This editorial piece considers what happens when we abandon the concept that models of social processes have global application in favor of a local approach in which context or the influence of \u27place\u27 has an important role. A brief history of this local approach to statistical modelling is given, followed by a consideration of its ramifications for understanding societal issues. The piece concludes with futures challenges and prospects in this area

Fotheringham, A. Stewart

English

University of Maine

JOURNAL OF SPATIAL INFORMATION SCIENCENumber 21 (2020), pp. 83–87 doi:10.5311/JOSIS.2020.21.726INVITED ARTICLELocal modeling:one size does not fit allA. Stewart FotheringhamSchool of Geographical Sciences and Urban Planning, Arizona State University, Tempe, AZ, USAReceived: November 4, 2020; accepted: November 17, 2020Abstract: This paper considers what happens when we abandon the concept that modelsof social processes have global application in favor of a local approach in which context orthe influence of ‘place’ has an important role. A brief history of this local approach to statis-tical modeling is given, followed by a consideration of its ramifications for understandingsocietal issues. The piece concludes with future challenges and prospects in this area.Keywords: local modeling, spatial context, process nonstationarity1 IntroductionA goal of research, in whatever discipline, is to discover something new about the wayour world, and our universe, works. We are typically prompted to ask questions such as“why are things like that?” or “what’s causing that to happen?” by observing data onsomething and noting that the values fluctuate over time or over space or both. Such aprocess goes back to the dawn of civilization when people became aware of the passage ofthe sun over the course of a day, the changing position of the sun at midday over the courseof a year, and the shifting pattern of stars in the night sky. Today, in the spatial realm, weare more likely to be prompted to ask such questions by looking at maps of data, remotelysensed imagery or graphical representations of human behavior. Whatever the source ofour questions, they remain basically the same: “why are some values high and some low?”or “why is something present in this location but absent in that one?” Essentially we areasking questions about processes which we cannot observe that have produced data whichwe can observe. The situation is encapsulated in Figure 1.We measure associations between data that we observe on the real world and fromthese associations we infer something about the processes that have produced the data weobserve. Inference is generally necessary in spatial analysis because the processes we arec© by the author(s) Licensed under Creative Commons Attribution 3.0 License CC©84 FOTHERINGHAMFigure 1: The relationship between spatial data and spatial processes.interested in are unobservable—we can only see or measure their impacts. Consequently,the subject matter of spatial information science can be divided into the visible and theinvisible. Problems specific to spatial data (the visible) include spatial dependency and themodifiable areal unit problem; a problem specific to spatial processes (the invisible) is thatof spatial nonstationarity and it is this latter problem that is the focus of the remainder ofthis paper.Traditionally, in the analysis of spatial data and spatial processes it was assumed, gen-erally without thought, that the processes that produced the data and associations betweendata were stationary over space. That is, models were calibrated using data gathered frommultiple locations and a single parameter estimate representing each process or conditionalrelationship was estimated. Consequently, if the processes that were being modeled variedover space, such a single parameter estimate would represent an average of the underlyingspatially varying processes, much the same as, say, an average annual rainfall value for acountry represents, but hides, the interesting spatial variation in local rainfall values.The advent of local models in the last couple of decades has altered this perspectiveby allowing for the possibility that the processes being modeled might vary across spaceand replacing the single parameters that represented these processes in traditional modelswith location-specific parameters. Such local modeling frameworks have arisen from dif-ferent academic and philosophical perspectives but have in common the relaxation of thewww.josis.orgLOCAL MODELING: ONE SIZE DOES NOT FIT ALL 85assumption that processes are stationary over space. Examples of local modeling frameworks include Bayesian spatially varying coefficients models [1,5,9], multiscale geograph-ically weighted regression [7, 8, 13] and eigenvector spatial filtering [10].2 Implications for research on societal issuesThe development of local statistical models has several profound implications for researchon critical societal issues. First and foremost is the recognition that, whatever the focus,be it on health-related issues, the study of societal ills, voting preferences, or consumerchoice, a one-size-fits-all mentality might not be the most appropriate way to understandsuch issues. In situations involving human behavior where we want to know either if wechange x by a given amount, what will the change in y be or what is the best way to achievea certain change in y, a traditional global approach may be misleading. Where responsesto a given stimulus vary over space, more informed decision-making will result from thecalibration of models yielding parameter estimates unique to each location.Secondly, the recognition that processes might vary spatially is a mixed blessing interms of understanding the transferability of models in the social sciences. On one hand,if we cannot replicate the results of a model calibrated when we use data from a differentlocation, this does not necessarily mean the model is wrong. It could mean the behaviorwe are modeling is not the same everywhere so the results of the calibration of a model inone location will not be transferable to another location. To aim for models with consistentparameter estimates over space may be a false goal of social science research. On the otherhand, if such is the case then models need to be calibrated locally which increases the costand effort of data collection.Thirdly, if human preferences and behavior are partly a result of where people live, thisraises several questions. To what extent does spatial context affect our decision-making—does it play a major role in our decisions or a minor role and does it vary by the type ofdecision-making? Over what spatial area does context apply to decision-making—is it atthe level of a household, a street, a neighborhood, a city, a region, or a country? How doescontext affect behavior? Is it local media, the influence of family and friends, local cus-toms, exposures to various societal norms or combinations of all these? Is context simplya shorthand for model misspecification and could we include its effects directly throughmeasurable attributes?Fourthly, does process spatial nonstationarity explain the modifiable areal unit prob-lem? Most applications of spatial analytical research related to societal problems use ag-gregated data because individual-level data are generally not available for confidentialityreasons. A well-known problem in dealing with aggregated spatial data is the modifiableareal unit problem (MAUP) which is that the inferences we draw from the analysis of spa-tially aggregated data can depend on the level to which the data are aggregated [6]. Inits most extreme form it is possible to draw completely different inferences from the sameunderlying data which have been aggregated to different levels. To date, the cause andsolution to the MAUP eludes us. However, it is possible to rethink the MAUP as a resultof process spatial nonstationarity. If the underlying processes being examined are constantover space, no matter what aggregation of the basic data we undertake, our results shouldbe similar. However, if the processes producing the basic data vary over space, then dif-ferent aggregations of data will produce different results, and in extreme cases, differentJOSIS, Number 21 (2020), pp. 83–8786 FOTHERINGHAMinferences about the processes that produced the data. Local modeling may well then alle-viate the issue of the MAUP as inferences will not be based on one set of averaged results.Fifthly, the calibration of local models such as MGWR which yield covariate-specificmeasures of the scale over which different geographic processes are relatively stable, caninform decision-makers of the geographic extent of areas where certain actions can producedesirable results. That is, rather than having to undertake a global-model-led ‘shotgun’ ap-proach and applying policies or actions across an entire area, local models can highlightlimited subareas where actions should be concentrated to achieve more desirable and effi-cient outcomes.3 The futureLocal spatial modeling has been part of the spatial analyst?s armory for two decades yetit is still very much in its infancy and evolving rapidly—the advent of MGWR in 2017being an example [8]. One suspects much more is yet to come. For instance, conceptuallyone could derive optimised bandwidths as indicators of scale which not only vary acrosscovariates but also vary over space. That is, the area over which a process is relatively stableover space may itself vary over space. Such models would be hugely complex to calibrateand would produce massive amounts of output which would need careful examinationfor robust interpretation. Inference in local models is a challenge given both the issue ofmultiple hypothesis testing and the dependency of the tests being conducted [4, 12, 14].However, the big challenge in local modeling is not statistical but mental. There needsto be a move away from traditional global thinking to an increased recognition that thesame action may produce consequences that vary locally. That is, a one-size-fits-all mentalityis not always the best. This is being recognised statistically with the advent of local formsof analysis other than regression [2, 3, 11] but it has also been recognised more broadly forexample in the way that some countries have dealt with the coronavirus pandemic by notimposing blanket restrictions on every part of the country equally but by spatially varyingthese restrictions. Identifying where limited resources may have the most impact is thekey to improving societal well-being and this can only be done through a mentality thatrecognises that the unobservable processes producing the observable outcomes we want tochange are not the same everywhere and need to be examined locally.References[1] BANERJEE, S., CARLIN, B. P., AND GELFAND, A. E. Hierarchical modeling and analysisfor spatial data. CRC Press, 2014.[2] BOOTS, B., AND OKABE, A. Local statistical spatial analysis: Inventory and prospect.International Journal of Geographical Information Science 21, 4 (2007), 355–375.[3] COMBER, A., BRUNSDON, C., CHARLTON, M., AND HARRIS, P. Geographicallyweighted correspondence matrices for local error reporting and change analyses:Mapping the spatial distribution of errors and change. Remote Sensing Letters 8, 3(2017), 234–243.www.josis.orgLOCAL MODELING: ONE SIZE DOES NOT FIT ALL 87[4] DA SILVA, A. R., AND FOTHERINGHAM, A. S. The multiple testing issue in geograph-ically weighted regression. Geographical Analysis, 48 (2015), 233–247.[5] FINLEY, A. O. Comparing spatially-varying coefficients models for analysis of ecolog-ical data with non-stationary and anisotropic residual dependence: Spatially-varyingcoefficients models. Methods in Ecology and Evolution 2, 2 (2011), 143–154.[6] FOTHERINGHAM, A., AND WONG, D. W.-S. The modifiable areal unit problem inmultivariate statistical analysis. Environment and Planning A 23, 7 (1991), 1025–1044.[7] FOTHERINGHAM, A. S., LI, Z., AND WOLF, L. Scale, context and heterogeneity: Aspatial analytical perspective on the 2016 US presidential election. Annals of the Amer-ican Association of Geographers (2020).[8] FOTHERINGHAM, A. S., YANG, W., AND KANG, W. Multi-scale geographicallyweighted regression. Annals of the American Association of Geographers 107, 6 (2017),1247–1265.[9] GELFAND, A. E., KIM, H.-J., SIRMANS, C. F., AND BANERJEE, S. Spatial modelingwith spatially varying coefficient processes. Journal of the American Statistical Associa-tion 98, 462 (2003), 387–396.[10] GRIFFITH, D. A. Spatial-filtering-based contributions to a critique of geographicallyweighted regression (GWR). Environment and Planning A 40, 11 (2008), 2751–2769.[11] KORDI, M., AND FOTHERINGHAM, A. Spatially weighted interaction models – SWIM.Annals of the American Association of Geographers 106, 5 (2016), 990–1012.[12] LI, Z., FOTHERINGHAM, A. S., OSHAN, T. M., AND WOLF, L. J. Measuring band-width uncertainty in multiscale geographically weighted regression using akaikeweights. Annals of the American Association of Geographers 110, 5 (2020), 1500–1520.[13] WOLF, L. J., OSHAN, T. M., AND FOTHERINGHAM, A. S. Single and multiscale mod-els of process spatial heterogeneity: Single and multiscale models. Geographical Anal-ysis 50, 3 (2018), 223–246.[14] YU, H., FOTHERINGHAM, A. S., LI, Z., OSHAN, T., KANG, W., AND WOLF, L. J.Inference in multiscale geographically weighted regression. Geographical Analysis 52,1 (2020), 87–106.JOSIS, Number 21 (2020), pp. 83–87

Local modelling: one size does not fit all

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Local modelling: one size does not fit all

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