Calculation Procedures for Testing Known Points (Global Test)¶
After the observations have been examined for blunders, and statistically detectable gross errors have been removed from the observation register, the question is whether the observations can reveal constraints in the existing control network.
Statistical Test¶
For this purpose, a statistical test which we call Global Test in this presentation is used. It is a classic F-test, and can be expressed as follows:
- Null hypothesis: No constraint in existing control network.
- Alternative hypothesis: Constraint in the control network.
The null hypothesis is rejected if:
\(\frac{\frac {\Sigma pvv_{constraint}-\Sigma pvv_{free}}{f_{constraint}-f_{free}}}{\frac {\Sigma pvv_{free}}{f_{free}}}>F_{\alpha,f_{free},f_{constraint}}\)
Where:
* \(m_0\) : Number of redundancies in free adjustment
\(T_{\frac{\alpha}{2},f}\) : Number of redundancies in constrained adjustment
\(q_{DD}\) : Weighted error square sum in free adjustment
\(\Sigma pvv_{constraint}\) : Weighted error square sum in constrained adjustment
\(F_{\alpha,f_{free},f_{constraint}}\) : Statistical F-value at error probability and \(f_{free}\) and \(f_{constraint}\) degrees of freedom
Localization of Constraint¶
If the null hypothesis is rejected, i.e., if constraint in the control network is asserted, the question is where in the network the constraint is located. The localization can be done by automatically releasing control points one by one, and calculating the influence each control point has on the error sum of squares. The point that reduces the error sum of squares the most is the "main suspect," and we perform a new Global Test with this point free.
If the null hypothesis is rejected again, we release control points one by one again, and find the one which, together with the previous one, reduces the error sum of squares the most. This process continues until the null hypothesis can no longer be rejected. If the system is still overdetermined, it continues until one more point is released (overcompensation).
Technical Calculation¶
The technique of releasing control points means that the reduced normal equation from the constrained adjustment must be expanded with two columns, and the constant term column with two terms.
At the start of the global test, the unreduced normal equation terms for all points were calculated and stored. We retrieve these and place them in the correct position in the extra columns and terms. We then reduce the new columns and terms as usual, and the new weighted error sum of squares becomes the old sum minus the sum of squares of the new terms in the constant term column, \((a^2 + b^2)\).
The figure illustrates extra columns and terms when releasing a control point. Weighted error sum of squares pvv is improved by \(a^2 + b^2\).
Overcompensation and Fixing¶
Some of the released control points may still be retained if more control points than necessary have been released through overcompensation.
Released points must therefore be fixed one by one, and the increase in weighted error sum of squares must be calculated. The point that increases the error sum of squares the least becomes a candidate for fixing, and we perform a Global Test with this point fixed. This process continues as long as the null hypothesis passes the test.
The technique of fixing points again means that the point's columns in the normal equations and terms in the constant term column are zeroed out one by one, and subsequent columns are reduced again. (Only terms below the zeroed ones are reduced in the constant term column.)
In deformed networks, one may end up with a constraint-free system where no additional points can be fixed.
Calculation Procedures for Testing Changes¶
For each released point, the width of the confidence interval is calculated. Since the interval is symmetrical about the zero point, we focus on half the width \(D\).
\(D=m_0 \cdot \sqrt {q_{DD}} \cdot T_{\frac{\alpha}{2},f}\)
Where: * \(m_0\) : Estimated standard deviation of the unit weight * \(T_{\frac{\alpha}{2},f}\) : T-value with f degrees of freedom and error probability, e.g., 0.05 (5%) * \(q_{DD}\) : Weight coefficient for the distance between new and old position
\(m_0\) is calculated in the usual way from the weighted error sum of squares and the number of degrees of freedom.
\(T_{\frac{\alpha}{2},f}\) is defined by its degrees of freedom and error probability.
\(q_{DD}\) emerges through the covariance propagation law:
\(q_{DD} = f \cdot N^{-1} \cdot f^T = f \cdot (R^{-1} )^T \cdot R^{-1} \cdot f^T\)
Where: * \(f\) = [0,0,...-cosr,-sinr...0,0] * \(r\) = The directional angle from new to old position * \(q_{DD}\) = Inverted reduced normal equation matrix
Multiplying by \(R^{-1}\) corresponds to reducing. We can therefore use the technique as before with adding an extra column. For the first released point, we add a column as shown on the previous page. This is reduced as usual and multiplied by itself. For the second released point, we do the same, but the coefficients \(f\) are different. In this way, we calculate the weight coefficient \(q_{DD}\) for the distance between new and old position.
Assessment of Position Change¶
For all released points, position change (S) in multiples of its standard deviation is calculated.
This can be expressed as:
\(\frac{S}{m_0 \cdot \sqrt{q_{DD}}}\)
Where: * \(S\) is the position change
The point with the smallest quotient is first subjected to the test of changes. If the result of the test is that this point must be fixed with its original coordinates, the normal equations must be updated and new coordinates and their standard deviations recalculated before the process continues. If, on the other hand, the result is that new coordinates are preferred, new coordinate values can be adopted for all released points, and the Test of changes is finished.