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This contribution analyzes a completely unknown 16th century drawing. In the document, the author identifies a lost project for a pulpit in the Cathedral of Cosenza, draw up by Andrea Maggiore, Niccolò Ciolli and Antonio Grasso. The reconstruction of the drawing’s origin is possible thanks to some valuable indications on the document itself. The discovery allows us to re-analyze the personality of Andrea Maggiore, and his first known work, in the context of the Calabrian renaissance. Andrea Maggiore is especially important for the introduction of coloured marble altars in the region. His works are framed in the general phenomenon of the introduction of a new architectural language in southern Italy with an important reference to the role of his clients: a powerful and sometimes foreign cultural élite.

We investigate sequential change point estimation and detection in univariate nonparametric settings, where a stream of independent observations from sub-Gaussian distributions with a common variance factor and piecewise-constant but otherwise unknown means are collected. We develop a simple CUSUM-based methodology that provably control the probability of false alarms or the average run length while minimizing, in a minimax sense, the detection delay. We allow for all the model parameters to vary in order to capture a broad range of levels of statistical hardness for the problem at hand. We further show how our methodology is applicable to the case in which multiple change points are to be estimated sequentially. [ABSTRACT FROM AUTHOR]

We study the multivariate nonparametric change point detection problem, where the data are a sequence of independent $p$ -dimensional random vectors whose distributions are piecewise-constant with Lipschitz densities changing at unknown times, called change points. We quantify the size of the distributional change at any change point with the supremum norm of the difference between the corresponding densities. We are concerned with the localization task of estimating the positions of the change points. In our analysis, we allow for the model parameters to vary with the total number of time points, including the minimal spacing between consecutive change points and the magnitude of the smallest distributional change. We provide information-theoretic lower bounds on both the localization rate and the minimal signal-to-noise ratio required to guarantee consistent localization. We formulate a novel algorithm based on kernel density estimation that nearly achieves the minimax lower bound, save possibly for logarithm factors. We have provided extensive numerical evidence to support our theoretical findings. [ABSTRACT FROM AUTHOR]

We introduce a new family of network models, called hierarchical network models, that allow us to represent in an explicit manner the stochastic dependence among the dyads (random ties) of the network. In particular, each member of this family can be associated with a graphical model defining conditional independence clauses among the dyads of the network, called the dependency graph. Every network model with dyadic independence assumption can be generalized to construct members of this new family. Using this new framework, we generalize the Erdös–Rényi and the β models to create hierarchical Erdös–Rényi and β models. We describe various methods for parameter estimation, as well as simulation studies for models with sparse dependency graphs. [ABSTRACT FROM AUTHOR]

We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size. [ABSTRACT FROM AUTHOR]

We consider the problem of deciding, based on a single noisy measurement at each vertex of a given graph, whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation. This problem is relevant to several applications such as surveillance, disease outbreak detection, biomedical imaging, environmental monitoring, etc. Since the activations in these problems often tend to be localized to small groups of vertices in the graphs, we model such activity by a class of signals that are elevated over a (possibly disconnected) cluster with low cut size relative to its size. We analyze the corresponding generalized likelihood ratio (GLR) statistics and relate it to the problem of finding a sparsest cut in the graph. We develop a convex relaxation of the GLR statistic based on spectral graph theory, which we call the graph Fourier scan statistic (GFSS). In our main theoretical result, we show that the performance of the GFSS depends explicitly on the spectral properties of the graph. To assess the optimality of the GFSS, we prove an information theoretic lower bound for the detection of anomalous activity on graphs. Because the GFSS requires the specification of a tuning parameter, we develop an adaptive version of the GFSS. Using these results, we are able to characterize in a very explicit form the performance of the GFSS on a few notable graph topologies. We demonstrate that the GFSS can efficiently detect a simulated Arsenic contamination in groundwater. [ABSTRACT FROM AUTHOR]

Persistent homology is a widely used tool in Topological Data Analysis that encodes multi-scale topological information as a multi set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we summarize persistent homology with a persistence landscape, introduced by Bubenik, which converts a diagram into a well-behaved real-valued function. We investigate the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence of the bootstrap. In addition, we introduce an alternate functional summary of persistent homology, which we call the silhouette, and derive an analogous statistical theory. [ABSTRACT FROM AUTHOR]

A metric graph is a 1-dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as street maps, neurons, networks of rivers and galaxies. We consider the statistical problem of reconstructing the topology of a metric graph embedded in RD from a random sample. We derive lower and upper bounds on the minimax risk for the noiseless case and tubular noise case. The upper bound is based on the reconstruction algorithm given in Aanjaneya et al. (2012). [ABSTRACT FROM AUTHOR]

Fiber tractography on diffusion imaging data offers rich potential for describing white matter pathways in the human brain, but characterizing the spatial organization in these large and complex data sets remains a challenge. We show that level set trees–which provide a concise representation of the hierarchical mode structure of probability density functions–offer a statistically-principled framework for visualizing and analyzing topography in fiber streamlines. Using diffusion spectrum imaging data collected on neurologically healthy controls (N = 30), we mapped white matter pathways from the cortex into the striatum using a deterministic tractography algorithm that estimates fiber bundles as dimensionless streamlines. Level set trees were used for interactive exploration of patterns in the endpoint distributions of the mapped fiber pathways and an efficient segmentation of the pathways that had empirical accuracy comparable to standard nonparametric clustering techniques. We show that level set trees can also be generalized to model pseudo-density functions in order to analyze a broader array of data types, including entire fiber streamlines. Finally, resampling methods show the reliability of the level set tree as a descriptive measure of topographic structure, illustrating its potential as a statistical descriptor in brain imaging analysis. These results highlight the broad applicability of level set trees for visualizing and analyzing high-dimensional data like fiber tractography output. [ABSTRACT FROM AUTHOR]

Graphs are the primary mathematical representation for networks, with nodes or vertices corresponding to units (e.g., individuals) and edges corresponding to relationships. Exponential Random Graph Models (ERGMs) are widely used for describing network data because of their simple structure as an exponential function of a sum of parameters multiplied by their corresponding sufficient statistics. As with other exponential family settings the key computational difficulty is determining the normalizing constant for the likelihood function, a quantity that depends only on the data. In ERGMs for network data, the normalizing constant in the model often makes the parameter estimation intractable for large graphs, when the model involves dependence among dyads in the graph. One way to deal with this problem is to approximate the likelihood function by something tractable, e.g., by using the method of pseudo-likelihood estimation suggested in the early literature. In this paper, we describe the family of ERGMs and explain the increasing complexity that arises from imposing different edge dependence and homogeneous parameter assumptions. We then compare maximum likelihood (ML) and maximum pseudo-likelihood (MPL) estimation schemes with respect to existence and related degeneracy properties for ERGMs involving dependencies among dyads. [ABSTRACT FROM AUTHOR]

Differential privacy is a rigorous cryptographically-motivated characterization of data privacy which may be applied when releasing summaries of a database. Previous work has focused mainly on methods for which the output is a finite dimensional vector, or an element of some discrete set. We develop methods for releasing functions while preserving differential privacy. Specifically, we show that adding an appropriate Gaussian process to the function of interest yields differential privacy. When the functions lie in the reproducing kernel Hilbert space (RKHS) generated by the covariance kernel of the Gaussian process, then the correct noise level is established by measuring the "sensitivity" of the function in the RKHS norm. As examples we consider kernel density estimation, kernel support vector machines, and functions in RKHSs. [ABSTRACT FROM AUTHOR]

High density clusters can be characterized by the connected components of a level set L(λ) = {x : p(x) > λ} of the underlying probability density function p generating the data, at some appropriate level λ ≥ 0. The complete hierarchical clustering can be characterized by a cluster tree Τ = ∪λ L(λ). In this paper, we study the behavior of a density level set estimate Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. (λ) and cluster tree estimate Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. based on a kernel density estimator with kernel bandwidth h. We define two notions of instability to measure the variability of Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. (λ) and Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. as a function of h, and investigate the theoretical properties of these instability measures. [ABSTRACT FROM AUTHOR]

We study the problem of optimal estimation of the density cluster tree under various smoothness assumptions on the underlying density. Inspired by the seminal work of Chaudhuri et al. (2014), we formulate a new notion of clustering consistency which is better suited to smooth densities, and derive minimax rates for cluster tree estimation under Hölder smooth densities of arbitrary degree. We present a computationally efficient, rate optimal cluster tree estimator based on simple extensions of the popular DBSCAN algorithm of Ester et al. (1996). Our procedure relies on kernel density estimators and returns a sequence of nested random geometric graphs whose connected components form a hierarchy of clusters. The resulting optimal rates for cluster tree estimation depend on the degree of smoothness of the underlying density and, interestingly, match the minimax rates for density estimation under the sup-norm loss. Our results complement and extend the analysis of the DBSCAN algorithm in Sriperumbudur and Steinwart (2012). Finally, we consider level set estimation and cluster consistency for densities with jump discontinuities. We demonstrate that the DBSCAN algorithm attains the minimax rate in terms of the jump size and sample size in this setting as well. [ABSTRACT FROM AUTHOR]

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Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the distance function (the distance of any point x to S). Given a sample from P we can infer the persistent homology using an empirical version of the distance function. However, the empirical distance function is highly non-robust to noise and outliers. Even one outlier is deadly. The distance-to-a-measure (DTM), introduced by Chazal et al. (2011), and the kernel distance, introduced by Phillips et al. (2014), are smooth functions that provide useful topological information but are robust to noise and outliers. Chazal et al. (2015) derived concentration bounds for DTM. Building on these results, we derive limiting distributions and confidence sets, and we propose a method for choosing tuning parameters. [ABSTRACT FROM AUTHOR]