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 ....

[Article contains additional citation context not shown here]

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.

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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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