# Analysing Interaction Data

John Stillwell (University of Leeds, United Kingdom) and Kirk Harland (University of Leeds, United Kingdom)
DOI: 10.4018/978-1-61520-755-8.ch004
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## Abstract

Large and complex interaction data sets present researchers with analytical challenges and this chapter attempts to identify and illustrate a number of ways to analyse origin-destination flows. Given the impossible task of providing a comprehensive review in such a limited space, certain analytical measures, modelling methods and visualisation techniques have been selected for inclusion, following an introduction to the notation commonly employed to represent interaction variables. Various Census and NHS patient register data sets are used to exemplify interaction measures, beginning with simple net balances and inflow/outflow ratios and moving onto indices of connectivity, inequality and distance moved. The multiplicative component framework is introduced as a particularly useful analytical approach. More sophisticated methods of modelling interaction data using statistical or mathematical calibration techniques are reviewed, examples of log-linear regression and spatial interaction model structure are highlighted in the context of historical calibration and a brief discussion of the use models for future projection is included. Maps that show patterns of geographical movement function as effective illustrative and research tools. Computerized mapping of geographical movement has evolved since the 1970s and 1980s and, in this chapter, we introduce a new method of mapping flows using vectors and illustrate this approach with micro data on pupils travelling to school. The chapter aims to provide a broad introduction to analysis methods for interaction data, many of which are subsequently applied in later chapters of the book.
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## Interaction Matrix And Notation

An interaction matrix (Figure 1) is a framework of rows and columns, representing origins and destinations respectively, such that the cell at the intersection of any one row i and any one column j contains a measure of the interaction flow, Iij, between that origin and destination pair. The flow may be a raw count of migrants or commuters although it may be a derived measure such as a rate, a probability or some other measure of interaction. The matrix may be symmetric with n origins and m destinations, in which case n=m, but this is often not the case and sometimes origins and destinations are entirely separate spatial entities, such as the residential addresses of school children and the schools they attend or the neighbourhoods where consumers live and the retail outlets they frequent. In a symmetric matrix, the diagonal (unshaded) cells of the matrix are the cells with intra-area flows (Iii and Ijj) whereas the off-diagonal cells contain all the inter-area counts. The sum of the cell values for each origin i is the outflow total, frequently represented as Oi:= Ii*(1)

Figure 1.

Symmetric flow matrix with marginal totals

whilst the sum of the inflows to one destination j is usually represented as Dj:

= I*j(2)

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