Spatial Aggregation in Gravity Models. 1. An Information-Theoretic Framework

1982 ◽  
Vol 14 (3) ◽  
pp. 377-405 ◽  
Author(s):  
M Batty ◽  
P K Sikdar
Keyword(s):  
Spatial Interaction ◽  
Spatial Aggregation ◽  
Gravity Models ◽  
Model Parameters ◽  
Data Sets ◽  
Density Level ◽  
Information Theoretic ◽  
Spatial Interaction Models ◽  
Spatial Entropy ◽  
Information Components

This is the first of four papers concerned with developing a comprehensive approach to the spatial aggregation problem in gravity models. A framework for exploring the problem is outlined in this paper, and will be applied to one-dimensional and two-dimensional spatial interaction models in the subsequent papers. Appropriate statistics for measuring changes in spatial variation due to spatial aggregation in data sets and in model predictions of spatial interaction are derived by use of information theory; and these statistics, such as spatial entropy, have excellent decomposition properties which can be readily exploited in the study of aggregation effects in data and models. These properties involve information components associated with density, level of detail or average zone size, dimension, and level of resolution. Use of the spatial entropy function, in particular, enables consistent relationships to be developed between information components and model parameters, and these relationships are examined in detail in subsequent papers.

Spatial Aggregation in Gravity Models: 3. Two-Dimensional Trip Distribution and Location Models

1982 ◽  
Vol 14 (5) ◽  
pp. 629-658 ◽  
Author(s):  
M Batty ◽  
P K Sikdar
Keyword(s):  
Spatial Interaction ◽  
Model Performance ◽  
Spatial Aggregation ◽  
Gravity Models ◽  
Canonical Forms ◽  
Two Dimensional ◽  
Variable Model ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
Spatial Entropy

This is the third of four papers and in it the methodology for analysing spatial aggregation in gravity models outlined in the first paper is further elaborated. In the second paper, the methodology was applied to one-dimensional spatial interaction models of the population density type, with some success; and here it is proposed to apply the methodology to two-dimensional spatial interaction models using the same data base, the Reading (UK) region. Accordingly, the methodology is first stated for linking information in data measured by spatial entropy to the parameters of models generated from spatial entropy. The family of four spatial interaction models due to Cordey-Hayes and Wilson is then derived, the canonical forms of their associated spatial entropy functions presented, and the analytic properties of such models explored. These four models are then fitted to spatial aggregations of the Reading region, and various empirical relationships between their entropies and parameters described. The results are not as regular as those of the models in the second paper because of more variable model performance, but nevertheless a means of approximating scale parameters from data based on the work of Kirby is outlined. This enables estimates of the dispersion parameters to be made through the canonical forms. Although the results are poor because of model performance, the methodology outlined here serves as a basis for the more fully fledged application to be discussed in the final paper.

Effects of Spatial System Design on Spatial Interaction Models. 1: The Spatial System Definition Problem

1989 ◽  
Vol 21 (1) ◽  
pp. 27-46 ◽  
Author(s):  
S H Putman ◽  
S-H Chung
Keyword(s):  
Goodness Of Fit ◽  
Spatial Interaction ◽  
Interaction Model ◽  
Spatial Aggregation ◽  
Model Parameters ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
Aggregation Methods ◽  
Future Paper ◽  
Spatial System

Rather little has been published about systematic empirical research on the problem of spatial aggregation and its effects on spatial interaction models. Of the work which has been published, all of it has dealt almost exclusively with single-parameter spatial interaction models. In this article five different aggregation procedures are examined. The experiments were based on the use of a multivariate multiparametric spatial interaction model. A first set of hypotheses tests was performed with respect to the sensitivity of model parameters to spatial aggregation methods. A second set was performed with respect to the sensitivity of model goodness-of-fit to the five spatial aggregation methods. Although questions remain, the results clearly show that the multiparametric model responds well to different aggregation algorithms. Some parameters showed substantial response, as they should, to different zonal aggregations, whereas others are shown to be much less responsive. Further, the results clearly indicate that systematic aggregation procedures generally produce better results than do random procedures. A future paper will continue with a discussion of zone definition criteria, and recommendations will be made with regard to aggregation algorithms.

A New Set of Spatial-Interaction Models: The Theory of Competing Destinations

1983 ◽  
Vol 15 (1) ◽  
pp. 15-36 ◽  
Author(s):  
A S Fotheringham
Keyword(s):  
Spatial Interaction ◽  
Model Misspecification ◽  
Distance Decay ◽  
Gravity Models ◽  
Parameter Estimates ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
The Family ◽  
Decay Parameters ◽  
Alternative Set

Members of the family of spatial-interaction models commonly referred to as gravity models are shown to be misspecified. One result of this misspecification is the occurrence of an undesirable ‘spatial-structure effect’ in estimated distance-decay parameters and this effect is examined in detail. An alternative set of spatial-interaction models is formulated from which more accurate predictions of interactions and more accurate parameter estimates can be obtained. These new interaction models are termed competing destinations models, and estimated distance-decay parameters obtained in their calibration are shown to have a purely behavioural interpretation. The implications of gravity-model misspecification are discussed.

Spatial Aggregation in Gravity Models: 2. One-Dimensional Population Density Models

1982 ◽  
Vol 14 (4) ◽  
pp. 525-553 ◽  
Author(s):  
M Batty ◽  
P K Sikdarfl
Keyword(s):  
Population Density ◽  
Spatial Information ◽  
Interaction Model ◽  
Theoretical Models ◽  
Spatial Aggregation ◽  
Gravity Models ◽  
Model Parameters ◽  
One Dimensional ◽  
Dimensional Interaction ◽  
Population Density Model

This paper, the second of four, is concerned with applying a methodology for analysing the spatial aggregation problem in gravity models outlined in the first paper. The methodology is based on a consistent framework for linking measures of pattern in interaction data to the derivation and estimation of related interaction models using spatial information theory. In this quest, a link is forged between information in data and the parameters of an associated model, and in part 1 it was suggested that if this link could be formalised then a means would be available for predicting changes in model parameters from different aggregations of the data, prior to the actual estimation of the models themselves. This relationship can be formalised for the case of the continuous one-dimensional interaction model such as the population density model, and this paper is concerned with demonstrating such an application to aggregations of zones in the Reading region. The framework is first described and two continuous models are presented. Then, the discrete model is estimated by means both of regression and of entropy techniques applied to various aggregations of the region, and the resulting parameters are related to the predicted and observed informations. Finally, the parameters approximated from observed information by use of the theoretical models are compared with the estimated parameters, and the approximation is deemed good, thus providing some confidence in the general concepts developed to handle these types of problem.

Proximate Aggregation-Estimation of Spatial Interaction Models

1984 ◽  
Vol 16 (4) ◽  
pp. 467-486 ◽  
Author(s):  
M Batty ◽  
P K Sikdar
Keyword(s):  
Spatial Interaction ◽  
Gravity Models ◽  
New Method ◽  
Entropy Measure ◽  
Interaction Models ◽  
Information Measures ◽  
Spatial Interaction Models ◽  
Entropy Measures ◽  
Log Linear ◽  
Parameter Values

In this paper the authors introduce a method of approximating the parameter values of gravity models from measures of information or entropy associated with the observed pattern of spatial interaction. The method builds on the previous work of the authors in which parameter values were estimated in a two-stage process which involved utilising the log-linear properties of entropy models through the canonical form of entropy, together with other approximations based on Kirby's method. Here the method is elaborated by adopting a consistent set of information measures to which the parameters of the model are related and this negates the need for other approximations. The original framework is first reviewed and then elaborated through the introduction of a weighted entropy measure. The traditional family of spatial interaction models is sketched and the new method developed for each of these models. The models are then applied to various aggregations of trip data in the Reading (United Kingdom) subregion, and estimates of parameter values based on the old, new, and conventional methods are compared. The new method is demonstrably superior to the old method and various extensions through the spatial disaggregation of entropy measures are noted in conclusion.

Spatial Aggregation in Gravity Models: 4. Generalisations and Large-Scale Applications

1982 ◽  
Vol 14 (6) ◽  
pp. 795-822 ◽  
Author(s):  
M Batty ◽  
P K Sikdar
Keyword(s):  
Large Scale ◽  
Spatial Information ◽  
Spatial Interaction ◽  
Model Performance ◽  
Spatial Aggregation ◽  
Gravity Models ◽  
Future Research ◽  
Data Set ◽  
Four Levels ◽  
Parameter Values

This paper is concerned with applying and extending a methodology for analysing spatial aggregation in gravity models developed in three earlier papers to a larger scale and hence more realistic example of spatial interaction than has been treated so far. Problems of model performance and the analysis of spatial aggregation effects identified in earlier papers are first described, and accordingly the spatial information (entropy) function used previously is then generalised to enable a wider set of models to be applied. Seven models are generated by means of spatial and generalised entropy functions subject to a standard set of model constraints, and the properties of the models in terms of their canonical forms are presented. The models are then applied to four levels of aggregation (234, 121, 58, and 22 zones) of the spatial interaction pattern in Edmonton, Alberta. As in previous papers, information in the data set at the four levels of aggregation is first measured and interpreted, the models are then fitted to these four levels, and relationships between information and parameter values sought. The approximation theory developed earlier is then used to predict parameter values of such models directly from observed spatial information. The results are only fair, better than those of part 3, but worse than those of part 2; although in terms of predicted parameter shift between levels of aggregation, the shifts associated with the doubly constrained model are accurately predicted. The various themes in these papers are then drawn together, conclusions with respect to the value of the insights gained are made, and speculations as to the most fruitful lines for future research outlined.

On the Relationship between Alonso's Theory of Movement and Wilson's Family of Spatial-Interaction Models

1981 ◽  
Vol 13 (2) ◽  
pp. 217-224 ◽  
Author(s):  
J Ledent
Keyword(s):  
Input Data ◽  
Spatial Interaction ◽  
System Of Equations ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
Standard Family ◽  
Original Contribution ◽  
Standard Models ◽  
The Relationship

This paper compares the system of equations underlying Alonso's theory of movement with that of Wilson's standard family of spatial-interaction models. It is shown that the Alonso model is equivalent to one of Wilson's four standard models depending on the assumption at the outset about which of the total outflows and/or inflows are known. This result turns out to supersede earlier findings—inconsistent only in appearance—which were derived independently by Wilson and Ledent. In addition to this, an original contribution of this paper—obtained as a byproduct of the process leading to the aforementioned result—is to provide an exact methodology permitting one to solve the Alonso model for each possible choice of the input data.

scholarly journals Do galactic bars depend on environment?: an information theoretic analysis of Galaxy Zoo 2

2020 ◽  
Vol 501 (1) ◽  
pp. 994-1001
Author(s):  
Suman Sarkar ◽  
Biswajit Pandey ◽  
Snehasish Bhattacharjee
Keyword(s):  
Spatial Distribution ◽  
Mutual Information ◽  
Local Density ◽  
Statistical Significance ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Host Galaxy ◽  
Data Sets ◽  
Data Set ◽  
Information Theoretic

ABSTRACT We use an information theoretic framework to analyse data from the Galaxy Zoo 2 project and study if there are any statistically significant correlations between the presence of bars in spiral galaxies and their environment. We measure the mutual information between the barredness of galaxies and their environments in a volume limited sample (Mr ≤ −21) and compare it with the same in data sets where (i) the bar/unbar classifications are randomized and (ii) the spatial distribution of galaxies are shuffled on different length scales. We assess the statistical significance of the differences in the mutual information using a t-test and find that both randomization of morphological classifications and shuffling of spatial distribution do not alter the mutual information in a statistically significant way. The non-zero mutual information between the barredness and environment arises due to the finite and discrete nature of the data set that can be entirely explained by mock Poisson distributions. We also separately compare the cumulative distribution functions of the barred and unbarred galaxies as a function of their local density. Using a Kolmogorov–Smirnov test, we find that the null hypothesis cannot be rejected even at $75{{\ \rm per\ cent}}$ confidence level. Our analysis indicates that environments do not play a significant role in the formation of a bar, which is largely determined by the internal processes of the host galaxy.

scholarly journals Theory and Applications of the Unit Gamma/Gompertz Distribution

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1850
Author(s):  
Rashad A. R. Bantan ◽  
Farrukh Jamal ◽  
Christophe Chesneau ◽  
Mohammed Elgarhy
Keyword(s):  
Stochastic Ordering ◽  
Real Data ◽  
Rate Function ◽  
The Other ◽  
Likelihood Method ◽  
Model Parameters ◽  
Data Sets ◽  
Gompertz Distribution ◽  
Probability And Statistics ◽  
Analytical Behavior

Unit distributions are commonly used in probability and statistics to describe useful quantities with values between 0 and 1, such as proportions, probabilities, and percentages. Some unit distributions are defined in a natural analytical manner, and the others are derived through the transformation of an existing distribution defined in a greater domain. In this article, we introduce the unit gamma/Gompertz distribution, founded on the inverse-exponential scheme and the gamma/Gompertz distribution. The gamma/Gompertz distribution is known to be a very flexible three-parameter lifetime distribution, and we aim to transpose this flexibility to the unit interval. First, we check this aspect with the analytical behavior of the primary functions. It is shown that the probability density function can be increasing, decreasing, “increasing-decreasing” and “decreasing-increasing”, with pliant asymmetric properties. On the other hand, the hazard rate function has monotonically increasing, decreasing, or constant shapes. We complete the theoretical part with some propositions on stochastic ordering, moments, quantiles, and the reliability coefficient. Practically, to estimate the model parameters from unit data, the maximum likelihood method is used. We present some simulation results to evaluate this method. Two applications using real data sets, one on trade shares and the other on flood levels, demonstrate the importance of the new model when compared to other unit models.

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