  We present a variational principle for the extraction of a time-dependent
orthonormal basis from random realizations of transient systems. The optimality
condition of the variational principle leads to a closed-form evolution
equation for the orthonormal basis and its coefficients. The extracted modes
are associated with the most transient subspace of the system, and they provide
a reduced description of the transient dynamics that may be used for
reduced-order modeling, filtering and prediction. The presented method is
matrix-free and relies only on the observables of the system and ignores any
information about the underlying system. In that sense, the presented reduction
is purely observation-driven and may be applied to systems whose models are not
known. The presented method has linear computational complexity and memory
storage requirement with respect to the number of observables and the number of
random realizations. Therefore, it may be used for a large number of
observations and samples. The effectiveness of the proposed method is tested on
three examples: (i) stochastic advection equation, (ii) a reduced description
of transient instability of Kuramoto-Sivashinsky, and (iii) a transient
vertical jet governed by incompressible Navier-Stokes equation. In these
examples, we contrast the performance of the time-dependent basis versus static
basis such as proper orthogonal decomposition, dynamic mode decomposition and
polynomial chaos expansion.
