Leverage correction

It should be mentioned that another validation technique, called leverage correction [1], is available in some software packages. This method, unlike cross validation, does not involve splitting of the calibration data into model and test sets, but is simply an altered calculation of the RMSEE fit error of a model. This alteration involves the weighting of the contribution of the root mean square error from each calibration... 

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. 

Nebs T, Ellekjaer MR, Cross-validation and leverage-correction revisited, NIR News, 1993, 4, 8-9. 

Leverage correction. Leverage is a concept applied to observations used in a calibration model. It is a value between 1/n and 1, where n is the number of observations. The leverage of an object indicates its importance to the structure of the model. Observations having high leverage are important to the model in that they contribute greatly to its structure. 

For calibration methods other than MLR, the cross-validation relation [equation (10.16)] does not hold, but Martens and Naes (1989) have suggested adjusted leverage-correction MSEP, msep estimators for PCR and PLSR ... 

To discuss the prediction error, one must validate the calibration model [2]. There are two sorts of validation. One method is based on a new set of objects (external prediction). It requires a large and representative set of objects which have to be kept apart from the calibration for testing purposes only. The other validation method is based on the calibration data themselves (internal validation). In most cases, internal validation methods such as cross-validation and leverage correction [2] give sensible results with valuable information about the prediction ability. Cross-validation seeks to validate the calibration model with independent test data, but contrary to external validation it does not use data for testing only. The cross-validation is performed a number of times, each time with the use of only a few calibration samples as a test set. From the validation set it is possible to compare the prediction ability for the models, expressed by the estimated prediction mean square error. 

When this method is used, the models are calculated only once, which significantly increases the speed of the analysis. However, the leverage correction is applied to the scores assuming that they will then be used in a regression. Although this is fine for PCR models, remember that PLS... 

The studentized residual is an indication of how well the calibration model estimates the analyte property in each sample. The studentized residual is similar to the Student s t-statistic the estimation error of each sample is converted to a distance in standard deviations away from zero. An additional term is often added to the calculation to correct for the weight each sample has in determining the calibration model. The studentized residual is increased for samples with a large leverage this is known as the studentized leverage corrected residuals. 

The exponential dependencies in Eq. (14-195) represent averages of values reported by a number of studies with particular weight given to Lefebvre [Atomization and Sprays, Hemisphere, New York, (1989)]. Since viscosity can vary over a much broader range than surface tension, it has much more leverage on drop size. For example, it is common to find an oil with 1000 times the viscosity of water, while most liquids fall within a factor of 3 of its surface tension. Liquid density is generally even closer to that of water, and since the data are not clear mat a liquid density correction is needed, none is shown in Eq. 

Looking at the blog entry that follows, this one is technically correct if rather peremptorily explained. Hysteretic controllers are actually quite simple. In Chapter 2 we discussed the leverage factor of a voltage divider. In this case the reference is 1.24V and Vom is 3.8V. So the leverage factor is 3.8/1.24 = 3.0645. So... 

Note the homogeneity of terms in the two equations. The new DICARB functions (plotted in Figure 2C) eliminated the inconsistency in the I[unsubst] terms characteristic of the Phase I models and minimized leverage in the "correcting I[5-phenyl] term of the Phase I models. Both the I[pivalate] and I[twopar] terms were validated. 

FIGURE 5.104. Studentized residuals versus sample leverage for three-factor PLS model for component A, corrected data.. ... 

Experience has shown that it is most efficient to intervene early in the discovery process. This is when discovery teams traditionally have the least amount of information available for making decisions. Informed decisions at this stage have the greatest leverage on productive downstream work. Correct choice of one or more lead series at this stage can make later work more successful. Conversely, lack of information at this stage can lead to investment of time and resources in a compound series that is later found to have a fatal flaw. This is demoralizing for the team, wasteful of resources, and leads to later entry into the clinical market. 

Bootstrapping residuals is not without its own problems. Particularly, that an appropriate model has been chosen and that e —> s at least asymptotically (in fact, OLS residuals typically underestimate s). For this latter reason, sometimes residuals corrected for leverage are used... 

Even if these projections of future dependence are believed to be correct, concerns about the resulting commercial and political leverage form only a small part of a security environment. It also includes a cluster of short- and long-term issues among which are resource availability, technical breakdown and accident, terrorist attack, political instability, and lack of timely investment, as well as disagreements in relation to existing and future supplies and prices, transit and facilities. 

Biological systems are weU-poUshed machines, providing effective, highly redundant, and error-correcting processes and mechanisms, so it only makes sense to leverage millions of years of Mother Nature s work toward developing CB defense systems. 

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