Least Squares Adjustment: Linear and Nonlinear Weighted Regression Analysis

Allan Aasbjerg Nielsen

AbstractThis note primarily describes the mathematics of least squares regression analysis as it is often used in geodesy including land surveying and satellite based positioning applications. In these fields regression is often termed adjustment. The note also contains a couple of typical land surveying and satellite positioning application examples. In these application areas we are typically interested in the parameters of the model (often 2- or 3-D positions) and their uncertainties and not in predictive modelling which is often the main concern in other regression analysis applications.

Adjustment is often used to obtain estimates of relevant parameters in an over-determined system of equations which may arise from deliberately carrying out more measurements than actually needed to determine the set of desired parameters. An example may be the determination of a geographical position based on information from a number of Global Navigation Satellite System (GNSS) satellites also known as space vehicles (SV). It takes at least four SVs to determine the position (and the clock error) of a GNSS receiver. Often more than four SVs are used and we use adjustment to obtain a better estimate of the geographical position (and the clock error) and to obtain estimates of the uncertainty with which the position is determined.

Regression analysis is used in many other fields of application both in the natural, the technical and the social sciences. Examples may be curve fitting, calibration, establishing relationships between different variables in an experiment or in a survey, etc. Regression analysis is probably one the most used statistical techniques around. Dr. Anna B. O. Jensen provided insight and data for the Global Positioning System (GPS) example.

Matlab code and sections that are considered as either traditional land surveying material or as advanced material are typeset with smaller fonts.

Comments in general or on for example unavoidable typos, shortcomings and errors are most welcome.
TypeLecture note
Year2013    Month September    pp. 55
PublisherDTU Compute
AddressRichard Petersens Plads, Building 321
SeriesTechnical University of Denmark
Electronic version(s)[pdf]
BibTeX data [bibtex]
IMM Group(s)Image Analysis & Computer Graphics, Geoinformatics