Derivation of the Negative Exponential Model by an Entropy Maximising Method

Abstract

All the inhabitants of a city who participate in the choice of a place of residence are assumed to have a propensity to visit the urban centre. The distribution of residential locations is represented by a probability density surface whose horizontal plane projection is coextensive with that of the city. A spatially continuous system is thus defined in which it is shown that, under conditions of maximum entropy and subject to specific normalisation and cost constraints, the population is distributed in accordance with the negative exponential model of urban population densities.

Population and Employment Densities: Structure and Change

“…The negatwe exponent:al ts the most commonly used density function m the monocentnc~ty hterature, and ~t is used m th~s paper In the case of population dens:ty, this spec~ficatlon can be derwed theoretacaUy e:ther from a utlhty-maxam~zmg model w:th a umtary value of the price elastlc~ty of housing servxces [28,29,31] or from entropy maxamlzatlon [3] In the case of employment dens:ty, the negatwe exponential form ~s derived by Malls [25] by assuming that the productlon functxons for product and transportation have the Cobb-Douglas form. and that the demand for product has a constant price elast~c:ty We gwe this negatwe exponentml form a multxphcatwe error structure, wh:ch ~s supported (relative to an ad&twe structure) by ewdence reported by Greene and Barnbrock [t4] To summarize, the vers:on of the monocentnc model that we examine empmcally :s…”

Spatial Entropy

“…AX-0 Equation (8) demonstrates that the discrete entropy of equation (1) increases without bound for the case in which the Ax,'s are equal. This argument can be easily generalized to the case where each Axiis different.…”

“…Information gain is calculated from equation ( can be assessed using The effect of changing the interval size Axi such statistics. For example, equation (11) provides a discrete approximation to equation (8) for given Axi, and a measure of information loss can be calculated by comparing such approximations with their continuous forms. Mathematically, the problem involves approximating an integral by a finite sum and in a geographical context, such interpreta-tions are helpful in partitioning a spatial system into zones.…”

Understanding the Fractal Dimensions of Urban Forms through Spatial Entropy

“…Equation (8) is identical in form to Equation (7). Comparing Equation (8) with Equation (7) suggests that fractal dimension D q is just the characteristic value of general entropy M q relative to the linear scale ε.…”