Background: Intramax is a hierarchical aggregation procedure for dealing with the multi-level specification problem and with the association issue of data set reduction, but it was used as a functional regionalization procedure many times in the past.
Objectives: In this paper, we analyse the simultaneous use of three different constraints in the original Intramax procedure, i.e. the contiguity constraint, the higher-inner-flows constraint, and the lower-variation-of-inner-flows constraint.
Methods/Approach: The inclusion of constraints in the Intramax procedure was analysed by a programme code developed in Mathematica 10.3 by the processing time, by intra-regional shares of total flows, by self-containment indexes, by numbers of singleton and isolated regions, by the number of aggregation steps where a combination of constraints was applied, by the number of searching steps until the combination of constraints was satisfied, and by surveying the results geographically.
Results: The use of the contiguity constraint is important only at the beginning of the aggregation procedure; the higher-inner-flows constraint gives singleton regions, and the lower-variation constraint forces the biggest employment centre as an isolated region up to a relatively high level of aggregation.
Conclusions: The original Intramax procedure (without the inclusion of any constraint) gives the most balanced and operative hierarchical sets of functional regions without any singletons or isolated regions.
1. Abler, R., Adams, J. S., Gould, P. (1972), “Spatial organization”, London, Prentice-Hall.
2. Andersen, A. K. (2002), “Are commuting areas relevant for the delimitation of administrative regions in Denmark?”, Regional Studies, Vol. 36, No. 8, pp. 833–844.
3. Ball, R. M. (1980), “The use and definition of Travel-to-Work areas in Great Britain: some problems”, Regional Studies, Vol. 14, No. 2, pp. 125–139.
4. Brown, P. J. B., Hincks, S. (2008), “A framework for housing market area delineation: principles and application”, Urban Studies, Vol. 45, No. 11, pp. 2225–2247.
5. Brown, L. A., Holmes, J. (1971), “The delimitation of functional regions, nodal regions, and hierarchies by functional distance approaches”, Journal of Regional Science, Vol. 11, No. 1, pp. 57–72.
6. Brown, P. J. B., Pitfield, D. E. (1990), “The Intramax derivation of commodity market structures from freight flow data”, Transportation Planning and Technology, Vol. 15, No. 1, pp. 59–81.
7. Casado-Díaz, J. M. (2000), “Local labour market areas in Spain: A case study”, Regional Studies, Vol. 34, No. 9, pp. 843–856.
8. Casado-Díaz, J. M., Coombes, M. G. (2011), “The delineation of 21st century local labour market areas: a critical review and a research agenda”, Boletín de la Asociación de Geógrafos Españoles, Vol. 57, pp. 7–32.
10. Coombes, M. G., Dixon, J. S., Goddard, J. B., Openshaw, S., Taylor, P. J. (1979), “Daily urban systems in Britain: from theory to practice”, Environment and Planning A, Vol. 11, No. 5, pp. 565–574.
11. Coombes, M. G., Green, A. E., Openshaw, S. (1986), “An efficient algorithm to generate official statistical reporting areas: the case of the 1984 travel-to-work-areas revision in Britain”, Journal of the Operational Research Society, Vol. 37, No. 10, pp. 943–953.
12. Cörvers, F., Hensen, M., Bongaerts, D. (2009), “Delimitation and coherence of functional and administrative regions”, Regional Studies, Vol. 43, No. 1, p. 19–31.
13. Daras, K. (2005), “An information statistics approach to zone design in the geography of health outcomes and provision”, PhD dissertation, Newcastle, University of Newcastle.
15. Drobne, S., Bogataj, M. (2012a), “A method to define the number of functional regions: An application to NUTS 2 and NUTS 3 levels in Slovenia” = “Metoda opredelitve števila funkcionalnih regij: Aplikacija na ravneh NUTS 2 in NUTS 3 v Sloveniji”, Geodetski vestnik, Vol. 56, No. 1, pp. 105–150.
16. Drobne, S., Bogataj, M. (2012b), “Evaluating functional regions”, in Babić, Z. et al. (Eds.), 14th International conference on operational research, Trogir, Croatia, September 26–28, 2012, Croatian operational research review, Vol. 3, pp. 14–26, available at http://hrcak.srce.hr/file/142254 (January 28, 2016).
17. Drobne, S., Bogataj, M. (2014), “Regions for servicing old people: case study of Slovenia”, Business systems research journal, Vol. 5, No. 3, pp. 19–36.
18. Drobne, S., Bogataj, M. (2015), “Optimal allocation of public service centres in the central places of functional regions”, IFAC-PapersOnLine Vol. 48, No. 3, pp. 2362–2367.
19. Drobne, S., Lakner, M. (2016), “Intramax and other objective functions”, Moravian Geographical Reports, Vol. 24, to be appeared.
20. Drobne, S., Lisec, A., Konjar, M., Zavodnik Lamovšek, A., Pogačnik, A. (2009), “Functional vs. administrative regions: case of Slovenia”, in Vujošević, M. (Ed.), Thematic Conference Proceedings, Vol. 1, Belgrade, Institute of architecture and urban & spatial planning of Serbia, pp. 395–416.
21. Drobne, S., Konjar, M., Lisec, A., Pichler Milanović, N., Zavodnik Lamovšek, A. (2010a), “Functional regions defined by urban centres of (inter)national importance – the case of Slovenia”, in Schrenk, M. (Ed.), Popovich, V. V. (Ed.), Zeile, P. (Ed.), Liveable, healthy, prosperous cities for everyone, Real Corp 2010, Proceedings 2010, 15th International conference on urban planning, regional development and information society, May 18 – 20, 2010, Wien, Austria, pp. 295–304.
22. Drobne, S., Konjar, M., Lisec, A. (2010b), “Razmejitev funkcionalnih regij Slovenije na podlagi analize trga dela = Delimitation of functional regions of Slovenia based on labour market analysis”, Geodetski vestnik, Vol. 54, No. 3, pp. 481–500.
23. Farmer, C. J. Q., Fotheringham, A. S. (2011), “Network-based functional regions”, Environment and Planning A, Vol. 43, No. 11, pp. 2723–2741.
24. Feldman, O., Simmonds, D., Troll, N., Tsang, F. (2005), “Creation of a system of functional areas for England and Wales and for Scotland”, paper presented at European Transport Conference, October 3–5, 2005, Strasbourg, France, available at: http://abstracts.aetransport.org/paper/index/id/2284/confid/11 (January 28, 2016).
25. Flórez-Revuelta, F., Casado-Díaz, J. M., Martínez-Bernabeu, L. (2008), “An evolutionary approach to the delineation of functional areas based on travel-to-work flows”, International Journal of Automation and Computing, Vol. 5, No. 1, pp. 10–21.
26. Fowlkes, E. B., Mallows, C. L. (1983), “A method for comparing two hierarchical clusterings”, Journal of the American Statistical Association, Vol. 78, No. 383, pp. 553–569.
27. Fukumoto, J., Okamoto, Y., Ujiie, A. (2013), “A modularity approach to the delineation of functional regions from spatial interaction data”, in Proceedings of The 13th World Conference on Transportation Research in Rio de Janeiro, Brazil, July 15–18, 2013, COPPE – Federal University of Rio de Janeiro, Brazil, available at http://www.wctrs-society.com/wp/wp-content/uploads/abstracts/rio/selected/3377.pdf (January 28, 2016)
29. Goodman, J. F. B. (1970), “The definition and analysis of local labour markets: some empirical problems”, British Journal of Industrial Relations, Vol. 8, No. 2, pp. 179–196.
30. Haggett, P. (1965), “Locational network analysis in human geography”, London, Arnold.
31. Hirst, M. A. (1977), “Hierarchical aggregation procedures for interaction data: a comment”, Environment and Planning A, Vol. 9, No. 1, pp. 99–103.
32. Jaegal, Y. (2013). “Delineating housing market areas in the Seoul metropolitan area using a geo-computational approach”, Journal of the Association of Korean Geographers, Vol. 2, No. 1, pp. 7–20.
33. Karlsson, C., Olsson, M. (2006), “The identification of functional regions: theory, methods, and applications”, The Annals of Regional Science, Vol. 40, No. 1, pp. 1–18.
34. Kim, H., Chun, Y., Kim, K. (2015), “Delimitation of functional regions using a p-regions problem approach”, International Regional Science Review, Vol. 38, No. 3, pp. 235–263.
35. Kohl, T., Brouver, A. E. (2014), “The development of trade blocs in an era of globalisation”, Environment and Planning A, Vol. 46, No. 7, pp. 1535–1553.
36. Koo, H. (2012), “Improved hierarchical aggregation methods for functional regionalization in the Seoul metropolitan area. Journal of the Korean Cartographic Association, Vol. 12, No. 2, p. 25–35.
37. Konjar, M., Lisec, A., Drobne, S. (2010), “Methods for delineation of functional regions using data on commuters”, in Painho, M. (Ed.), Santos, M. Y. (E.), Pundt, H. (Ed.), Geospatial thinking, Proceedings of the 13th AGILE International Conference on Geographic Information Science, May 10–14, 2010, Guimarães, Portugal, pp. 1–10, available at http://www.agile-online.org/Conference_Paper/CDs/agile_2010/ShortPapers_PDF/93_DOC.pdf (January 28, 2016).
38. Koo, H. (2012), “Improved hierarchical aggregation methods for functional regionalization in the Seoul metropolitan area”, Journal of the Korean Cartographic Association, Vol. 12, No. 2, pp. 25–35.
40. Landré, M., Håkansson, J. (2013), “Rule versus interaction function: evaluating regional aggregations of commuting flows in Sweden”, European Journal of Transport and Infrastructure Research, Vol. 13, No. 1, pp. 1–19.
41. Manley, E. (2014), “Identifying functional urban regions within traffic flow”, Regional Studies, Regional Science, Vol. 1, No. 1, pp. 40–42.
42. Masser, I., Brown, P. J. B. (1975), “Hierarchical aggregation procedures for interaction data”, Environment and Planning A, Vol. 7, No. 5, pp. 509–523.
43. Masser, I., Brown, P. J. B. (1977), “Spatial representation and spatial interaction”, Papers of the Regional Science Association, Vol. 38, No. 1, pp. 71–92.
44. Masser, I., Scheurwater, J. (1980), “Functional regionalisation of spatial interaction data: an evaluation of some suggested strategies”, Environment and Planning A, Vol. 12, No. 12, pp. 1357–1382.
45. Mitchell, W., Stimson, R. (2010), “Creating a new geography of functional economic regions to analyse aspects of labour market performance in Australia”, Working Paper No. 10-09, Centre of Full Employment and Equity, Newcastle, November 2010.
46. Mitchell, W., Watts, M. (2010), “Identifying functional regions in Australia using hierarchical aggregation techniques”, Geographical Research, Vol. 48, No. 1, pp. 24–41.
47. Mitchell, W., Bill, A., Watts, M. (2007), “Identifying functional regions in Australia using hierarchical aggregation techniques”, Working Paper No. 07-06, Centre of Full Employment and Equity, Newcastle, November 2007.
49. Nel, J. H., Krygsmany, S. C., de Jong, T. (2008), “The identification of possible future provincial boundaries for South Africa based on an Intramax analysis of journey-to-work data”, Orion, Vol. 24, No. 2, pp. 131–156.
51. Poon, J. P. (1997), “The cosmopolitanization of trade regions: global trends and implications, 1965–1990”, Economic Geography, Vol. 73, No. 4, pp. 390–404.
52. Slater, P. B. (1975), “A hierarchical regionalisation of Russian administrative units using 1965-1969 migration data”, Soviet Geography, Vol. 16, No. 7, pp. 453–465
53. SMARS (2016), Digital data on territorial units of Slovenia, Data on administrative services, Surveying and Mapping Authority of the Republic Slovenia, Ljubljana, available at http://www.gu.gov.si/en/ (January 30, 2016).
54. Smart, M. W. (1974), “Labour market areas: uses and definition”, Progress in Planning, Vol. 2, No. 4, pp. 239–353.
57. Ullman, E. L. (1980), “Geography as spatial interaction”, Seattle, University of Washington Press.
58. Van der Laan, L., Schalke, R. (2001), “Reality versus policy: the delineation and testing of local labour market and spatial policy areas”, European Planning Studies, Vol. 9, No. 2, pp. 201–221.
59. Vinh, N. X. (2010), “Information theoretic methods for clustering with applications to microarray data”, PhD dissertation, Sydney, The University of New South Wales.
60. Vinh, N. X., Epps, J., Bailey, J. (2010), “Information theoretic measures for clusterings comparison: variants, properties, normalization and correction for chance”, Journal of Machine Learning Research, Vol. 11, pp. 2837–2854.
61. Wallace, D. L. (1983), “A method for comparing two hierarchical clusterings: comment”, Journal of the American Statistical Association, Vol. 78, No. 383, pp. 569–576.
62. Watts, M. (2013), “Assessing different spatial grouping algorithms: an application to the design of Australia’s new statistical geography”, Spatial Economic Analysis, Vol. 8, No. 1, pp. 92–112.