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...time intervals (e.g., monthly volumes) can be used. Due to a fine seasonality of sales such brute force approaches rarely work. in multivariate statistics principal components analysis (PCA) is used (=-=[MKB80]-=-, [Jo186]), but this approach is problematic since it leads to clusters with poor interpretability. Singular value decomposition (SVD) based techniques are used to reduce dimensionality in information...

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...l C reports the results of multivariate F-tests on equality of coefficientssacross regressions, which are appropriate when different dependent variablessare regressed on common independent variables (=-=Mardia et al., 1979-=-). l° Thestests on the slope coefficient for the difference in sensitivity to negative andspositive returns,/~1, indicate that the coefficient in the regression for CFOI isssignificantly smaller than ...

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...outline for the proof of the asymptotic result discussed heresis given in the Appendix. A referee pointed out the simi-slarity between this result and the result of a likelihood ratiostest in MANOVA (=-=Mardia et al. 1979-=-, p. 342; further con-snection between SIR and sec. 12.5.4 of that reference cansalso be drawn). We remind the reader, however, that thesfundamental assumption about the error distribution insMANOVA i...

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...•) distances from group centroids to overall centroid; (q), overall centroid; (h), group centroid; (d), individual observation. Table 1c). Note that this additive partitioning using a geometric approach yields one value for each of SSW, SSA and SST as sums of squared Euclidean distances. This geometric approach gives sums of squares equivalent to the sum of the univariate sums of squares (added across all variables) described in the previous paragraph. This differs from the traditional MANOVA approach, where partitioning is done for an entire matrix of sums of squares and cross-products (e.g. Mardia et al. 1979; Table 1d). The key to the non-parametric method described here is that the sum of squared distances between points and their centroid is equal to (and can be calculated directly from) the sum of squared interpoint distances divided by the number of points. This important relationship is illustrated in Fig. 2 for points in two dimensions. The relationship between distances to centroids and interpoint distances for the Euclidean measure has been known for a long time (e.g. Kendall & Stuart 1963; Gower 1966; Calinski & Harabasz 1974; Seber 1984; Pillar & Orlóci 1996; Legendre & Legendre 1998; ...

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...tion, PCA is reviewed as background for the current developments. The NLPCA technique is then developed. Artificial neural networks are not reviewed here; the reader is referred to Rumelhart et al. (1986), Hoskins and Himmelblau (1988), or Venkatasubramanian and Chan (1989) for background information. Finally, NLPCA is applied to test problems, and results comparing PCA and NLPCA are presented. Principal Component Analysis PCA is a technique for mapping multidimensional data into lower dimensions with minimal loss of information. Since PCA has been described frequently by other authors (e.g., Mardia et al., 1980), only a brief summary is given here. Let Irepresent a n x rn table of data ( n = number of observations, g= number of variables). PCA is an optimal factorization of 1 into two matrices, _T called the scores matrix ( n x f, and Pca l l ed the loadings matrix (rn x f l , plus a matrix of residua% E - (n x rn): 234 February 1991 where f is the number of factors Cf<m). The Zondition of optimality on the factorization is that the Euc1id:an norm of the residual matrix, IIEll, must be minimized for the given number of factors. Tosatisfy this criterion, it i. known that the columns of _P are the eige...