  A new methodology is proposed for generating realizations of a random vector
with values in a finite-dimensional Euclidean space that are statistically
consistent with a data set of observations of this vector. The probability
distribution of this random vector, while a-priori not known, is presumed to be
concentrated on an unknown subset of the Euclidean space. A random matrix is
introduced whose columns are independent copies of the random vector and for
which the number of columns is the number of data points in the data set. The
approach is based on the use of (i) the multidimensional kernel-density
estimation method for estimating the probability distribution of the random
matrix, (ii) a MCMC method for generating realizations for the random matrix,
(iii) the diffusion-maps approach for discovering and characterizing the
geometry and the structure of the data set, and (iv) a reduced-order
representation of the random matrix, which is constructed using the
diffusion-maps vectors associated with the first eigenvalues of the transition
matrix relative to the given data set. The convergence aspects of the proposed
methodology are analyzed and a numerical validation is explored through three
applications of increasing complexity. The proposed method is found to be
robust to noise levels and data complexity as well as to the intrinsic
dimension of data and the size of experimental data sets. Both the methodology
and the underlying mathematical framework presented in this paper contribute
new capabilities and perspectives at the interface of uncertainty
quantification, statistical data analysis, stochastic modeling and associated
statistical inverse problems.
