Local Polynomial Regression with Application on Lidar Measurements
Centre for Mathematical Sciences
Lund Institute of Technology
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.