  In our previous work, a reduced order model (ROM) for a stochastic system was
made, where noisy data was projected onto principal component analysis
(PCA)-derived basis vectors to obtain an accurate reconstruction of the
noise-free data. That work used techniques designed for deterministic data, PCA
was used for the basis function generation and $L_2$ projection was used to
create the reconstructions. In this work, probabilistic approaches are used.
The probabilistic PCA (PPCA) is used to generate the basis, which then allows
the noise in the training data to be estimated. PPCA has also been improved so
that the derived basis vectors are orthonormal and the variance of the basis
expansion coefficients over the training data set can be estimated. The
standard approach assumes a unit variance for these coefficients. Based on the
results of the PPCA, model selection criteria are applied to automatically
choose the dimension of the ROM. In our previous work, a heuristic approach was
used to pick the dimension. Lastly, a new statistical approach is used for the
projection step where the variance information obtained from the improved PPCA
is used as a prior to improve the projection. This gives improved accuracy over
$L_2$ projection when the projected data is noisy. In addition, the noise
statistics for the projected data are not assumed to be the same as that of the
training data, but are estimated in the projection process. The entire approach
gives a fully stochastic method for computing a ROM from noisy training data,
determining ideal model selection, and projecting noisy test data, thus
enabling accurate predictions of noise-free data from data that is dominated by
noise.
