Least median of squares

Rousseeuw, P. J. J. Amer. Stat. Assoc. 79, 1984, 871-880. Least median of squares regression. 

Some methods, therefore, take as the best pair of regression coefficients that which gives the least median of squares ... 

For multiple regression, the least median of squares (LMS) of Rousseeuw [7] and the LTS described previously were the first equivariant methods to attain a 50% breakdown value. Their low finite-sample efficiency can be improved by carrying out a one-step reweighed least-squares fit (Equation 6.9) afterward. Another approach is to compute a one-step M-estimator starting from LMS or LTS, which also maintains the breakdown value and yields the same efficiency as the corresponding M-estimator. In order to combine these advantages with those of the bounded-influence approach, it was later proposed to follow the LMS or LTS by a one-step GM-estimator [35],... 

Rutan and Carr have recently compared five algorithms with respect to their abilities to deal with outliers within small data sets during calibration. They concluded that the least-median of squares approach or the zero-lag adaptive Kalman filter methods were superior because these two methods generated slope values that were less than 1% in error for small data sets with outlier points. 

Multivariate models using neural networks, support vector machines and least median squares regression have been used to predict hERG activity [96-98]. These types of models function more as computational black box assays. 

One important enhancement is that events with additional constraints on their locations, such as e.g. the explosions produced when developing a tuimel underground, are used to constrain the hybrid localizations. Another enhancement is that the median of the distribution of residuals is used to weight the data. Experience gained by Andersen [2001] suggests that use of median corrections is both more stable and more accurate even while the localization of individual events is based on weighted least-squares minimization. Additionally, the data recorded from closer sensors is given... 

Linear least squares treatments of plots of the logarithm of the vapor pressure versus the reciprocal temperature were performed. The second-law enthalpy and entropy of sublimation at the median temperature are proportioned to the slope and... 

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. 

If L(e) e, l.e. the loss Is proportional to the squared error, the least square criterion Is apparent, and the best estimator P(x) Is the conditional expectation defined In (3). Note that this estimator Is usually different from that provided by ordinary krlglng for the simple fact that expression (3) Is usually non-llnear In the N data values. If L(e) e, l.e. the loss Is proportional to the absolute value of the error, the best estimator Is the conditional median, l.e. the value ... 

Similar biomonitoring studies in Belgium and China conducted with adolescents (14—16 years) and children and students (3-24 years), respectively, showed a high detection frequency of more than 90% [154, 156]. However, mean and median values were lower compared to the US data. In the Chinese study, females had a statistically higher least square geometric mean concentration than males. They also observed a decreasing tendency with age in the 7-24 age group [156]. 

This is an approximation to the least-squares method, and is fairly good within the range of the data studied The accuracy can be improved by more precise construction Since medians are used instead of least squares for fitting the line, some bias is expected, which will increase with the amount of error in the data. 

Frake and co-workers " extensively evaluated numerous chemometric techniques for the NIRS prediction of mass median particle size determination of lactose monohydrate. Models evaluated in zero order (untreated) and second derivative were MLR, PLS (partial least squares), and ANN (artificial neural network). The researchers concluded that there is more than one way to treat data and achieve a good calibration model. The group also confirms previous observations that derivitization of data does not remove particle size effects (previously thought to contribute to baseline shift). 

From the total sample set (48 samples), 45 samples were used as calibration samples. The three samples excluded from the calibration set were selected on the basis of a representative variation of their active ingredient concentrations, and finally used as unknown test samples to predict the content of their active ingredients. Partial least squares (PLS) models for each active ingredient were developed with the Unscrambler Software (version 9.6 CAMO Software AS, Oslo, Norway) from the MSC-pretreated median spectra of all pixels of each of the 45 calibration sample images. Based on these calibration models, the predictions of the active ingredient content for each pixel of the imaging data of the three test samples and their evaluation as histograms, contour plots and RGB plots was performed with Matlab v. 7.0.4 software (see below). 

Using a method of weighed least-squares, a log-normal function describing the aerosol mass and activity size distribution with respect to aerodynamic (resistance) diameter has been routinely fitted to the data. It is also possible to approximate a log-normal function for the mass and activity distribution with the use of log-probability paper. The distribution is then describable by a mass or activity median aerodynamic diameter, MMAD or AMAD, and associated geometric standard deviation, Og. 

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