262,843 research outputs found
Gradient bounds for a thin film epitaxy equation
We consider a gradient flow modeling the epitaxial growth of thin films with
slope selection. The surface height profile satisfies a nonlinear diffusion
equation with biharmonic dissipation. We establish optimal local and global
wellposedness for initial data with critical regularity. To understand the
mechanism of slope selection and the dependence on the dissipation coefficient,
we exhibit several lower and upper bounds for the gradient of the solution in
physical dimensions
Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases
We investigate the optimality for model selection of the so-called slope
heuristics, -fold cross-validation and -fold penalization in a
heteroscedastic with random design regression context. We consider a new class
of linear models that we call strongly localized bases and that generalize
histograms, piecewise polynomials and compactly supported wavelets. We derive
sharp oracle inequalities that prove the asymptotic optimality of the slope
heuristics---when the optimal penalty shape is known---and -fold
penalization. Furthermore, -fold cross-validation seems to be suboptimal for
a fixed value of since it recovers asymptotically the oracle learned from a
sample size equal to of the original amount of data. Our results are
based on genuine concentration inequalities for the true and empirical excess
risks that are of independent interest. We show in our experiments the good
behavior of the slope heuristics for the selection of linear wavelet models.
Furthermore, -fold cross-validation and -fold penalization have
comparable efficiency
Adaptive estimation in circular functional linear models
We consider the problem of estimating the slope parameter in circular
functional linear regression, where scalar responses Y1,...,Yn are modeled in
dependence of 1-periodic, second order stationary random functions X1,...,Xn.
We consider an orthogonal series estimator of the slope function, by replacing
the first m theoretical coefficients of its development in the trigonometric
basis by adequate estimators. Wepropose a model selection procedure for m in a
set of admissible values, by defining a contrast function minimized by our
estimator and a theoretical penalty function; this first step assumes the
degree of ill posedness to be known. Then we generalize the procedure to a
random set of admissible m's and a random penalty function. The resulting
estimator is completely data driven and reaches automatically what is known to
be the optimal minimax rate of convergence, in term of a general weighted
L2-risk. This means that we provide adaptive estimators of both the slope
function and its derivatives
Adaptive estimation of linear functionals in functional linear models
We consider the estimation of the value of a linear functional of the slope
parameter in functional linear regression, where scalar responses are modeled
in dependence of random functions. In Johannes and Schenk [2010] it has been
shown that a plug-in estimator based on dimension reduction and additional
thresholding can attain minimax optimal rates of convergence up to a constant.
However, this estimation procedure requires an optimal choice of a tuning
parameter with regard to certain characteristics of the slope function and the
covariance operator associated with the functional regressor. As these are
unknown in practice, we investigate a fully data-driven choice of the tuning
parameter based on a combination of model selection and Lepski's method, which
is inspired by the recent work of Goldenshluger and Lepski [2011]. The tuning
parameter is selected as the minimizer of a stochastic penalized contrast
function imitating Lepski's method among a random collection of admissible
values. We show that this adaptive procedure attains the lower bound for the
minimax risk up to a logarithmic factor over a wide range of classes of slope
functions and covariance operators. In particular, our theory covers point-wise
estimation as well as the estimation of local averages of the slope parameter
Sensor Selection to Improve Estimates of Particulate Matter Concentration from a Low-Cost Network
Deployment of low-cost sensors in the field is increasingly popular. However, each sensor requires on-site calibration to increase the accuracy of the measurements. We established a laboratory method, the Average Slope Method, to select sensors with similar response so that a single, on-site calibration for one sensor can be used for all other sensors. The laboratory method was performed with aerosolized salt. Based on linear regression, we calculated slopes for 100 particulate matter (PM) sensors, and 50% of the PM sensors fell within ±14% of the average slope. We then compared our Average Slope Method with an Individual Slope Method and concluded that our first method balanced convenience and precision for our application. Laboratory selection was tested in the field, where we deployed 40 PM sensors inside a heavy-manufacturing site at spatially optimal locations and performed a field calibration to calculate a slope for three PM sensors with a reference instrument at one location. The average slope was applied to all PM sensors for mass concentration calculations. The calculated percent differences in the field were similar to the laboratory results. Therefore, we established a method that reduces the time and cost associated with calibration of low-cost sensors in the field
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