This chapter covers principal component analysis and low-rank models, which are popular techniques to process high-dimensional datasets with many features. We begin by defining the mean of random vectors and random matrices. Then, we introduce the covariance matrix which encodes the variance of any linear combination of the entries in a random vector, and explain how to estimate it from data. We model the geographic location of Canadian cities as a running example. Next, we present principal component analysis (PCA), a method to extract the directions of maximum variance in a dataset. We explain how to use PCA to find optimal low-dimensional representations of high-dimensional data and apply it to a dataset of human faces. Then, we introduce low-rank models for matrix-valued data and describe how to fit them using the singular-value decomposition. We show that this approach is able to automatically identify meaningful patterns in real-world weather data. Finally, we explain how to estimate missing entries in a matrix under a low-rank assumption and apply this methodology to predict movie ratings via collaborative filtering.

This chapter covers the analysis of static systems under probabilistic input uncertainty. The first part of the chapter is devoted to analyzing linear and nonlinear static systems when the first and second moments of the input vector are known, and it provides techniques for characterizing the first and second moments of the state vector. For the linear case, the techniques provide the exact moment characterization, whereas for the nonlinear case, the characterization, which is based on a linearization of the system model, is approximate. The second part of the chapter provides techniques for the analysis of both linear and nonlinear static systems when the pdf of the input vector is known. The techniques included provide exact characterizations of the state pdf for both linear and nonlinear systems. In both cases, the inversion of the input-to-state mapping is required, which in the linear case involves the computation of the inverse of a matrix; however, for the nonlinear, it involves obtaining an analytical expression for the input-to-state mapping. The chapter concludes by utilizing the techniques developed to study the power flow problem under active power injection uncertainty.