Accuracy estimation

Babuska, I. Zlenklewlcz, O. C. Gago, J. Oliveira, E. R. Accuracy Estimates and Adaptive Refinements in Finite Element Computations John Wiley and Sons New York, 1986. 

The molten salt standard program was initiated at Rensselaer Polytechnic Institute (RPI) in 1973 because of difficulties being encountered with accuracy estimates in the NBS-NSRDS molten salt series. The density, surface tension, electrical conductivity, and viscosity of KNO3 and NaCl were measured by seven laboratories over the world using samples distributed by RPI. The data from these round-robin measurements of raw properties were submitted to RPI for evaluation. Their recommendations are summarized in Table 2. 

The analytical results for each sample can again be pooled into a table of precision and accuracy estimates for all values reported for any individual sample. The pooled results for Tables 34-7 and 34-8 are calculated using equations 34-1 and 34-2 where precision is the root mean square deviation of all replicate analyses for any particular sample, and where accuracy is determined as the root mean square deviation between individual results and the Grand Mean of all the individual sample results (Table 34-7) or as the root mean square deviation between individual results and the True (Spiked) value for all the individual sample results (Table 34-8). The use of spiked samples allows a better comparison of precision to accuracy, as the spiked samples include the effects of systematic errors, whereas use of the Grand Mean averages the systematic errors across methods and shifts the apparent true value to include the systematic error. Table 34-8 yields a better estimate of the true precision and accuracy for the methods tested. 

Accuracy is a measure of how close a measurement is to the "true" value. While it is impossible to determine absolute accuracy it is possible to obtain an accuracy estimate using several techniques. 

The internal basis can still be sizeable. Therefore the calculation of Coulomb matrix present in Vc, can be computationally very demanding. At the present time there are several possibilities implemented which will give different (in computational effort and accuracy) estimates of the Coulomb matrix. 

AA—atomic absorption spectrophotometry precision and accuracy estimates in relative %. 

The em quantities shift the spin-free excitation energies (Em1 - E ), calculated at a lower level of correlation treatment, to the exact values (Em - Ex) or at least to higher accuracy estimates. Herein, X denotes a common reference state, in general the electronic ground state. 

Absorption, HSC>3, 180 Accommodation coefficients calculations, 111 measurements, 104-105 ozone, 109-116 sulfur dioxide, 109-116,154 Accuracy, estimation for an instrument, 188 Acetate ions, introduction into precipitation, 219-227 Acetic acid... 

The square matrix T with elements T(ij)(rs) has m(m — 1) rows (in accordance with the number of ordered pairs (ij) or parameters ay) and determines the parameter sensitivity of the azeotrope value towards the accuracy estimation of the reactivity ratios. Really, when their errors are the same, the deviation 8x 2 (4.18) of the theoretically predicted location of azeotrope will more or less depend on the values of elements of matrix T. The calculation of such elements have no principal difficulties since an explicit dependence of x on parameters ay is known. In the case of the rather strong parameter sensitivity, when derivatives of xf with the respect to ay are large, even comparatively small errors in 8ay may result in substantial errors in calculations of x making it quite impossible to predict theoretically the existence or absence of an azeotrope in the given system. The examples of such systems were discussed earlier [125, 132, 135, 139] but as far as the author knows nobody has yet carried out the quantitative consideration of the parameter sensitivity by means of the expressions (4.18). 

The recognition accuracy estimation described above faces one very important problem what is the best choice for the threshold value 0 To solve this problem, statistical decision theory is used. ° The basis for this is an analysis of the so-called the Received Operating Characteristic (ROC) curve. By tradition, ROC is plotted as a function of true positive rate TPj TP + FN) (or sensitivity) versus false positive rate FPj TN+FP) (or 1-Specificity) for all possible threshold values 0. Figure 6.5 presents an example of such a ROC curve for the results obtained with our computer program PASS in predicting antineoplastic activity. 

However, despite Q3 being the most popular and widespread measure it suffers serious problems in terms of providing a reliable and significant accuracy estimate. The main Q3 drawback is that it does not take in account under- and over-predictions failing to capture the real significance of the results. For example, if we predict all the residues as being coil in the test database, an average Q3 value of 48.19% is obtained but correlation coefficients and information values will be null. 

Enthalpy of Fusion The enthalpy (heat) of fusion AH is the difference between the molar enthalpies of the equilibrium liquid and solid at the melting temperature and 1.0 atm pressure. There is no generally applicable, high-accuracy estimation method for AH , but the GC method of Chickos can be used to obtain approximate results if the melting temperature is known. 

Limited numbers of full-scale, test house studies have been conducted to provide validation data for lAQ models in order to improve their accuracy. Estimates of k and k based on test house studies are provided in Table 3. 

For estimation of the performance of an algorithm or a recognition function we will use several quality measures [42, 46, 47]. True positives (TP) is the number of correctly predicted and false positives (Fp) is the number of falsely predicted authentic sites. True negatives (Tn) is the number of correctly predicted and false negative (FN) is the number of falsely predicted non-sites. Sensitivity (S ) measures the fraction of the true examples that are correctly predicted Sn = Tp/TP + FN. Specificity (Sp) measures the fraction of the predicted examples that are correct Sp = Tp/Tp + Fp. Only consideration of both Sn and Sp values makes sense when we aim at providing accuracy information. If we want to concentrate on a single value for accuracy estimation the average of the correctly predicted number of sites and non-sites AC = 0.5 (TP + TN) is a suitable measure. However, this... 

Pitzer s equations and available ion-pair parameters allow calculation of mean-ion activity coefficients Y+ in complex, concentrated electrolyte solutions with an accuracy estimated to be better than + 25% in the range 25° - 55°C. The accuracy of calculated activity coefficients is limited to about the same degree by uncertainties in the estimated parameters and by simplifications introduced in the theory both to reduce the number of parameters to be estimated and to reflect the uncertainties of the estimates. Because activity coefficients are determined to quite an extent by the form of Pitzer s equations and are not extremely sensitive to the exact values of parameters, ion-pair parameters only have to be estimated within a reasonable range. 

The mass flux of the dispersed phase is in fact an essential measurement quantity in many experimental investigations. The accuracy of mass flux measurements will depend not only on the instrumentation, but also on the flow field and the size distribution of the dispersed phase, so that a general accuracy estimate is not feasible. In simple spray flows however, an accuracy of 10% on the local mass flux can be expected (Sommerfeld and Qiu 1995, Mundo 1996). 

Another study focusing on the comparison between theoretical and experimental densities is that of Tsirelson el al. on MgO.133 Here precise X-ray and high-energy transmission electron diffraction methods were used in the exploration of p and the electrostatic potential. The structure amplitudes were determined and their accuracy estimated using ab initio Hartree-Fock structure amplitudes. The model of electron density was adjusted to X-ray experimental structure amplitudes and those calculated by the Hartree-Fock model. The electrostatic potential, deformation density and V2p were calculated with this model. The CPs in both experimental and theoretical model electron densities were found and compared with those of procrystals from spherical atoms and ions. A disagreement concerning the type of CP at ( , 0) in the area of low,... 

THE USE OF BASE CALL ACCURACY ESTIMATES OR CONFIDENCE VALUES... 

Divide data into k subsets on k occasions, leave one subset from the data and compare the k replicate estimates (may be inefficient because of variation in accuracy estimation). 

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