This chapter provides a focused examination of spatio-temporal analysis using multilayer networks in which each layer represents the instantiation of a spatial network at a particular time of observation. The nodes in all layers may be the same with the only differences being of edges among layers (a multiplex network) or the nodes may change or move between layers and times. Multilayer characteristics such as versatility (multilayer centrality) and spectral properties are introduced. Several examples are described and reviewed as model studies for future ecological applications.

Some of the key messages of this book are reviewed here in the format of ’reminders’ to clarify the concerns of past misunderstandings and to emphasize solutions to perceived challenges. The importance of basic fundamentals, such as visual assessment, awareness of assumptions and potential numerical solutions is described and then the complementarity of the many statistics and their bases is reviewed. The exciting potential of ongoing developments is summarized, featuring hierarchical Bayesian analysis, spatial causal inference, applications of artificial intelligence (AI), knowledge graphs (KG), literature-based discovery (LBD) and geometric algebra. A quick review of future directions concludes this chapter and the book.

Sets of points can be analysed from their positions in space and line segments can be studied separately for their own spatial arrangements and relationships. Combining points and lines as the nodes and edges of a spatial graph provides a flexible and powerful approach to spatial analysis. Such graphs and their network versions are studied by Graph Theory, a branch of mathematics that quantifies their properties, with or without additional features such as labels, weights and functions associated with the nodes and edges. Some relevant graph theory terms are introduced, including connectivity, connectedness, modularity and centrality. Networks are graphs with additional features, usually representing an observed system of interest, whether aspatial like a food web or spatial like a metacommunity. Key concepts for the latter example are connectivity, migration and network flow.

The spatial patterns of point events in the plane can exist at several different scales in a single data set. The assessment of point patterns can be based on the distances between neighbour events, on the counts of events in quadrats or on counts of events in point-centred circles of changing size. Ripley’s K function evaluates simple point patterns and can be modified for different spatial dimensions, for bi- and multi-variate variables and for non-homogeneous data. Quadrat-based quantitative data are usually analysed by one of many related ’quadrat variance’ methods that assess variance or covariance as a function of spatial scale and which can also be modified for different conditions, such as bi- or multi-variate data. There are related methods from other traditions to be considered, including spectral analysis and wavelets. These approaches share a conceptual basis of comparing the data with spatial templates and we provide a summary of their relationships and differences.

Spatial structure is key to understanding diversity in ecological systems, being affected by both location and scale. The effects of scale are often dealt with as the hierarchy of alpha (local area), beta (between areas) and gamma (largest areas) diversity. All have spatial aspects, but beta diversity may be most interesting for spatial analysis because it involves complex responses such as intermediate-scale nestedness and species turnover with or without environmental gradients. In addition to species diversity within communities, the diversity of species composition or combinations as a function of location is an important characteristic of ecological assemblages. Many aspects of spatial diversity are best understood by spatial graphs, with sites as nodes and edges quantifying inter-site relationships. Temporal information, when available, can provide crucial insights about spatial diversity through understanding the dynamics of the system.

Spatial analysis originated in a broad range of disciplines, producing a diverse set of concepts and terminologies. Ecological processes take place in space and time, and the spatio-temporal structure that results takes different forms that produce spatial dependence at all scales. That dependence has major effects, even when ecological data are abstracted from the spatial context. Not all dependence exhibits a smooth decay with increasing separation, but it can vary with scale, stationarity or its absence and direction (anisotropy versus isotropy). A key factor in spatial analysis is the ability to determine neighbour events for points or patches and we present various algorithms to create networks of neighbours. We discuss a range of spatial statistics and related randomization tests, including a ’Markov and Monte Carlo’ approach. The chapter provides a detailed conceptual background for the technical aspects presented in subsequent chapters.

We start the explanation of analyzing spatial sample data with join-count statistics for regular (lattice) and irregular (spatial network) samples, leading to methods for spatial autocorrelation and variography or geostatistics. The latter provides spatial interpolation methods that estimate variables at unsampled locations, based on the values at measured samples. There are a range of such methods based on different assumptions and the types of data analysed. For quantitative data, Kriging estimates interpolated values at unsampled locations and their associated errors. In these applications, as elsewhere, there is an important distinction between global and local statistics and their estimates.

The analysis of spatio-temporal data is critical for understanding change in ecological systems. Spatio-temporal methods are the natural extensions of spatial statistics incorporating change over time. This chapter covers spatio-temporal approaches such as join counts, scan statistics, cluster and polygon change and the analysis of movement, cyclic phenomena and synchrony. In all these applications, we must consider and account for multi-dimensional autocorrelation in the data.

This chapter examines the related objectives of defining spatial clusters and delineating spatial boundaries in discontinuous data. The former often proceeds by grouping together adjacent locations when they have the most similar characteristics; the latter proceeds by estimating boundaries between locations that are most different. For this, there are several methods available that suggest ’boundary elements’ as possible components of a final division or complete boundary, depending on the kind of data (e.g. binary versus qualitative versus continuous quantitative) and the arrangement of the measured locations (e.g. regular lattice versus irregular spatial network). Once boundaries have been established, statistics are available to evaluate them, including boundary overlap measures. Clusters and boundaries represent two aspects of the same phenomenon, with the same challenge of formalizing similarity and difference in continuous spatial data.

The presence of autocorrelation in data violates the usual assumption of independence in the data for evaluating inferential statistics. We describe several models of autocorrelation in spatial data (both positive and negative). Given two serial variables, x and y, autocorrelation observed in y can be due to inherent autoregression in the variable itself, autoregression induced by its dependence on x, which has its own autocorrelation, or doubly autoregressive, with autocorrelation in both variables. This effect can be addressed by estimating the effective sample size (number of independent observations equivalent in information content to the n that are autocorrelated). We present the calculation of the effective sample size for many inferential statistics, including correlation, partial correlation, t-tests and ANOVA. The use of restricted randomization is explained as a method for testing when other approaches are not available. We also provide recommendations for sampling and experimental design in the presence of spatial autocorrelation.