There has been rapid development of novel and advanced statistical methods and computational algorithms for genomic, epigenomic, physiological, and imaging data analysis in the past several years. These methods and algorithms involve advanced statistics and convex optimization methods. To lay solid mathematical foundation for modern genomic and epigenomic analysis, this chapter introduces (1) sparsity-inducing norms, dual norms, and Fenchel Conjugate; (2) subdifferential calculus; (3) convex optimization and proximal methods; (4) matrix calculus; (5) principal component analysis; (6) functional principal component analysis; (6) canonical correlation analysis; and (7) functional canonical correlation analysis.
Various norms, subdifferential and subgradient, and matrix calculus are powerful tools for developing efficient statistical methods and computational algorithms in genomic, epigenomic, and imaging data analysis. The coverage of these materials in this book will facilitate the training of a new generation of statistical geneticists and computational biologists with modern mathematics. Proximal methods have been recently developed for convex optimization, which provide foundations for regularized statistical methods, low-rank models, machine learning, and causal inferences. Both multivariate and functional principal component analysis and canonical correlation analysis, which are major high-dimension data reduction methods, are elucidated in detail.