Top

Journal of Geographical Systems

Published in:

01-07-2013 | Original Article

Testing spatial autocorrelation in weighted networks: the modes permutation test

Author: François Bavaud

Published in: Journal of Geographical Systems | Issue 3/2013

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In a weighted spatial network, as specified by an exchange matrix, the variances of the spatial values are inversely proportional to the size of the regions. Spatial values are no more exchangeable under independence, thus weakening the rationale for ordinary permutation and bootstrap tests of spatial autocorrelation. We propose an alternative permutation test for spatial autocorrelation, based upon exchangeable spatial modes, constructed as linear orthogonal combinations of spatial values. The coefficients obtain as eigenvectors of the standardized exchange matrix appearing in spectral clustering and generalize to the weighted case the concept of spatial filtering for connectivity matrices. Also, two proposals aimed at transforming an accessibility matrix into an exchange matrix with a priori fixed margins are presented. Two examples (inter-regional migratory flows and binary adjacency networks) illustrate the formalism, rooted in the theory of spectral decomposition for reversible Markov chains.

previous article Editorial: Advances in the statistical modelling of spatial interaction data

next article Effects of scale in spatial interaction models

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Here the notations match the higher-order discrete time extensions of the exchange matrix, resulting (under weak regularity conditions) from the iteration of the Markov transition matrix as

$$ E^{(r)}:=\Uppi W^r\quad E^{(0)}=\Uppi\quad E^{(2)}=E\Uppi^{-1}E \quad E^{(\infty)}=ff^{\prime} . $$

Aldous D, Fill J (2002) Reversible Markov chains and random walks on graphs. Draft chapters, online version available at http://www.stat.berkeley.edu/users/aldous/RWG/book.html

Anselin L (1988) Spatial econometrics: methods and models. Kluwer, DordrechtCrossRef

Anselin L (1995) Local indicators of spatial association—LISA. Geograph Anal 27:93–115CrossRef

Arbia G (2006) Spatial econometrics: statistical foundations and applications to regional convergence. Springer, Berlin

Assunção RM, Reis EA (1999) A new proposal to adjust Moran’s I for population density. Stat Med 18:2147–2162CrossRef

Barbu VS, Limnios N (2008) Semi-Markov chains and hidden semi-Markov models toward applications: their use in reliability and DNA analysis. Springer, Berlin

Bavaud F (1998) Models for spatial weights: a systematic look. Geograph Anal 30:153–171CrossRef

Bavaud F (2002) The quasisymmetric side of gravity modelling. Environ Plan A 34:61–79CrossRef

Bavaud F (2008) Local concentrations. Papers Reg Sci 87:357–370CrossRef

Bavaud F (2008) The endogenous analysis of flows, with applications to migrations, social mobility and opinion shifts. J Math Sociol 32:239–266CrossRef

Bavaud F (2010) Multiple soft clustering, spectral clustering and distances on weighted graphs. In: Proceedings of the ECML PKDD’10. Lecture notes in computer science, Springer, vol 6321, pp 103–118

Berger J, Snell JL (1957) On the concept of equal exchange. Behav Sci 2:111–118CrossRef

Besag J, Diggle PJ (1977) Simple Monte Carlo tests for spatial pattern. J R Stat Soc (Ser C Appl Stat) 26:327–333

Bivand RS, Müller W, Reder M (2009) Power calculations for global and local Moran’s I. Comput Stat Data Anal 53:2859–2872CrossRef

Bivand R (2009) Applying Measures of spatial autocorrelation: computation and simulation. Geograph Anal 41:375–384CrossRef

Chun Y (2008) Modeling network autocorrelation within migration flows by eigenvector spatial filtering. J Geogr Syst 10:317–344CrossRef

Chung FRK (1997) Spectral graph theory. CBMS regional conference series in mathematics 92. American Mathematical Society, Washington

Çinlar E (1975) Introduction to stochastic processes. Prentice Hall, New York

Cliff AD, Ord JK (1973) Spatial autocorrelation. Pion, London

Cliff AD, Ord JK (1981) Spatial processes: models and applications. Pion, London

Corcoran CD, Mehta CR (2002) Exact level and power of permutation, bootstrap, and asymptotic tests of trend. J Mod Appl Stat Methods 1:42–51

Cressie NAC (1993) Statistics for spatial data. Wiley, Hoboken, NJ

Dray S (2011) A new perspective about Moran’s coefficient: spatial autocorrelation as a linear regression problem. Geograph Anal 43:127–141CrossRef

Fotheringham AS, O’Kelly ME (1989) Spatial interaction models: formulations and applications. Kluwer, Dordrecht

Geary R (1954) The contiguity ratio and statistical mapping. Incorporated Stat 5:115–145CrossRef

Goodchild MF, Smith TR (1980) Intransitivity, the spatial interaction model, and US migration streams. Environ Plan A 12:1131–1144CrossRef

Griffith DA (2000) Eigenfunction properties and approximations of selected incidence matrices employed in spatial analyses. Linear Algebra Appl 321:95–112CrossRef

Griffith DA (2003) Spatial autocorrelation and spatial filtering: gaining understanding through theory and scientific visualization. Springer, BerlinCrossRef

Griffith DA, Peres-Neto PR (2006) Spatial modeling in ecology: the flexibility of eigenfunction spatial analyses. Ecology 87:2603–2613CrossRef

Haining RP (2003) Spatial data analysis: theory and practice. Cambridge University Press, CambridgeCrossRef

Janssen A, Pauls T (2005) A Monte Carlo comparison of studentized bootstrap and permutation tests for heteroscedastic two-sample problems. Comput Stat 20:369–383CrossRef

Kijima M (1997) Markov processes for stochastic modeling. Chapman & Hall, London

Lebart L (1969) Analyse statistique de la contiguïté. Publication de l’Institut de Statistiques de l’Université de Paris 18:81–112

Leenders RTAJ (2002) Modeling social influence through network autocorrelation: constructing the weight matrix. Soc Netw 24:21–47CrossRef

LeSage JP, Pace RK (2009) Introduction to spatial econometrics. Chapman & Hall, LondonCrossRef

Li F, Calder CA, Cressie N (2007) Beyond Moran’s I : testing for spatial dependence based on the spatial autoregressive model. Geograph Anal 39:357–375

von Luxburg U (2007) A tutorial on spectral clustering. Stat Comput 17:395–416CrossRef

Moran PAP (1950) Notes on continuous stochastic phenomena. Biometrika 37:17–23

Sen A, Smith T (1995) Gravity Models of spatial interaction behavior. Springer, BerlinCrossRef

Thioulouse J, Chessel D, Champely S (1995) Multivariate analysis of spatial patterns: a unified approach to local and global structures. Environ Ecol Stat 2:1–14CrossRef

Tiefelsdorf M, Boots B (1995) The exact distribution of Moran’s I. Environ Plan A 27:985–999CrossRef

Tiefelsdorf M, Griffith DA (2007) Semiparametric filtering of spatial autocorrelation: the eigenvector approach. Environ Plan A 39:1193–221CrossRef

Upton G, Fingleton B (1985) Spatial data analysis by example. Wiley, Hoboken, NJ

Waldhör T (1996) The spatial autocorrelation coefficient Moran’s I under heteroscedasticity. Stat Med 15:887–892CrossRef

Willekens FJ (1983) Specification and calibration of spatial interaction models: a contingency-table perspective and an application to intra-urban migration in Rotterdam. Tijdschrift voor Economische en Sociale Geografie 74:239–252

Title: Testing spatial autocorrelation in weighted networks: the modes permutation test
Author: François Bavaud
Publication date: 01-07-2013
Publisher: Springer-Verlag
Published in: Journal of Geographical Systems / Issue 3/2013
Print ISSN: 1435-5930
Electronic ISSN: 1435-5949
DOI: https://doi.org/10.1007/s10109-013-0179-2

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 3/2013

The effects of World Heritage Sites on domestic tourism: a spatial interaction model for Italy

Effects of scale in spatial interaction models

Constrained variants of the gravity model and spatial dependence: model specification and estimation issues

Social networks and trade of services: modelling interregional flows with spatial and network autocorrelation effects

A spatial interaction model with spatially structured origin and destination effects

Editorial: Advances in the statistical modelling of spatial interaction data

Premium Partner