Linear dependence in data

Observing a process, scientists and engineers frequently record several variables. For example, (ref. 20) presents concentrations of all species for the thermal isomerization of a-pinene at different time points. These species are ct-pinene (yj), dipentene ( 2) allo-ocimene ( 3), pyronene (y ) and a dimer product (y5). The data are reproduced in Table 1.3. In (ref. 20) a reaction scheme has also been proposed to describe the kinetics of the process. Several years later Box at al. (ref. 21) tried to estimate the rate coefficients of this kinetic model by their multiresponse estimation procedure that will be discussed in Section 3.6. They run into difficulty and realized that the data in Table 1.3 are not independent. There are two kinds of dependencies that may trouble parameter estimation  

Multiplying (1.104) by X we have (X X)u = 0, and thus there exists an affine linear dependence of the form (1.103) among the columns of Y if and only if the matrix X X has a i = 0 eigenvalue. It is obvious that Xmin will equal not zero, but some small number because of the roundoff errors. 

To obtain a similar upper bound on xmin in the case (i), when there are only roundoff errors present, Box at al. (ref. 21) suggested to assume that the rounding error is distributed uniformly with range -0.5 to +0.5 of the last digit reported in the data. The rounding error 

We used the program given in Example 1.6 to calculate the eigenvalues and eigenvectors of X X, where X is the centered observation matrix from the data of Table 1.3. The program output is as follows. 

Since X4, 5 4.2, and both are close to the threshold min = 0.006, we expect to find two exact linear dependences in the data. From an exemination ofthe original paper (ref. 20) Box at al. (ref. 21) found that 4 had been not measured because of experimental difficulties, but rather had been assumed to constitute . of the total conversion of a-pinene (y ). That is, it was assumed that 4 = 0.031100-y ), which gives the exact affine linear relationship 

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