MLR models

Multiple linear regression (MLR) models a linear relationship between a dependent variable and one or more independent variables. 

NN models for the three datasets contained the same number of descriptors as the MLR models, yet no more than two descriptors in each model were the same in both NN and MLR models. No descriptor was found in common with all models, although, each model contained a descriptor that relied on H-bonding in some manner. Nonlinear modeling from the NN approach gave better representation of the data than the linear models from MLR the value for the three datasets was 0.88, 0.98 and 0.90, respectively. 

The quality of fit of an MLR model can be assessed nsing both the RMSEE and the correlation coefficient, discussed earlier (Equations 12.11 and 12.10). 

Figure 12.10 The fit of the MLR model for c s-butadiene content in styrene-butadiene copolymers that uses four selected x-variables. The four variables were chosen by the stepwise method.

Table 12.5 The model fit (RMSEE) values of 4 different MLR models for predicting cis-butadiene content in styrene-butadiene copolymers by NIR spectroscopy using 1, 2, 3 and 4 x variables that were selected in a stepwise manner...

In the last several years, a set of BBB QSAR models have been developed. One of the top models, developed by Abraham et al. in 2006 [44], reached the predictive limit obtainable from the data set they used. The experimental errors of the logBB measurements were estimated to be 0.3. Their model utilized linear free energy relationship (LFER) as descriptors. For the 328-molecule data set, r2 and RMSE of the MLR model were 0.75 and 0.3 log units, respectively. Interestingly, the RMSE for their test set (n = 164) was even lower (0.25 log units). 

Zhao and coworkers [53] also constructed a linear model using the Abraham descriptors. The MLR model possesses good correlation and predictability for external data sets. In this equation, E is an excess molar refraction (cm3/mol/ 10.0) and S the dipolarity/polarizability, A and B are the hydrogen bond acidity and basicity, respectively, and V is the McGowan characteristic volume (cm3/ mol/100). The large coefficients of A and B indicate too polar molecules having poor absorption. 

FIGURE 5,81. Predicted versus known caustic concentration for the validation samples, three-variable MLR model. 

The advantage of estimating a model with stepwise MLR rather than with the full-spectrum techniques (e.g., PLS and PCR, Section 5-3-2) is that the MLR model is simple. It does not add variables whose variability is described by previously entered variables or that are not linearly related to the anal te of interest (e.g., have large contributions from interfering species). With the full-spectrum techniques, all sources of variation are implicitly accounted for in the model. This is a more complicated way of dealing with the variation not related to the anahte of interest. 

TABLE 5.10. MLR Model for Component A Using One Variable Plus an Intercept. ... 

FIGURE 5.87. Preprocessed prediction samples, with the vertical line indicating the variables used in the MLR model. 

Y value. The only relevant measure of how well the MLR model performs is provided by the Y variances. Residual Y variance is the variance of the Y residuals. It expresses how much variation remains in the observed response after the modeled part is removed. It is an overall measure of misfit, that is, the error made when fitted... 

Only a small number of applications combining multiple linear regression and atomic spectrometry have been published. Most of them have been covered in two reviews [23,24], which readers are strongly encouraged to consult. Since most applications of MLR models have been studied together with the use of PLS models, the practical examples will be referred to in Chapter 4, in order to avoid unnecessary repetitions here. 

The MLR model is simply an extension of the linear regression model (Equation 8.6), and is given below ... 

When MLR is used to provide a predictive model that uses multiple analyzer signals (e.g. wavelengths) as inputs and one property of interest (e.g. constituent concentration) as output, it can be referred to as an inverse linear regression method. The word inverse arises from the spectroscopic application of MLR because the inverse MLR model represents an inverted form of Beer s Law, where concentration is expressed as a function of absorbance rather than absorbance as a function of concentration 1,38... 

Once the regression coefficients of the inverse MLR model are determined (according to Equation 8.24), the property (Y-value) of an unknown sample can be estimated from the values of its selected X-variables (Xse] p) ... 

The fit of an MLR model can be assessed using both the RMSEE and the correlation coefficient, discussed earlier (Equations 8.10 and 8.11). The correlation coefficient has the advantage that it takes into account the range of the Y-data, and the RMSEE has the advantage that it is in the same units as the property of interest. 

Aside from univariate linear regression models, inverse MLR models are probably the simplest types of models to construct for a process analytical application. Simplicity is of very high value in PAC, where ease of automation and long-term reliability are critical. Another advantage of MLR models is that they are rather easy to communicate to the customers of the process analytical technology, since each individual X-variable used in the equation refers to a single wavelength (in the case of NIR) that can often be related to a specific chemical or physical property of the process sample. 

E-state indices, counts of atoms determined for E-state atom types, and fragment (SMF) descriptors. Individual structure-complexation property models obtained with nonlinear methods demonstrated a significantly better performance than the models built using MLR. However, the consensus models calculated by averaging several MLR models display a prediction performance as good as the most efficient nonlinear techniques. The use of SMF descriptors and E-state counts provided similar results, whereas E-state indices led to less significant models. For the best models, the RMSE of the log A- predictions is 1.3-1.6 for Ag+and 1.5-1.8 for Eu3+. 

The accordance of the modeled values of Cd and Zn with the experimentally determined values is sufficient. So the effects of binding of Cd and Zn to fulvic acid in water are quantitatively interpretable. The error of PLS modeling is smaller than that of MLR modeling because the interactions between the analyzed heavy metals have also been taken into consideration. 

R2 and Q2Ext were 0.81 and 0.79, respectively. Recently, the four-variable MLR model was applied to 122 various organic solvents and provide even more reliable predictions than the investigation performed on much smaller pool of solvents [73]. 

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