Abstract
A major focus of applications of doubly constrained spatial interaction models has been population flows. The solutions to such models yield measures of the (constraint) multipliers and also their derivatives with respect to the impedance parameter (see O’Kelly et al. in J Geogr Syst 14:357–387, 2012; O’Kelly in Geogr Anal 42(4):472–487, 2010). These measures are derived by imposing a numeraire after solving the first order conditions of the model. The relative magnitudes of both sets of measures are shown to be robust because they are unaffected by the choice of numeraire. However the author challenges the conceptual and analytical claims about the interpretation of the solutions which are made in the papers listed above and also recommends that the emissivity and attraction concepts be redefined.
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Notes
A similar technique is used in the transportation problem which is solved via linear programming. In that case one dual variable is set to an arbitrary value.
Likewise, flow ratios to a common destination can be respecified in terms of \(\frac{B_k D_k }{B_i D_i }=\exp \left( {\frac{\mu _{\mathrm{k}} }{\mu _{\mathrm{i}} }}\right) \). This ratio is also invariant to the choice of numeraire.
Martyn Senior was in private correspondence with Alan Wilson (see Wilson 2010, p. 385). In addition, O’Kelly (2012, pp.7–8) notes that ‘\(\lambda \) and \(\mu \) are increasing functions of the size of origins and destinations respectively, so that emissivity is correlated with the size (as well as location) of the place’.
Also the adjusted emissivity for origin i in log form, log(A\(_{i})\) = \(\lambda _{i}\) – log(O\(_{i})\), is clearly invariant in relative terms, i.e. log(A\(_{i})\) \(-\) log(A\(_{k})\) to the choice of numeraire.
It could be argued that the differentiation of the Lagrange multipliers, \(\lambda _i\) and \(\mu _j \), by the multiplier associated with the average distance constraint, \(\beta ,\) is unsatisfactory, since \(\beta \) represents part of the solution to the DCSI model, rather than an exogenous parameter. However Roy and Thill (2004, pp. 349–350) note that the DCSI can be respecified, via an invariant Legendre transform, as a maximization model subject to the chosen value(s) of \(\beta \) and will yield the identical (sets of) flow T \(_{ij}\)s, as the original DCSI model. Thus, these derivatives are meaningful analytical constructs.
Evidently these data demands typically imply that the actual JTW matrix is known, whereas the DCSI model can be applied when the actual flows are unknown, but row and column flow sums are known (see, for example O’Kelly and Lee 2005).
The (numeraire invariant) DCSI model based estimates of the average trip lengths to and from the origins and destinations are now replaced by the actual trip lengths, since the iterative approach to deriving the balancing variables and row and column exponents can yield model based computations of average trip lengths which are arbitrarily close to the actual average trip lengths.
We ignore the subscript ‘k’ in O’Kelly et al. (2012) which denotes different groups of workers.
O’Kelly et al. (2012: 366) incorrectly state that the right hand side of their equation (14) is the average distance, rather than the total distance, commuted.
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Watts, M. Analytical and conceptual issues in the interpretation of doubly constrained spatial interaction models. Lett Spat Resour Sci 9, 189–200 (2016). https://doi.org/10.1007/s12076-015-0151-5
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DOI: https://doi.org/10.1007/s12076-015-0151-5