Robust Estimation Approaches

FIGURE 3 Reconciliation results for variable a 3 with Cauchy distribution and 10% ofoutliers(, measured value o, reconciled value). 

In robust statistics, rather than assuming an ideal distribution, an estimator is constructed that will give unbiased results in the presence of this ideal distribution, but that will try to minimize the sensitivity to deviations from ideality. Several approaches are described here  

Robust system identification and estimation has been an important area of research since the 1990s in order to get more advanced and robust identification and estimation schemes, but it is still in its initial stages compared with the classical identification and estimation methods (Wu and Cinar, 1996). With the classical approach we assume that the measurement errors follow a certain statistical distribution, and all statistical inferences are based on that distribution. However, departures from all ideal distributions, such as outliers, can invalidate these inferences. In robust statistics, rather than assuming an ideal distribution, we construct an estimator that will give unbiased results in the presence of this ideal distribution, but will be insensitive to deviation from ideality to a certain degree (Alburquerque and Biegler, 1996). 

Consider a nonlinear system with g inputs and I outputs described by the following model  

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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. 

Experience with the use of the robust estimator approach in the fitting of spectroscopic data is limited. The robust estimator procedures presently in use appear to require more computer time than least square procedures, but the reduced incidence of trapping makes robust estimators (and hybrid least squares/robust schemes, e.g., Ruckstuhl, et al., 2001) very attractive. 

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. 

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. 

Methods of robust PCA are less sensitive to outliers and visualize the main data structure one approach for robust PCA uses a robust estimation of the covariance matrix, another approach searches for a direction which has the maximum of a robust variance measure (projection pursuit). 

The B score (Brideau et al., 2003) is a robust analog of the Z score after median polish it is more resistant to outliers and also more robust to row- and column-position related systematic errors (Table 14.1). The iterative median polish procedure followed by a smoothing algorithm over nearby plates is used to compute estimates for row and column (in addition to plate) effects that are subtracted from the measured value and then divided by the median absolute deviation (MAD) of the corrected measures to robustly standardize for the plate-to-plate variability of random noise. A similar approach uses a robust linear model to obtain robust estimates of row and column effects. After adjustment, the corrected measures are standardized by the scale estimate of the robust linear model fit to generate a Z statistic referred to as the R score (Wu, Liu, and Sui, 2008). In a related approach to detect and eliminate systematic position-dependent errors, the distribution of Z score-normalized data for each well position over a screening run or subset is fitted to a statistical model as a function of the plate the resulting trend is used to correct the data (Makarenkov et al., 2007). 

Traditional approaches used in the estimation of TPAR have been compared with the PpbB approach and the recently proposed RS approach. The traditional approaches—independent time points and naive data averaging approaches—are inferior to the samphng/resampling approaches. The RS approach performed better than the PpbB approach because of its unique algorithm. Also, fewer rephcations are required for robust estimation of TPAR. The computer intensive methods provide estimates of TPAR with measures of dispersion and uncertainty. The RS approach is the method of choice for obtaining robust estimates of TPAR, when analyzing extremely sparsely sampled data. 

H.-M. Chu and E. I. Ette, A random samphng approach for robust estimation of tissue to plasma ratio from extremely sparse data. AAPS J 7(1) E249-E258 (2005). 

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