PCA-based approximation of a class of distributed parameter systems: classical vs. neural network approach

In this article, an approximation of the spatiotemporal response of a distributed parameter system (DPS) with the use of the principal component analysis (PCA) is considered. Based on a data obtained by the numerical solution of a set of partial differential equations, a PCA-based approximation procedure is performed. It consists in the projection of the original data into the subspace spanned by the eigenvectors of the data covariance matrix, corresponding to its highest eigenvalues. The presented approach is carried out using both the classical PCA method as well as two different neural network structures: two-layer feed-forward network with supervised learning (FF-PCA) and single-layer network with unsupervised, generalized Hebbian learning rule (GHA-PCA). In each case considered, the effect of the approximation model structure represented by the number of eigenvectors (or, in the neural case, units in the network projection layer) on the mean square approximation error of the spatiotemporal response and on the data compression ratio is analysed. As shown in the paper, the best approximation quality is obtained for the classical PCA method as well as for the FF-PCA neural approach. On the other hand, an adaptive learning method for the GHA-PCA network allows to use it in e.g. an on-line identification scheme.