Robust Estimators

For the robust estimation of the pair potentials, some obstacles had to be overcome. There are a huge number of different triples (si, Sk,i — k), and to find densities, we needed a way to group them in a natural way together into suitable classes. A look at the cumulative distribution functions (cdf s) of the half squared distances Cjfc at residue distance d = i — k (w.l.o.g. >0), displayed in Figure 1, shows that the residue distances 8 and higher behave very similarly so in a first step we truncated all residue distances larger than 8 to 8. 

The mean is the most common estimator of central tendency. It is not considered a robust estimator, however, because extreme measurements, those much larger or smaller than the remainder of the data, strongly influence the mean s value. For example, mistakenly recording the mass of the fourth penny as 31.07 g instead of 3.107 g, changes the mean from 3.117 g to 7.112 g ... 

As shown by Examples 4.1 and 4.2, the mean and median provide similar estimates of central tendency when all data are similar in magnitude. The median, however, provides a more robust estimate of central tendency since it is less sensitive to measurements with extreme values. For example, introducing the transcription error discussed earlier for the mean only changes the median s value from 3.107eto3.112e. 

The first task considered is the robust estimation of fitting parameters. Following to Peter Huber, the consideration is built at the assumption that the density function of the experimental random errors (8) can be presented in the following form ... 

Further primary and secondary research is required to provide robust estimates of the formal and informal care costs associated with the new dmgs and the value of health improvements to patients and carers. 

If basic assumptions concerning the error structure are incorrect (e.g., non-Gaussian distribution) or cannot be specified, more robust estimation techniques may be necessary. In addition to the above considerations, it is often important to introduce constraints on the estimated parameters (e.g., the parameters can only be positive). Such constraints are included in the simulation and parameter estimation package SIMUSOLV. Beeause of numerical inaccuracy, scaling of parameters and data may be necessary if the numerical values are of greatly differing order. Plots of the residuals, difference between model and measurement value, are very useful in identifying systematic or model errors. 

Along the mantle solidus this ratio is approximately 10 ". Dsa itself is not well constrained for garnets because of difficulties in analysing such trace quantities. The most robust estimates are probably those of Beattie (1993b), which are around 10 . Suffice to say Z)Ra in garnet is vanishingly small. 

Weighted regression of U- " U- °Th- Th isotope data on three or more coeval samples provides robust estimates of the isotopic information required for age calculation. Ludwig (2003) details the use of maximum likelihood estimation of the regression parameters in either coupled XY-XZ isochrons or a single three dimensional XYZ isochron, where A, Y and Z correspond to either (1) U/ Th, °Th/ Th and... 

Because the current estimates of the interaction parameters are used when solving the above equations, convergence problems are often encountered when these estimates are far from their optimal values. It is therefore desirable to have, especially for multi-parameter equations of state, an efficient and robust estimation procedure. Such a procedure is presented next. 

Finally, approaches are emerging within the data reconciliation problem, such as Bayesian approaches and robust estimation techniques, as well as strategies that use Principal Component Analysis. They offer viable alternatives to traditional methods and provide new grounds for further improvement. 

In Chapter 11 some recent approaches for dealing with different aspects of the data reconciliation problem are discussed. A more general formulation in terms of a probabilistic framework is first introduced and its application in dealing with gross error is discussed in particular. In addition, robust estimation approaches are considered, in which the estimators are designed so they that are insensitive to outliers. Finally, an alternative strategy that uses Principal Component Analysis is reviewed. 

Chen, J., Bandoni, A., and Romagnoli, J. A. (1997). Robust estimation of measurement error variance/ covariance from process sampling data. Comput. Chem. Eng. 21, 593-600. 

One type of common robust estimator is the so-called M-estimator or generalized maximum likelihood estimator, originally proposed by Huber (1964). The basic idea of an M-estimator is to assign weights to each vector, based on its own Mahalanobis distance, so that the amount of influence of a given point decreases as it becomes less and less characteristic. 

To investigate the performance of the proposed robust estimator, Monte Carlo studies were performed on the two previous examples used for the case without outliers. 

Pr from the robust estimator still gives the correct answer, as expected. However, the conventional approach fails to provide a good estimate of the covariance even for the case when only one outlier is present in the sampling data. 

The robust estimator still provides a correct estimation of the covariance matrix on the other hand, the estimate J>C> provided by the conventional approach, is incorrect and the signs of the correlated coefficients have been changed by the outliers. 

Huber, R J. (1964). A robust estimation of a location parameter. Ann. Math. Stat. 35,73-101. 

If the errors are normally distributed, the OLS estimates are the maximum likelihood estimates of 9 and the estimates are unbiased and efficient (minimum variance estimates) in the statistical sense. However, if there are outliers in the data, the underlying distribution is not normal and the OLS will be biased. To solve this problem, a more robust estimation methods is needed. 

Robust estimation approaches were then considered in this case, the estimators are designed so they are insensitive to outliers. That is, they will give unbiased results in the presence of the ideal distribution, but will try to minimize the sensitivity to deviations from ideality. 

Forsythe, A. B. (1972). Robust estimation of straight line regression coefficients by minimizing p-th power deviations. Technometrics 14, 159-166. 

Wang, D., and Romagnoli, J. A. (1998). Wavelet Based Robust Estimation, Internal Rep. PSE-I-No. 5. University of Sydney, Laboratory of Process Systems Engineering, Sydney, Australia. 

This is a very robust estimator which does not assume normality, linearity, or minimal error of measurement. 

There are several other possibilities for robustly estimating the central value. Well known are M-estimators for location (Huber 1981). The basic idea is to use a function iji that defines a weighting scheme for the objects. The M-estimator is then the solution of the implicit equation... 

Depending on the choice of the i/i-function, the influence of outlying objects is bounded, resulting in a robust estimator of the central value. Moreover, theoretical properties of the estimator can be computed (Maronna et al. 2006). 

MAD is based on the median xM as central value the absolute differences x — xM are calculated and MAD is defined as the median of these differences. In the case of a normal distribution MAD can be used for a robust estimation of the theoretical standard deviation cr by... 

Robust estimations of the variance are squared standard deviations as obtained from IQR or MAD, (siqr)2, and (smad)2 respectively. 

There exist other estimators for robust covariance or correlation, like S-estimators (Maronna et al. 2006). In general, there are restrictions for robust estimations of the... 

For identifying outliers, it is crucial how center and covariance are estimated from the data. Since the classical estimators arithmetic mean vector x and sample covariance matrix C are very sensitive to outliers, they are not useful for the purpose of outlier detection by taking Equation 2.19 for the Mahalanobis distances. Instead, robust estimators have to be taken for the Mahalanobis distance, like the center and... 

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