Principle of least squares pdf

See book for derivation. In the weighted case, the weighted squares of the residuals must be minimized. Technically the weighted form shown assumes that the measurements are independent, but we can handle the general case involving covariance.

Then solve the two equations. These equations simplify to the following normal equations. Note: m is the number of observations and n is the number of unknowns. We will need the Jacobian matrix and a set of initial approximations. Example 2 - Continued Take partial derivatives and form the Jacobian matrix.

Add the corrections to get new approximations and repeat. Add the new corrections to get better approximations. Question: What about x-values?

Are they observations? Fitting a Parabola to a Set of Points. Need more than 3 points for a redundant solution.

Note that the answer is the same as that obtained with condition equations. Simple Method for Angular Closure Given a set of angles and associated variances and a misclosure, C, residuals can be computed by the following:.

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Explore Ebooks. Bestsellers Editors' Picks All Ebooks. This handbook summarises knowledge from experts and empirical studies. It provides guidelines that can be applied in fields such as economics, sociology, and psychology. Includes a comprehensive forecasting dictionary. The Sixth Edition of this influential best-selling book delivers the most up-to-date and comprehensive text and reference yet on the basis of the finite element method FEM for all engineers and mathematicians.

Since the appearance of the first edition 38 years ago, The Finite Element Method provides arguably the most authoritative. Practically every scholar who is concerned with the work of Francis Ysidro Edgeworth feels compelled to preface discussion with some sort of apologia or rationalization.

This tendency first surfaced in the context of an abortive attempt to get him elected to the British Royal Society, and things have not. General Motors Engineering Journal by Anonim. All observations have errors so any practical set of observations will not perfectly fit any chosen set of coordinates for the unknown points.

Some observations will be of a better quality than others. For example, an angle observed with a 1'' theodolite should be more precise than one observed with a 20'' instrument. The weight applied to an observation, and hence to its residual, is a function of the previously assessed quality of the observation.

In the above example the angle observed with a 1" theodolite would have a much greater weight than one observed with a 20" theodolite. How weights are calculated and used will be described later. The principle of least squares applied to surveying is that the sum of the squares of the weighted residuals must be a minimum. A locus line is the line that a point may lie on and may be defined by a single observation. Figure1 a , b and c show the locus lines associated with an angle observed at a known point to an unknown point, a distance measured between a known point and an unknown point and an angle observed at an unknown point between two known points respectively.

In each case the locus line is the dotted line. In each case all that can be concluded from the individual observation is that the unknown point lies somewhere on the dotted line, but not where it lies. In the following, the coordinates of new point P are to be determined from horizontal angles observed at known points A, B, C and D as in Figure 2 a.

Each observation may be thought of as defining a locus line. For example, if only the horizontal angle at A had been observed then all that could be said about P would be that it lies somewhere on the locus line from A towards P and there could be no solution for the coordinates of P. The two lines cross at a unique point and if the observations had been perfect then the unique point would be exactly at P.

But since observations are never perfect when the horizontal angles observed at C and D are added to the solution the four locus lines do not all cross at the same point and the mismatch gives a measure of the overall quality of the observations. Figure 2 b shows the detail at point P where the four lines intersect at six different points. The cross is at the unique point where the sum of the squares of the residuals is a minimum.

By far the easiest way to handle the enormous amounts of data associated with least squares estimation is to use matrix algebra. So, for example, in a two-dimensional network of 10 points where there are a total of 50 observations there would be a set of 50 simultaneous equations in 20 unknowns.