Multiscale Bayesian rectification of data from linear steady-state and dynamic systems without accurate models

A common situation in chemical processes is that the measured data come from a dynamic process, but the available accurate process models only represent steady-state behavior. Furthermore, process data usually contain multiscale features due to different localizations in time and frequency. Existing methods for rectifying dynamic data require an accurate dynamic process model and are best for rectifying single-scale data. This paper presents a multiscale Bayesian approach for rectification of measurements from linear steady-state or dynamic processes with a steady-state model or without a model. This approach exploits the ability of wavelets to approximately decorrelate many autocorrelated stochastic processes and to extract deterministic features in a signal. The decorrelation ability results in wavelet coefficients at each scale that contain almost none of the process dynamics. Consequently, these wavelet coefficients can be rectified without a model or with a steady-state process model. The dynamics are captured in the wavelet domain by the scale-dependent variance of the wavelet coefficients and the last scaled signal. The proposed approach uses a scale-dependent prior for rectifying the wavelet coefficients and rectifies the last scaled signal without a model. In addition to more accurate rectification than existing methods, the multiscale Bayesian approach can eliminate the less relevant scales from the rectification before actually rectifying the data, resulting in significant savings in computation. This paper focuses on the rectification of Gaussian errors, but the approach is general and can be easily extended to other types of error distributions.