Convolutional Neural Networks can be effectively used only when data are endowed with an intrinsic concept of neighbourhood in the input space, as is the case of pixels in images. We introduce here Ph-CNN, a novel deep learning architecture for the classification of metagenomics data based on the Convolutional Neural Networks, with the patristic distance defined on the phylogenetic tree being used as the proximity measure. The patristic distance between variables is used together with a sparsified version of MultiDimensional Scaling to embed the phylogenetic tree in a Euclidean space. Ph-CNN is tested with a domain adaptation approach on synthetic data and on a metagenomics collection of gut microbiota of 38 healthy subjects and 222 Inflammatory Bowel Disease patients, divided in 6 subclasses. Classification performance is promising when compared to classical algorithms like Support Vector Machines and Random Forest and a baseline fully connected neural network, e.g. the Multi-Layer Perceptron. Ph-CNN represents a novel deep learning approach for the classification of metagenomics data. Operatively, the algorithm has been implemented as a custom Keras layer taking care of passing to the following convolutional layer not only the data but also the ranked list of neighbourhood of each sample, thus mimicking the case of image data, transparently to the user.

In this work, we deal with a robust fitting of a wrapped normal model to multivariate circular data. Robust estimation is supposed to mitigate the adverse effects of outliers on inference. Furthermore, the use of a proper robust method leads to the definition of effective outlier detection rules. Robust fitting is achieved by a suitable modification of a classification-expectation-maximization algorithm that has been developed to perform a maximum likelihood estimation of the parameters of a multivariate wrapped normal distribution. The modification concerns the use of complete-data estimating equations that involve a set of data dependent weights aimed to downweight the effect of possible outliers. Several robust techniques are considered to define weights. The finite sample behavior of the resulting proposed methods is investigated by some numerical studies and real data examples.

The classical Tukey–Huber contamination model (CCM) is a commonly adopted framework to describe the mechanism of outliers generation in robust statistics. Given a dataset with n observations and p variables, under the CCM, an outlier is a unit, even if only one or a few values are corrupted. Classical robust procedures were designed to cope with this type of outliers. Recently, a new mechanism of outlier generation was introduced, namely, the independent contamination model (ICM), where the occurrences that each cell of the data matrix is an outlier are independent events and have the same probability. ICM poses new challenges to robust statistics since the percentage of contaminated rows dramatically increase with p, often reaching more than 50% whereas classical affine equivariant robust procedures have a breakdown point of 50% at most. For ICM, we propose a new type of robust methods, namely, composite robust procedures that are inspired by the idea of composite likelihood, where low-dimension likelihood, very often the likelihood of pairs, are aggregated to obtain a tractable approximation of the full likelihood. Our composite robust procedures are built on pairs of observations to gain robustness in the ICM. We propose composite τ-estimators for linear mixed models. Composite τ-estimators are proved to have a high breakdown point both in the CCM and ICM. A Monte Carlo study shows that while classical S-estimators can only cope with outliers generated by the CCM, the estimators proposed here are resistant to both CCM and ICM outliers. Supplementary materials for this article are available online.