The second dimension of spatial association
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A reasonable and adequate understanding of spatial association between geographical variables is the basis of spatial statistical inference and geocomputation, such as spatial prediction. Most of the current models for exploring spatial association of variables are constructed using data at sample locations. In this study, approaches for exploring spatial association using observations at sample locations are defined as the first dimension of spatial association (FDA). However, geographical information outside sample locations is usually missing in current models. To address this issue, this study proposes the concept of the second dimension of spatial association (SDA), which is an approach that extracts geographical information at locations outside samples for exploring spatial association. Based on the concept of SDA, three SDA models, including SDA-based multivariate linear regression, machine learning (i.e., random forest), and geostatistical models (i.e., random forest kriging), are developed for examining spatial association and predicting spatial distributions of trace elements, including Cr and Cu, in a mining region in Western Australia. Model accuracy is evaluated by comparing with corresponding FDA models. A new R package “SecDim” is developed to conduct SDA models. Results show that SDA models have a series of advantages in examining spatial association compared with FDA models. First, the accuracy of spatial prediction can be critically improved by SDA compared with the FDA, although identical explanatory variables and models are used for the modeling and prediction. Second, SDA can effectively indicate the multi-scale effects and diverse information within explanatory variables of the geographical environment at local ranges using the second dimension variables. Third, SDA can avoid underestimating high values and overestimating low values in the general FDA-based statistical models, machine learning, and geostatistical models. Finally, SDA models provide more smooth spatial predictions across space than that predicted by FDA models and avoid massive fluctuations at local ranges. The concept of SDA provides new insight into geographical information-based spatial association. SDA and multiple types of SDA models have great potential for more accurate and effective spatial statistical inference and geocomputation in diverse fields.
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