Predictive residual sum of squares

Another measure for the precision of multivariate calibration is the so-called PRESS-value (predictive residual sum of squares, see Frank and Todeschini [1994]), defined as... 

No. of Factors Prediction Residual Sum of Squares (Reconstruction of Orieinal Data) % of Total Spectral Variance of Data Set... 

In any case, the cross-validation process is repeated a number of times and the squared prediction errors are summed. This leads to a statistic [predicted residual sum of squares (PRESS), the sum of the squared errors] that varies as a function of model dimensionality. Typically a graph (PRESS plot) is used to draw conclusions. The best number of components is the one that minimises the overall prediction error (see Figure 4.16). Sometimes it is possible (depending on the software you can handle) to visualise in detail how the samples behaved in the LOOCV process and, thus, detect if some sample can be considered an outlier (see Figure 4.16a). Although Figure 4.16b is close to an ideal situation because the first minimum is very well defined, two different situations frequently occur ... 

This work was supported by a grant from the National Science Foundation, t Abbreviations used are as follows. FTIR Fourier transform infrared spectroscopy, ATR attenuated total reflectance, IRE internal reflection element, SATR solution ATR FTIR, FSD Fourier self-deconvolution, PLS partial least-squares analysis, PRESS prediction residual sum of squares from PLS. SECV standard error of calibration values from PLS, PLSl PLS analysis in which each component is predicted independently, PLS2 PLS analysis in which all components are predicted simultaneously. 

For PLS solution basis sets, bulk spectra were generated as described above. Standard error of calibration values (SECV) were determined from prediction residual sum of squares (PRESS) analyses of various permutations of the amide I, II, and III bands (always including amide I) from both Ge and ZnSe spectra. After determination of the effects of different types of normalization on the results, these bands were individually normalized to an area of 100 absorbance units before PLS 1 training. 

Predictive Residual Sum of Squares, PRESS. The sum of squared differences between the observed and estimated response by validation techniques ... 

Standard Deviation Error of Prediction, SDEP (. standard error in prediction, SEP). A function of the predictive residual sum of squares, defined as ... 

For each reduced data set, the model is calculated, and responses for the deleted objects are predicted from the model. The squared differences between the true response and the predicted response for each object left out are added to PRESS (predictive residual sum of squares). From the final PRESS, the (or R cv) and SDEP (standard deviation error of prediction) values are usually calculated [Cruciani et ah, 1992]. 

A commonly used measure of quality for a P-matrix analysis is the predicted residual sum of squares (PRESS) value computed by... 

Pratt measure —> statistical indices ( concentration indices) precision —> classification parameters prediction error sum of squares —> regression parameters predictive residual sum of squares —> regression parameters predictive square error regression parameters predictor variables = independent variables —> data set prime ID number ID numbers... 

Error types can be e.g. root mean square error of cross validation (RMSECV), root mean square error of prediction (RMSEP) or predictive residual sum of squares (PRESS). 

In this section, we will describe three regression criteria relevant to Section 3.5. These criteria can be used to assess how well a model is performing. The three criteria are - the residual sum of squares (RSS), the R-squared (R ) measure and the predictive residual sum of squares (PRESS). The residual sum of squares and R-squared criteria both measure how well the model fits the data. These criteria are respectively defined... 

Analogous to the correlation coefficient in eq. (17.10), we want a measure of the quality of fit produced by a given correlation model. Two commonly used quantities are the Predicted REsidual Sum of Squares (PRESS) and the correlation coefficient R defined by the normalized PRESS value and the variance of the y data (c/). 

To assess the predictive ability of a QSAR in the frame of MTD method the cross-validation technique is used, in which one supposes that one or more of the known experimental values are in fact unknown . The analysis is repeated, excluding the temporarily unknown compotmds. The resulting equations are used to predict the experimental measurements for the omitted compound(s), and the resulting individual squared errors of prediction are accumulated. The cross-validation cycle is repeated, leaving one out (LOO) or more (LMO) different compotmd(s), until each compound has been excluded and predicted exactly once. The result of cross-validation is the predictive discrepancy sum of squares, sometimes called PRESS (Predictive REsidual Sum of Squares) ... 

Note that this sum of squares looks similar to the residual sum of squares (RSS) given by eqn (6.12) but is different in eqn (6.12) the j/i is predicted from an equation that includes that data point here the y, is not in the model hence the term predictive residual sum of squares. The difference in predictive ability of two PLS models can be evaluated by comparison of their PRESS values. 

The OUTPUT stat ent specifies that each prediction residual (PRESS = PRES) be listed in the output. The PROC REG specifies that the following prediction residual sum of squares statistic also be generated. 

After developing a model, the deleted data are used as a test set, and the differences between actual and predicted Y values are calculated for the test set. The sum of squares (SS) of these differences is computed and collected from all the parallel models to form the predictive residual sum of squares (PRESS), which is a measure of the predictive ability of the model. 

An alternative error measure is the PRESS (predicted residual sums of squares) ... 

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