Multiscale Bayesian data rectification

Existing methods for data rectification with process models including, maximum likelihood and Bayesian methods, are inherently single-scale in nature, since they represent the data at the same resolution everywhere in time and frequency. The multiscale Bayesian data rectification method developed in this section combines the benefits of Bayesian rectification and multiscale filtering using orthonormal wavelets. 

The methodology for multiscale Bayesian rectification is a special case of the general multiscale analysis and modeling methodology shown in Fig. 2. Each 

The linear process model used at each scale remains unchanged. 

Like the single-scale Bayesian approach, the multiscale Bayesian approach with Gaussian error and prior also provides an estimate of the covariance of the error of approximation at each scale as 

Decomposition of the variables on orthonormal wavelets decomposes the error covariance at all scales as shown in Eq. (2) since. 

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B.R. Bakshi, M.N. Nounou. P.K. Goel. X. Shen. Multiscale Bayesian Data Rectification with Linear Steady-State Models, /ml. Eng. CItein. Res., submitted (1999). 

A multiscale Bayesian approach for data rectification of Gaussian errors with linear steady-state models was also presented in this chapter. This approach provides better rectification than maximum likelihood rectification and single-scale Bayesian rectification for measured data where the underlying signals or errors are multiscale in nature. Since data from most chemical and manufacturing processes are usually multiscale in nature due to the presence of deterministic and stochastic features that change over time and/or frequency, the multiscale Bayesian approach is expected to be beneficial for rectification of most practical data. 

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