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We derive normal approximation results for a class of stabilizing functionals of binomial or Poisson point process, that are not necessarily expressible as sums of certain score functions. Our approach is based on a flexible notion of the add-one cost operator, which helps one to deal with the second-order cost operator via suitably appropriate first-order operators. We combine this flexible notion with the theory of strong stabilization to establish our results. We illustrate the applicability of our results by establishing normal approximation results for certain geometric and topological statistics arising frequently in practice. Several existing results also emerge as special cases of our approach.

This work proposes and investigates a novel method for anomaly detection and shows it to be competitive in a variety of Euclidean and non-Euclidean situations. It is based on an extension of the depth quantile functions (DQF) approach. The DQF approach encodes geometric information about a point cloud via functions of a single variable, whereas each observation in a data set is associated with a single such function. Plotting these functions provides a very beneficial visualization aspect. This technique can be applied to any data lying in a Hilbert space. The proposed anomaly detection approach is motivated by the geometric insight of the presence of anomalies in data being tied to the existence of antimodes in the data generating distribution. Coupling this insight with novel theoretical understanding into the shape of the DQFs gives rise to the proposed adaptive DQF (aDQF) methodology. Applications to various data sets illustrate the DQF and aDQF's strong anomaly detection performance, and the benefits of its visualization aspects., Comment: 24 pages, 13 figures

International audience; We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is "topologically smooth".

The extraction of filamentary structure from a point cloud is discussed. The filaments are modeled as ridge lines or higher dimensional ridges of an underlying density. We propose two novel algorithms, and provide theoretical guarantees for their convergences. We consider the new algorithms as alternatives to the Subspace Constraint Mean Shift (SCMS) algorithm that do not suffer from a shortcoming of the SCMS that is also revealed in this paper., 41 pages, 8 figures

In statistical network analysis it is common to observe so called interaction data. Such data is characterized by actors forming the vertices and interacting along edges of the network, where edges are randomly formed and dissolved over the observation horizon. In addition covariates are observed and the goal is to model the impact of the covariates on the interactions. We distinguish two types of covariates: global, system-wide covariates (i.e. covariates taking the same value for all individuals, such as seasonality) and local, dyadic covariates modeling interactions between two individuals in the network. Existing continuous time network models are extended to allow for comparing a completely parametric model and a model that is parametric only in the local covariates but has a global non-parametric time component. This allows, for instance, to test whether global time dynamics can be explained by simple global covariates like weather, seasonality etc. The procedure is applied to a bike-sharing network by using weather and weekdays as global covariates and distances between the bike stations as local covariates.

It is well known that the empirical likelihood ratio confidence region suffers finite sample under-coverage issue, and this severely hampers its application in statistical inferences.} The root cause of this under-coverage is an upper limit imposed by the convex hull of the estimating functions that is used in the construction of the profile empirical likelihood. For i.i.d data, various methods have been proposed to solve this issue by modifying the convex hull, but it is not clear how well these methods perform when the data are no longer independent. In this paper, we propose an adjusted blockwise empirical likelihood that is designed for weakly dependent multivariate data. We show that our method not only preserves the much celebrated asymptotic $\chi^2-$distribution, but also improves the finite sample coverage probability by removing the upper limit imposed by the convex hull. Further, we show that our method is also Bartlett correctable, thus is able to achieve high order asymptotic coverage accuracy., Comment: Typos corrected

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process $ (r,s) \mapsto n^{-1/2} (\beta^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[\beta^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$. So far, pointwise limit theorems have been established in different settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime, see Yogeshwaran et al. [2017] and Hiraoka et al. [2018]. In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich [2001] of persistent Betti numbers and generalize the existing results on the asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that the multivariate asymptotic normality holds for all pairs $(r,s)$, $0\le r\le s

The topic of this paper is modeling and analyzing dependence in stochastic social networks. Using a latent variable block model allows the analysis of dependence between blocks via the analysis of a latent graphical model. Our approach to the analysis of the graphical model then is based on the idea underlying the neighborhood selection scheme put forward by Meinshausen and B\"{u}hlmann (2006). However, because of the latent nature of our model, estimates have to be used in lieu of the unobserved variables. This leads to a novel analysis of graphical models under uncertainty, in the spirit of Rosenbaum et al. (2010), or Belloni et al. (2017). Lasso-based selectors, and a class of Dantzig-type selectors are studied.

We present an approach to model time series data from resting state fMRI for autism spectrum disorder (ASD) severity classification. We propose to adopt kernel machines and employ graph kernels that define a kernel dot product between two graphs. This enables us to take advantage of spatio-temporal information to capture the dynamics of the brain network, as opposed to aggregating them in the spatial or temporal dimension. In addition to the conventional similarity graphs, we explore the use of L1 graph using sparse coding, and the persistent homology of time delay embeddings, in the proposed pipeline for ASD classification. In our experiments on two datasets from the ABIDE collection, we demonstrate a consistent and significant advantage in using graph kernels over traditional linear or non linear kernels for a variety of time series features., Under review as a conference paper to BHI '17

In this paper we penetrate and extend the notion of local constancy in graphical models that has been introduced by Honorio et al. (2009). We propose Neighborhood-Fused Lasso, a method for model selection in high-dimensional graphical models, leveraging locality information. Our approach is based on an extension of the idea of node-wise regression (Meinshausen-B\"{u}hlmann, 2006) by adding a fusion penalty. We propose a fast numerical algorithm for our approach, and provide theoretical and numerical evidence for the fact that our methodology outperforms related approaches that are ignoring the locality information. We further investigate the compatibility issues in our proposed methodology and derive bound for the quadratic prediction error and $l_1$-bounds on the estimated coefficients.

Based on observation vectors (Xt,Yt) from a strong mixing stochastic process,we estimate the conditional distribution of Y given X = x by means of aNadaraya-Watson-type estimator. Using this, we study the asymptotics ofa conditional empirical process indexed by classes of sets.Under assumptions on the richness of the indexing class in terms ofmetric entropy with bracketing, we have established uniform convergence, and asymptotic normality. The key technical result gives rates of convergences for the sup-norm of the conditional empirical process over a sequenceof indexing classes with decreasing maximum Lt-norm.The results are then applied to derive Bahadur-Kiefer type approximationsfor a generalized conditional quantile process which is closelyrelated to the minimum volume sets. The potential applications in the areas ofestimation of level sets and testing for unimodality of conditionaldistributions are discussed.

Motivated by interval/region prediction in nonlinear timeseries, we propose a minimum volume predictor (MV-predictor) for astrictly stationary process. The MV-predictor varies with respect tothe current position inthe state space and has the minimum Lebesgue measure amongall regions with the nominal coverage probability.We have established consistency, convergence rates, andasymptotic normality for both coverage probability and Lebesguemeasure of the estimated MV-predictor under the assumption thatthe observations are taken from a strong mixing process.Applications with both real and simulated data sets illustrate theproposed methods.

By using empirical process theory we study a method addressed totesting for multimodality and estimating density contour clusters in higherdimensions. The method is based on the so-called excess mass. Given aprobability measure F and a class of sets in the d-dimensional Euclidean space, the excess mass is defined as the maximal difference between theF-measure and l times the Lebesgue measure of sets in the given class. The excess mass can be estimated by replacing F by the empirical measure. Thecorreponding maximizing sets can be used for estimating density contourclusters. Comparing excess masses over different classes yields informationabout the modality of the underlying probability measure. This can be usedto construct tests for multimodality. The asymptotic behaviour of theconsidered estimators and test statistics is studied for different classesof sets, including the classes of balls, ellipsoids and convex sets.