| Analytical instrumentation that produces multivariate data is now commonplace in chemical laboratories. Such data includes first-order tensors ( e.g. a spectrum), second-order tensors (e.g. chromatography with multichannel detection), and third- and higher order tensors ( e.g. fluorescence excitation-emission-lifetime measurements). From a data analysis perspective, the "cubes" of data that form third-order tensors, or three-way data, offer unique advantages not observed for lower order measurements. In particular, data that exhibit a trilinear structure can be decomposed in such a way that unique underlying factors are extracted, without the rotational ambiguity that exists when bilinear data are used. Common data analysis tools employed to carry out this type of decomposition include the well-known Parallel Factor Analysis (PARAFAC) and Direct Trilinear Decomposition (DTLD) algorithms. The application of these tools to trilinear data has tremendous potential to extract fundamental information such as spectra, concentration profiles, rate constants, and equilibrium constants from complex mixtures with little or no prior information. However, this potential is mitigated by the fact that these methods do not optimally accommodate the complex error structures commonly found in three-way data.; In this work, the development and application of Maximum Likelihood Parallel Factor Analysis (MLPARAFAC) is described. This approach is designed to incorporate prior measurement error information, including information about heteroscedascity and correlation of errors, into the decomposition procedure. Although MLPARAFAC is an extension of maximum likelihood methods for two-way data, the application to three-way data greatly expands the types of error interactions that can be observed, the size of matrices produced, and complexity of the algorithms involved. The principles behind the generalized MLPARAFAC algorithm, as well as several simplifications based on different measurement error structures and pre-compression of the data, are described. These algorithms are applied to simulated data and three fluorescence data sets to demonstrate their statistical validity, computational efficiency and estimation accuracy. Further implementation of MLPARAFAC in conjunction with the Direct Exponential Curve Resolution Algorithm (DECRA) is also examined using simulated data and spectral data from two widely studied reaction systems. |
