We don’t always have a single response variable, and disciplines like community ecology or the new “omics” bring rich datasets. Chapters 14–16 introduce the treatment of these multivariate data, with multiple variables recorded for each unit or “object.” We start with how we measure association between variables and use eigenanalysis to reduce the original variables to a smaller number of summary components or functions while retaining most of the variation. Then we look at the broad range of measures of dissimilarity or distance between objects based on the variables. Both approaches allow examination of relationships among objects and can be used in linear modeling when response and predictor variables are identified. We also highlight the important role of transformations and standardizations when interpreting multivariate analyses.

The components or functions derived from an eigenanalysis are linear combinations of the original variables. Principal components analysis (PCA) is a very common method that uses these components to examine patterns among the objects, often in a plot termed an ordination, and identify which variables are driving those patterns. Correspondence analysis (CA) is a related method used when the variables represent counts or abundances. Redundancy analysis and canonical CA are constrained versions of PCA and CA, respectively, where the components are derived after taking into account the relationships with additional explanatory variables. Finally, we introduce linear discriminant function analysis as a way of identifying and predicting membership of objects to predefined groups.

This chapter elaborates some initial efforts in establishing a tractable method, namely Formal Analysis (FA), for assessing the stability of networked microgrids under uncertainties from heterogeneous sources including DERs. Both centralized and distributed formal methods are established for computing the bounds of all possible trajectories and estimating the stability margin for the entire networked microgrid system.