Linear operator‐based statistical analysis: A useful paradigm for big data

Abstract

In this article we lay out some basic structures, technical machineries, and key applications, of Linear Operator‐Based Statistical Analysis, and organize them toward a unified paradigm. This paradigm can play an important role in analyzing big data due to the nature of linear operators: they process large number of functions in batches. The system accommodates at least four statistical settings: multivariate data analysis, functional data analysis, nonlinear multivariate data analysis via kernel learning, and nonlinear functional data analysis via kernel learning. We develop five linear operators within each statistical setting: the covariance operator, the correlation operator, the conditional covariance operator, the regression operator, and the partial correlation operator, which provide us with a powerful means to study the interconnections between random variables or random functions in a nonparametric and comprehensive way. We present a case study tracing the development of sufficient dimension reduction, and describe in detail how these linear operators play increasingly critical roles in its recent development. We also present a coordinate mapping method which can be systematically applied to implement these operators at the sample level. The Canadian Journal of Statistics 46: 79–103; 2018 © 2017 Statistical Society of Canada

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