Generalized Variance

Abstract

Generalized variance (GV), proposed by Wilks, is an one‐dimensional measure of multidimensional scatter. It plays an important role in both theoretical and applied research. GV has been extended to Standardized GV (SGV) to enable comparison of scatters in differing dimensions. Interesting results on the distribution, estimation, and testing of GVs and SGVs have been emerging. Also, these measures have been finding novel applications in a variety of applied problems, ranging from such age‐old areas as agriculture and sample surveys to the modern developments in signal processing and Bioinformatics.

Preserving projection properties when regular two-level designs are blocked

“…The 12 blocks obtaining the highest minimum D s -efficiency, highest average D s -efficiency and a maximum D s -efficiency of 1 were regarded as the best. These correspond to columns 9,10,11,12,13,14,15,16,50,52,54 and 56 in R. They all obtained the same frequencies of D s -efficiencies when testing all combinations of three active factors, as shown in table 4.42. The blocks can be found along with the design matrix in columns b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 and b12 in the tables 6.21 and 6.22 in Appendix A.…”

“…They define a D-optimal design X D as the design which minimises the generalised variance of the parameter estimates. The generalised variance is defined as the determinant of the covariance matrix, according to Gupta [13]. The covariance matrix of β is σ 2 (X T X) −1 , so minimising its determinant equals maximising the value of |X T X|.…”

“…Considering four blocks instead, the 2 8−3 IV design was made by rearranging M2 in the order 32, 4,5,25,30,18,3,15,12,24,17,13,10,6,23,27,28,8,1,29,26,22,7,11,16,20,21,9,14,2,19 and 31. The columns 3, 7, 29, 4 and 15 were used to define factors A, B, C, D and E, respectively.…”

“…Among the candidate blocks, 16 obtained both the highest minimum and the highest average D s -efficiency. They were given by the column combinations (9,19), (9,23), (10,19), (10,23), (11,31), (11,32), (12,31), (12,32), (13,25), (13,26), (14,25), (14,26), (18,25), (18,26), (22,25) and (22,26). The columns are shown in table 4.33.…”

Proof that the Kalman gain minimizes the generalized variance

“…Another scalar measure of dispersion is the generalized variance, introduced by Samuel S. Wilks, which is defined as the determinant of the covariance matrix. The generalized variance has been used in a wide variety of fields as a measure of multidimensional scatter [9,4,2].…”