Linear grouping using orthogonal regression

Abstract

A new method to detect different linear structures in a data set, called Linear Grouping Algorithm (LGA), is proposed.LGA is useful for investigating potential linear patterns in data sets, that is, subsets that follow different linear relationships.LGA combines ideas from principal components, clustering methods and resampling algorithms. It can detect several different linear relations at once. Methods to determine the number of groups in the data are proposed. Diagnostic tools to investigate the results obtained from LGA are introduced. It is shown how LGA can be extended to detect groups characterized by lower dimensional hyperplanes as well. Some applications illustrate the usefulness of LGA in practice.

A review of robust clustering methods

“…Although Dasgupta and Raftery's approach is clearly related to the TCLUST approach, it does not exactly fit within it because of the type of constraints on the eigenvalues. Moreover, there are problems where the multivariate normality assumption for the groups is too rigid because it also forces normality for the data point projections onto the linear structures (see Van Aelst et al 2006).…”

“…The Robust Linear Grouping Algorithm (RLGA) there introduced is a combination of the "self-trimming" methodology behind trimmed k-means with the Linear Grouping Algorithm (LGA) in Van Aelst et al (2006) is the subspace minimizing the sum of orthogonal square distances for the observations in each H j . This subspace is determined through solving a Principal Components (PCA) problem for the observations in H j .…”

tclust: AnRPackage for a Trimming Approach to Cluster Analysis

“…The use of this type of discriminant factors was already suggested in Van Aelst et al (2006) in a clustering problem without trimming. "Silhouette" plots (Rousseeuw 1987) can be used for summarizing the obtained ordered discriminant factors.…”

Grouping Around Different Dimensional Affine Subspaces

“…There exist many references about clustering around affine subspaces with equal dimensions within the statistical literature (see, e.g., [10] and [6] and the references therein). We can distinguish between two different approaches: "clusterwise regression" and "orthogonal residuals methods".…”

“…Throughout this work, we will be assuming that no privileged outcome variables do exist. Other model-based approaches have been already proposed based on fitting mixtures of multivariate normals assuming that the smallest groups' covariances eigenvalues are small (see, e.g, [3]) but they are not directly aimed at finding clusters around linear subspaces (see [10]). …”