Maximum likelihood rectification

To alleviate these assumptions the maximum likelihood rectification (MLR) technique was proposed by Johnston and Kramer. This approach incorporates the prior distribution of the plant states, P x, into the data reconciliation process to obtain the maximum likely rectified states given the measurements. Mathematically the problem can be stated (Johnston and Kramer, 1995) as... 

This approach is equivalent to maximum likelihood rectification for data contaminated by Gaussian errors. The likelihood function is proportional to the probability of realizing the measured data, yj, given the noise-free data, yi. 

Unlike maximum likelihood rectification, Bayesian rectification can remove errors even in the absence of process models. Another useful feature of the Bayesian approach is that if the probability distributions of the prior and noise are Gaussian, the error of approximation between the noise-free and rectified measurements can be estimated before rectifying the data as,... 

The mean and standard deviation of the MSE for 500 realizations of the 2048 measurements per variable are summarized in Table 1, and are similar to those of Johnston and Kramer. The average and standard deviation of the mean-squared errors of single-scale and multiscale Bayesian rectification are comparable, and smaller than those of maximum likelihood rectification. The Bayesian methods perform better than the maximum likelihood approach, since the empirical Bayes prior extracts and utilizes information about the finite range of the measurements. In contrast, the maximum likelihood approach implicitly assumes all values of the measurements to be equally likely. If information about the range of variation of the rectified values is available, it can be used for maximum likelihood rectification, leading to more accurate results. For this example, since the uniformly distributed uncorrelated measurements are scale-invariant in nature, the performance of the single-scale and multiscale Bayesian methods is comparable. 

Fig. 11 Data rectification of signal with deterministic features. Dashed line is noisy data, (a) Original and noisy data, (h) Wavelet thresholding, (c) Wavelet thresholding after maximum likelihood rectification, (d) Multi.scale Bayesian rectification.

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. 

Lanteri, H., Roche, M., Cuevas, O., Aime, C., 2001, A general method to devise maximum-likelihood signal restoration multiplicative algorithms with nonnegativity constraints. Signal Processing 81, 945 Lucy, L.B., 1974, An iterative technique for the rectification of observed distributions, ApJ 79, 745... 

Johnston, L., and Kramer, M. A. (1995). Maximum likelihood data rectification Steady state systems. AIChE J. 41,2415-2426. 

Johnson, L.P.M. Kramer, M.A. Maximum likelihood data rectification steady-state systems. AIChE J. 1995, 41 (1), 2415-2426. 

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 multiscale rectification approach may be used for both maximum likelihood and Bayesian methods by solving the corresponding optimization... 

Gaussian errors with standard deviations 1, 4, 4, 3, and 1, respectively. The performance of maximum likelihood, single-scale Bayesian, and multiscale Bayesian rectification are compared by Monte-Carlo simulation with 500 realizations of 2048 measurements for each variable. The prior probability distribution is assumed to be Gaussian for the single-scale and multiscale Bayesian methods. The normalized mean-square error of approximation is computed as,... 

L.P.M. Johnston, M.A. Kramer. Maximum Likelihood Data Rectification Steady-State Systems, AIChE Journal, 41 (1995), 2415. 

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