Maximum likelihood multivariate methods in analytical chemistry

The development of analytical methods demands a reliable method for modeling the instrumental response function. This is particularly true for multivariate measurements and is closely related to the measurement error characteristics of the method. This work presents new methods for extracting information from univariate and multivariate data sets based on the principle of maximum likelihood. These maximum likelihood techniques allow for the incorporation of measurement errors in the modeling process and therefore yield a more reliable representation of the true underlying model.; The potential of these techniques is first evaluated in two-dimensions where it was shown that these methods have properties that make them statistically desirable. A maximum likelihood analog to principal component analysis (PCA) is then developed for the multivariate case. The theoretical foundations of maximum likelihood principal component analysis (MLPCA) are initially established using a regression model and then extended to the framework of PCA and singular value decomposition (SVD). The proposed technique also allows for the incorporation of correlated errors and intercept terms. Simulated and experimental data are used to evaluate the performance of the new algorithm. In all cases, models determined by MLPCA are found to be superior to those obtained by PCA when non-uniform error distributions are present, although the level of improvement depends on the error structure of the particular data set.; To demonstrate the practical implications of MLPCA, this technique was applied to problems in multivariate calibration, the modeling of incomplete data sets, and calibration transfer. Two new calibration methods, maximum likelihood principal component regression (MLPCR) and maximum likelihood latent root regression (MLLRR), are developed which exhibit superior performance over conventional multivariate calibration methods when there is a non-uniform error structure. MLPCA is also shown to be useful in handling incomplete data sets in a reliable and simple manner by assigning large uncertainties to missing measurements. This approach is extended to the general problem of multivariate calibration transfer.