Predicted Residual Error

We can also examine these results numerically. One of the best ways to do this is by examining the Predicted Residual Error Sum-of-Squares or PRESS. To calculate PRESS we compute the errors between the expected and predicted values for all of the samples, square them, and sum them together. 

PRESS for validation data. One of the best ways to determine how many factors to use in a PCR calibration is to generate a calibration for every possible rank (number of factors retained) and use each calibration to predict the concentrations for a set of independently measured, independent validation samples. We calculate the predicted residual error sum-of-squares, or PRESS, for each calibration according to equation [24], and choose the calibration that provides the best results. The number of factors used in that calibration is the optimal rank for that system. 

Fortunately, since we also have concentration values for our samples, We have another way of deciding how many factors to keep. We can create calibrations with different numbers of basis vectors and evaluate which of these calibrations provides the best predictions of the concentrations in independent unknown samples. Recall that we do this by examing the Predicted Residual Error Sum-of Squares (PRESS) for the predicted concentrations of validation samples. 

Just as we did for PCR, we must determine the optimum number of PLS factors (rank) to use for this calibration. Since we have validation samples which were held in reserve, we can examine the Predicted Residual Error Sum of Squares (PRESS) for an independent validation set as a function of the number of PLS factors used for the prediction. Figure 54 contains plots of the PRESS values we get when we use the calibrations generated with training sets A1 and A2 to predict the concentrations in the validation set A3. We plot PRESS as a function of the rank (number of factors) used for the calibration. Using our system of nomenclature, the PRESS values obtained by using the calibrations from A1 to predict A3 are named PLSPRESS13. The PRESS values obtained by using the calibrations from A2 to predict the concentrations in A3... 

The Predicted Residual Error Sum of Squares (PRESS) is simply the sum of the squares of all the errors of all of the samples in a sample set. 

MSE is preferably used during the development and optimization of models but is less useful for practical applications because it has not the units of the predicted property. A similar widely used measure is predicted residual error sum of squares (PRESS), the sum of the squared errors it is often applied in CV. 

PRESS Predicted residual error sum of squares (sum of squared prediction errors). rjk (Pearson) correlation coefficient between variables j and k r2 is the... 

Sometimes the question arises whether it is possible to find an optimum regression model by a feature selection procedure. The usual way is to select the model which gives the minimum predictive residual error sum of squares, PRESS (see Section 5.7.2) from a series of calibration sets. Commonly these series are created by so-called cross-validation procedures applied to one and the same set of calibration experiments. In the same way PRESS may be calculated for a different sets of features, which enables one to find the optimum set . 

The PLS model is calculated without these values. The omitted values are predicted and then compared with the original values. This procedure is repeated until all values have been omitted once. Therefore an error of prediction, in terms of its dependence on the number of latent variables, is determined. The predicted residual error sum of squares (PRESS) is also the parameter which limits the number of latent vectors u and t ... 

D is the matrix of deviations between true and calculated. Since B is dependent on the rank A, also D will be dependent on it. A useful expression is the prediction residual error sum of squares (PRESS) ... 

This procedure is then calculated [100/(y%)] times, ensuring that a given data grouping is only deleted once. The predicted residual error sum of squares (PRESS) is then... 

Fig. 4.19. Schematic comparison of methods of multivariate data analysis using the classical and inverse approach. BLUE, best linear unbiased estimator PRESS, predictive residual error...

The influence of observations on a regression model can be assessed by its leverage. If the prediction error is to be estimated on the basis of the caUbration sample set, k samples are left out to calculate new prediction residual errors, e y By using the elements of the hat matrbc, sometimes also called leverages of case k, a good approximation of the estimated prediction error can be obtained according to... 

The selection of an optimal number of factors (loadings) is a central point in PCR and PLS. In both methods the so-called prediction residual error sum of squares (PRESS) is calculated... 

An increasingly popular stopping rule for variable selection involves calculating a statistic referred to as PRESS. PRESS stands for predicted residual error sum of squares. To calculate PRESS, the data must be split into at least... 

The prediction performance can be validated by using a cross-validation ( leave-one-out ) method. The values for the first specimen (specimen A) are omitted from the data set and the values for the remaining specimens (B-J) are used to find the regression equation of, e.g., Cj on Ay A2, etc. Then this new equation is used to obtain a predicted value of Cj for the first specimen. This procedure is repeated, leaving each specimen out in turn. Then for each specimen the difference between the actual and predicted value is calculated. The sum of the squares of these differences is called the predicted residual error sum of squares or PRESS for short the closer the value of the PRESS statistic to zero, the better the predictive power of the model. It is particularly useful for comparing the predictive powers of different models. For the model fitted here Minitab gives the value of PRESS as 0.0274584. 

One of the biggest problems in using PCA spectral decomposition for discriminant analysis is identifying the correct number of factors to use for the models. In the case of quantitative analysis methods, there is always a set of secondary benchmarks to compare the quality of the model the primary calibration data. By performing a prediction residual error sum of squares (PRESS) analysis, it is very easy to determine the number of factors by calculating the prediction error of the constituent values at every factor. The smaller the error, the better the model. 

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