Least absolute deviations

EX 182 1.8.2 Fitting a line - least absolute deviations see EX12... 

Fitting a line by the method of least absolute deviations... 

A. Bloomfield and W. Steiger, Least absolute deviations curve fitting,... 

Though the maximum likelihood principle is not less intuitive than the least squares method itself, it enables the statisticans to derive estimation criteria for any known distribution, and to generally prove that the estimates have nice properties such as asymptotic unbiasedness (ref. 1). In particular, the method of least absolute deviations introduced in Section 1.8.2 is also a maximum likelihood estimator assuming a different distribution for the error. 

Many techniques have been proposed for fitting the parameters of a model to data. Some of these seek to minimize some function of the residuals of differences between observed and fitted values. In the early nineteenth century, the methods of least absolute deviations and of least squares were used. These sought to minimize... 

Cg pH-sensitive film had a dynamic range from 6 to 8 and Cg-film responded at higher pH values (8-10).The membranes showed good reproducibility, reversibility and a short response time (<10 s). They also can be used for at least 3 months without any considerable absolution deviations. These sensors can be used for direct determination of pH in drinking water detergent and dishwasher liquid that have good agreement with pH meter data. 

Firstly, the krlglng estimator is optimal only for the least square criterion. Other criteria are known which yield no more complicated estimators such as the minimization of the mean absolute deviation (mAD), E P(2c)-P (3c), yielding median-type regression estimates. 

BP, BLYP, EDFl and B3LYP density functional models all lead to significant improvements over both Hartree-Fock and local density models, at least in terms of mean absolute deviations. While most reactions are better described, there are exceptions. Most notable among these is the bond separation reaction for tetrachloromethane. All four models show a highly exothermic reaction in contrast with both G3 and experimental results which show a nearly thermoneutral reaction. Similar, but somewhat smaller, effects are seen for isobutane and trimethylamine. As was the case with Hartree-Fock calculations. 

The older, Pauli-Hamiltonian based QR method has recently been compared to the more modem ZORA approach (10). The comparison has been done based on methane derivatives. We have summarized the results in Table II. Both the ZORA and QR methods agree well for molecules containing atoms no heavier than Cl. This shouldn t be surprising since relativistic effects are still small in such molecules. However, at least for this sample of molecules, the ZORA method is clearly superior for molecules containing heavy nuclei like Br or I. This is reflected in the mean absolute deviation between theory and experiment of 9.2 ppm (ZORA) and 15.6 ppm (QR), respectively (10). Note that some of the same systems have also been studied by other authors (35-37). 

Variations of G3 Theory At least two variations of G3 theory have been proposed. The first does the basis set extensions at the second-order Mailer-Plesset level. This method, referred to as G3(MP2) theory,97 has an average absolute deviation from experiment of 1.30 kcal/mol for the G2/97 test set and 1.18 kcal/mol for the subset of 148 neutral enthalpies (see Table 5). This is a significant improvement over the related G2(MP2) theory. The new method provides significant savings in computational time compared to G3 theory (see Figure 2). The modification to step 4 in G3 theory is shown in Table 6, along with the new higher level correction parameters. 

Table 7.8 compares with experiment dipole moments calculated by B3LYP/6-31G, M06/6-31G, AMI (as a check on this fast method), and MP2(fc)/6-31G, for ten molecules. The two DFT methods give the same mean unsigned error, 0.11 D, three times smaller than the error of 0.31 D from the slowest method, MP2 (at least for this small selection of molecules), and the very fast AMI moments lie in-between, 0.22 D. None of these methods consistently gives values accurate to within 0.1 D. Very accurate dipole moments (mean absolute deviation 0.06-0.07 D) can be obtained with gradient-corrected DFT and very large basis sets [74]. 

Without an accompanying statement of traceability, individual participant results (and the consensus derived therefrom) may or may not relate to the same measu-rand and may or may not be traceable to common references. However, provided that the consensus (median) agrees with at least one (traceable) independent measure of the assigned value, it can be used, in conjunction with the other independent measure(s), to set the (traceable) assigned value. The (reproducibility) uncertainty of the (consensus) median of participants results can be calculated from the median of absolute deviations, MAD [3]. 

Li and Lin (2002) used a form of penalized least squares with the smoothly clipped absolute deviation penalty proposed by Fan and Li (2001). This method estimates the parameters, /3, by minimizing not the usual residual sum of squares, but... 

We report the number of data (AO, mean absolute deviation (MAD), data range, and average absolute error and average percent error between calculated and experimental values. Next, we give the correlation coefficient (R ), slope, and intercept of the least square line. 

The values of k reported in Tables VII and VIII for the systems studied were determined by applying a linear least-squares calculation to the experimental data. The average absolute deviation (A.A.D.) of logio(tts/ ) was determined at fixed intervals of z from the individual... 

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