Root mean square error of cross validation RMSECV

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... 

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.

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. 

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. 

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. 

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... 

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.

RMSEcv root mean square error of cross validation... 

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.

The relative efficiency of both unfold-PLS and N-PLS has been compared in order to understand how the nature of data, that is the different source of variability could influence the performance of a model. The use of N-PLS model led to a more parsimtMiious (1 LV) model and showed, in general, lower root mean square error in cross validation (RMSECV) values and a higher value of explained variance in prediction. 

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.

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... 

Figure 6.7 The root mean square error of calibration (RMSEC), leave-one-out cross validation (RMSECV) and prediction (RMSEP) are plotted as a function of the signal-to-noise ratio (SNR). While the intrinsic SNR amounts to 3000, random noise was artificially added to mid-IR spectra of 247 serum samples (which decreases the SNR) and the concentration of glucose was recalculated by means of partial least squares (PLS) based on the noisy spectra. The open symbols refer to assessing the quality of quantification within the teaching set, while the filled symbols relate to an external validation set. The data show that the noise can be increased by more than an order of magnitude before the prediction accuracy of the independent external validation set (RMSEP) is affected. In addition, it can clearly be observed that the RMSEC is a poor measure of accuracy since it suggests delivering seemingly better results for lower SNRs, while in fact the calibration simply tends to fit the noise for low values of SNR (see section 6.7).

In GA-PLS, model vahdation was achieved through leave-one-out cross-validation (LOO CV) to find the best number of latent variables (Lv) to be used in calibration and prediction. External validation (for a test set), and the predictive ability was statistically evaluated through the root mean square errors of calibration (RMSEC) and vahdation (RMSECV). The results indicate that four latent variables are the best number to make a model. The following equation represents the best model achieved by GA-PLS ... 

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. 

FIGURE 17 Sugar data set. Selection of the optimal number of N-PLS components root mean square error in modelling (RMSEC) and in cross-validation (RMSECV) as a function of the number of factors. The black circle indicates the complexity of the final model (six components). 

---