Local Polynomial Regression with Application on Lidar Measurements

Torgny Lindström

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology

ISBN 91-628-6194-8

This thesis deals with the problem of estimating a function or one of its derivatives from a set of measurements, mainly of a bivariate or spatial nature which is so common in environmental applications. In this work particular attention has been on the lidar (light detection and ranging) application which is a versatile technique for measurement of among other things atmospheric trace gases. In lidar measurements the information about the concentration is carried by the derivative of the mean-function.
The exclusive tool that is used for estimation of a function or its derivatives in this thesis is local polynomial regression. However, other nonparametric techniques might be possible to use and some of the results presented here are indeed of a more general nature. The thesis consists of four papers of which the first and the last are applied to lidar measurements. The other two papers are methodology based with the intention to contribute to and also improve the statistical evaluation of the lidar process.
In the first paper, Paper A, lidar measurements are considered by adopting a univariate model with a non-constant variance-function. Evaluation is based on local polynomial regression with automatically selected local bandwidths, both for the derivatives of the mean-function and for the variance-function. Paper B presents a method for estimation of spatial covariance fields. Estimation is based on nonparametric techniques and considers covariances as functions of the location with fixed displacements. Paper C considers the problem of selecting local bandwidth matrices for bivariate local polynomial regression. In this paper an automatic bandwidth selector, EBBSdep, is developed which allows for correlated errors. Also, a set of MATLAB files for bivariate local polynomial regression based on EBBSdep -selected bandwidth matrices is developed. Finally, in Paper D, the method in Paper C is used to construct estimates of 2-D concentration maps of atomic mercury from fields of lidar measurements.
Bivariate estimation; differential absorption; heteroscedasticity; lidar; local bandwidth selection; local polynomial regression; nonparametric; spatial dependence; variance-function estimation.