We tackle the issue of anomaly detection for multivariate functional data in a supervised setting. Deep learning applied to multivariate time series has become common nowadays, especially for medical data such as electrocardiogram (ECG). There are not many explanability techniques that can handle multivariate time series. In this paper, we propose a model to understand abnormal class features on multivariate time series. We present a new variational autoencoder (VAE) training method that focuses on dividing the latent space into general and class-based features using supervised contrastive learning. The Contrastive VAE produces a well-organized latent space that enables us to modify only the class-based features and to use the generative part of the VAE to produce counterfactual examples. This method is able to easily provide plausible counterfactual observations, which highlights the differences between pathological and non-pathological data. We demonstrate the superiority of our approach over other counterfactual methods in terms of validity and performance.

We propose two novel nonparametric two-sample kernel tests based on the Maximum Mean Discrepancy (MMD). First, for a fixed kernel, we construct an MMD test using either permutations or a wild bootstrap, two popular numerical procedures to determine the test threshold. We prove that this test controls the probability of type I error non-asymptotically. Hence, it can be used reliably even in settings with small sample sizes as it remains well-calibrated, which differs from previous MMD tests which only guarantee correct test level asymptotically. When the difference in densities lies in a Sobolev ball, we prove minimax optimality of our MMD test with a specific kernel depending on the smoothness parameter of the Sobolev ball. In practice, this parameter is unknown and, hence, the optimal MMD test with this particular kernel cannot be used. To overcome this issue, we construct an aggregated test, called MMDAgg, which is adaptive to the smoothness parameter. The test power is maximised over the collection of kernels used, without requiring held-out data for kernel selection (which results in a loss of test power), or arbitrary kernel choices such as the median heuristic. We prove that MMDAgg still controls the level non-asymptotically, and achieves the minimax rate over Sobolev balls, up to an iterated logarithmic term. Our guarantees are not restricted to a specific type of kernel, but hold for any product of one-dimensional translation invariant characteristic kernels. We provide a user-friendly parameter-free implementation of MMDAgg using an adaptive collection of bandwidths. We demonstrate that MMDAgg significantly outperforms alternative state-of-the-art MMD-based two-sample tests on synthetic data satisfying the Sobolev smoothness assumption, and that, on real-world image data, MMDAgg closely matches the power of tests leveraging the use of models such as neural networks.

We propose a novel procedure for outlier detection in space telemetries, in a semi-supervised framework. As the data is functional, we reduce its dimension by considering the coefficients obtained after projecting the observations onto orthonormal bases. A multiple testing procedure based on the two-sample test is defined in order to highlight the levels of the coefficients on which the outliers appear as significantly different from the nominal data. The Local Outlier Factor is computed on the selected coefficients to highlight the outliers. This procedure for selecting the features is applied on simulated data that mimic the behavior of space telemetries and on a real telemetry and then compared with existing dimension reduction techniques.

To overcome the cost of numerical simulations of transient phenomena, the goal is to construct a robust spatio-temporal reduced model capable of long-term in time predictions. The construction proposed in this article has to deal with several constraints: the reference model is a black box with high dimensional inputs and outputs, long-term in time prediction, few learning samples available, non-linear behaviour and the construction time must remain reasonable while the prediction time must be negligible. Recurrent neural networks are predictive models adapted to this dynamic framework. The improvements of the construction methodology detailed in this paper are the weights optimization through a multilevel optimization approach, a robust construction based on cross-validation and an application of sensitivity analysis in order to reduce the input dimension of the network. Finally, this construction is validated on an industrial test case predicting the temperature of an electronic equipment located in the avionic bay and subjected to fluctuations of its boundary conditions.