Multiple linear regression calibration model

FIGURE 4.24 PLS as a multiple linear regression method for prediction of a property y from variables xi,..., xm, applying regression coefficients b1,...,bm (mean-centered data). From a calibration set, the PLS model is created and applied to the calibration data and to test data. 

An extension of linear regression, multiple linear regression (MLR) involves the use of more than one independent variable. Such a technique can be very effective if it is suspected that the information contained in a single dependent variable (x) is insufficient to explain the variation in the independent variable (y). In PAT, such a situation often occurs because of the inability to find a single analyzer response variable that is affected solely by the property of interest, without interference from other properties or effects. In such cases, it is necessary to use more than one response variable from the analyzer to build an effective calibration model, so that the effects of such interferences can be compensated. 

Like MLR, PCR [63] is an inverse calibration method. However, in PCR, the compressed variables (or PCs) from PCA are used as variables in the multiple linear regression model, rather than selected original X variables. In PCR, PCA is first done on the calibration x data, thus generating PCA scores (T) and loadings (P) (see Section 12.2.5), then a multiple linear regression is carried out according to the following model ... 

ILS is a least-squares method that assumes the inverse calibration model given in eqn (3.4). For this reason it is often also termed multiple linear regression (MLR). In this model, the concentration of the analyte of interest, k, in sample i is regressed as a linear combination of the instrumental measurements at J selected sensors [5,16-19] ... 

For evaluation of the PLS model and for comparison with multiple linear regression the independent parameters were varied in the calibration range and predictions were made. Tab. 8-15 illustrates the comparison of the predicted and the measured values. 

In many chemical studies, the measured properties of the system can be regarded as the linear sum of the fundamental effects or factors in that system. The most common example is multivariate calibration. In environmental studies, this approach, frequently called receptor modeling, was first applied in air quality studies. The aim of PCA with multiple linear regression analysis (PCA-MLRA), as of all bilinear models, is to solve the factor analysis problem stated below ... 

Near-infrared (NIR) spectroscopy is becoming an important technique for pharmaceutical analysis. This spectroscopy is simple and easy because no sample preparation is required and samples are not destroyed. In the pharmaceutical industry, NIR spectroscopy has been used to determine several pharmaceutical properties, and a growing literature exists in this area. A variety of chemoinfometric and statistical techniques have been used to extract pharmaceutical information from raw spectroscopic data. Calibration models generated by multiple linear regression (MLR) analysis, principal component analysis, and partial least squares regression analysis have been used to evaluate various parameters. 

PLSR nowadays is a reference method for multivariate calibration and its utilization has overcome limitations in the use of multiple linear regressions. In the PLSR approach, the full spectrum is used to establish a linear regression model, where the significant information contained in the near-infrared spectra is concentrated in a few latent variables that are optimized to produce the best correlation with the desired property to be determined. 

Ayyalasomayajula et employed LIBS for the analysis of slurry samples. Three calibration models were developed using univariate calibration, multiple linear regression (MLR) and PLS. The LIBS analytical results obtained from the PLS model best fit the results obtained from inductively coupled plasma optical emission spectroscopy (ICP-OES) analysis. 

The previous section alludes to the most common problems in quantitative Raman spectroscopic calibrations Most models require that all components in a system to be known and modeled in the calibration data to accurately predict any one component. Inverse calibration techniques such as inverse multiple linear regression (inverse MLR), principal component regression (PCR) and partial least squares (PLS also known as principal latent structures) avoid this problem by forcing the calibration steps to utilize only the spectral features which are either changing (PCR) or directly correlated to the property of interest (PLS). More so, not all components in a sample need to be known to perform an inverse calibration. The basic form of an inverse calibration centers around an equation of the form... 

An important aspect of all methods to be discussed concerns the choice of the model complexity, i.e., choosing the right number of factors. This is especially relevant if the relations are developed for predictive purposes. Building validated predictive models for quantitative relations based on multiple predictors is known as multivariate calibration. The latter subject is of such importance in chemo-metrics that it will be treated separately in the next chapter (Chapter 36). The techniques considered in this chapter comprise Procrustes analysis (Section 35.2), canonical correlation analysis (Section 35.3), multivariate linear regression... 

This method can be considered a calibration transfer method that involves a simple instrument-specific postprocessing of the calibration model outputs [108,113]. It requires the analysis of a subset of the calibration standards on the master and all of the slave instmments. A multivariate calibration model built using the data from the complete calibration set obtained from the master instrument is then applied to the data of the subset of samples obtained on the slave instruments. Optimal multiplicative and offset adjustments for each instrument are then calculated using linear regression of the predicted y values obtained from the slave instrument spectra versus the known y values. 

Quantitative analysis for one or more analytes through the simultaneous measurement of experimental parameters such as molecular UV or infrared absorbance at multiple wavelengths can be achieved even where clearly defined spectral bands are not discernible. Standards of known composition are used to compute and refine quantitative calibration data assuming linear or nonlinear models. Principal component regression (PCR) and partial least squares (PLS) regression are two multivariate regression techniques developed from linear regression (Topic B4) to optimize the data. 

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