Matrix of parameters

Again, let X be the matrix of parameter coefficients defined by the model to be fit and the coordinates of the experiments in factor space. Let Y be the response matrix associated with those experiments. Let B be the matrix of parameters, and let a new matrix R be the matrix of residuals. Equation 5.25 may now be rewritten in matrix notation as... 

It is not possible to fit this model using matrix least squares techniques The matrix of parameter coefficients, X, does not exist - it is a 0x0 matrix and has no elements because there are no parameters in the model. However, the matrix of residuals, R, is defined. It should not be surprising that for this model, R = Y that is, the matrix of residuals is identical to the matrix of responses. 

The effect on the variance-covariance matrix of two experiments located at different positions in factor space can be investigated by locating one experiment at X, = 1 and varying the location of the second experiment. The first row of the matrix of parameter coefficients for the model y,- = + p,jc, + r, can be made to... 

Although it is true that the first three columns of plus and minus signs in Table 14.3 are equivalent to the abbreviated coded experimental design matrix D, the signs in Table 14.3 are used for a slightly different purpose than they were Table 14.2. In fact, as we will see, the eight columns of signs in Table 14.3 are equivalent to the matrix of parameter coefficients, X. 

Let us fit the probabilistic model, = P0 + ru, to the same data (see Figure 5.10). If the least squares approach to the fitting of this model is employed, the appropriate matrices and results are exactly those given in Section 5.2 where the same model was fit to the different factor levels xu = 3, yn = 3, xl2 = 6, yl2 = 5. This identical mathematics should not be surprising the model does not include a term for the factor xx and thus the matrix of parameter coefficients. A", should be the same for both sets of data. The parameter / 0 is again estimated to be 4, and ar2 is estimated to be 2. 

An indicative and preliminary statistical analysis of the parameters obtained by regression analysis can be done by regarding two important quantities the variances of the parameters and the correlation coefficients between them. These quantities are calculated from the objective function (Q) and the matrix of parameter derivatives A, given by eq. (10.45). 

The species list for the Mabira Forest Reserve was obtained from the 1996 biodiversity report (Davenport et al., 1996). A total of 284 species of trees and shrubs were recorded by the inventory and a further 28 species were reported to be known from previous studies (total 312). It was not possible to create a full data matrix of parameters for all 312 species as the information had to be collated from printed Floras. As we were conducting a functional analysis to aid species selection for restoration ecology, we concentrated on useful trees and shrubs as defined in the Floras of Katende et al. (1995) and Maundu and Tengnas (2005). The analysis was conducted on 131 species from the Mabira Forest Reserve (42%). A total of the 65 most abimdant species were selected for study in the Mau Forest Complex, Kenya, on the basis of their abundance in the forest. A Flora published by the Museum of Kenya (Beentje, 1994) was also utilised to obtain information on the trees from the Mau Forest Complex. 

In the examples above the L/D ratio remains constant but the screw length can also be changed to retain extra degrees of freedom. This leads to a three-dimensional matrix of parameters, but that is outside the scope of this work. 

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