By K. Bley, J. Brunvoll, R. Carlson, B. N. Cyvin, S. J. Cyvin, B. Gruber, E. Hladka, M. Knauer, J. Koca, M. Kratochvil, V. Kvasnicka, L. Matyska, A. Nordahl, J. Pospichal, V. Potucek, N. Stein, I. Ugi (auth.)
1. R. Carlson, A. Nordahl: Exploring natural artificial Experimental tactics 2. S.J. Cyvin, B.N. Cyvin, J. Brunvoll: Enumeration of Benzenoid Chemical Isomers with a learn of Constant-Isomer sequence three. E.Hladka, J. Koca, M. Kratochvil, V. Kvasnicka, L. Matyska, J. Pospichal, V. Potucek: The Synthon version and this system PEGAS for machine AssistedOrganic Synthesis four. ok. Bley, B. Gruber, M. Knauer, N. Stein, I. Ugi: New parts within the illustration of the Logical constitution of Chemistry byQualitative Mathematical versions and Corresponding information buildings
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2 Central Composite Designs A central composite design consists of three parts: A factorial or fractional factorial design One or several experiments at the centre point (average settings of the experimental variables A set of experiments symmetrically displaced along the variable axes. The distributions of experimental points for designs in two and three variables are shown in Fig. 6. A response surface model can be used for predictions and simulations by entering values of variable settings in test points to the model.
To estimate these coefficients, a design with twelve experiments was assumed to be sufficient. To span a maximum of variation the experimental design should include combinations of the variable setting at their high and low levels. With six variables at two levels, this gives a total of 2 6 = 64 possible combinations. /(52! 28 x 1012 different ways to select twelve experiments from these 64 combinations a n d it is evident that a r a n d o m selection runs the risk of being a very p o o r choice.
7) (X - X)' (X - X) = P L P ' (42) 37 Rolf Carlson and Ake Nordahl The eigenvectors of (X - X)' (X - X) are the principal component vectors p~ and the eigenvalues X~describe how much of the total variance is accounted for by p~. The diagonal eigenvalue matrix L can be written as the product of two matrices T ' T in which the columns tl are mutually orthogonal and have their scalar product t'ih = X~. If we let T be a (n x k) matrix which obeys these criteria, then T ' T = L. We can therefore write (X -- X)' (X - X) = P T ' T P ' (43) which is equivalent to (X -- X)' (X - X) = (TP')' (TP') (44) (X -- X) = TP' (45) which yields It is seen that any (n x k) matrix can be factorized into a (n x k) score matrix T and an orthogonal eigenvector matrix.