Root mean square error cross validation

Root mean square (RMS) granularity, 19 264 Root-mean-squared error of cross-validation (RMSECV), 6 50-51 Root-mean-squared error of calibration (RMSEC), 6 50-51... 

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... 

Root mean square error of cross-validation for PCA plot CRMSECVJ CA vs. number of PCs)... 

K-MSECV PCA, see Root mean square error, of cross-validation... 

Root mean square error of calibration (RMSEC). 255 of cross validation for PC.A (RMSEC PCA). 93-94 of prediction IRMSEP) in DCLS. 200- 201 idealized behavior. 2SS-289 in MLR, 255 in PLS, 287-290 Row space, 58-59 Rsquare. 253 adjusted. 253... 

Root Mean Square Error of Cross Validation for PCA Plot (Model Diagnostic) As described above, the residuals from a standard PCA calculation indicate how the PCA model fits the samples that were used to construction the PCA model. Specifically, they are the portion of the sample vectors that is not described by the model. Cross-validation residuals are computed in a different manner, A subset of samples is removed from the data set and a PCA model is constructed. Then the residuals for the left out samples are calculated (cross-validation residuals). The subset of samples is returned to the data set and the process is repeated for different subsets of samples until each sample has been excluded from the data set one time. These cross-validation residuals are the portion of the left out sample vectors that is not described by the PCA model constructed from an independent sample set. In this sense they are like prediction residuals (vs. fit). 

A common approach to cross-validation is called leave-one-out" cross-validation. Here one sample is left out, a PC model with given number of factors is calculated using the remaining samples, and then the residua of the sample left out is computed. This is repeated for each sample and for models with 1 to n PCs. The result is a set of cross-validation residuals for a given number of PCs. The residuals as a function of the number of PCs can be examined graphically as discussed above to determine the inherent dimensionality. In practice, the cross-validation residuals are summarized into a single number termed the Root Mean Squared Error of Cross Validation for PCA (RMSECV PCA), calculated as follows ... 

Root Mean Square Error of Cross Validation for PCA Plot (Model Diagnostic) The RMSECV PGV vs. number of principal components for a leavc-one-out cross-validation displayed in Figure 4.66 indicates a rank of 3. 

Root Mean Square Error of Prediction (RMSEP) Plot (Model Diagnostic) Prediction error is a useful metric for selecting the optimum number of factors to include in the model. This is because the models are most often used to predict the concentrations in future unknown samples. There are two approaches for generating a validation set for estimating the prediction error internal validation (i.e., cross-validation with the calibration data), or external validation (i.e., perform prediction on a separate validation set). Samples are usually at a premium, and so we most often use a cross- validation approach. 

The model was also validated by full cross-validation. Four PCs were necessary to explain the most variation in the spectra (99.9%), which best described the molecular weight. The root mean square error of prediction... 

In the following sections, we review the application of Raman spectroscopy to glucose sensing in vitro. In vitro studies have been performed using human aqueous humor (HAH), filtered and unfiltered human blood serum, and human whole blood, with promising results. Results in measurement accuracy are reported in root mean squared error values, with RMSECV for cross-validated and RMSEP for predicted values. The reader is referred to Chapter 12 for discussion on these statistics. 

An important issue in PCR is the selection of the optimal number of principal components kopt, for which several methods have been proposed. A popular approach consists of minimizing the root mean squared error of cross-validation criterion RMSECV,. For one response variable (q = 1), it equals... 

Figures 11 and 12 illustrate the performance of the pR2 compared with several of the currently popular criteria on a specific data set resulting from one of the drug hunting projects at Eli Lilly. This data set has IC50 values for 1289 molecules. There were 2317 descriptors (or covariates) and a multiple linear regression model was used with forward variable selection the linear model was trained on half the data (selected at random) and evaluated on the other (hold-out) half. The root mean squared error of prediction (RMSE) for the test hold-out set is minimized when the model has 21 parameters. Figure 11 shows the model size chosen by several criteria applied to the training set in a forward selection for example, the pR2 chose 22 descriptors, the Bayesian Information Criterion chose 49, Leave One Out cross-validation chose 308, the adjusted R2 chose 435, and the Akaike Information Criterion chose 512 descriptors in the model. Although the pR2 criterion selected considerably fewer descriptors than the other methods, it had the best prediction performance. Also, only pR2 and BIC had better prediction on the test data set than the null model.

The two regression models have lower prediction accuracy than the random-function model when assessed using cross-validated root mean squared error of prediction . This quantity is simply the root mean of the squared cross-validated residuals y( (,)) - for i = 1,..., n in the numerator of (20). The cross-... 

Repeat questions 2 and 3, but instead of MLR use PLS1 (centred) for the prediction of the concentration of A retaining die first three PLS components. Note that to obtain a root mean square error it is best to divide by 21 rather than 25 if three components are retained. You are not asked to cross-validate the models. Why are the predictions much better ... 

The results with PCR and PLS regression include the number of PCs obtained by leave-one-out cross-validation procedure, the values of regression coefficients for X variables, the value of R, and the root mean square error of calibration (RMSE C ) and the root mean square error of prediction by cross-validation proce-... 

NC = number of components selected by cross-validation, = determination coefficient, RMSEC = Root Mean Square Error of Calibration, RMSEP = Root Mean Square Error of Prediction... 

Table 4.1. Leave-one-out cross-validation results from Tuckerl- and A-PLS on sensor) data for one to four components ( LV) for prediction of salt content. The percentage o variation explained (sum-squared residuals versus sum-squared centered data) is shown foi fitted modes (Fit) and for cross-validated models (Xval) for both X (sensory data) and Y (salt). The root mean squared error of cross-validation (RMSECV) of salt (weight %) is also provided.

The residuals may also be obtained from cross-validation. With the methods described in this book, both the X part and the y part of Equation (7.7) are modeled and hence have associated residuals. The residual ey of y is usually summed to a number of regression statistics such as percentage variance explained, the coefficient of determination, root mean squared error of prediction etc. Diagnostics based on y-residuals are well covered in standard regression literature [Atkinson 1985, Beebe et al. 1998, Cook 1996, Cook Weisberg 1980, Cook Weisberg 1982, Martens Ntes 1989, Weisberg 1985],... 

Figure 10.36. Cross-validation results of X(ixjK). Legend RMSECV stands for root-mean-squared error of cross-validation and represents the prediction error in the same units as the original measurements.

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). 

Table 1. Comparison of three PLS models in the Slurry-Fed Ceramic Melter data set. The variance in both blocks of data and the Root Mean Square Error of Calibration (RMSEC), Cross-validation (RMSECV) and Prediction (RMSEP) are compared.

Another example of applying chemometrics to separations data is depicted in Figures 8 and 9. Here, interval PLS (iPLS) was applied to blends of oils in order to quantify the relative concentration of olive oil in the samples (de la Mata-Espinosa et al., 2011b). iPLS divides the data into a number of intervals and then calculates a PLS model for each interval. In this example, the two peak segments which presented the lower root mean square error of cross validation (RMSECV) were used for building the final PLS model. 

PLSR was used to develop a prediction model in the entire wave range from 4000 cm" to 10000 cm-i. Cross validation was applied to the calibration set. Each time, one sample was taken out from the calibration set. A calibration model was established for the remaining samples and the model was then used to predict the sample left out. Thereafter, the sample was placed back into the calibration set and a second sample was taken out. The procedure was repeated until all samples have been left out once. The root mean square error of cross validation (RMSEcv) was calculated for each of all wavelength combinations. The best principal component (PC) number with the highest Rev (correlation coefficient of cross validation) and lowest RMSEcv value was selected. 

The total sample sets were separated into calibration set and validation set. Cross validation was first used in calibration sample set to find the optimal principle component number. From figure 4 we can see the best principle component nmnber to be 10 with corresponding highest Rev of 0.91 and lowest RMSEcv of 0.41. Model accuracy was then evaluated on the validation set using the root mean square error of prediction (RMSEP), correlation... 

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