Root mean square error plots

B Interpreting Root-Mean-Square-Error Plots 208... 

Figure 16 Root-mean-squared error progression plot for Fletcher nonlinear optimization and back-propagation algorithms during training.

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

Root Mean Square Error of Calibration (RMSEC) Plot (Model Diagnostic) The RMSEC as a function of the number of variables included in the model is shown in Figure 5-77. It decreases as variables are added to the model and the largest decrease is observed between a one- and two-variable model. The reported error in the reference caustic concentration is approximately 0.033 vrt.% (la). The tentative conclusion is that four variables are appropriate because the RMSEC is less than the reference concentration error after five variables are included in the model. 

Root Mean Square Error of Prediction (RMSEP) Plot (Model Diagnostic) The validation set is employed to determine the optimum number of variables to use in the model based on prediction (RMSEP) rather than fit (RMSEO- RM-SEP as a function of the number of variables is plotted in Figure 5.7S for the prediction of the caustic concentration in the validation set, Tlie cuive levels off after three variables and the RMSEP for this model is 0.053 Tliis value is within the requirements of the application (lcr= 0.1) and is not less than the error in the reported concentrations. 

Root Mean Square Error of Prediction (RMSEP) Plot (Model Diagnostic) The new RMSEP plot in Figure 5-100 is more well behaved than the plot shown in Figure 5-93 (with the incorrect spectrum 3). A minimum is found at 3 factors with a corresponding RMSEP that is almost two orders of magnitude smaller than the minimum in Figure 5-93- The new RMSEP plot shows fairly ideal behavior with a sharp decrease in RMSEP as factors are added and then a slight increase when more than three factors are included. 

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). 

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. 

Root Mean Square Error of Prediction (RMSEP) Plot (Model Diagnostic) The RMSEP versus number of factors plot in Figure 5.113 shows a break at three factors and a leveling off after six factors. Tlie RMSEP value with six factors (0,04) is comparable to the estimated error in the reported concentrations (0.033), indicating the model is predicting well At this point we tentatively choose a rank six model. The rank three model shows an RMSEP of 0.07 and may well have been considered to be an adequate model, depending on how well the reference values are known. 

Root Mean Square Error of Prediction (RMSEP) Plot (Model Diagnostic) The RMSEP plot for the MCB model is shown in Figure 5.127. Although the shape of this RMSEP plot is not ideal, it does not exhibit erratic behavior. Tlie first minimum in this plot is at four factors with a lower minimum at six factors. In Section 5.2.1.2, nonlinear behavior was suspected as the root cause of the failure of the DCLS method. Tlicreforc, it is reasonable that a PLS model re-... 

Fig. 8 Root-mean-square-error (RMSE) plots for the results of Fig. 7. Traces (a)-(d) are for no smoothing, 5-point quadratic smoothing, 13-point quartic smoothing, and multismoothing, respectively, corresponding to the results shown in traces (c)-(f) of Fig. 7.

Fig. 16 Root-mean-square-error (RMSE) plots for the deconvolutions of Fig. 15. The response-function widths are (a)7, (b) 7.5, (c) 8, (d) 8.5, (e) 9, and (f) 9.5 points.

Calculate the six correlation coefficients between the observed and predicted variables and plot a graph of the percentage root mean square error obtained in question 3 against the correlation coefficient, and comment. 

In the model of question 2, plot a graph of predicted against true concentrations. Determine the root mean square error both in mM and as a percentage of the average. Comment. 

Plot graphs of predicted versus known concentrations for one, two and three PLS components, and calculate the root mean square errors in mM. 

Within each of the assays A, B, C, and D, least squares linear regression of observed mass will be regressed on expected mass. The linear regression statistics of intercept, slope, correlation coefficient (r), coefficient of determination (r ), sum of squares error, and root mean square error will be reported. Lack-of-fit analysis will be performed and reported. For each assay, scatter plots of the data and the least squares regression line will be presented. 

Figure 10.38. Residual plots of (a) the stationary phases and (b) all mobile-phase/solute combinations. Legend RMSE stands for root-mean-squared error (see text) and the dotted line indicates the mean RMSE for the whole data set.

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.

FIGURE 34.5 Cross-validated predicted versus measured plots for rapeseed methyl ester (RME) in jet fuel (ppm). The root mean square error (RMSE) is 2.2. ppm. Adapted from Eide et al. [13]. 

Fig. 8.5 PLS validation plots showing for each predicted variable (i.e., sensory descriptor) the root mean squared error of prediction (RMSEP) over the first five model dimensions. RMSEP values were obtained from a leave-one-out bootstrapping algorithm, and both the cross-validated estimate black solid line) and the bias-adjusted eross-validation estimate ned doited line) are shown [38]...

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).

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