Chris Brunsdon

Geographically Weighted Regression

In regression models where the cases are geographical locations, sometimes regression coefficients do not remain fixed over space. A technique for exploring this phenomenon, geographically weighted regression is introduced. A related Monte Carlo significance test for spatial non-stationarity is also considered. Finally, an example of the method is given, using limiting long-term illness data from the 1991 UK census.

Geographically Weighted Regression: A Natural Evolution of the Expansion Method for Spatial Data Analysis

Geographically weighted regression and the expansion method are two statistical techniques which can be used to examine the spatial variability of regression results across a region and so inform on the presence of spatial nonstationarity. Rather than accept one set of ‘global’ regression results, both techniques allow the possibility of producing ‘local’ regression results from any point within the region so that the output from the analysis is a set of mappable statistics which denote local relationships. Within the paper, the application of each technique to a set of health data from northeast England is compared. Geographically weighted regression is shown to produce more informative results regarding parameter variation over space.

Estimating probability surfaces for geographical point data: an adaptive kernel algorithm

The statistical analysis of spatially referenced information has been acknowledged as an important component of geographical data processing. With the arrival of GIS there has been a need to devise statistical methods that are compatible with, and relevant to, GIS-based methodologies. Here an algorithm is presented which estimates a “risk surface” from a set of point-referenced events. Such a surface may be viewed as an object embedded in three-dimensional space, or as a contour map. In addition to this view, it is possible to incorporate these surfaces into a broader based GIS framework, allowing the mapping of these patterns in conjunction with other data, overlay analysis, and spatial query. The technique is adaptive, in the sense that parameters which control the surface estimation are adjusted over geographic space, allowing for local variations in point pattern characteristics. The paper is concluded with an example based on probabilistic mapping using data taken from Californian Redwood seedling data.

Spatial Nonstationarity and Autoregressive Models

Until relatively recently, the emphasis of spatial analysis was on the investigation of global models and global processes. Recent research, however, has tended to explore exceptions to general processes, and techniques have been developed which have as their focus the investigation of spatial variations in local relationships. One of these techniques, known as geographically weighted regression (GWR), developed by the authors is used here to investigate spatial variations in spatial association. The particular framework in which spatial association is examined here is the spatial autoregressive model of Ord, although the technique can easily be applied to any form of spatial autocorrelation measurement. The conceptual and theoretical foundations of GWR applied to the Ord model are followed by an empirical example which uses data on owner-occupation in the housing market of Tyne and Wear in northeast England where the problems of relying on global models of spatial association are demonstrated. This empirical investigation of spatial variations in spatial autocorrelation prompts a further discussion of several issues concerning the statistical technique.

A spatial analysis of variations in health access: linking geography, socio-economic status and access perceptions

BackgroundThis paper analyses the relationship between public perceptions of access to general practitioners (GPs) surgeries and hospitals against health status, car ownership and geographic distance. In so doing it explores the different dimensions associated with facility access and accessibility.MethodsData on difficulties experienced in accessing health services, respondent health status and car ownership were collected through an attitudes survey. Road distances to the nearest service were calculated for each respondent using a GIS. Difficulty was related to geographic distance, health status and car ownership using logistic generalized linear models. A Geographically Weighted Regression (GWR) was used to explore the spatial non-stationarity in the results.ResultsRespondent long term illness, reported bad health and non-car ownership were found to be significant predictors of difficulty in accessing GPs and hospitals. Geographic distance was not a significant predictor of difficulty in accessing hospitals but was for GPs. GWR identified the spatial (local) variation in these global relationships indicating locations where the predictive strength of the independent variables was higher or lower than the global trend. The impacts of bad health and non-car ownership on the difficulties experienced in accessing health services varied spatially across the study area, whilst the impacts of geographic distance did not.ConclusionsDifficulty in accessing different health facilities was found to be significantly related to health status and car ownership, whilst the impact of geographic distance depends on the service in question. GWR showed how these relationships were varied across the study area. This study demonstrates that the notion of access is a multi-dimensional concept, whose composition varies with location, according to the facility being considered and the health and socio-economic status of the individual concerned.

Principal Component Analysis on Spatial Data: An Overview

This article considers critically how one of the oldest and most widely applied statistical methods, principal components analysis (PCA), is employed with spatial data. We first provide a brief guide to how PCA works: This includes robust and compositional PCA variants, links to factor analysis, latent variable modeling, and multilevel PCA. We then present two different approaches to using PCA with spatial data. First we look at the nonspatial approach, which avoids challenges posed by spatial data by using a standard PCA on attribute space only. Within this approach we identify four main methodologies, which we define as (1) PCA applied to spatial objects, (2) PCA applied to raster data, (3) atmospheric science PCA, and (4) PCA on flows. In the second approach, we look at PCA adapted for effects in geographical space by looking at PCA methods adapted for first-order nonstationary effects (spatial heterogeneity) and second-order stationary effects (spatial autocorrelation). We also describe how PCA can be used to investigate multiple scales of spatial autocorrelation. Furthermore, we attempt to disambiguate a terminology confusion by clarifying which methods are specifically termed “spatial PCA” in the literature and how this term has different meanings in different areas. Finally, we look at a further three variations of PCA that have not been used in a spatial context but show considerable potential in this respect: simple PCA, sparse PCA, and multilinear PCA.

Visualising space and time in crime patterns: A comparison of methods

Abstract Previous research exploring space–time patterns has focused on the relative merits and drawbacks of the effectiveness of static maps vis-a-vis interactive dynamic visualisation techniques. In particular, they have tended to concentrate on the role of animation in interpretation of patterns and the understanding of underlying factors influencing such patterns. The aim of this paper is to broaden this debate out to consider the effectiveness of a wider range of visualisation techniques in permitting an understanding of spatio-temporal trends. The merits of three visualisation techniques, (map animation, the comap and the isosurface) are evaluated on their ability to assist in the exploration of space–time patterns of crime disturbance data. We conclude that each technique has some merit for crime analysts charged with studying such trends but that further research is needed to apply the techniques to other sources of crime data (and to other sectors such as health) to permit a comprehensive evaluation of their respective strengths and limitations as exploratory visualisation tools.

Geographically weighted summary statistics — a framework for localised exploratory data analysis

Geographical kernel weighting is proposed as a method for deriving local summary statistics from geographically weighted point data. These local statistics are then used to visualise geographical variation in the statistical distribution of variables of interest. Univariate and bivariate summary statistics are considered, for both moment-based and order-based approaches. Several aspects of visualisation are considered. Finally, an example based on house price data is presented.

Non-linearities in drip water hydrology: an example from Stump Cross Caverns, Yorkshire

Drip rate data have been collected at 15 min intervals at six locations in Stump Cross Caverns, N England, since 1998. The different drip sites cover a wide range of drip rates from ∼2 drips/s to 2 drips/h, and in general the variability of drip rate increases with mean drip rate. In our continuous data sampling we observe rapid discharge increases which appear to be synchronous between drips sites, and which can be explained by flow switching of the water overlying the cave during times of high infiltration rate, such as intense rain storms or rapid snowmelt. A test for non-linearity (White test) in the drip series provides very strong evidence that many of the drip sequences are non-linear. We conclude that at our drip sites there is a non-linear input (weather) and non-linearities within the karst system leading to non-linear dripping, which is independent of drip rate. Our results have implications for stalagmite palaeoclimatology, where such widespread non linearities have not been taken account of.

The use of spatial analytical techniques to explore patterns of fire incidence : A South Wales case study

The application of mapping and spatial analytical techniques to explore geographical patterns of crime incidence is well established. In contrast, the analysis of operational incident data routinely collected by fire brigades has received relatively less research attention, certainly in the UK academic literature. The aim of this paper is to redress this balance through the application of spatial analytical techniques that permit an exploration of the spatial dynamics of fire incidents and their relationships with socio-economic variables. By examining patterns for different fire incident types, including household fires, vehicle fires, secondary fires and malicious false alarms in relation to 2001 Census of Population data for an area of South Wales, we demonstrate the potential of such techniques to reveal spatial patterns that may be worthy of further contextual study. Further research is needed to establish how transferable these findings are to other geographical settings and how replicable the findings are at different geographical scales. The paper concludes by drawing attention to the current gaps in knowledge in analysing trends in fire incidence and proposes an agenda to advance such research using spatial analytical techniques.