Ery Arias-Castro

Large-scale absence of sharks on reefs in the greater-Caribbean: a footprint of human pressures.

Background In recent decades, large pelagic and coastal shark populations have declined dramatically with increased fishing; however, the status of sharks in other systems such as coral reefs remains largely unassessed despite a long history of exploitation. Here we explore the contemporary distribution and sighting frequency of sharks on reefs in the greater-Caribbean and assess the possible role of human pressures on observed patterns. Methodology/Principal Findings We analyzed 76,340 underwater surveys carried out by trained volunteer divers between 1993 and 2008. Surveys were grouped within one km2 cells, which allowed us to determine the contemporary geographical distribution and sighting frequency of sharks. Sighting frequency was calculated as the ratio of surveys with sharks to the total number of surveys in each cell. We compared sighting frequency to the number of people in the cell vicinity and used population viability analyses to assess the effects of exploitation on population trends. Sharks, with the exception of nurse sharks occurred mainly in areas with very low human population or strong fishing regulations and marine conservation. Population viability analysis suggests that exploitation alone could explain the large-scale absence; however, this pattern is likely to be exacerbated by additional anthropogenic stressors, such as pollution and habitat degradation, that also correlate with human population. Conclusions/Significance Human pressures in coastal zones have lead to the broad-scale absence of sharks on reefs in the greater-Caribbean. Preventing further loss of sharks requires urgent management measures to curb fishing mortality and to mitigate other anthropogenic stressors to protect sites where sharks still exist. The fact that sharks still occur in some densely populated areas where strong fishing regulations are in place indicates the possibility of success and encourages the implementation of conservation measures.

On the Fundamental Limits of Adaptive Sensing

Suppose we can sequentially acquire arbitrary linear measurements of an n -dimensional vector x resulting in the linear model y = A x + z, where z represents measurement noise. If the signal is known to be sparse, one would expect the following folk theorem to be true: choosing an adaptive strategy which cleverly selects the next row of A based on what has been previously observed should do far better than a nonadaptive strategy which sets the rows of A ahead of time, thus not trying to learn anything about the signal in between observations. This paper shows that the folk theorem is false. We prove that the advantages offered by clever adaptive strategies and sophisticated estimation procedures-no matter how intractable-over classical compressed acquisition/recovery schemes are, in general, minimal.

Global testing under sparse alternatives: ANOVA, multiple comparisons and the higher criticism

Testing for the significance of a subset of regression coefficients in a linear model, a staple of statistical analysis, goes back at least to the work of Fisher who introduced the analysis of variance (ANOVA). We study this problem under the assumption that the coefficient vector is sparse, a common situation in modern high-dimensional settings. Suppose we have p covariates and that under the alternative, the response only depends upon the order of p^(1−α) of those, 0 ≤ α ≤ 1. Under moderate sparsity levels, that is, 0 ≤ α ≤ 1/2, we show that ANOVA is essentially optimal under some conditions on the design. This is no longer the case under strong sparsity constraints, that is, α > 1/2. In such settings, a multiple comparison procedure is often preferred and we establish its optimality when α ≥ 3/4. However, these two very popular methods are suboptimal, and sometimes powerless, under moderately strong sparsity where 1/2 1/2. This optimality property is true for a variety of designs, including the classical (balanced) multi-way designs and more modern “p > n” designs arising in genetics and signal processing. In addition to the standard fixed effects model, we establish similar results for a random effects model where the nonzero coefficients of the regression vector are normally distributed.

Searching for a Trail of Evidence in a Maze

Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following hypothesis testing problem: under the null, the random variables have common distribution N(0,1) while under the alternative, there is an unknown path along which random variables have distribution N(�, 1), � > 0, and distribution N(0,1) away from it. For which values of the mean shiftcan one reliably detect and for which values is this impossible? This paper develops detection thresholds for two types of common graphs which exhibit a different behavior. The first is the usual regular lattice with vertices of the form {(i, j) : 0 ≤ i, −i ≤ j ≤ i and j has the parity of i} and oriented edges (i, j) → (i+1, j+s) where s = ±1. We show that for paths of length m start- ing at the origin, the hypotheses become distinguishable (in a minimax sense) ifm ≫ √ log m, while they are not ifm ≪ log m. We derive equivalent results in a Bayesian setting where one assumes that all paths are equally likely; there the asymptotic threshold ism ≈ m 1/4 . We obtain corresponding results for trees (where the threshold is of order 1 and independent of the size of the tree), for distributions other than the Gaussian, and for other graphs. The concept of predictability profile, first introduced by Benjamini, Pemantle and Peres, plays a crucial role in our analysis.

Detection of correlations

We consider the hypothesis testing problem of deciding whether an observed high-dimensional vector has independent normal components or, alternatively, if it has a small subset of correlated components. The correlated components may have a certain combinatorial structure known to the statistician. We establish upper and lower bounds for the worst-case (minimax) risk in terms of the size of the correlated subset, the level of correlation, and the structure of the class of possibly correlated sets. We show that some simple tests have near-optimal performance in many cases, while the generalized likelihood ratio test is suboptimal in some important cases.

Compressive binary search

In this paper we consider the problem of locating a nonzero entry in a high-dimensional vector from possibly adaptive linear measurements. We consider a recursive bisection method which we dub the compressive binary search and show that it improves on what any nonadaptive method can achieve. We also establish a non-asymptotic lower bound that applies to all methods, regardless of their computational complexity. Combined, these results show that the compressive binary search is within a double logarithmic factor of the optimal performance.

Detecting Positive Correlations in a Multivariate Sample

We consider the problem of testing whether a correlation matrix of a multivariate normal population is the identity matrix. We focus on sparse classes of alternatives where only a few entries are nonzero and, in fact, positive. We derive a general lower bound applicable to various classes and study the performance of some near-optimal tests. We pay special attention to computational feasibility and construct near-optimal tests that can be computed efficiently. Finally, we apply our results to prove new lower bounds for the clique number of high-dimensional random geometric graphs.

The Normalized Graph Cut and Cheeger Constant: from Discrete to Continuous

Let M be a bounded domain of with a smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set of M.

Detection and feature selection in sparse mixture models

We consider Gaussian mixture models in high dimensions and concentrate on the twin tasks of detection and feature selection. Under sparsity assumptions on the difference in means, we derive information bounds and establish the performance of various procedures, including the top sparse eigenvalue of the sample covariance matrix and other projection tests based on moments, such as the skewness and kurtosis tests of Malkovich and Afifi (1973), and other variants which we were better able to control under the null.

Cluster detection in networks using percolation

We consider the task of detecting a salient cluster in a sensor network, that is, an undirected graph with a random variable attached to each node. Motivated by recent research in environmental statistics and the drive to compete with the reigning scan statistic, we explore alternatives based on the percolative properties of the network. The first method is based on the size of the largest connected component after removing the nodes in the network with a value below a given threshold. The second method is the upper level set scan test introduced by Patil and Taillie [Statist. Sci. 18 (2003) 457-465]. We establish the performance of these methods in an asymptotic decision- theoretic framework in which the network size increases. These tests have two advantages over the more conventional scan statistic: they do not require previous information about cluster shape, and they are computationally more feasible. We make abundant use of percolation theory to derive our theoretical results, and complement our theory with some numerical experiments.