Introduction to Least Squares

Least Squares – Original and Advancement

In history....original method

1795 : Karl Friederich Gauss & Adrien Marie Legendre (refer : Gauss, K.F. (1963). Theory of Motion of the Heavenly Bodies. Dover, New York

Advancements made to the original method
1905 : Kruger (refer : Kruger, L. (1905). Uber die Ausgleichung von bedingten Beobachtungen in zwei Gruppen, Postdam.

1930 : Tobey (refer : Tobey, W.M. (1930). The Differential Adjustment of Normally Distributed Observations. Geodetic Survey of Canada Pub. 27. Ottawa

1950 : Tienstra (refer : Tienstra, J.M. (1956). Theory of the Adjustment of Normally Distributed Observations. Argus, Amsterdam

- Tienstra explains that a least squares problem can be rigorously solved in “phases” – by treating the already estimated parameter and corrected observations of a previous phase as quasi-observation in the subsequent phase.

Advancement – Kalman Filtering
1960 : Kalman (refer : Kalman, R.E. (1960). A New Approach to Linear Filtering and Prediction. Journal of Basic Engineering, ASME, 82D

- extended to i) adding a second model – vector parameters to vary with time, ii) inserting covariance matrix in second model and iii) solving the problem in part

Advancement – Bayesian Procedure

1972 : Morrison ( refer : Morrison, N. (1972). Introduction to Sequential Smoothing and Prediction. McGraw-Hill Book Company, New York

Advancement – Sequential Procedure

1965 : Schmid & Schmid (refer : Schmid, H.H. & Schmid, E. (1965). A Generalized Least Squares Solution for a Hybrid Measuring System. The Canadian Surveyor, XIX, No. 1, Ottawas

- formulated with weight ranging from zero to infinity and rigorously solved “sequentially”. Sequential procedure achieved by updating the original estimate by a corrective term.

Advancement – Collocation Procedure

1969 & 1970 : Krarup ( refer : Krarup, L. (1969). A Contribution to the Mathematical Foundation of Physical Geodesy. Publication No. 44 of the Danish Geodetic Institute, Copenhagen & Krarup, L. (1970). The Method of Least Square Collocation. Studi Geophysica, No. 2, Roc. 14.

1972 : Moritz (refer : Moritz, H. (1972). Advanced Least Squares Method. Report of the Department id Geodetic Science, No. 75, The Ohio State University, Columbus.

Advancement – Stepwise & Sequential Procedure

1973 : Moritz (refer : Moritz, H. (1973). Stepwise and Sequential Collocation. Report of the Department of Geodetic Science. No. 203, The Ohio State University, Columbus