The purpose of this paper is to show in regression clustering how to choose the most relevant solutions, analyze their stability, and provide information about best combinations of optimal number of groups, restriction factor among the error variance across groups and level of trimming. The procedure is based on two steps. First we generalize the information criteria of constrained robust multivariate clustering to the case of clustering weighted models. Differently from the traditional approaches which are based on the choice of the best solution found minimizing an information criterion (i.e. BIC), we concentrate our attention on the so called optimal stable solutions. In the second step, using the monitoring approach, we select the best value of the trimming factor. Finally, we validate the solution using a confirmatory forward search approach. A motivating example based on a novel dataset concerning the European Union trade of face masks shows the limitations of the current existing procedures. The suggested approach is initially applied to a set of well known datasets in the literature of robust regression clustering. Then, we focus our attention on a set of international trade datasets and we provide a novel informative way of updating the subset in the random start approach. The Supplementary material, in the spirit of the Special Issue, deepens the analysis of trade data and compares the suggested approach with the existing ones available in the literature.

We assess the performance of state-of-the-art robust clustering tools for regression structures under a variety of different data configurations. We focus on two methodologies that use trimming and restrictions on group scatters as their main ingredients. We also give particular care to the data generation process through the development of a flexible simulation tool for mixtures of regressions, where the user can control the degree of overlap between the groups. Level of trimming and restriction factors are input parameters for which appropriate tuning is required. Since we find that incorrect specification of the second-level trimming in the Trimmed CLUSTering REGression model (TCLUST-REG) can deteriorate the performance of the method, we propose an improvement where the second-level trimming is not fixed in advance but is data dependent. We then compare our adaptive version of TCLUST-REG with the Trimmed Cluster Weighted Restricted Model (TCWRM) which provides a powerful extension of the robust clusterwise regression methodology. Our overall conclusion is that the two methods perform comparably, but with notable differences due to the inherent degree of modeling implied by them.

The forward search (FS) is a general method of robust data fitting that moves smoothly from very robust to maximum likelihood estimation. The regression procedures are included in the MATLAB toolbox FSDA. The work on a SAS version of the FS originates from the need for the analysis of large datasets expressed by law enforcement services operating in the European Union that use our SAS software for detecting data anomalies that may point to fraudulent customs returns. Specific to our SAS implementation, the fsdaSAS package, we describe the approximation used to provide fast analyses of large datasets using an FS which progresses through the inclusion of batches of observations, rather than progressing one observation at a time. We do, however, test for outliers one observation at a time. We demonstrate that our SAS implementation becomes appreciably faster than the MATLAB version as the sample size increases and is also able to analyse larger datasets. The series of fits provided by the FS leads to the adaptive data-dependent choice of maximally efficient robust estimates. This also allows the monitoring of residuals and parameter estimates for fits of differing robustness levels. We mention that our fsdaSAS also applies the idea of monitoring to several robust estimators for regression for a range of values of breakdown point or nominal efficiency, leading to adaptive values for these parameters. We have also provided a variety of plots linked through brushing. Further programmed analyses include the robust transformations of the response in regression. Our package also provides the SAS community with methods of monitoring robust estimators for multivariate data, including multivariate data transformations.

We contribute to the discussion of an article where Andrea Cerioli, Marco Riani, Anthony Atkinson and Aldo Corbellini review the advantages of analyzing multivariate data by monitoring how the estimated model parameters change as the estimation parameters vary. The focus is on robust methods and their sensitivity to the nominal efficiency and breakdown point. In congratulating with the authors for the clear and stimulating exposition, we contribute to its discussion with an overview of what we experienced in applying the monitoring in our application domain.

Supplementary Figure 7 PDF file - 205K, This Figure shows that pharmacological inhibition of FAK results in suppression of NSCLC in vivo