Linear regression design matrix

The set of selected wavelengths (i.e. the experimental design) affects the variance-covariance matrix, and thus the precision of the results. For example, the set 22, 24 and 26 (Table 41.5) gives a less precise result than the set 22, 32 and 24 (Table 41.7). The best set of wavelengths can be derived in the same way as for multiple linear regression, i.e. the determinant of the dispersion matrix (h h) which contains the absorptivities, should be maximized. 

F design matrix for linear regression (mxnp or nsxnp)... 

The coefficients in the model equation 3.4 may be estimated as before, as linear combinations or contrasts of the experimental results, taking the columns of the effects matrix as described in section III.A.5 of chapter 2. Alternatively, they may be estimated by multi-linear regression (see chapter 4). The latter method is more usual, but in the case of factorial designs both methods are mathematically equivalent. 

On the other hand, atomic emission spectra are inherently well suited for multivariate analysis due to the fact that the intensity data can be easily recorded at multiple wavelengths. The only prerequisite is that the cahbration set encompasses all likely constituents encountered in the real sample matrix. Calibration data are therefore acquired by a suitable experimental design. Not surprisingly, many of the present analytical schemes are based on multivariate calibration techniques such as multiple linear regression (MLR), principal components regression (PCR), and partial least squares regression (PLS), which have emerged as attractive alternatives. 

A five-level Doehlert design provided mxfitivariate insight into the complex relations between interfering effects and matrix composition. Models were developed through multiple linear regression. 

Calibration Most process analyzers are designed to monitor concentration and/or composition. This requires a calibration of the analyzer with a set of prepared standards or from well-characterized reference materials. The simple approach must always be adopted first. For relatively simple systems the standard approach is to use a simple linear relationship between the instrument response and the analyte/ standard concentration [27]. In more complex chemical systems, it is necessary to adopt either a matrix approach to the calibration (still relying on the linearity of the Beer-Lambert law) using simple regression techniques, or to model the concentration and/or composition with one or more multivariate methods, an approach known as chemometrics [28-30]. 

According to the Hadamard matrix, a 22 factorial design was built. The complete linear models were fitted by regression for each response, reflecting the compression behaviour and dissolution kinetics. 

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