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In our previous work, a reduced order model (ROM) for a stochastic system wasmade, where noisy data was projected onto principal component analysis(PCA)-derived basis vectors to obtain an accurate reconstruction of thenoise-free data. That work used techniques designed for deterministic data, PCAwas used for the basis function generation and $L_2$ projection was used tocreate the reconstructions. In this work, probabilistic approaches are used.The probabilistic PCA (PPCA) is used to generate the basis, which then allowsthe noise in the training data to be estimated. PPCA has also been improved sothat the derived basis vectors are orthonormal and the variance of the basisexpansion coefficients over the training data set can be estimated. Thestandard approach assumes a unit variance for these coefficients. Based on theresults of the PPCA, model selection criteria are applied to automaticallychoose the dimension of the ROM. In our previous work, a heuristic approach wasused to pick the dimension. Lastly, a new statistical approach is used for theprojection step where the variance information obtained from the improved PPCAis 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 noisestatistics for the projected data are not assumed to be the same as that of thetraining data, but are estimated in the projection process. The entire approachgives a fully stochastic method for computing a ROM from noisy training data,determining ideal model selection, and projecting noisy test data, thusenabling accurate predictions of noise-free data from data that is dominated bynoise.

Improved Probabilistic Principal Component Analysis for Application to Reduced Order Modeling