Description
This book demonstrates the application of covariance matrices through cutting-edge models and practical applications, as well as extensions induced by multivariate data and other related subjects. In data analysis, when studying the relationships among a set of variables, the covariance matrix plays an important role. It has been commonly and widely used across many fields, including agriculture, biology, business, communications, economics, engineering, finance, marketing, mathematics, medicine, data science, and social science, regardless of whether the data is dense or sparse, low-dimension or high-dimension, time series or non-time series, structured or unstructured, fixed or random, and training (learning) data or testing data. The covariance matrix is fundamental for extracting valuable information from multivariate data, such that this classical tool can be influential in modern data science and innovative statistical models.
Specifically, this book utilizes the covariance matrix to comprehensively unify classical multivariate methods (e.g., principal components and factor analysis) and innovative models and algorithms (e.g., spatial autoregressive and network autocorrelation models, matrix factor models, tensor covariance models, deep learning, and transfer learning). In so doing, it surveys statistical and data science techniques for estimation, selection, prediction, inference, and decision making. As a result, the book provides a unique approach for readers to understand how the traditional and modern techniques in data analysis, such as multivariate analysis and machine learning, can be unified with different features but the same foundation, which is the covariance matrix. This book is suitable for graduate students and researchers across various quantitative disciplines.
Chapter 1 Introduction.- Chapter 2 Covariance Matrices, Precision (Concentration) Matrices, Estimations, and Tests.- Chapter 3 Structured Covariance Matrices and Unconstrained Parameterizations.- Chapter 4 Covariance Regression Models.- Chapter 5 Covariance-Mean Regression Models.- Chapter 6 Fixed and Random Covariance Models.- Chapter 7 Spatial and Network Autoregressive Models.- Chapter 8 Factor Models and Covariance Matrices.- Chapter 9 Machine Learning and Covariance Matrices.- Chapter 10 Tensor Analysis and Covariance Matrices.
Wei Lan is Professor in the School of Statistics and Data Science and Center of Statistical Research, Southwestern University of Finance and Economics. His research interests include High Dimensional Data Analysis, Social Network Data Analysis, and the Application of Statistics to Finance (e.g., Empirical Asset Pricing, Risk Management and Portfolio Optimization).
Chih-Ling Tsai is Distinguished Professor of Management (Emeritus), University of California, Davis. His research interests include Regression Analysis, Model Selection, High Dimensional Data, Machine Learning, Time Series, Biostatistics, and the Application of Statistics to Business. Dr. Tsai is a Fellow of the American Association for the Advancement of Science and American Statistical Association, and an Elected Member of International Statistical Institute. He is also a co-author of the book Regression and Time Series Model Selection (World Scientific, 1998).
