Multi-linear

The simplest function with several predictors is a multi-linear function of the form... 

This step involves calibration of the apparatus which will serve as a reference. It consists of analysing the greatest number possible (minimum 50) wines or must samples containing different and accurately known concentrations of each analyte. The concentration points should be uniformly distributed over the probable scale of measure for each analyte. The matrices should mimic as accurately as possible the wines and musts destined for analysis using that particular instrument. For each calibration sample, a measurement is carried out at a maximum number of wavelengths in the infra-red. Multi-linear regression is then carried out on the results which enables the following relationship to be established ... 

Fig. 5 Chargeability factor A/can be predicted by a Fig. 6 Permeability prediction from electrical multi-linear model composed by different behaviour and structures parameters of porous parameters formation factor F, water porosity O, solids, k Katz and Thompson model Hg-specific surface Asp and water permeability k for kjsc Johnson, Schwartz and co-workers different textures. model.

Chargeability factor M depends on the brine/gas saturation of porous solids. Figure 3 gives the relationship between the chargeability and brine saturation for two samples. We noted that the M decreases hardly with the decrease of the brine saturation. The presence of vugs and karsts pore types (sample 9-LS8) seems to speed up the decrease of the M Chargeability factor M can be explained by a multi-linear model composed of different structures parameters such as the formation resistivity factor, water porosity, Hg-specific surface area and water permeability, e.g.. Fig. 5. 

O Galan, A Palazoglu, and JA Romagnoli. Robust Hoo control of nonlinear plants based on multi-linear models - An application to a bench scale pH neutralization reactor. Chem. Engg. Sci., 55 4435-4450, 2000. 

O Galan, JA Romagnoli, A Palazoglu, and Y Arkun. The gap metric concept and implications for multi-linear model-based controller design. Ind. Eng. Chem. Research, 42 2189-2197, 2003. 

In the present study, two ANN methods - the FFBP with the Levenberg-Marquardt algorithm and the radial basis functions (RBF) - were employed to estimate the air pollution parameters measured at a station in Istanbul on chosen episode days, with the focus on the particulate matter. The results were compared with those obtained with the multi-linear regression (MLR) method. 

It is instructive to have an understanding of why the Khatri-Rao product is useful for multi-linear models. Starting with a bilinear two-way model... 

Bro R, Multiway calibration. Multi-linear PLS, Journal of Chemometrics, 1996,10, 47-61. 

Bro R, Smilde AK, de Jong S, On the difference between low-rank and subspace approximation improved model for multi-linear PLS regression, Chemometrics and Intelligent Laboratory Systems, 2001, 58, 3-13. 

Another factorial design, used for studying solubility in mixed micelles, introduces and demonstrates multi-linear regression and analysis of variance. It is then extended, also in chapter 5, to a central composite design to illustrate the estimation of predictive models and their validation. 

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. 

As before, the coefficients may be estimated either by linear combinations (contrasts) corresponding to the 16 columns of the model matrix, or by multi-linear regression. These estimates are listed in table 3.9 and plotted in figure 3.8. 

Estimates of the statistical significance of the coefficients can and frequently should be obtained by other means - in particular by replicated experiments (usually centre points) followed by multi-linear regression of the data, and analysis of variance, as developed in chapter 4. The methods we have described above are complementary to those statistical methods and are especially useful for saturated designs of 12 to 16 or more experiments. For designs of only 8 experiments, the results of these analyses should be examined with caution. 

Another solution, by far the more general one, is to carry out the experiments in a random order. If there are the same number of experiments in the design as there are parameters in the postulated model, the effect of time is to perturb the different estimations in a random fashion. If the number of experiments exceeds the number of parameters to be estimated then the experimental error estimated by multi-linear regression (see chapter 4) includes the effect of time and is therefore overestimated with respect to its true value. 

We may take the results of the 2 factorial study of an effervescent table formulation reported earlier, and select the data corresponding to the 12 experiments of table 3.31. Estimates of the coefficients obtained either by contrasts or by the usual method of multi-linear regression are very close to those estimated from the... 

In the same way as for the % x 2 design, we can select the data for these 12 experiments from the experimental results of table 3.8 (2 effervescent tablet factor study) and estimate the coefficients by multi-linear regression. (The linear combinations method is not applicable here.) The results, given in table 3.36, are almost identical to those found with the full design and reported in table 3.9. 

---