Bruno Pelletier

Retrieving of particulate matter from optical measurements: A semiparametric approach

[1] The fine particle abundance, i.e., particle matter (PM) concentration, is one of the indicators of air quality and is therefore subject to ground-based measurements. Complementary satellite aerosol remote sensing techniques provide one with maps of the aerosol optical thickness (AOT), which is sensitive to particle abundance. This paper investigates the problem of retrieving the PM concentration from the AOT, both on daily average values, on the basis of a large data set where data from the air quality networks are combined with ground-based measurements of the AOTs. It is found that a linear model fails at explaining the data well but that the performance may be significantly improved when such a linear relationship is conditioned on auxiliary parameters, mainly meteorological variables. The proposed model is expressed as an additive varying coefficient model (AVCM), which is defined as a linear model where the coefficients are additive functions of the auxiliary parameters. The model is represented using penalized smoothing splines, allowing for a proper control of the overall number of degrees of freedom via multiple smoothness parameters selection. The methodology is applied to data collected around Lille (France). The PM 10 concentrations are retrieved with an average uncertainty of less than 20%, leading to a correlation coefficient of 0.87 between fitted and expected PM 10 .

Non-parametric regression estimation on closed Riemannian manifolds

The non-parametric estimation of the regression function of a real-valued random variable Y on a random object X valued in a closed Riemannian manifold M is considered. A regression estimator which generalizes kernel regression estimators on Euclidean sample spaces is introduced. Under classical assumptions on the kernel and the bandwidth sequence, the asymptotic bias and variance are obtained, and the estimator is shown to converge at the same L 2-rate as kernel regression estimators on Euclidean spaces.

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.

Informative barycentres in statistics

Barycentres of a discrete probability measure on a dually flat statistical manifold are introduced. They are shown to be unique and to behave as barycentres in Euclidean space. The estimation of these barycentres is studied. Potential applicative usefulness of informative barycentres include the problem of interpolating a statistical manifold valued map and the problem of model merging, which consists in merging several statistical models into a unique one. The results are illustrated on the exponential family, for which a projection theorem is proved.

Estimation of density level sets with a given probability content

Given a random vector X valued in ℝ d with density f and an arbitrary probability number p∈(0; 1), we consider the estimation of the upper level setf≥t (p)of f corresponding to probability content p, that is, such that the probability that X belongs tof≥t (p)is equal to p. Based on an i.i.d. random sample X 1, …, X n drawn from f, we define the plug-in level set estimate , where is a random threshold depending on the sample and [fcirc] n is a nonparametric kernel density estimate based on the same sample. We establish the exact convergence rate of the Lebesgue measure of the symmetric difference between the estimated and actual level sets.

Approximation by ridge function fields over compact sets

We study the approximation of a continuous function feld over a compact set T by a continuous field of ridge approximants over T, named ridge function fields. We first give general density results about function fields and show how they apply to ridge function fields. We next discuss the parameterization of sets of ridge function fields and give additional density results for a class of continuous ridge function fields that admits a weak parameterization. Finally, we discuss the construction of the elements in that class.

Remote sensing of phytoplankton chlorophyll-a concentration by use of ridge function fields

A methodology is presented for retrieving phytoplankton chlorophyll-a concentration from space. The data to be inverted, namely, vectors of top-of-atmosphere reflectance in the solar spectrum, are treated as explanatory variables conditioned by angular geometry. This approach leads to a continuum of inverse problems, i.e., a collection of similar inverse problems continuously indexed by the angular variables. The resolution of the continuum of inverse problems is studied from the least-squares viewpoint and yields a solution expressed as a function field over the set of permitted values for the angular variables, i.e., a map defined on that set and valued in a subspace of a function space. The function fields of interest, for reasons of approximation theory, are those valued in nested sequences of subspaces, such as ridge function approximation spaces, the union of which is dense. Ridge function fields constructed on synthetic yet realistic data for case I waters handle well situations of both weakly and strongly absorbing aerosols, and they are robust to noise, showing improvement in accuracy compared with classic inversion techniques. The methodology is applied to actual imagery from the Sea-Viewing Wide Field-of-View Sensor (SeaWiFS); noise in the data are taken into account. The chlorophyll-a concentration obtained with the function field methodology differs from that obtained by use of the standard SeaWiFS algorithm by 15.7% on average. The results empirically validate the underlying hypothesis that the inversion is solved in a least-squares sense. They also show that large levels of noise can be managed if the noise distribution is known or estimated.

Retrieval of the Aerosol Vertical Distribution from Atmospheric Radiance

The feasibility of retrieving aerosol vertical distribution from the ratio of atmospheric radiance in, and out of, the oxygen A-band is investigated. Two typical cases of aerosol vertical profiles are considered, namely an exponential profile (aerosols concentrated near the surface), and a Gaussian profile (aerosols concentrated in altitude). The problem is expressed as a linear inverse problem with a compact operator, and a Tikhonov regularization scheme is implemented for its inversion. It is found that the exponential profile can be reconstructed accurately and in a stable manner, while this is not the case for the profile with aerosols concentrated in altitude. These results are explained by the spectral properties of the operator. Information on profile shape and/or utilization of spectral ratios more sensitive to upper layers would improve reconstruction when aerosols are located in altitude.

Remote sensing of phytoplankton with neural networks

In this paper, a neural network model, realizing a function assigning an estimate of the phytoplankton content in the ocean to several remote sensing acquisitions, is presented. This inverse problem is first shown to be a family of inverse subproblems, all of the same kind and continuously parameterized by the geometrical parameters defining the viewing geometry, thus allowing a two-steps modeling process. The central point of the method is that reflectances and geometrical parameters are processed in a different way. The first ones are considered as random variables while the seconds play the role of deterministic parameters. First, a set of local regression phytoplankton concentration estimators, i.e. small size neural networks, is constructed, locality being defined in the geometrical parameters space. Under some non restrictive hypotheses, each of those local models is shown to be optimal. Further, a lower bound on the expected accuracy is given. Secondly, a global model is constructed from a set of local models which in fact amounts to be a neural network, the parameters of which are continuous functions of the geometrical parameters. The model has been tested on a wide simulated data set of about 7 million points for different geometrical configurations, different atmospheric conditions and several wind speed and direction values. It has shown very good results for a large set of geometrical configurations. Moreover, many much results have been obtained with this model than with global approaches based on multilayer perceptrons and radial basis functions neural networks. The presented methodology is also a promising direction for the elaboration of complex models from a set of simpler ones.

Remember the curse of dimensionality: the case of goodness-of-fit testing in arbitrary dimension

ABSTRACT Despite a substantial literature on nonparametric two-sample goodness-of-fit testing in arbitrary dimensions, there is no mention there of any curse of dimensionality. In fact, in some publications, a parametric rate is derived. As we discuss below, this is because a directional alternative is considered. Indeed, even in dimension one, Ingster, Y. I. [(1987). Minimax testing of nonparametric hypotheses on a distribution density in the l_p metrics. Theory of Probability & Its Applications, 31(2), 333–337] has shown that the minimax rate is not parametric. In this paper, we extend his results to arbitrary dimension and confirm that the minimax rate is not only nonparametric, exhibits but also a prototypical curse of dimensionality. We further extend Ingsters work to show that the chi-squared test achieves the minimax rate. Moreover, we show that the test adapts to the intrinsic dimensionality of the data. Finally, in the spirit of Ingster, Y. I. [(2000). Adaptive chi-square tests. Journal of Mathematical Sciences, 99(2), 1110–1119], we consider a multiscale version of the chi-square test, showing that one can adapt to unknown smoothness without much loss in power.