  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. Here we introduce a framework
for extending techniques of multivariate analysis to such settings. The
proposed continuous-time multivariate analysis (CTMVA) framework rests on the
assumption that the curves can be represented as linear combinations of basis
functions such as $B$-splines, as in 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 present continuous-time extensions of the classical MVA
methods of covariance and correlation estimation, principal component analysis,
Fisher's linear discriminant analysis, and $k$-means clustering. We show that
CTMVA can improve on the performance of classical MVA, in particular for
correlation estimation and clustering, and can be applied in some settings
where classical MVA cannot, including variables observed at disparate time
points. CTMVA is illustrated with a novel perspective on a well-known Canadian
weather data set, and with applications to data sets involving international
development, brain signals, and air quality. The proposed methods are
implemented in the publicly available R package \texttt{ctmva}.
