from stochastic complementation for nite state space Markov chains [23] and from simulated annealing approaches in optimization [20] in a way comparable to hierarchical annealing structures also used in the macrostate dissection approach for thermodynamical integrals [3] It should be ....

H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29(2):111-138, 1961.

....into its metastable sets. The key idea of UCMC is to regard metastable sets as almost invariant sets w.r.t. some propagation operator corresponding to the Markov chain. Furthermore it combines aspects from simulated annealing approaches in optimization [19] aggregation disaggregation techniques [28] and stochastic complementation techniques [22] for nite state space Markov chains. A hierarchical annealing structure is also used in the macrostate dissection approach for thermodynamical integrals [3] UCMC essentially di ers from these approaches by the consequent iterative decomposition into ....

H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29(2):111-138, 1961.

....efficiency characteristics is proposed. Applications in queueing models are also considered. We mention that the aggregation technique for finite Markov chains is studied in various papers. Mostly these results are devoted to the analysis of a steady state ( 11,16] or transient probabilities ([10,14]) see also references therein) Our approach is based on the results about the weak asymptotic merging of a state space for Markov systems as well as limit theorems for switching processes ( 2 5] and gives the possibility to study not only the behaviour of state probabilities but also a weak ....

H. A. Simon and A. Ando, Aggregation of Variables in Dynamic Systems, Econometrica, 29, 2 (1961) 111-138.

....systems the relevant time scale for users may be seconds, while the relevant time scale for system transactions may be milliseconds or microseconds. In stochastic models, di#erent time scales can be represented by nearlycompletely decomposable (NCD) Markov chains; e.g. see Simon and Ando [38], Courtois [12] Latouche [26] Philippe, Saad and Stewart [34] Chang and Nelson [3] and Latouche and Schweitzer [28] In an NCD Markov chain, the state space can be decomposed into subsets such that the chain usually tends to move around within each subset and only rarely moves from one subset ....

Simon, H. A. and Ando, A. (1961) Aggregation of variables in dynamic systems. Econometrica 29, 111--138.

....densities in the uncoupling step for ecient approximations of ratio of normalizing constants [12] which arise as components of C. The UCMC scheme combines aspects from simulated annealing approaches in optimization [19] aggregation disaggregation techniques evolved from the Simon Ando theory [28] and stochastic complementation techniques investigated by Meyer [23] on nite state space Markov chains. A hierarchical annealing structure is also used by Church et al. in the macrostate dissection approach for thermodynamical integrals [3] there the strategy is followed to dissect the state ....

....to T , so that we end up with T as a reversible stochastic matrix. The existence of almost invariant sets and its relation to the spectral structure is already treated in the Simon Ando theory for nite Markov chains and investigated there under the term nearly completely decomposable systems [23, 28]. However, therein only such systems are studied for which a suitable decomposition is known in advance. In contrast, an algorithmic approach has to determine the almost invariant sets for given eigenvalues and corresponding eigenvectors. This is done in an identi cation algorithm proposed by ....

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H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29(2):111-138, 1961.

....flow through membranes on a time scale of minutes, growth requires days, and surrounding ecological processes may occur on a time scale of months or years. Given the time scale of interest for a question, any influence that causes significant change only on a slower time scale is insignificant [24, 28, 51]. For example, to answer the question concerning the effect of decreasing soil moisture on a plant s transpiration rate, a time scale of hours is most appropriate; since the effects of growth are significant only on a time scale of days or longer, they are insignificant for purposes of answering ....

....in this paper do not depend on this particular criterion for significance. In the future, we plan to incorporate other criteria as well, as discussed in Section 6.1.1. Still, time scale is an important significance criterion in many domains, including biology [19, 49] ecology [2, 40] economics [51], and many branches of engineering 5 In qpt, activity preconditions are called quantity conditions. 6 [26, 48] Moreover, empirical results (described in Section 5) show that this criterion is capable of pruning many irrelevant phenomena from models. 2.1.6 Validity Preconditions Many ....

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H.A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29:111--138, 1961.

....at the same level and lower interactions between other components. Near decomposability has been observed in other domains than computing: in economics, in biology, genetics, social sciences. The pioneers in this domain are Simon and Ando who studied several study cases in economics and in physics [11]. What they stated is that aggregation of variables in a nearly decomposable system must separate the analysis of the short term and long term dynamics. They proved two major theorems. The first says that a nearly decomposable system can be analyzed by a completely decomposable system if the ....

.... I ) are the elements of Z(i I ) The properties of Z(i I ) are given in [10] Similarly for A , we have A = P Gamma1 P 1 (1)P N X I=2 (1 I )P Gamma1 P I (1)P N X I=1 n(I) X i=2 (i I )P Gamma1 P I (i)P (14) Here we will give the first theorem of Simon and Ando [11] without demonstration: Theorem 1 For an arbitrary positive number , there exists a number ffl such that for ffl ffl , max k;l jz kl (i I ) Gamma z kl (i I )j (15) with 2 i n(I) 1 I N; 1 k; l n Let us now focus our attention on the implication of this theorem. The ....

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H. Simon and A. Ando, "Aggregation of variables in dynamic systems," Econometrica, no. 29, 1961. 14

....of this kind are used to model systems whose states can be grouped into aggregates that are loosely connected to one another. They have been 1 Research supported by Alexander von Humboldt Foundation of Germany and Natural Sciences Foundation of China. 1 addressed by many authors, see e.g. [1, 2, 3, 4, 8, 9, 10, 12, 13]. One reason why nearly uncoupled Markov chains receive so much attention is that their stationary distributions are very sensitive to the perturbations in the transition matrices. Let T and b T be stationary distributions of transition matrices P and b P = P F , respectively; that is, ....

H.A. Simon and A. Ando, Aggregation of variables in dynamic systems, Econometrica, 29(1961), pp.111-138.

.... is generation of a model of a coarse temporal grain size from a model of a finer grain size by making assumptions about the relative adjustment speeds of the equations in the model (Iwasaki 1988a) The other technique is aggregation of a dynamic system model when the system is nearly decomposable (Simon Ando 1961, Iwasaki Bhandari 1988) The two techniques are briefly described in Section 2. Section 3 discusses the effects of model abstraction elaboration on causal ordering in various cases. 2 Model Abstraction Techniques The types of abstraction discussed in this paper is temporal abstraction, where ....

....long run behavior of the entire system in terms of these subsets instead of individual variables, treating each subset as a black box. Simon and Ando proved that this was indeed true for the case of a nearly decomposable dynamic matrix with one significant characteristic root for each subsystem (Simon Ando 1961). Consider a self contained dynamic system M and its matrix P of coefficients such that P is almost block diagonal except for small (less than e for some small e elements outside the diagonal blocks. P looks like, P = P 1 2 N P P .e e 5 where the elements of P outside of the ....

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Simon, H. A. and Ando, A. Aggregation of Variables in Dynamic Systems. Econometrica 29, 1961.

....in the tool s methodology. 4 The state explosion problem is due to the need to model real systems which often result in extremely large models and sometimes infinite ones (as explained below) Numerous research efforts have focused on development of methodologies (including our own work) [3, 17, 6] that address this state explosion problem. However, what is needed (and currently lacking) in order to apply existing methodologies as well as facilitate development of new ones, is a tool that is able to easily incorporate such approaches. For reasons explained below, we believe that a tool ....

H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29:111-- 138, 1961.

....QAM systems see [3] The general techniques of this paper can be applied to such modified QAM systems with clustered state values in order to reduce the computationalcomplexity of demodulation. Much work has been carried out into grouping states associated with Markov chains (see for example, [7]) Techniques such as stochastic complementation [8] are sophisticated methods of producing reduced complexity representations of Markov chains which have large numbers of states. They have been used mainly to evaluate steady state probability distributions [9] and reduced order controllers for ....

H. A. Simon and A. Ando, "Aggregation of variables in dynamic systems," Econometrica, vol. 29, no. 2, pp. 111--138, 1961.

....distributions by phase type distributions leads to an enormous growth of the state space. Various approaches to attack the state space explosion problem have been pursued so far. Without aiming to be exhaustive we mention product form solutions [BCMP75] decomposition based solution [SA61,CS85] tensor based representations [PA91,Buc94,Don93,Sie94b] symmetry exploitation [CDFH93,HR98,Sie94a] and partial order representations [BKLL95] In the field of design and verification of digital circuits, state space explosion is a similarly omnipresent phenomenon. During the last decade, ....

H.A. Simon and A. Ando. Aggregation of Variables in Dynamic Systems. Econometrica, 29:111--138, 1961.

....of the simulation. Furthermore, the behavior of each component is reasoned about separately from the interactions between components. This architecture allows other decomposition methods to be applied to provide alternative abstraction techniques. Iwasaki and Bhandari [1988] build upon Simon s [1961] techniques for variable aggregation in dynamic structures. They provide a formal analysis of how variables within a system of equations can be partitioned and aggregated based upon a quantitative analysis of the equations and their roots. Some of these techniques could be extended to address the ....

H. A. Simon and A. Ando. Aggregation of Variables in Dynamic Systems. In Econometrica, Vol 29, 1961.

....modeling, our research has been guided by the practices of human modelers in biology, ecology, economics, and engineering. While human modelers rarely offer operational advice for automating their task, their textbooks (e.g. 4, 14, 31, 34, 45, 49, 59, 75, 76] and journal articles (e.g. [32, 38, 39, 40, 50, 69, 78, 79, 82]) often reveal the modeling alternatives they consider and the criteria for their choices. 1.10 Summary of Contributions and Results Our research provides three types of contributions. First, we formulated declarative criteria that specify when a model is adequate for answering a prediction ....

....of this process are irrelevant, the modeler can simply treat the level of water as an instantaneous function of the level of solutes, and this functional dependence can be represented with a functional influence. Quasi static approximations are important in many branches of science and engineering [14, 46, 77, 78, 80, 82]. In fact, several branches of engineering, notably circuit theory and equilibrium thermodynamics, rest on such approximations [13, 83] A functional influence that represents a quasi static approximation is called an equilibrium influence. An equilibrium influence summarizes the net effect of ....

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H.A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29:111--138, 1961.

....of the whole state space at once is avoided. Only one partition at a time is held in memory. The accuracy of the results is excellent for systems with NCD structure. The algorithm is based on existing work on SPNs [8] and goes back to the decomposition aggregation scheme of Simon and Ando [61]. Of course, there are some drawbacks, in particular due to the restriction that the partitions need to be solvable by steady state analysis, i.e. they have to be irreducible. If this is not the case, an additional error is introduced. Response Time Approximation Response time approximation (RTA) ....

H.A. Simon and A. Ando. Aggregation of Variables in Dynamic Systems. Econometrica, 29:111--138, 1961.

....to one another. Since the system spends a relatively large amount of time in a group before passing on to another, it seems natural that a numc would achieve steady states within groups rather quickly and a steady state between groups more slowly. This behavior was first noted by Simon and Ando [10] and has been the subject of much subsequent research (see, e.g. 2, 11, 1, 6] One practical difficulty with numcs is that it is difficult to determine the off diagonal elements accurately, at least empirically. This is because transitions between blocks are rare events. Consequently, the ....

H. A. Simon and A. Ando (1961). "Aggregation of Variables in Dynamic Systems." Econometrica, 29, 111--138.

.... after one application (because T z = T z ) In this paper we shall analyze only the Gauss Seidel step, since the Rayleigh Ritz step has been analyzed elsewhere [4] The latter is closely related to methods of aggregation, which were first proposed in this connection by Simon and Ando [5]. The idea is to determine a vector z such that T z is a good approximation to T y and then apply a Rayleigh Ritz step to approximate y. The diagonals of T z are usually found by solving eigenvalue problems associated with with the A ii . For example, Stewart [8] takes the z i to be the ....

....: C l ) O(ffl) 2:21) From Regularity Condition 2, the eigenvalues of the C i are bounded away from one. Consequently, A s 3 approaches zero more swiftly than A s 2 ; that is, Y 3 corresponds to a fast transient. This behavior of nearly uncoupled chains was first noted by Simon and Ando [5]. We note in passing that a condition different from Regularity Condition 2 was used by the authors of this paper to establish the results in [4] Two Stage Iteration 9 However, that condition is implied by Regularity Condition 2 along with (2.20) and (2.21) The spaces associated with the ....

H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29:111--138, 1961.

.... unit eigenvalue implies a slow rate of convergence for standard matrix iterative methods [12] IAD methods do not suffer from these limitations [7] 23] The idea in IAD methods is to observe the system in isolation in each of the diagonal blocks as if the system is completely decomposable (see [15]) and to compute the stationary probability distribution of each diagonal block. However, there are two problems with this approach. First, since the diagonal blocks are substochastic, the off diagonal probability mass must somehow be incorporated into the diagonal blocks. Second, the ....

H. Simon and A. Ando, Aggregation of variables in dynamic systems, Econometrica, 29 (1961), pp. 111--138.

....of the whole state space at once is avoided. Only one partition at a time is held in memory. The accuracy of the results is excellent for systems with NCD structure. The algorithm is based on existing work on SPNs [6] and goes back to the decomposition aggregation scheme of Simon and Ando [44]. Of course, there are some drawbacks, in particular due to the restriction that the partitions need to be solvable by steady state analysis, i.e. they have to be irreducible. If this is not the case, an additional error is introduced. Figure 8: Compositional reduction by selecting parts of the ....

H.A. Simon and A. Ando. Aggregation of Variables in Dynamic Systems. Econometrica, 29:111--138, 1961.

....goals, and it is indicated how this can lead to fully parallel algorithms for the determination of the stationary distribution vector of the original chain. Finally, in the third part of this survey, the application of stochastic complementation to the classical Simon Ando theory developed in [18] for nearly completely reducible chains is presented. It is demonstrated how to apply the concept of stochastic complementation in order to develop the theory for nearly completely reducible systems in a unified, clear, and simple manner while simultaneously sharpening some results and ....

....the limiting vectors s i = lim n## s (n) i are easily coupled according to the rules of Theorem 4.1 in order to produce the stationary distribution vector for P as # = # 1 s 1 # 2 s 2 # k s k ) 7. The Simon Ando theory for nearly completely reducible systems. Simon and Ando [18] provided the classical theory for nearly completely reducible systems, and Courtois [3] along with others who followed his pioneering work) applied the theory and helped develop numerical aspects associated with queueing networks. The contribution of Simon and Ando [18] was to provide ....

[Article contains additional citation context not shown here]

H. A. Simon and A. Ando, Aggregation of variables in dynamic systems, Econometrica, 29(1961), pp. 111--138.

....It is apparent that the assumption that the subsystems are independent and can therefore be solved separately does not hold. Consequently an error arises. This error will be small if the assumption is approximately true. The pioneering work on NCD systems was performed by Simon and Ando, [51], in investigating the dynamic behavior of linear systems as they apply to economic models. The concept was later extended to Markov chains and the performance analysis of computer systems by Courtois, 11] The technique is founded on the idea that it is easy to analyse large systems in which all ....

H.A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica 29, pp. 111-138, 1961.

....1994) for which the computational complexity is in the order of the square of the number of parameters. The challenge is to achieve reductions in the computational complexity of such algorithms. Much work has been carried out into grouping states associated with Markov chains (see for example, Simon and Ando (1961)) Techniques such as stochastic complementation (Meyer 1989) are sophisticated methods of producing reduced complexity representations of Markov chains which have large numbers of states. They have been used mainly to evaluate steady state probability distributions (Cao and Stewart 1985) and ....

Simon, H. A. and A. Ando (1961). Aggregation of variables in dynamic systems. Econometrica 29(2), 111--138.

....that is, one which, after a suitable reordering of the states, is almost block diagonal. For the case of three blocks, such a chain has the form P = 0 B P 11 E 12 E 13 E 21 P 22 E 23 E 31 E 32 P 33 1 C A ; where the matrices E ij are small. Such chains were introduced by Simon and Ando [11], and have been studied extensively since (e.g. see [1, 13] Since the E ij are small, each of the matrices P ii has an eigenvalue near one. Consequently the entire matrix, in addition to an eigenvalue of one, has k Gamma 1 eigenvalues near one, where k is the number of blocks. Consequently, ....

H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29:111--138, 1961.

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Simon, H. and Ando, J. (1961). Aggregation of variables in dynamic systems. Econometrica, 29:111--138.

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H. A. Simon and A. Ando. 1961. Aggregation of variables in dynamic systems. Econometrica 29: 111-138.

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H.A. Simon and A. Ando, `Aggregation of variables in dynamic systems', Econometrica, vol.29, pp. 111-138, 1961.

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H.A. Simon and A. Ando, \Aggregation of variables in dynamic systems", Econometrica, v.29, pp.111-138, 1961.

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Herbert A. Simon and Albert Ando. Aggregation of variables in dynamic systems. Econometrica, 29:111--138, 1961. 19

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H. A. Simon and A Ando. Aggregation of variables in dynamic systems. Econometrica, 29:111--138, 1961.

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H.A. Simon and A. Ando, \Aggregation of variables in dynamic systems", Econometrica, v.29, pp.111-138, 1961.

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H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, pages 29: 111--138, 1961.

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H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29:111--138, 1961.

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H.A. Simon and A. Ando, \Aggregation of variables in dynamic systems", Econometrica, v.29, pp.111-138, 1961.

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H.A. Simon, A. Ando, Aggregation of variables in dynamic systems, Econometrica 29 (1961) 111--138.

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Herbert A. Simon and Albert Ando. Aggregation of variables in dynamic systems. Econometrica, 29(2):111--138, April 1961.

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H.A. Simon and A. Ando. Aggregation of Variables in Dynamic Systems. Econometrica 29:111-138, 1961.

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H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29(2):111-138, 1961.

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H.A. Simon and A. Ando. Aggregation of Variables in Dynamic Systems. Econometrica, 29:111--138, 1961.

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H. A. Simon and A. Ando : "Aggregation of Variables in Dynamic Systems," Econometrica, Vol. 729, p.111--138, (1963).

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H. A. Simon and A. Ando. 1961. Aggregation of variables in dynamic systems. Econometrica 29: 111-138.

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Report # 10018701 H. A. Simon and A. Ando [1961] Aggregation of variables in dynamic systems, Econometrica, Vol.

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H. A. Simon and A. Ando : "Aggregation of Variables in Dynamic Systems," Econometrica, Vol. 729, p.111--138, (1963).

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