A Fast Algorithm for Calculation of Th\^eo1

TL;DR

This paper introduces a faster algorithm for calculating Tho1, a frequency stability statistic, reducing computation time significantly while maintaining accuracy with proper numerical precision adjustments.

Contribution

A novel recurrence-based algorithm reduces Tho1 calculation complexity to that of the Allan variance, enabling faster processing of large datasets.

Findings

Computation time reduced by orders of magnitude.

Algorithm maintains accuracy with increased numerical precision.

Applicable to related statistics ThoBr and ThoH.

Abstract

Th\^eo1 is a frequency stability statistic which is similar to the Allan variance but can provide stability estimates at longer averaging factors and with higher confidence. However, the calculation of Th\^eo1 is significantly slower than the Allan variance, particularly for large data sets, due to a worse computational complexity. A faster algorithm for calculating the `all-$\tau$' version of Th\^eo1 is developed by identifying certain repeated sums and removing them with a recurrence relation. The new algorithm has a reduced computational complexity, equal to that of the Allan variance. Computation time is reduced by orders of magnitude for many datasets. The new, faster algorithm does introduce an error due to accumulated floating point errors in very large datasets. The error can be compensated for by increasing the numerical precision used at critical steps. The new algorithm can also be used to increase the speed of Th\^eoBr and Th\^eoH which are more sophisticated statistics derived from Th\^eo1.