Fast and simple methods for the optimization of kurtosis used as a projection pursuit index

As a powerful method for exploratory data analysis, projection pursuit (PP) often outperforms principal component analysis (PCA) to discover important data structure. PP was proposed in 1970s but has not been widely used in chemistry largely because of the difficulty in the optimization of projection indices. In this work, new algorithms, referred as "quasi-power methods", are proposed to optimize kurtosis as a projection index. The new algorithms are simple, fast, and stable, which makes the search for the global optimum more efficient in the presence of multiple local optima. Maximization of kurtosis is helpful in the detection of outliers, while minimization tends to reveal clusters in the data, so the ability to search separately for the maximum and minimum of kurtosis is desirable. The proposed algorithms can search for either with only minor changes. Unlike other methods, no optimization of step size is required and sphering or whitening of the data is not necessary. Both univariate and multivariate kurtosis can be optimized by the proposed algorithms. The performance of the algorithms is evaluated using three simulated data sets and its utility is demonstrated with three experimental data sets relevant to analytical chemistry.