The forward search: Theory and data analysis

Abstract

a b s t r a c tThe Forward Search is a powerful general method, incorporating flexible data-driven trimming, for the detection of outliers and unsuspected structure in data and so for building robust models. Starting from small subsets of data, observations that are close to the fitted model are added to the observations used in parameter estimation. As this subset grows we monitor parameter estimates, test statistics and measures of fit such as residuals. The paper surveys theoretical development in work on the Forward Search over the last decade. The main illustration is a regression example with 330 observations and 9 potential explanatory variables. Mention is also made of procedures for multivariate data, including clustering, time series analysis and fraud detection.

Asymptotic Analysis of the Forward Search

“…Indeed, the t-approximation discussed in Section 2.4 is of this form. Figure 1 of Atkinson, Riani and Cerioli (2010b) indicates than in small samples (n = 200) the 50% and 99% pointwise bands ofr (m) andẑ (m) =^ (m) are nearly indistinguishable. Here we have given a distribution theory forẑ (m) =^ (m) in terms of empirical distribution functions.…”

“…This approach has been suggested, analysed and applied for regression models in the monograph Atkinson and Riani (2000), see also Atkinson, Riani and Cerioli (2010a) for a recent overview, while R and matlab code is freely available from www.riani.it. Riani, Atkinson and Cerioli (2009) discuss the application of the Forward Search to multivariate location-scale models.…”

Robust Bayesian regression with the forward search: theory and data analysis

“…The results of the FS are typically presented through a forward plot of quantities of interest as a function of m. As a result, it is possible to connect individual observations with changes in residuals and parameter estimates, thus identifying outliers and systematic failures of the fitted model. (See Atkinson et al (2010) for a general survey of the FS, with discussion). In addition, since the method is based on the repeated use of least squares, it is relatively straightforward to introduce prior information into the search.…”

“…The procedure continues until no further outliers are identified. Such procedures, described in the books of Cook and Weisberg (1982) and Atkinson (1985), are prone to the effect of masking. Illustrations of this effect for several different models are in Atkinson and Riani (2000) and demonstrate the failure of the method to identify outliers.…”

A New Algorithm for Detecting Outliers in Linear Regression

“…Satman (2012) proposed a genetic algorithm (GA) based modification on the method given in Rousseeuw and van Driessen (2006) and showed that the GA based search obtains smaller objective values in reasonable CPU times. Torti et al (2012) performed a simulation study to compare powers of fast robust regression estimators including forward seach (Atkinson, 2010). Shortly, as the level of technology increases, we can collect more data; as we collect more data, the need for the technology increases.…”