Multi-block PLS modeling for L-shape data structures with applications to mixture modeling

In some situations blocks of data are available from different sources. These data matrices often have no common dimension and are connected in a T- or L-shape structure. The example treated in this paper involves mixture modeling where blocks of data are available on the final product properties (Y), on the process conditions used to manufacture the products (Z), on the ratios of raw materials used in the formulations (R) and on the properties of each of those raw materials (X). In previous research [K. Muteki, J.F. MacGregor, T. Ueda, On the rapid development of new polymer blends: The optimal selection of materials and blend ratios, Industrial Engineering Chemical Research, Published on-line, May 25 (in press)] T- or L-shaped data matrices were combined using ideal or other known mixing rules to achieve a block data structure with a single common dimension as required in regression modeling. In this paper, we present an alternative multi-block PLS regression approach that does not require the use of such mixing rules to combine the data. Rather, it directly builds a multi-block PLS model for the T- or L-shaped structure. The approach is very flexible and provides increased interpretability on the relationships among the blocks. Furthermore, the method is extended to allow for nonlinear modeling between any of the adjacent data blocks, thereby allowing for non-ideal situations where ideal mixing rules do not hold or are unknown, and where the process is nonlinear. The approach is illustrated using data from an industrial coke making operation where many coals having different properties are blended and processed in coke ovens.