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.

GWmodel: An R Package for Exploring Spatial Heterogeneity Using Geographically Weighted Models

Geographically weighted regression (GWR) is an important local technique for exploring spatial heterogeneity in data relationships. In fitting with Tobler’s first law of geography, each local regression of GWR is estimated with data whose influence decays with distance, distances that are commonly defined as straight line or Euclidean. However, the complexity of our real world ensures that the scope of possible distance metrics is far larger than the traditional Euclidean choice. Thus in this article, the GWR model is investigated by applying it with alternative, non-Euclidean distance (non-ED) metrics. Here we use as a case study, a London house price data set coupled with hedonic independent variables, where GWR models are calibrated with Euclidean distance (ED), road network distance and travel time metrics. The results indicate that GWR calibrated with a non-Euclidean metric can not only improve model fit, but also provide additional and useful insights into the nature of varying relationships within the house price data set.

Purpose: This review of sediment source fingerprinting assesses the current state-of-the-art, remaining challenges and emerging themes. It combines inputs from international scientists either with track records in the approach or with expertise relevant to progressing the science.Methods: Web of Science and Google Scholar were used to review published papers spanning the period 2013–2019, inclusive, to confirm publication trends in quantities of papers by study area country and the types of tracers used. The most recent (2018–2019, inclusive) papers were also benchmarked using a methodological decision-tree published in 2017.Scope: Areas requiring further research and international consensus on methodological detail are reviewed, and these comprise spatial variability in tracers and corresponding sampling implications for end-members, temporal variability in tracers and sampling implications for end-members and target sediment, tracer conservation and knowledge-based pre-selection, the physico-chemical basis for source discrimination and dissemination of fingerprinting results to stakeholders. Emerging themes are also discussed: novel tracers, concentration-dependence for biomarkers, combining sediment fingerprinting and age-dating, applications to sediment-bound pollutants, incorporation of supportive spatial information to augment discrimination and modelling, aeolian sediment source fingerprinting, integration with process-based models and development of open-access software tools for data processing.Conclusions: The popularity of sediment source fingerprinting continues on an upward trend globally, but with this growth comes issues surrounding lack of standardisation and procedural diversity. Nonetheless, the last 2 years have also evidenced growing uptake of critical requirements for robust applications and this review is intended to signpost investigators, both old and new, towards these benchmarks and remaining research challenges for, and emerging options for different applications of, the fingerprinting approach.

Principal components analysis (PCA) is a widely used technique in the social and physical sciences. However in spatial applications, standard PCA is frequently applied without any adaptation that accounts for important spatial effects. Such a naive application can be problematic as such effects often provide a more complete understanding of a given process. In this respect, standard PCA can be (a) replaced with a geographically weighted PCA (GWPCA), when we want to account for a certain spatial heterogeneity; (b) adapted to account for spatial autocorrelation in the spatial process; or (c) adapted with a specification that represents a mixture of both (a) and (b). In this article, we focus on implementation issues concerning the calibration, testing, interpretation and visualisation of the location-specific principal components from GWPCA. Here we initially consider the basics of (global) principal components, then consider the development of a locally weighted PCA (for the exploration of local subsets in attribute-space) and finally GWPCA. As an illustration of the use of GWPCA (with respect to the implementation issues we investigate), we apply this technique to a study of social structure in Greater Dublin, Ireland.