Root mean square error of 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... 

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 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. 

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... 

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.

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). 

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. 

This method was in fact carried out around two decades ago [30, 31]. However, it was applied only in the fermentation of pure microbial cultures. In a recent report by Acros-Hernandez and coworkers [32], infrared spectroscopy was applied to quantify the PHA produced in microbial mixed cultures. Around 122 spectra from a wide range of production systems were collected and used for calibrating the partial least squares (PLS) model, which relates the spectra with the PHA content (0.03-0.58 w/w) and 3-hydroxyvalerate monomer (0-63 mol%). The calibration models were evaluated by the correlation between the predicted and measured PHA content (R ), root mean square error of calibration, root mean square error of cross validation and root mean square error of prediction (RMSEP). The results revealed that the robust PLS model, when coupled with the Fourier-Transform infrared spectrum, was found to be applicable to predict the PHA content in microbial mixed cultures, with a low RMSEP of 0.023 w/w. This is considered to be a reliable method and robust enough for use in the PHA biosynthesis process using mixed microbial cultures, which is far more complex. 

Figure 6 Measured versus predicted plots from the PLSR analyses. M/G ratio measured by NMR without (a) and with (b) water suppression (zgpr) versus the M/G ratio predicted from the Raman spectra. RMSECV = root mean square error of cross validation. Both models are based on two PLS components.

In Figure 4 the root mean square error (RMS) and the root mean square error of cross validation (RMSECV) of different data processing methods and parameters are shown. As expected, the RMSECV is larger than the RMS for each method. The larger errors of the IHM are due to the non-perfect description of the pure spectra. Interestingly, CPR shows for a set of ranks (number of components used for description of the spectra) and power coefficients the lowest errors. In this example of the mixture of water and oil, this is attributed to the fact, that CPR not only considers the correlation, but also the variance with a power coefficient. 

Optimal root mean squared error of cross validation calculated from the calibration set. 

The quality of developed PLS model was evaluated by cross-validation technique. The values of root mean square error of cross validation obtained are relatively low, 1.68% and 1.32% (volume/volume), respectively for com oil and sunflower. Based on this result, the method developed has a good ability to estimate the percentage of com oil and sunflower as oil adulterants in virgin eoconut oil samples. 

It is often necessary to include at least 50 samples in the calibration and prediction sets. Sometimes, measurement of the primary analytical data of so many samples is excessively time consuming. The number of samples can be approximately halved, at the cost of computation time, by using only one calibration set and calculating the root-mean-square error of cross validation (RMSECV), as described in Section 9.9. In general, however, it is preferable to use an independent prediction set to investigate the validity of the calibration but the leave-one-out method significantly reduces the number of samples for which primary analytical data are required. 

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