# Least Squares Theory

(This functionality is available with the Adjustment module)

The following includes extracts from a paper on "A review of Least Squares Theory applied to traverse adjustment" by Rod E. Deakin from the Department of Land Information, RMIT - Victoria University of Technology.

"Least Squares is an adjustment technique founded on well accepted principles of measurements and their errors and is regarded as superior to all other methods of adjustment."

"The variance of a measurement is its standard deviation squared."

"In any least squares adjustment it is assumed that the apriori estimates of the standard deviations are available" and are used to determine the variance matrix. The inverse of the apriori variance matrix "is commonly known as the weight matrix."

NOTE: The values in the standard deviation columns of the adjustment spread sheet are the apriori estimates that the system uses in its adjustment. They are initially based on the settings in the Field Transfer/Settings/Instrument command, but may be edited.

The variance factor (shown in the Adjustment report) is based on the assumption that the expected mean of the residuals is zero, that is, there is no bias in the observations. This is unlikely if additional constraints (see note below) are placed on the network or if the degrees of freedom (that is the number of redundant observations) is statistically small. However it is still the best estimate of "the relationship between the "true" measurement variances and the apriori estimates."

NOTE: Applying additional constraints will cause an increase in the variances "and an apparent loss of precision in the adjusted co-ordinates. Constraints should be carefully chosen so as not to distort the network or concentrate residuals along particular lines".

Example:

Apriori estimates

Distances 0.10m
Angles 10"

Variance factor computed as 6.55

This result indicates that better estimates of the true standard deviations would have been

Distances 0.26m (i.e. 0.10 * sqrt(6.55))
Angles 26" (i.e. 0.10 * sqrt(6.55))

Ideally the computed variance factor should be close to 1.0

A large variance factor may be due to:

1. undetected blunders in the observations
2. the apriori estimates of the standard deviations are incorrect
3. the constraints are tending to distort the adjustment.
4. the degrees of freedom is statistically small.

Apriori estimates used

The apriori estimates used by the system are based on the instrument settings. For each measurement they are modified using the standard propagation of errors (e.g. although generally minimal, the apriori estimate of standard deviation of a distance also includes a component based on the error in reading the vertical angle)

Apriori estimates of directions are also modified reflecting the number of pointing if rounds are read. Please note that the values of the standard deviations of the angles and distances from the rounds summary are NOT used for the computation of the priori estimates of directions, as they are generally derived from an extremely small sample of observations and are therefore an extremely biased value. They are often zero which is obviously not possible.

It is common to underestimate the precision of measurements. While the estimates of the standard deviation of the measurements themselves may be reasonable, users give no consideration to the fact that their estimates do not consider other errors like setting up errors, target placement error, errors in the actual control used etc. These errors can become very significant component of any observation.