Multigroup discrimination of river, beach, and dune sands using axial-ratio sample statistics of quartz grains

Optimal discrimination among several groups can be achieved by simultaneous diagonalization of pooled within-group, W, and among-group, A, sums of squares and cross-product matrices formed by utilizing axial-ratio sample statistics of quartz grains belonging to different sieve grades. This method maximizes the ratio of among-group cross products to within-group cross product quadratic forms (V'AV/V'WV)and simultaneously yields discriminant scores whose correlation coefficients are zero for group means as well as for within each group. This procedure enables a simple Euclidean distance measure for partitioning the discriminant space for assignment. Although W^–1and Amatrices are symmetric, the W^–1 Amatrix needed for multigroup discrimination is asymmetric and hence the eigenstructure of W^–1 Ais obtained by simultaneous diagonalization of Wand Amatrices. The first four sample statistics (mean, standard deviation, skewness, kurtosis) of normalized axial-ratios are required for discrimination, although the mean and standard deviation are the most important discriminators.