Regression robust

An extensive introduction into robust statistical methods is given in Ref. 134 a discussion of non-linear robust regression is found in Ref. 135. An example is worked in Section 3.4. 

Phillips, G. R., and Eyring, E. M., Comparison of Conventional and Robust Regression in Analysis of Chemical Data, Anal. Chem. 55, 1983, 1134-1138. 

Koscielniak, R, Non-linear Robust Regression Procedure for Calibration in Flame Atomic Absorption Spectrometry, Analytica Chimica Acta 278, 1993, 177-187. 

Note the close analogy with the Lineweaver-Burk form of the simple Michaelis-Menten equation. In a diagram representing MV against MX one obtains a line which has the same intercept as in the simple case. The slope, however, is larger by a factor (1 + YIK-) as shown in Fig. 39.17b. Usually, one first determines and in the absence of a competitive inhibitor (F = 0), as described above. Subsequently, one obtains A" from a new set of experiments in which the initial rate V is determined for various levels of X in the presence of a fixed amount of inhibitor Y. The slope of the new line can be obtained by means of robust regression. 

Siegel AF (1982) Robust regression using repeated medians. Biometrika 69 242-244... 

Rousseeuw PJ, Leroy AM (1987) Robust regression and outlier detection. Wiley, New York... 

Penrose R (1955) A generalized inverse for matrices. Proc Cambridge Phil Soc 51 406 Rousseeuw PJ, Leroy AM (1987) Robust regression and outlier detection. Wiley, New York Sachs L (1992) Angewandte Statistik. Springer, Berlin Heidelberg New York Sharaf MA, Illman DL, Kowalski BR (1986) Chemometrics. Wiley, New York... 

Rousseeuw, P. J., Leroy, A. M. Robust Regression and Outlier Detection. Wiley, New York, 1987. 

Simple OLS, robust regression PLS, PCR, multiple OLS, robust regression, Ridge regression, Lasso regression PLS 2, CCA... 

The least-squares approach can become very unreliable if outliers are present in the data (see Section 4.4). In this case, it is more advisable to minimize another function of the errors which results in more robust regression estimates. Although with the OLS approach the Equations 4.22 and 4.23 can always be applied, it is advisable to use the following assumptions for obtaining reliable estimates ... 

FIGURE 4.15 Comparison of OLS and robust regression on data with an outlier in the y-variable (left) and in the x-variable (right). 

For robust regression, the objective function is changed. While for OLS regression, the sum of all squared residuals is minimized in robust regression, another function of the residuals is minimized. Three methods for robust regression are mentioned here ... 

Typical robust regression methods are linear methods. 

As an example for robust regression, we consider data from incineration of biomass. The problem is to model the softening temperature (SOT) of ash by the elemental... 

FIGURE 4.16 Summary statistics of the result of robust regression for the ash data. 

FIGURE 4.17 Diagnostic plots from robust regression on the ash data. 

FIGURE 4.18 Response versus fitted values for the ash data. Left OLS for original data the symbol I indicated outliers identified with robust regression. Right OLS regression for cleaned data where outliers are excluded. 

In situations where robust regression cannot be applied (colhnearity, n < 2m) the x-variables could be summarized by robust principal components (Section 3.5). Then the procedure as mentioned above can be applied (see Section 4.6). 

QSPR models have been developed by six multivariate calibration methods as described in the previous sections. We focus on demonstration of the use of these methods but not on GC aspects. Since the number of variables is much larger than the number of observations, OLS and robust regression cannot be applied directly to the original data set. These methods could only be applied to selected variables or to linear combinations of the variables. 

The aim of multivariate calibration methods is to determine the relationships between a response y-variable and several x-variables. In some applications also y is multivariate. In this chapter we discussed many different methods, and their applicability depends on the problem (Table 4.6). For example, if the number m of x-variables is higher than the number n of objects, OLS regression (Section 4.3) or robust regression (Section 4.4) cannot be applied directly, but only to a selection... 

Outliers or inhomogeneous data can affect traditional regression methods, hereby leading to models with poor prediction quality. Robust methods, like robust regression (Section 4.4) or robust PLS (Section 4.7.7), internally downweight outliers but give full weight to objects that support the (linear) model. Note that to all methods discussed in this chapter robust versions have been proposed in the literature. 

Shapiro Wilks W-test for normal data Shapiro Wilks W-test for exponential data Maximum studentlzed residual Median of deviations from sample median Andrew s rho for robust regression Classical methods of multiple comparisons Multivariate methods... 

The method is not as useful with multisubstrate systems however, Cornish-Bowden and EndrenyE have presented a robust regression method to treat data with more parameters. 

To show the performance of the concept of robust regression let us assume three cases with three pairs of (xh yl) each, where the outliers are underlined ... 

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