Combinatorial projection pursuit analysis for exploring multivariate chemical data

Kurtosis-based projection pursuit analysis (kPPA) has demonstrated the ability to visualize multivariate data in a way that complements other exploratory data analysis tools, such as principal components analysis (PCA). It is especially useful for partitioning binary data sets (2^k classes) with a balanced design. Since kPPA is not a variance-based method, it can often provide unsupervised class separation where other methods fail. However, when multiple classifications are possible (e.g. by gender, age, disease state, etc.), the projection provided by kPPA (corresponding to the global minimum kurtosis) will not necessarily be the one of greatest interest to the researcher. Fortunately, the optimization algorithm for kPPA allows for interrogation of projections obtained from numerous local minima. This strategy provides the basis of a new method described here, referred to as combinatorial projection pursuit analysis (CombPPA) because it presents alternative combinations of class separation. The method is truly exploratory in that it allows the landscape of interesting projections to be more fully probed. The approach uses Procrustes rotation to map local minima among the kPPA solutions, whereupon the researcher can visualize different projections. To demonstrate the new method, the clustering of grape juice samples using visible spectroscopy is presented as a model problem. This problem is well-suited to this type of study because there are eight classes of samples symmetrically partitioned into two classes by type (organic/non-organic) or four classes by brand. Results presented show the different combinations of projections that can be obtained, including the desired partitions. In addition, this work describes new enhancements to the kPPA algorithm that improve the orthogonality of solutions obtained.