This chapter introduces the mathematics of data through the example of clustering, a fundamental technique in data analysis and machine learning. The chapter begins with a review of essential mathematical concepts, including matrix and vector algebra, differential calculus, optimization, and elementary probability, with practical Python examples. The chapter then delves into the k-means clustering algorithm, presenting it as an optimization problem and deriving Lloyd's algorithm for its solution. A rigorous analysis of the algorithm's convergence properties is provided, along with a matrix formulation of the k-means objective. The chapter concludes with an exploration of high-dimensional data, demonstrating through simulations and theoretical arguments how the "curse of dimensionality" can affect clustering outcomes.

Chapter 2 explores the fundamental concept of least squares, covering its geometric, algebraic, and numerical aspects. The chapter begins with a review of vector spaces and matrix inverses, then introduces the geometry of least squares through orthogonal projections. It presents the QR decomposition and Householder transformations as efficient methods for solving least-squares problems. The chapter concludes with an application to regression analysis, demonstrating how to fit linear and polynomial models to data. Key topics include the normal equations, orthogonal decomposition, and the Gram–Schmidt algorithm. The chapter also addresses the issue of overfitting in polynomial regression, highlighting the importance of model selection in data analysis. The chapter includes practical Python implementations and numerical examples to reinforce the theoretical concepts.

Covers differentiation and integration, higher derivatives, partial derivatives, series expansion, integral transforms, convolution integrals, Laplace transforms, linear and time-invariant systems, linear ordinary differential equations, periodic functions, Fourier series and transforms, and matrix algebra.

Chapter 2 is methodological, offering a primer on multimodal network analysis. It proceeds by quickly reviewing 1-mode network analysis, paying special attention to summarizing several measures of network centrality and how they relate to power. Often, relational data that are 2-mode or multimodal are “projected” into one of the node sets. Ties are then defined by their shared relations to the second-mode nodes so that 1-mode measures of centrality and algorithms for community detection can be employed. We discuss the loss of information on structure and agency that projection entails and argue that, in many cases, projection is neither helpful nor necessary. We then proceed to detailed discussions of methods for 2-mode and 3-mode network analysis, from first principles of matrix algebra to centrality measures and core-periphery analysis; faction analysis and community detection; as well as structural/regular equivalence and blockmodeling. We conclude with a brief introduction to recent advances in statistical network modelling that facilitate inferences about multimodal networks.

Abadir and Magnus (2002, Econometric Theory) proposed a standard for notation in econometrics. The consistent use of the proposed notation in our volumes shows that it is in fact practical. The notational conventions described here mainly apply to the material covered in this volume. Further notation will be introduced, as needed, as the Series develops.