Validation of Dynamic Models in the Time-Scale Domain

Model validation is the procedure whereby the fidelity of a model is evaluated. The traditional approaches to dynamic model validation consider model outputs and observations as time series and use their similarity to assess the closeness of the model to the process. A common measure of similarity between the two time series is the cumulative magnitude of their difference, as represented by the sum of squared (or absolute) prediction error. Another important measure is the similarity of shape of the time series, but that is not readily quantifiable and is often assessed by visual inspection. This paper proposes the continuous wavelet transform as the framework for characterizing the shape attributes of time series in the time-scale domain. The feature that enables this characterization is the multiscale differential capacity of continuous wavelet transforms. According to this feature, the surfaces obtained by certain wavelet transforms represent the derivatives of the time series and, hence, can be used to quantify shape attributes, such as the slopes and slope changes of the time series at different times and scales (frequencies). Three different measures are considered in this paper to quantify these shape attributes: (i) the Euclidean distance between the wavelet coefficients of the time series pairs to denote the cumulative difference between the wavelet coefficients, (ii) the weighted Euclidean distance to discount the difference of the wavelet coefficients that do not coincide in the time-scale plane, and (iii) the cumulative difference between the markedly different wavelet coefficients of the two time series to focus the measure on the pronounced shape attributes of the time series pairs. The effectiveness of these measures is evaluated first in a model validation scenario where the true form of the process is known. The proposed measures are then implemented in validation of two models of injection molding to evaluate the conformity of shapes of the models’ pressure estimates with the shapes of pressure measurements from various locations of the mold.