Although my research interests have meandered between spatial interaction/spatial choice modelling and spatial statistics, a common theme to much of my work has been an interest in identifying and understanding differences across space rather than similarities. The development of spatial disaggregations of global statistics was the subject of my PhD on origin-specific distance-decay parameters in spatial interaction models (still a prime interest) and it is the subject of recent research at the University of Newcastle on Geographically Weighted Regression. The concern for the local encompasses the dissection of global statistics into their local constituents; the concentration on local exceptions rather than the search for global regularities; and the production of local or mappable statistics rather than on ‘whole-map’ values. This trend is important not only because it brings issues of space to the fore in analytical methods, but also because it refutes the rather naive criticism that quantitative geography is unduly concerned with the search for global generalities and ‘laws’
Obviously, local forms of spatial analysis are important to GIS because they result in geocoded output that can be mapped. It could also be claimed that some of the impetus for the development of local statistics derives from the growing interest in integrating advanced forms of spatial analysis and GIS (Fotheringham and Charlton, 1994; Fotheringham, 1994; Fotheringham and Rogerson, 1993).
The theme of much of my research has been that when analysing spatial data, it may be incorrect to assume that the results obtained from the whole data set apply equally to all parts of the study area. Interesting insights might be obtained from investigating spatial variations in the results. Simply reporting one ‘average’ set of results and ignoring any possible spatial variations in those results is as limiting as reporting a mean value of a spatial distribution without seeing a map of the data
2. The Nature of Local Variations in Relationships
There are at least three reasons to question the assumption of stationarity in spatial data analysis and to allow variations in observed relationships, as measured for example by parameter estimates in a regression model. The first and simplest is that there will inevitably be spatial variations in observed relationships caused by random sampling variations. The contribution of this source of spatial non-stationarity is not usually of great interest but it does need to be accounted for by significance testing. That is, we are only interested in relatively large variations in parameter estimates which are unlikely to be caused by random sampling and which therefore constitute interesting spatial non-stationarity.
The second is that, for whatever reasons, some relationships are intrinsically different across space. Perhaps, for example, there are spatial variations in people’s attitudes or preferences or there are different administrative, political or other contextual issues that produce different responses to the same stimuli over space. This idea that human behaviour can vary intrinsically over space is consistent with post-modernist beliefs on the importance of place and locality as frames for understanding such behaviour.
The third reason why relationships might exhibit spatial non-stationarity is that the model from which the relationships are measured is a gross misspecification of reality and that one or more relevant variables are either omitted from the model or are represented by an incorrect functional form. This view, rather more in the positivist school of thought, runs counter to that discussed above in that it assumes a global statement of behaviour can be made but that the structure of our model is not sufficiently well-formed to allow us to make it. In this case mapping local statistics is useful in order to understand the nature of the model misspecification more clearly. For what parts of the study region does the model replicate observed data less accurately and does the spatial distribution of these parts suggest the addition of an extra explanatory variable to the model?
3. Attempts to Measure Local Variations in Relationships
Within the last several years, there has been a relatively flurry of academic work reflecting the calls of Fotheringham and Rogerson (1993), Fotheringham (1992) and Openshaw (1993) for greater attention to be given to local or mappable statistics. Four areas are now described where progress has been made in this direction
3.1 Local Point Pattern Analysis
The analysis of spatial point patterns has long been an important concern in geographical enquiry but until relatively recently most applications of spatial point pattern analysis involved the calculation of some global statistic that described the whole point pattern and from which a conclusion was reached related to the clustered, dispersed or random nature of the whole pattern. Clearly, such an analysis is potentially flawed in that interesting spatial variations in the point pattern are subsumed in the calculation of the average or global statistic. In many instances, particularly in the study of disease, such an approach would appear to be contrary to the purpose of the study, namely to identify any interesting local clusters.
Amongst the first example of a local point pattern analysis technique was the Geographical Analysis Machine (GAM) developed by Openshaw et al (1987) and updated by Fotheringham and Zhan (1996). The basic idea is very simple and serves to demonstrate the interest in the local quite well. Within the study region containing a spatial point pattern, randomly select a location and then randomly select a radius of a circle to be centred at that location. Within this random circle count the number of points and compare this observed value with an expected value based on an assumption about the process generating the point pattern (usually that it is random). Ideally, the population-at-risk should be used as a basis for generating the expected value, as shown in Fotheringham and Zhan (1996) who use a Poisson probability model with the observed mean and the population-at-risk within each circle. Once the statistical significance of the observed count within a circle has been examined the circle is drawn on a map of the region if it contains a statistically significant cluster of points. The process is repeated many times until a map is produced containing a set of circles centred on parts of the region where interesting clusters of points appear to be located.
3.2 The Local Measurement of Univariate Spatial Relationships
Much of the work undertaken in exploratory graphical analysis is essentially concerned with identifying local exceptions to general trends in either data or relationships. Hence, techniques such as linked windows and brushing allow data to be examined interactively so that points appearing as outliers in various statistical displays can be located on a map automatically. Usually this type of graphical interrogation takes place with univariate distributions so that histograms or box-and-whisker displays form the basis of the graphics although scatterplots can also be linked to a map display and even 3-D spin plots can be used. No matter which exploratory technique is used, however, the aim of the analysis is generally to identify unusual data points and the focus is on the exceptions rather than the general trend. More formally, local versions of global univariate statistics have recently been developed by Getis and Ord (1992), Ord and Getis (1995) and by Anselin (1995).
3.3 The Local Measurement of Multivariate Spatial Relationships
The increasing availability of large and complex spatial datasets has led to a greater awareness that the univariate statistical methods described above are of limited application and that there is a need to understand local variations in more complex relationships. In response to this recognition, several attempts have been made to produce localised versions of traditionally global multivariate techniques, with the greatest challenge being to produce local versions of regression analysis.
Perhaps the best-known attempt to do this is the expansion method (Casetti, 1972; Jones and Casetti, 1992) which attempts to measure parameter ‘drift’. In this framework, parameters of a global model are expanded in terms of other attributes. If the parameters of the regression model are made functions of geographic space, trends in parameter estimates over space can then be measured (Fotheringham and Pitts, 1995; Eldridge and Jones 1991). Whilst this is a useful and easily applicable framework in which improved models can be developed, it is essentially a trend-fitting exercise in which complex patterns of parameter estimates will be missed. The output from spatial variants of the expansion method is thus a second-order set of relationships when what is required is information on the first-order relationships.
More recently, Geographically Weighted Regression (GWR) (Brunsdon et
al. 1996; 1998; Fotheringham et al 1996; 1998) has been developed to extend
the traditional regression framework by allowing local rather than global
parameters to be estimated. That is, the model to be estimated has
the general form:
where y represents the dependent variable, represents the kth independent variable, represents an error term and is the value of the kth parameter at location i. In the calibration of this model it is assumed that observed data near to point i have more influence in the estimation of the s than do data located farther from point i. In essence, the equation measures the relationships inherent in the model around each point i. To calibrate the model, a modified weighted least squares approach is taken so that the data are weighted according to their proximity to point i. Thus the weighting of any point is not constant but varies with i. Data from observations closer to i are weighted more heavily than those from farther away. Hence the estimator for the parameters in GWR is:
where is an n by n matrix whose off-diagonal elements are zero and whose diagonal elements denote the geographical weighting of observed data for point i. It should be noted that as well as producing localised parameter estimates, the GWR technique described above will produce localised versions of all standard regression diagnostics including goodness-of-fit measures such as r-squared. The latter can be particularly informative in understanding the application of the model being calibrated and in exploring the possibility of adding additional explanatory variables to the model. A list of recent GWR publications and code is provided at the GWR web site: www.ncl.ac.uk/ngeog/nmec/GWR
3.4 The Mathematical Modelling of Flows
Perhaps one of the earliest, yet still misunderstood, examples of providing local information on relationships rather than simply reporting global results is the spatial disaggregation of spatial flow or spatial interaction models (Fotheringham and O’Kelly, 1989). The reason for calibrating spatial flow models is to obtain information, via the estimated parameters of the models, on how individuals make choices amongst spatial alternatives. By far the most important attribute in many spatial choice contexts is the spatial separation between the individual and the alternative - individuals are less likely to choose an alternative that is farther way, ceteris paribus - and an important aspect of the calibration of spatial choice/spatial interaction models is to obtain information on the rate of this ‘distance-decay’. It was recognised quite early in the spatial interaction modelling literature that localised distance-decay parameters would yield more useful information on the spatial choice process than simply estimating a global interaction model (see Fotheringham, 1981 for a review). From an accumulation of empirical examples of origin-specific parameter estimates, it has proven possible to map trends in parameter estimates that have led to the identification of a severe misspecification bias in the general spatial interaction modelling formula (Fotheringham, 1984; 1986). It is worth stressing that such misspecification only came to light through an investigation of spatial variations in localised parameters and would have been missed in the calibration of a global model.
The relative explosion of attention to the ‘local’ rather than the ‘global’ in quantitative geography is interesting for several reasons. It belies the criticism that the quantitative approach is only concerned with the search for broad generalisations and not with identifying local exceptions. It links quantitative geography with the powerful visual display environments of various GIS and statistical graphics packages where the all-important display is the map. It also allows quantitative geographers to explore relationships in different ways as a guide to a better understanding of spatial processes and finally it affords the exciting opportunity of developing new statistical approaches to spatial data analysis
Anselin, L. 1995: Local Indicators of Spatial Association - LISA Geographical Analysis 27, 93-115.
Brunsdon, C.F., Fotheringham A.S. and Charlton, M.E. 1996: Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity, Geographical Analysis, 28: 281-298
Brunsdon C.F., Fotheringham A.S. and Charlton, M.E. 1998 Spatial nonstationarity and autoregressive models Environment and Planning A 30: 957-973
Casetti, E. 1972: Generating Models by the Expansion Method: Applications to Geographic Research, Geographical Analysis 4, 81-91.
Cleveland, W. S. 1979: Robust Locally Weighted Regression and Smoothing Scatterplots Journal of the American Statistical Association 74, 829-836.
Eldridge, J.D. and J.P. Jones III 1991: Warped Space: a Geography of
Distance Decay Professional Geographer 43, 500-511.
Fotheringham, A.S. 1981: Spatial Structure and Distance-Decay Parameters, Annals of the Association of American Geographers, 71, 425-436.
Fotheringham, A.S. 1984: Spatial Flows and Spatial Patterns, Environment and Planning A, 16, 529-543.
Fotheringham, A.S. 1986: Modelling Hierarchical Destination Choice, Environment and Planning A, 18, 401-418.
Fotheringham, A.S. 1992: Exploratory Spatial Data Analysis and
and Planning A, 24, 1675-1678.
Fotheringham, A.S. 1994: On the Future of Spatial Analysis: The Role of GIS Environment and Planning A Anniversary Issue: 30-34.
Fotheringham, A.S. and Charlton, M.E. 1994: GIS and Exploratory Spatial Data Analysis: An Overview of Some Research Issues, Geographical Systems, 1, 315-327.
Fotheringham, A.S. and O’Kelly, M.E. 1989: Spatial Interaction Models:
and Applications London: Kluwer.
Fotheringham, A.S. and Pitts, T.C. 1995: Directional Variation in Distance-Decay Environment and Planning A 27, 715-729.
Fotheringham, A.S. and Rogerson, P.A. 1993: GIS and Spatial Analytical Problems” International Journal of Geographic Information Systems 7, 3-19.
Fotheringham, A.S. and Zhan, F. 1996: A Comparison of Three Exploratory Methods for Cluster Detection in Spatial Point Patterns Geographical Analysis in press.
Fotheringham, A.S., Brunsdon, F.C. and Charlton, M.E. 1997: Measuring Spatial Variations in Relationships with Geographically Weighted Regression, Chapter 4 pp 60-82 in Recent Developments in Spatial Analysis, Spatial Statistics, Behavioural Modelling and Computational Intelligence eds. M.M. Fischer and A. Getis, Berlin: Springer-Verlag.
Fotheringham, A.S., Brunsdon, C.F., and Charlton, M.E. 1999 Geographically Weighted Regression: A Natural Evolution of the Expansion Method for Spatial Data Analysis, Environment and Planning A 30: 1905-1927.
Getis, A. and J.K. Ord, 1992: The Analysis of Spatial Association by Use of Distance Statistics, Geographical Analysis 24, 189- 206.
Jones, J.P. and Casetti, E. 1992: Applications of the Expansion Method London: Routledge.
Openshaw, S. 1993: Exploratory Space-Time-Attribute Pattern Analysers, in Spatial Analysis and GIS , eds. Fotheringham A.S. and Rogerson, P.A, 147-163, London: Taylor and Francis.
Openshaw, S., Charlton, M.E., Wymer, C. and Craft, A.W. 1987: A Mark I Geographical Analysis Machine for the Automated Analysis of Point data Sets, International Journal of Geographical Information Systems 1, 359-377.
Ord, J.K. and A. Getis, 1995: Local Spatial Autocorrelation Statistics:
Distributional Issues and an Application, Geographical Analysis 27,
Professor Fotheringham’s research interests include the mathematical modelling of spatial choice, spatial cognition, the integration of spatial analysis and GIS, spatial statistics, exploratory spatial data analysis, retailing and migration. He has published several books and over 60 refereed journal articles. He has been Chair of both the Quantitative Methods Research Group of the Royal Geographical Society and the Quantitative Methods and Mathematical Modeling Speciality Group of the Association of American Geographers; he is also an editor of Transactions in GIS and the UK editor of Geographical and Environmental Modelling.
'Phone: +44 (0)191 222 6434
Fax: +44 (0)191 222 5421
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