Robust estimation for multivariate wrapped models

Abstract

A weighted likelihood technique for robust estimation of multivariate Wrapped distributions of data points scattered on a $$p-$$ p - dimensional torus is proposed. The occurrence of outliers in the sample at hand can badly compromise inference for standard techniques such as maximum likelihood method. Therefore, there is the need to handle such model inadequacies in the fitting process by a robust technique and an effective downweighting of observations not following the assumed model. Furthermore, the employ of a robust method could help in situations of hidden and unexpected substructures in the data. Here, it is suggested to build a set of data-dependent weights based on the Pearson residuals and solve the corresponding weighted likelihood estimating equations. In particular, robust estimation is carried out by using a Classification EM algorithm whose M-step is enhanced by the computation of weights based on current parameters’ values. The finite sample behavior of the proposed method has been investigated by a Monte Carlo numerical study and real data examples.

Robust Fitting of a Wrapped Normal Model to Multivariate Circular Data and Outlier Detection

“…Robust estimation is supposed to mitigate the adverse effects of outliers on estimation and inference. Outliers are unexpected anomalous values that exhibit a different pattern with respect to the rest of the data, as in the case of data orientated towards certain rare directions [3,8,9]. In circular data modeling, in the univariate case, the data can be represented as points on the circumference of the unit circle.…”

“…There have been several attempts to deal with outliers in circular data analysis, mainly focused on the Von Mises distribution and univariate problems [2,[11][12][13]. A first general attempt to develop a robust parametric technique in the multivariate case can be found in [9]: The authors focused on weighted likelihood estimation and considered outliers from a probabilistic point of view, as points that are unlikely to occur under the assumed model. A different approach, based on computing a local measure of outlyingness for a circular observation with respect to the distance from its k nearest neighbors, has been suggested in [14].…”

“…In this paper, we propose a novel robust estimation scheme. The key idea is that outlyingness is not measured directly on the torus as in [9] but only after unwrapping the multivariate circular data from the p-dimensional torus onto a hyperplane. This approach allows one to search for outliers based on their geometric distance from the robust fit.…”

“…This approach allows one to search for outliers based on their geometric distance from the robust fit. In other words, the main difference between the proposed technique and that discussed in [9] lies in the downweighting strategy: Here, weights are evaluated over the data unwrapped onto R p , whereas in [9], the weights are computed directly on the circular data over the torus and the fitted wrapped model.…”

“…It is worth noting that to the extent of our knowledge, M-estimation and hard trimming procedures have never been considered in robust estimation for circular data, neither in univariate problems, nor in the multivariate case. In addition, the weighted likelihood approach adopted here differs from that employed in [9], according to the comments above.…”