Statistical prediction

This is a recurrent theme of quantum theory. Many quantum systems can be formulated exactly in terms of a wave equation and the behaviour of the system will be described exactly by the wavefunction, the solution to the wave equation. What is not always appreciated is that this is a mathematical description only, which does not ensure understanding of the event in terms of a comprehensible physical model. The problem lies therein that the description is only possible in terms of a wave formalism. Understanding of the physical behaviour however, requires reduction to a particle model. The wave description is no more than a statistically averaged picture of the behaviour of many particles, none of which follows the actual statistically predicted course. The wave description is non-classical, and the particle model is classical. Mechanistic understanding is possible only in terms of the classical approach, and a mathematically precise description only in terms of the wave formalism. The challenge of quantum theory is to reconcile the two points of view. 

In electroluminescent applications, electrons and holes are injected from opposite electrodes into the conjugated polymers to form excitons. Due to the spin symmetry, only the antisymmetric excitons known as singlets could induce fluorescent emission. The spin-symmetric excitons known as triplets could not decay radiatively to the ground state in most organic molecules [65], Spin statistics predicts that the maximum internal quantum efficiency for EL cannot exceed 25% of the PL efficiency, since the ratio of triplets to singlets is 3 1. This was confirmed by the performance data obtained from OLEDs made with fluorescent organic... 

Antony, A. C., and Miller, M. E. (1994). Statistical prediction of the locus of endoproteo-lytic cleavage of the nascent polypeptide in glycosylphosphatidylinositol-anchored proteins. Biochem. J. 298, 9-16. 

The statistical prediction error does not account for biases in concentration or pathlei h changes. The (SS ) matrix depends only on the pure spectra, and the re ual spectra only depends on how well dS can find a linear combination of pures to fit the sample spectrum. In other words, the statistical prediction error is a measure of precision, not accuracy. 

The statistical prediction errors for the validation samples in this example are plotted in Figure 5.11. The maximum statistical prediction error for component A (Figure 5.1 la) is 0.025, which must be compared to the precision requirements of the application. For component B (Figure 5.life), maximum statistical prediction error is —0.019. In addition, no samples appear to have unusually large statistical prediction errors which would indicate an outlier. 

Root Mem Square Error of Prediction (RMSEP) (Model Diagnostic) The RMSEP is anciaer diagnostic for examining the errors in the predicted concentrations. Whie the statistical prediction error discussed earlier quantifies preci-... 

Statistic prediction errors (statisffical prediction error rs. sample number)... 

The statistical prediction errors are the uncertainty in the predicted concentrations which reflects the quality of the model. 

Statistical Prediction Errors The statistical prediction errors for the unknowns are shown in Table 5.3. These are an estimate of the precision of the predicted concentrations (Equation 5.14). These error estimates are different for the two components even though the same spectral residuals are used in the calculation (s in Equation 5.14). This is because there are different elements on the diagonal of the (SS )" matrix. Component B has a smaller... 

This is expected given that the statistical prediction errors were comparable to those fomd from the validation samples. 

The predicted concentrations for an unknown are not reliable if the statistical prediction errors are significantly larger than observed for the calibration samples. 

Statistical Prediction Errors (Model and Sample Diagnostic) Figure 5-28 shows the stss tical prediction errors for all four components for the samples in the validat n set. For MCB and ODCB the maximum value is —0.004 and for EB and C5IM the maximum value is —0.01. These errors are small compared to thecsncentration ranges. 

One approach for determining whether the statistical prediction errors are reasonable isi compare them to statistical prediction errors computed using an estimate ofs from replicate analyses. Analysis of the replicate center point spectra provafe an estimate of sr equal to 1.2e-07 AU. The diagonal elements of ared.37, 4.9, 0.52, 5.7, and using Equation 5.14 to estimate statisti-... 

FIGURE 5.60. Statistical prediction errors for the prediction samples, with the maximum from valcSation indicated by the horizontal line. 

Statistical Prediction Error vs. Sample Number Plot (Sample Diagnostic) A statistic is available for the predicted values using Equation 5.28. We will refer to this as me statistical prediction error to distinguish it from the observed concentration residuals. 

Maximum statistical prediction error = 0.006 Expectederror in prediction = 0.03... 

FIGURE 5.74. Statistical prediction errors for the unknowns for component A with the horizontal Sne indicating the maximum from model validation. 

FIGURE 5.84. Statistical prediction error for the validation samples, three-variable... 

Statistical Prediction Errors The statistical prediction errors are plotted in Figure 5.86, with the maximum from the model validation denoted by the horizontal line. All prediction samples fall below this line, indicating that there are no unusual samples. 

To estimate the model (St), the pure component spectra (with any augmentations for baseline) are supplied to the computer. For validation, the R and C matrices are submitted. The resulting output includes the statistical prediction errors, estimated concentrations, concentration residuals, and spectral residuals. For this example, no baselines are estimated only the pure component Spectra are used for S. 

Statistical Prediction Errors (Model and Sample Diag Jostic) Uncertainties in the concentrations can be estimated because the predicted concentrations are regression coefficients from a linear regression (see Equations 5.7-5.10). These are referred to as statistical prediction errors to distinguish them from simple concentration residuals (c — c). Tlie statistical prediction errors are calculated for one prediction sample as... 

The statistical prediction error is in concentration units and represents the uncertainty in the predicted concentrations due to deviations from the model assumptions, measurement noise, and degree of overlap of the pure spectra. As the system deviates from the underlying assumptions of CLS, the residual... 

Statistical prediction errors (statistical prediction error vs. sample nnmber)... 

Statistical prediction errors for a sample that are quite different than the other validation samples indicate a problem with that sample. 

TABLE 5.3. Statistical Prediction Errors for Four Unknowns... 

Unknown Component A Statistical Prediction Error Component B Statistical Prediction Error... 

The statistical prediction errors for the unknowns are compared to the maximum statistical prediction error found from model validation in order to assess the reliability of the prediction. Prediction samples which have statistical prediction errors that are significantly larger than this criterion are investigated funher. In the model validation, the maximum error observed for component A is 0.025 (Figure 5-1 In) and 0.019 for component B (Figure 5.11b). For unknown 1, the statistical prediction errors are within this range. For the other unknowns, the statistical prediction errors are much larger. Therefore, the predicted concentrations should not be considered valid. 

Measurement Residual Plot The statistical prediction errors indicate which samples have large spectral residuals. It can be instructive to then plot the residuals to diagnose the problem. In practice, only samples with large statistical prediction errors are examined, but all four will be plotted here. Tlte residuals for unknowns 1-4 are shown in Figures 5.20-5.23, respectively. Also shown are the measured and predicted responses. The residual for unknown 1 in Figure 5.20 resembles the model validation residuals shown in Figure 5.18. 

Measurement Residuals Plot (Sample and Variable Diagnostic) There is nonrandom behavior in the spectral residuals, indicating inadequacies in the model (see Figure 5-31). This is consistent with the statistical prediction errors being an order of magnitude larger than the ideal value. Several preprocessing... 

---