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UBC Theses and Dissertations

Mathematical methods for automated flow injection analysis Lee, Oliver


Presently, the development of automated flow injection analyzers for use in both chemical research and process control is an active area of research in this laboratory. For these systems to operate for extended periods of time without human supervision, it is imperative that they behave in an intelligent manner. It is also desirable that they produce the best possible data. In this thesis, advanced mathematical strategies have been devised to enhance the robustness, analytical reliability and performance of automated flow injection analyzers. The mathematical methods developed are based on signal representation by orthogonal polynomials. Any arbitrary function can be expanded as a weighted linear combination of orthogonal functions via a generalized Fourier expansion; together, the weights or coefficients form a spectrum. The most familiar of these functions is the complex exponential set associated with the classical Fourier series expansion. In addition to this set, the Gram and Meixner polynomial families have been employed here. Since the latter are transients, they are particularly suited for representing flow injection data. A peak shape analysis strategy is presented for automatic identification or classification of both physicochemical and mechanical faults during analyzer operation. The coefficients from decomposition of the flow injection peak into orthogonal functions were used as general descriptors of peak shape. Each set of functions generated a different spectrum and thus each offered a different view of the peak. The effects of white noise on the reproducibility of the individual coefficients was quantified. The Meixner polynomials are orthogonal over a semi-infinite interval. In practice, these functions must be truncated, and orthogonality is lost between all functions in the set. However, a time scale parameter is available to stretch or compress these functions. It was found that for robust identification, the time scale should be set at the highest value possible at which orthogonality was still observed numerically over the subset of functions chosen for identification. An empirical model was formulated to allow direct computation of this time scale. The capability of these descriptors for peak classification was demonstrated with simulated data and principal components analysis. A simple method based on the Fisher weight was developed to optimize the number of coefficients to use for a given classification problem. From the results obtained with this method, 5 to13 coefficients are recommended for most FIA systems; this applies to each of the three sets of orthogonal functions used. Orthogonal function representation was also employed for digital filtering. A comparison between two implementations: the finite impulse response filter and the indirect filter, was conducted with the Gram polynomials. The latter was found to be better suited for filtering typical (highly skewed) flow injection peaks. The efficacy of the three basis functions for indirect filtering of peak-shaped transient signals was subsequently compared. The Meixner filter was found to be best for highly skewed peaks, the Fourier filter was best for more symmetric peaks and the Gram filter performed somewhere in between. The problem of determining the optimal filter cut-off order was cast in terms of hierarchical model selection. Two solutions are presented: the Akaike information criterion and cross-validation. Near-optimal filtering can be achieved with either method and hence, automatic filtering is facilitated. This was demonstrated on both simulated and real data.

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