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written in Python-3.10.0
LBNP is the non-centrality parameter fulfilling the power function of F-distribution for the desired or used significance level (α), power of the test (γ), numerator degrees of freedom (h), and denominator degrees of freedom (r).
LBNP is iteratively computed with the program originally written in Python after you enter four input parameters:
Significance Level within the interval <0.00001-0.9999>
Power of the Test within the interval <Significance level-0.9999>
1st Degrees of Freedom within the interval <1-10000>
2nd Degrees of Freedom within the interval <1-1e8>
Remarks:
For the LBNP related to the chi-square distribution, please enter 1e8 for the 2nd degrees of freedom.
When the first degrees of freedom is much bigger than the second, the LBNP may take a value bigger than 10^5. Such an exceptional case may slow down the convergence speed. The upper limit for LBNP is 10^6. For bigger LBNPs, the program results in "out of capacity".
The implicit power function is defined as follows:
Power of the test=1-CDF(alpha, h,r, LBNP)
where CDF stands for the cumulative distribution function of non-central F-distribution. The solution of the above function for the non-centrality parameter yields the LBNP. This value is used in geodetic problems for two aims:
To optimize a geodetic network or experiment such that the predicted non-centrality parameter is greater than or equal to LBNP,
To figure out the minimal detectable bias (MDB), minimum detectable displacement (MDD), minimum detectable velocity (MDV), minimum detectable amplitude (MDA) (or any parameters to be tested with F-test) in the designed geodetic network or experiment.
Please refer to the following studies for further reading:
Aydin C (2012). Power of global test in deformation analysis, Journal of Surveying Engineering (ASCE), 10.1061/(asce)su.1943-5428.0000064
Aydin C and Demirel H (2005). Computation of Baarda's lower bound of the non-centrality parameter, Journal of Geodesy, 10.1007/s00190-004-0406-1
Aydin C and Gunes O (2024). Power function of F-distribution: Revisiting its computation and solution for geodetic studies, Journal of Geodesy, 98, 10.1007/s00190-024-01905-7