Polycert, Institute of Chemical Physics RAS, 4 Kosygin Street, 117977,
Moscow, Russia; Email: polycert@chph.ras.ru
Successive Bayesian Estimation (SBE) technique is effective for complex nonlinear regression systems [1]. This method converts the whole problem to a sequence of small regressions. It was implemented for estimating of reaction rate constants from spectral data. The model for kinetics of spectral data is a function of time t and wavelength x depending on unknown rate constants k.
Here Y is the (n´m) measured data matrix; C is the (n´l) concentration matrix, and P is the (l´m) unknown matrix of pure component spectra, with l being the number of reaction components. Besides, the (n´m) error matrix E is involved in the model. If pure component spectra matrix P is known, we obtain a rather simple OLS problem. However, if one or more pure spectra are unknown – which is the usual case – the situation changes dramatically. Practically, it is very difficult to find the minimum of the sum with respect to the p+m´l unknown parameters owing to the problem with ill-posed matrices.
The SBE method can be applied to this problem. The main concept is to split the whole data set into several parts. Then estimating of parameters is performed successively – fraction by fraction – with MLM. It is important, that the results obtained on the previous step are used as a priori values for the next part. The initial fraction is processed by OLS method without any a priori information. During this procedure the sequence of the parameter estimates is produced and its last term is the ultimate estimate. For linear regression, it was shown that SBE gives the same estimates as the traditional approach and these values do not depend on the order of the fractions. In non-linear case, all these properties are the same, asymptotically. This method was validated by a simulated data set and by a real-world example of reaction of two-step epoxidation.
SBE is the method of general nature that can be used for any kind of kinetic models. This method is Bayesian only in its form but not in its concept because no subjective a priori data are actually used. Each a priori information element yields from the experimental data processed at the previous step and only the way of its application is dictated by the Bayes theorem. No extra assumptions (number of PCs, a time-shift, pseudo first-order) are needed for this method.
Bystritskaya E. V., Pomerantsev A. L., Rodionova O. Ye., J. Chemometrics, 14, (2000), 667-692.