Squared prediction error

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

Several statistics from the models can be used to monitor the performance of the controller. Square prediction error (SPE) gives an indication of the quality of the PLS model. If the correlation of all variables remains the same, the SPE value should be low, and indicate that the model is operating within the limits for which it was developed. Hotelling s 7 provides an indication of where the process is operating relative to the conditions used to develop the PLS model, while the Q statistic is a measure of the variability of a sample s response relative to the model. Thus the use of a multivariate model (PCA or PLS) within a control system can provide information on the status of the control system. 

In any case, the cross-validation process is repeated a number of times and the squared prediction errors are summed. This leads to a statistic [predicted residual sum of squares (PRESS), the sum of the squared errors] that varies as a function of model dimensionality. Typically a graph (PRESS plot) is used to draw conclusions. The best number of components is the one that minimises the overall prediction error (see Figure 4.16). Sometimes it is possible (depending on the software you can handle) to visualise in detail how the samples behaved in the LOOCV process and, thus, detect if some sample can be considered an outlier (see Figure 4.16a). Although Figure 4.16b is close to an ideal situation because the first minimum is very well defined, two different situations frequently occur ... 

The goal of the RMSECV, statistic is twofold. It yields an estimate of the root mean squared prediction error E(y-y)2 when k components are used in the model, whereas the curve of RMSECV, for k = 1,. .., k,ma is a popular graphical tool to choose the optimal number of components. 

Figure 11. Performance of competing criteria the number of descriptors in the model, for various criteria versus the root mean squared prediction error (RMSEP) in forward selection. (Reproduced with permission from the author.)...

Calculate the percentage root mean square prediction errors for each of the six variables as follows, (i) Calculate residuals between predicted and observed, (ii) Calculate the root mean square of these residuals, taking care to divide by 5 rather than 8 to account for the loss of three degrees of freedom due to the PLS components and the centring, (iii) Divide by the sample standard deviation for each parameter and multiply by 100 (note that it is probably more relevant to use the standard deviation than the average in this case). 

In contrast to optimal design, Hamprecht and co-workers recently introdnced a space-filling design techniqne for compound selection. This stochastic method nses the best linear unbiased estimator, in the form of Kriging, " to constrnct selection designs that optimize the integrated mean-square prediction error, or entropy. This... 

Physiologic model-physiologically based pharmacokinetic model (PB/PK) A physiologically based model for Gl transit and absorption in humans is presented. The model can be used to study the dependency of the fraction dose absorbed (Fabs) of both neutral and ionizable compounds on the two main physico-chemical input parameters [the intestinal permeability coefficient (Pint) and the solubility in the intestinal fluids (Sint)] as well as the physiological parameters, such as the gastric emptying time and the intestinal transit time. For permeability-limited compounds, the model produces the established sigmoidal dependence between Fabs and Pnt. In case of solubility-limited absorption, the model enables calculation of the critical mass-solubility ratio, which defines the onset of nonlinearity in the response of fraction absorbed to dose. In addition, an analytical equation to calculate the intestinal permeability coefficient based on the compound s membrane affinity and MW was used successfully in combination with the PB-PK model to predict the human fraction dose absorbed of compounds with permeability-limited absorption. Cross-validation demonstrated a root-mean-square prediction error of 7% for passively absorbed compounds. 

The most commonly used parameter to assess predictability for the bootstrap has been the squared prediction error (SPE). The SPE refers to the square of the difference between a future response and its prediction from the model ... 

The structural model, So, was fit to each bootstrap data set. That is. So was fit to Di to Dsos, resulting in models 1 to 505 (Mi to M505). When each of these models for the bootstrap data sets (Di to D505) were estimated, the squared prediction error (SPE) for each concentration was estimated. 

This optimism represented the underestimation of the squared prediction error that was expected to occur when the model was applied to the data from which it was derived. In a final step, the average optimism across all bootstrap iterations was estimated and added to the SPE estimated when the Mo was applied to Do. This resulted in an improved estimate of the absolute prediction error (SPEimp). 

Squared Prediction Error (SPE) charts show deviations from NO based on variations that are not captured by the model. Recall Eq.3.2 that can be rearranged to compute the prediction error (residual) E... 

The squared prediction error (SPE) can be calculated for the X and the Y block models... 

Multivariate monitoring charts based on Hotelling s statistic (T ) and squared prediction errors SPEx and SPEy) are constructed using the PLS models. Hotelling s statistic for a new independent t vector is [298]... 

In this section, a calibration model will be constructed using the principal components analysis (PC A) (see Chapter 3). A common approach to isolate process disturbances is to check the contribution of each sensor to the sum of squared prediction error (SPE) of the calibration model. 

As shown above, the residuals will not be independent of the data in the case of component modeling when individual samples are left out during the cross-validation. In regression though, the y and yco are statistically independent and Equation (7.3) is an estimator of the summed squared prediction error of y using a certain model. 

Resubstitution of the benzanthracene concentrations reveals perfect fit, that is, the calibration mean squared error (Eq. (6.106)) is almost zero. Estimation of predictions from a single tree by leave-one-out cross-validation reveals a mean squared prediction error of 0.428. A plot of the recovery function for the individual predictions is given in Eigure 6.16a. Further improvement of the model is feasible if again ensemble methods (cf Tree-Based Classification Section) are applied. Figure 6.16b shows the recovery function for a bagged model with a smaller prediction error of 0.337. 

An application of squared prediction errors of time domain averaging across all scales on gearbox fault detection based on vibration signals... 

In this paper, the building of a statistical model to diagnose the fault condition of gearboxes based on a framework of Time Domain Averaging across all Scales - IDAS, proposed by HALIM et al. (2008), is undertaken. This model pays special attention to statistical analysis using the concept of squared prediction errors, which allows the most likely condition of gearboxes to be estimated. 

In order to overcome this limitation and extend the TDAS application, a model using the concept of Squared Prediction Error - SPE was proposed and a case study was conducted to show the applicability of the methodology proposed. 

The contribntion of this paper is making use the Squared Prediction Error - SPE for comparisons of TDAS matrixes. SPE is defined as ... 

The purpose of this paper is to make use of squared prediction error (SPE) so as to compare current data in which the condition is unknown with all patterns of fault data in which the condition is known. The condition with the lowest SPE/ represents the most likely condition of the gearbox. The squared prediction error (SPE) indicates how well a sample fits each reference condition. 

Finally, in order to obtain a diagnosis of gearboxes, the Squared Prediction Error (SPE) was calculated. In order to verify the performance of SPE, it is assumed that the one of data sets, for the same operational conditions, represents previously collected data for each fault condition, the defects of which were artificially introduced. While the two other data sets represent the actual data which suppose that the condition is unknown. 

Figure 7. Squared prediction error of TDAS (2000 rpm -data set 2).

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