Calculation Procedures for Data Snooping¶
Standardized Residuals¶
In data snooping, we test standardized residuals (Wi). For example, in HMK Stommätning 2021, normal distribution is assumed and values above 2 are marked as suspected gross errors.
\(|w_i|= \frac{|v_i|}{S(v_i)}=\frac{|v_i|}{S(l_i)\sqrt k_i}\)
Where:
* \(v_i\) : Residual (adjustment correction)
\(S(l_i)\) : Observation's assumed standard deviation (calculated from instrument)
\(k_i\) : Observation's controllability value (redundancy)
It is recommended to correct the measurement uncertainty for observations with \(S_0\) (standard deviation of unit weight). This is done by multiplying/reducing the assumed measurement uncertainty, after which the adjustment is repeated with more accurate assumptions.
\(S_{corrected}=S_0 S_{assumed} (l_i)\)
\(|w_i|= \frac{|v_i|}{S_0S(l_i) \sqrt k_i}\)
Relationship Between Adjustment Correction and Measurement Error¶
The following approximate relationship exists between an observation's adjustment correction and the underlying measurement error:
\(v_i \approx -k_i e_i\)
Where:
* \(v_i\) : Adjustment correction (residual)
\(k_i\) : k-value (controllability)
\(e_i\) : Measurement error
This gives us insight into the following situations:
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If \(k_i\) is large (approximately 1), gross errors will significantly affect the residuals and should be easy to find.
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If \(k_i\) is small (approximately 0), gross errors will have little effect on the residuals and will be difficult to find.
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If \(k_i=0\), the observation is not controllable.
Reliability¶
Reliability is an alternative to controllability for expressing a network's sensitivity to gross measurement errors.
Internal Reliability¶
A network where we can detect fairly small gross errors through data-snooping is said to have high internal reliability.
Minimal Detectable Bias (MDB) is calculated as:
\(MDB = {\frac{\delta_0} {\sqrt k_i}} S(l_i)\)
External Reliability¶
If the undetected gross errors only affect the adjustment result to a small degree, the network is said to have high external reliability.
\(ER = {\frac{\delta_0 (1-k_i)}{\sqrt{k_i}}} S(l_i)\)