Least Squares

A Review of Least Squares Theory Applied to Traverse Adjustment
A Technical paper published in two parts in The Australian Surveyor, Vol. 36, No. 3, pp. 245-253 and Vol. 36, No. 4, pp. 281-290, 1991.

Combined Least Squares
Notes on the combined least squares adjustment model, including the derivation of equations, covariance matrices (propagation of covariances) and the connection with parametric least squares (21 pages)
Combined Least Squares.pdf

Constrained Least Squares
Notes on the addition of constraint equations in parametric least squares (7 pages).
Constrained Least Squares.pdf

Free Net Level Adjustment
Notes on the application of inner constraints to overcome datum deficiency problems in level network adjustments (6 pages).
Free Net Level Adjustment.pdf

Inner Constraints in Least Squares Adjustment
Notes on the development of inner constraint equations for 3D survey network adjustments (12 pages)
Inner Constraints.pdf

The Kalman Filter: A Look Behind The Scene
Paper presented at the Victorian Regional Survey Conference, Mildura, 23-25 June, 2006 (12 pages).
Kalman Filter Mildura Conference.pdf

Kalman Filter and Surveying Applications
Technical paper (30 pages) explaining the Kalman Filter with surveying application. Includes Matlab functions.
Kalman Filter and Surveying Applications.pdf

Least Squares and Kalman Filtering
Technical document covering Least Squares and Kalman Filtering from a surveying/geodesy perspective (91 pages). Notes include a concise explanation of Combined Least Squares with general formula for solutions of particular cases. Examples of different problems are given with solutions and MATLAB functions given in an Appendix. The usual least squares approach (minimizing sum of squares of weighted residuals) is extended Primary and Secondary (or dynamic) measurement models at discrete time intervals to obtain the Kalman Filter equations (the derivation of equations is set out). Examples are given (EDM measurement, Position and Velocity of a Ship in a Navigation channel, global warming: estimating trends from temperature anomalies) and MATLAB functions given in an Appendix.
Least Squares and Kalman Filtering.pdf

Level Network Adjustments
Notes (including worked examples) of different methods of adjusting a level network using least squares (21 pages)
Yarra Bend Level Net Adjustment Exercise.pdf

Linear Regression (Line of Best Fit using Least Squares)
Fitting a straight line through X,Y data is a common problem in estimation.  Using a data plot and a ruler, the problem is solved by slowly moving the ruler to a position that visually minimizes the perpendicular distances between the data points and the ruler.  A mathematical solution can be determined using the theory of least squares that supposes the most probable ‘answer’ is one that minimizes the sum of the weighted squares of residuals, where residuals are small corrections to the X,Y data and weights are numbers reflecting the precision of measurements.
York (1966, 1968) solved this problem for uncorrelated and correlated data, but unfortunately, his solution is not well known and scientists and engineers often use inappropriate methods embedded in software products and calculators.  This paper will show, in detail, how York solved this problem of estimation.
A MATLAB function and ASCII text data are also provided.
Fitting a straight line.pdf

Notes on Least Squares
Lecture notes on the topic of least squares with plane surveying applications (222 pages). These notes contain 10 chapters plus an appendix on matrix algebra.
Chapter 1.pdf
Chapter 2.pdf
Chapter 3.pdf
Chapter 4.pdf
Chapter 5.pdf
Chapter 6.pdf
Chapter 7.pdf
Chapter 8.pdf
Chapter 9.pdf
Chapter 10.pdf

Rank of a Matrix
Notes on matrix Rank including methods of elimination to determine matrix rank (8 pages).
Rank of a Matrix.pdf