Continuous-time multivariate analysis

The starting point for much of multivariate analysis (MVA) is an $n\times p$ data matrix whose $n$ rows represent observations and whose $p$ columns represent variables. Some multivariate data sets, however, may be best conceptualized not as $n$ discrete $p$-variate observations, but as $p$ curves or functions defined on a common time interval. We introduce a framework for extending techniques of multivariate analysis to such settings. The proposed framework rests on the assumption that the curves can be represented as linear combinations of basis functions such as B-splines. This is formally identical to the Ramsay-Silverman representation of functional data; but whereas functional data analysis extends MVA to the case of observations that are curves rather than vectors -- heuristically, $n\times p$ data with $p$ infinite -- we are instead concerned with what happens when $n$ is infinite. We describe how to translate the classical MVA methods of covariance and correlation estimation, principal component analysis, Fisher's linear discriminant analysis, and $k$-means clustering to the continuous-time setting. We illustrate the methods with a novel perspective on a well-known Canadian weather data set, and with applications to neurobiological and environmetric data. The methods are implemented in the publicly available R package \texttt{ctmva}.