Gravity‐Type Interactive Markov Models—Part I: A Programming Formulation of Steady States

Smith, Tony; Hsieh, Shang-Hsing

doi:10.1111/0022-4146.00074

Cited by 2 publications

(4 citation statements)

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“…Quasi-symmetry amounts to the reversibility of weights w jk (theorem 1 below; see also Bavaud, 1998), that is, to the phenomenological identity between the Markov process and its time-inverted associated process. Analogous results also hold for more general interactive' Markov processes (Smith and Hsieh, 1997a), where the stability of the stationary flows is guaranteed again by quasi-symmetry (Smith and Hsieh, 1997b). Also, quasi-symmetric weights possess real eigenvalues, which considerably simplifies the study of their time evolution, not addressed here.…”

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confidence: 57%

The Quasi-Symmetric Side of Gravity Modelling

Bavaud

2002

Environ Plan A

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323) have observed various spatial flows, mobility tables, or opinion-shift data n jk to be well described by the quasi-symmetric model, n jk a j b k g jk , with g jk g kj .As a matter of fact (theorem 1 below), quasi-symmetry precisely enables us to parameterize flows as fng fr, g, ug, where frg are the trivial size effects, fgg are the size-independent accessibilities quantifying the distance-deterrence effect, the symmetric component of migration, and fug are the size-independent utilities quantifying the places' attractivities or push^pull drive, the antisymmetric component of migration.The above ingredients are those of gravity modelling, or more precisely constitute its endogenous aspect (the quasi-symmetry condition). Its exogenous aspect relates (typically by a multiple regression) the accessibilities fgg to travel costs between places, on one hand, and the utilities fug to local socioeconomic conditions, on the other hand.In this paper, I seek to derive or to reveal utility and accessibility parameters from the observation of flows, in the same sense that the sole observation of purchases for a given budget level reveals consumers' utility. As such, this approach is thus also of potential relevance for other nonspatial domains such as the study of mobility tables or opinion shifts. My study is endogenous in that I limit myself to study the theoretical properties of the parameters fg, ug so defined. I do not address the predictive, exogenous part of the program, namely the confrontation of the parameters with socioeconomic and effective distance data.More general, non-quasi-symmetric gravity models can indeed be considered, as, for example, in the work of Sen and Smith (1995), where the symmetry assumption on the accessibilities fgg is dropped. Although perfectly acceptable in a traditional predictive approach, and in particular testable (Smith, 1984), such an enlargement of the class of models under consideration is not feasible in the revealed approach: without quasi-symmetry, the`revealed' parameterization fr, g, ug of flows does not hold any more, and the question of how to construct a mathematically rigorous, behaviorally interpretable parameterization in the general, non-quasi-symmetric case remains open.Abstract. Many properties of gravity models are sole consequences of the quasi-symmetry condition or its avatars. We investigate here quasi-symmetry per se, in contrast to geographical tradition, which has been more focused on the exogenous socioeconomic and spatial conditions. In particular, thè size^utility^accessibility' parameterization of migration counts turn out to rely exclusively upon the quasi-symmetry of flows. Various facets of quasi-symmetry are presented and put in correspondence with Markov chains theory, Bradley^Terry^Luce decision theory, the Weidlich^Haag model, and alternative classical statistical models (marginal homogeneity, symmetry, independence). Existing as well as presumably new estimation and model selection procedures (maximum likelihood, minimum discrimination informati...

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confidence: 57%

The Quasi-Symmetric Side of Gravity Modelling

Bavaud

2002

Environ Plan A

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“…These positive solutions thus yield a well-defined function, ?Ira,f: P : -S, which we now designate as the flow-correspondence mapping for Ma,f. The following result, proved in Smith and Hsieh (1996), shows that the mapping ?ya,f satisfies the desired correspondence properties, together with certain additional mapping properties. In particular, if the range of Va,f is restricted to the image set, [Wf] -1(P) = P(-I),…”

Section: (16)mentioning

confidence: 87%

“…The next result shows that such a neighborhood can always be reached. In particular, it can be verified that (see Smith and Hsieh, 1996):…”

Section: @(Pi = [W(ayal+(p)mentioning

confidence: 98%

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Gravity‐Type Interactive Markov Models—Part II: Lyapunov Stability of Steady States

Smith

Hsieh

1997

Journal of Regional Science

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In Part I of this paper (Smith and Hsieh, 1997) a programming formulation of steady states was developed for gravity-type interactive Markov chains in terms of their associated spatial-flow chains. These results are here applied to analyze the stability properties of interactive Markov chains. In particular, the objective function for this programming formulation is shown to constitute a Lyapunov function for an appropriately defined continuous-time version of spatial-flow chains. The Lyapunov stability properties of these spatial flows are then shown to yield corresponding stability properties for the continuous-time versions of interactive Markov chains. In particular, these processes always exhibit global convergence to steady states. Finally, it is shown that when steady states are unique, these convergence results are inherited by those interactive Markov chains that are 'sufficiently close' to their continuous-time versions.

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Gravity‐Type Interactive Markov Models—Part I: A Programming Formulation of Steady States

Cited by 2 publications

References 12 publications

The Quasi-Symmetric Side of Gravity Modelling

The Quasi-Symmetric Side of Gravity Modelling

Gravity‐Type Interactive Markov Models—Part II: Lyapunov Stability of Steady States

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