Absorbing phase transitions are an important class of nonequilibrium phase transitions. They are characterized by one or more absorbing states, defined as microscopic states from which the system cannot escape. The most famous case with one absorbing state is called directed percolation (a sort of driven version of the usual, isotropic percolation) and it represents, for example, the spreading of a disease through a contact process: If the infection rate is large enough with respect to the recovery rate, the asymptotic state shows a finite fraction of infected individuals. Models with one absorbing state, local dynamics, and no additional symmetries typically fall within the directed percolation universality class. We also provide a short introduction to self-organized criticality, devoting a section to the Bak–Tang–Wiesenfeld model.

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.

The first part of the chapter is a not-so-small presentation of equilibrium phase transitions, which allows us to introduce key concepts for both equilibrium and nonequilibrium phase transitions. The lattice gas, that is, the Ising model with a conserved order parameter, is an appropriate model to analyze how an equilibrium model can be brought out of equilibrium and to highlight the importance of boundary conditions in nonequilibrium phase transitions. The driven lattice gas, introduced by Katz, Lebowitz, and Spohn around 40 years ago, allows to define the totally asymmetric simple exclusion (TASEP) model and subsequently also the BRIDGE model. The latter is a one-dimensional model displaying a nonequilibrium phase transition with a symmetry breaking between two equivalent classes of particles. This result, considering the short-range character of interactions, would not be possible at equilibrium. In an equally unexpected way, an external breaking of the symmetry (equivalent to the application of a magnetic field to the Ising model) does not make the phase transition disappear.