×

Lumpability in compartmental models. (English) Zbl 0709.93073

In order to reduce the number of states for a system, lumping of states into classes represents a natural and useful procedure. The present paper discusses an alternative to the classical statistical approach: the use of fuzzy set theory in lumping methodologies A fuzzy entropy compares successive lumpings, as a measure of the intrinsic loss of information by the lumping process.
Section 2 exposes the stochastic (Markov chain-based) approach to lumping for compartmental models of systems: the probabilistic structure of the random system with complete connections (RSCC) model is exposed. Section 3 analyses the particular type of lumping corresponding to systems with a limited number of observed compartments. Section 4 contains the numerical results obtained from applying the RSCC probabilistic model to such a system. Section 5 exposes the (new) proposed fuzzy model of lumping. Defining the fuzzy relations of similitude and the fuzzy entropies, the probabilities of the stochastic model are replaced by the (fuzzy) similitude degrees. The fuzzy lumping algorithm is exposed, its correctness is discussed. A numerical example, in Section 6, provides evidence on the usefulness of the proposed approach.
Since the study of entropy for lumping in stochastic and fuzzy models represents a difficult problem, in practical situations it is recommended to choose that lumping technique which gives smaller loss of entropy. The fuzzy approach seems to offer better results when the available data are insufficient or unreasonable, i.e. having a noise-tolerant better behaviour.
Reviewer: S.Curteanu

MSC:

93E12 Identification in stochastic control theory
93E10 Estimation and detection in stochastic control theory
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
93C42 Fuzzy control/observation systems
93A30 Mathematical modelling of systems (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aoki, M., Control of large scale systems by aggregation, IEEE Trans. Automat. Control, 13, 246 (1968)
[2] Aström, K. J., Optimal control of Markov processes with incomplete state information, J. Math. Anal. Appl., 10, 174-205 (1965) · Zbl 0137.35803
[3] Kemeny, J. G.; Snell, J. L., Finite Markov Chains (1960), Van Nostrand: Van Nostrand New York · Zbl 0112.09802
[4] D. BlackwellAcad. Sci.; D. BlackwellAcad. Sci.
[5] Grigorescu, S.; Iosifescu, M., (Stiintificǎşi Enciclopedicǎ, Dependence with Complete Connections (1982)), [in Romanian]
[6] Zadeh, L., Probability measures of fuzzy events, J. Math. Anal. Appl., 23, 421-427 (1968) · Zbl 0174.49002
[7] De Luca, A.; Termini, S., A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory, Inform. and Control, 20, 301-312 (1972) · Zbl 0239.94028
[8] Gibilaro, L. G.; Kropholler, H. W.; Spiking, D. T., Solution of a mixing model due to van de Vusse by a simple probability method, Chem. Eng. Sci., 22, 517-523 (1967)
[9] Iosifescu, M.; Theodorescu, R., Random Processes and Learning (1969), Wiley: Wiley New York · Zbl 0194.51101
[10] Negoiţǎ, C. V.; Ralescu, D. R., (Tehnicǎ, Fuzzy Sets and their Applications (1974)), (in Romanian) · Zbl 0326.94001
[11] (English ed., Birkhäuser, Basel/Stuttgart, 1975).; (English ed., Birkhäuser, Basel/Stuttgart, 1975).
[12] Trillas, E.; Riera, T., Entropy in finite fuzzy sets, Inform. Sci., 15, 159-168 (1978) · Zbl 0436.94012
[13] Kaijser, T., A limit theorem for partially observed Markov chains, Ann. Probab., 3, 677-696 (1975) · Zbl 0315.60038
[14] Iordache, O., Polystochastic Models in Chemical Engineering (1987), V.N.U. Science: V.N.U. Science Utrecht
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.