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Asymptotic normality in density support estimation
 Electron. J. Probab
"... c i E l e c t r o n J o u r n a l o f ..."
The Normalized Graph Cut and Cheeger Constant: from Discrete to Continuous
, 2012
"... Let M be a bounded domain of W ' with a smooth boundary. We relate the Cheeger eonstant of M and the eonductance 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 normal ..."
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Let M be a bounded domain of W ' with a smooth boundary. We relate the Cheeger eonstant of M and the eonductance 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.
Adaptation to lowest density regions with application to support recovery
, 2014
"... Adaptation to lowest density regions with ..."
Estimation of the support of the density and its boundary using Random Polyhedron
, 2013
"... WeconsiderrandomsamplesinR d drawnfromanunknowndensity. When the support is assumed to be convex and with sharp boundary, the convex hull is an estimator of the support that converges to S with a rate of n −2/(d+1). When the boundary of the support is sharp but the support is no longer assumed to be ..."
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WeconsiderrandomsamplesinR d drawnfromanunknowndensity. When the support is assumed to be convex and with sharp boundary, the convex hull is an estimator of the support that converges to S with a rate of n −2/(d+1). When the boundary of the support is sharp but the support is no longer assumed to be convex, the usual support estimators convergeswith a rate of n −1/d or (ln(n)/n) −1/d. This paper is devoted to presenting some new estimators of the support of the density, which are based on some local convexity criteria and converge to S with a rate of (n/lnn) −2/(d+1) (and their boundary converges toward ∂S with the same rate) when the support is assumed to have a sharp C 2 boundary. The convergence rate is also given when the sharpness hypothesis is relaxed (and it is close to the optimal rate when the dimension is two). key words: Delaunay complex, polyhedron, support estimation, topological data analysis, geometric inference.
Asymptotic Normality in Density Support Estimation
"... Let X1,..., Xn be n independent observations drawn from a multivariate probability density f with compact support Sf. This paper is devoted to the study of the estimator ˆ Sn of Sf defined as the union of balls centered at the Xi and with common radius rn. Using tools from Riemannian geometry, and u ..."
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Let X1,..., Xn be n independent observations drawn from a multivariate probability density f with compact support Sf. This paper is devoted to the study of the estimator ˆ Sn of Sf defined as the union of balls centered at the Xi and with common radius rn. Using tools from Riemannian geometry, and under mild assumptions on f and the sequence (rn), we prove a central limit theorem for λ(Sn∆Sf), where λ denotes the Lebesgue measure on R d and ∆ the symmetric difference operation.
A Weighted kNearest Neighbor Density Estimate for Geometric Inference
"... Motivated by a broad range of potential applications in topological and geometric inference, we introduce a weighted version of the knearest neighbor density estimate. Various pointwise consistency results of this estimate are established. We present a general central limit theorem under the light ..."
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Motivated by a broad range of potential applications in topological and geometric inference, we introduce a weighted version of the knearest neighbor density estimate. Various pointwise consistency results of this estimate are established. We present a general central limit theorem under the lightest possible conditions. In addition, a strong approximation result is obtained and the choice of the optimal set of weights is discussed. In particular, the classical knearest neighbor estimate is not optimal in a sense described in the manuscript. The proposed method has been implemented to recover level sets in both simulated and reallife data. Index Terms — Geometric inference, level sets, density estimation, knearest neighbor estimate, weighted estimate, consistency, rates of convergence, central limit theorem, strong approximation.
support and
, 2012
"... Using the k−nearest neighbor restricted Delaunay polyhedron to estimate the density ..."
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Using the k−nearest neighbor restricted Delaunay polyhedron to estimate the density
Local Convex Hull support and boundary estimation
, 2014
"... In this paper we study a new estimator for the support of a multivariate density. It is defined as a union of convexhulls of observations contained in small balls. We study the asymptotic behavior of this “local convex hull ” as an estimator of the support and the asymptotic behaviors of its boun ..."
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In this paper we study a new estimator for the support of a multivariate density. It is defined as a union of convexhulls of observations contained in small balls. We study the asymptotic behavior of this “local convex hull ” as an estimator of the support and the asymptotic behaviors of its boundary as an estimator of the boundary of the support. We analyze as well its ”topologypreserving ” properties.
BAYESIAN METHODOLOGY FOR OCEANCOLOR REMOTE SENSING
, 2013
"... The inverse ocean color problem, i.e., the retrieval of marine reflectance from topofatmosphere (TOA) reflectance, is examined in a Bayesian context. The solution is expressed as a probability distribution that measures the likelihood of encountering specific values of the marine reflectance given ..."
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The inverse ocean color problem, i.e., the retrieval of marine reflectance from topofatmosphere (TOA) reflectance, is examined in a Bayesian context. The solution is expressed as a probability distribution that measures the likelihood of encountering specific values of the marine reflectance given the observed TOA reflectance. This conditional distribution, the posterior distribution, allows the construction of reliable multidimensional confidence domains of the retrieved marine reflectance. The expectation and covariance of the posterior distribution are computed, which gives for each pixel an estimate of the marine reflectance and a measure of its uncertainty. Situations for which forward model and observation are incompatible are also identified. Prior distributions of the forward model parameters that are suitable for use at the global scale, as well as a noise model, are determined. Partitionbased models are defined and implemented for SeaWiFS, to approximate numerically the expectation and covariance. The illposed nature of the inverse problem is illustrated, indicating that a large set of ocean and atmospheric states, or preimages, may correspond to very close values of the satellite signal. Theoretical performance is good globally, i.e., on average over all the geometric and geophysical situations considered, with negligible biases and standard deviation decreasing from 0.004 at 412 nm to 0.001 at 670 nm. Errors are smaller