Regression coefficients, calculating

Scheme I Schematic representation of (a) PLS model (b) UVE-PLS model and (c) matrix B, containing regression coefficients calculated by the leave-one-out cross-validation procedure and their mean values, standard deviations, and stability.

For the spectroscopic identification probability I/>(5p) regression coefficients calculated from the X un and Xmax values, as well as the ratios of relative adsorbance (D) have to be high. Based on experimental results for the I/>(chr) and both lp(Sp) values empirical categories, high, medium, and low identification assurance levels can be established. In planar chromatography, two compounds (the reference and the substance to be identified) have to be identical with adequate assurance if the chromatographic and spectroscopic identification probabilities have minimum medium levels, as is demonstrated in Figure 12. 

The widespread availabihty and utihzation of digital computers for distillation calculations have given impetus to the development of analytical expressions for iregression equation and accompanying regression coefficients that represent the DePriester charts of Fig. 13-14. Regression equations and coefficients for various versions of the GPA convergence-pressure charts are available from the GPA. 

This will cause CLS to calculate an additional pure component spectrum for the G s. It will also give us an additional row of regression coefficients in our calibration matrix, Kc , which we can, likewise, discard. 

We calculate the calibration (regression) coefficients on a rank-by-rank basis using linear regression between the projections of the spectra on each individual spectral factor with the projections of the concentrations on each corresponding concentration factor of the same rank. 

Linear regression coefficients should be calculated for the ratio of analyte to internal standard area or height plotted versus the ratio of analyte to internal standard concentration in the calibration standards. The data from the analytical standards should then be fitted to the linear model... 

In this least squares method example the object is to calculate the terms /30, A and /J2 which produce a prediction model yielding the smallest or least squared differences or residuals between the actual analyte value Cj, and the predicted or expected concentration y To calculate the multiplier terms or regression coefficients /3j for the model we can begin with the matrix notation ... 

Like PCR, a PLS model can be condensed into a set of regression coefficients (bpLs)- For the NIPALS algorithm discnssed above, the regression coefficients can be calculated by the following equation ... 

Finally, the regression coefficient matrix S is calculated as a function of A, B, C and NCOMP... 

The optimal number of components from the prediction point of view can be determined by cross-validation (10). This method compares the predictive power of several models and chooses the optimal one. In our case, the models differ in the number of components. The predictive power is calculated by a leave-one-out technique, so that each sample gets predicted once from a model in the calculation of which it did not participate. This technique can also be used to determine the number of underlying factors in the predictor matrix, although if the factors are highly correlated, their number will be underestimated. In contrast to the least squares solution, PLS can estimate the regression coefficients also for underdetermined systems. In this case, it introduces some bias in trade for the (infinite) variance of the least squares solution. 

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