Leverage prediction

It is often helpful to examine the regression errors for each data point in a calibration or validation set with respect to the leverage of each data point or its distance from the origin or from the centroid of the data set. In this context, errors can be considered as the difference between expected and predicted (concentration, or y-block) values for the regression, or, for PCA, PCR, or PLS, errors can instead be considered in terms of the magnitude of the spectral... 

For this example, the prediction sample leverages are shown in Figure 5.109 with a horizontal line denoting three times the average leverage of the calibration samples. In this case, all of the samples are below the calibration maximum and are comparable with each other except for sample 13. This is another indication that this sample is outside the calibration range. 

FIGURE 5.109. Sample leverage for the prediction data. The horizontal line represents three times the average leverage from calibration. 

Leverage is a measure of the location of a prediction sample in the calibration measurement row space. A high leverage indicates a sample that has an unusual score vector relative to the calibration samples. 

I niike calibration samples, the leverage values for prediction samples are not constrained to be less than 1. 

Sample Leverage A plot of the leverage for the prediction samples is shown in Figure 5.124. All of the samples have leverage values less than three times the average leverage from the calibration (denoted by the horizontal line). Tliis is an indication that the model is not being used to extrapolate. 

R. Marbach and H. M. Heise, Calibration modeling by partial least-squares and principal component regression and its optimisation using an improved leverage correction for prediction testing, Chemom. Intell. Lab. Syst., 9(1), 1990, 45-63. 

The /z-statistic, or leverage, appears naturally when the predictions are calculated from the model. The leverage is calculated on the scores subspace and, thus, hi = 1// + [53a=i [23], where the letters... 

As the validation set is used to test the model under real circumstances , it may occur that one or several samples are different from the calibration samples. When we use experimental designs to set the working space, this is not frequent, but it may well happen in quality control of raw materials in production, environmental samples, etc. Therefore, it is advisable to apply the Hotelling s T or the leverage diagnostics to the validation set. This would prevent wrong predictions from biasing the RMSEP and, therefore, the final decision on the acceptance of the PLS model. 

Such real-time evaluation of process samples can be done by developing a PCA model of the calibration data, and then using this model in real time to generate prediction residuals (RESp) and leverages for each sample.3 Given a PCA model of the analytical profiles in the calibration data (conveyed by T and P), and the analytical profile of the prediction sample (xp), the scores of the prediction sample can be calculated ... 

The prediction leverage (LEVp) can then be calculated using the scores of the prediction sample, along with the scores of the PCA model ... 

Figure 8.23 Example of the use of an on-line PCA model application to identify outliers during prediction. Solid line ratio of prediction residual to average sample residual of the calibration samples dotted line ratio of prediction leverage to the average sample leverage of the calibration samples.

There are several methods that can be used to select well-distributed calibration samples from a set of such happenstance data. One simple method, called leverage-based selection, is to run a PCA analysis on the calibration data, and select a subset of calibration samples that have extreme values of the leverage for each of the significant PCs in the model. The selected samples will be those that have extreme responses in their analytical profiles. In order to cover the sample states better, it would also be wise to add samples that have low leverage values for each of the PCs, so that the center samples with more normal analytical responses are well represented as well. Otherwise, it would be very difficult for the predictive model to characterize any non-linear response effects in the analytical data. In PAC, where spectroscopy and chromatography methods are common, it is better to assume that non-linear effects in the analytical responses could be present than to assume that they are not. 

In a series of articles in 2001, Butler, Samdani, and McNish described the increase in LBOs in the chemical industry over the second half of the 1990s (Butler, P. Samdani, G. S. et al.). They labeled this phenomenon the Alchemy of Leveraged Buyouts and predicted a natural convergence between traditional chemical corporations and their new financial competitors in terms of management procedures and skills as well as how they created value. 

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