A ‘Total’ Imputation Algorithm that Fills Gaps in Time Series Measurements for ADEV and Phase Noise Characterizations of Power-law Noise Models

In this paper, we introduce an algorithm that uses the ‘Total’ approach as a means of filling gaps between live measurements. The Total approach extends a T-length data run by a backward, inverted replica before and after it, which in essence triples the data length to 3T as in Fig. 1. This operation is a way of obtaining long-term ADEV(τ) beyond its τ = T/2 limit with better overall confidence [1]. Ostensibly, the extensions convert the linear convolution of ADEV(τ) to circular but with an additional nuance: Total extensions are iid, i.e., independent, identically distributed while furthermore preserving intrinsic clock noise models [2]. We aptly call the gap-filling algorithm a ‘Total imputer.’ It has been tested on sets of 513-point simulated data runs in which 150 points (30%) were removed halfway into the runs. In Monte Carlo trials of ADEV(τ) plots for each of five f
^α
, integer power-law noise models, i.e., where −2 ≤ α ≤ +2, the imputation algorithm proved successful in yielding consistent ADEV(τ) responses in recovering the 150-point gaps with 90% confidence between original and recovered data [3]. We provide well-developed Python code and example data that enables analysts to test the algorithm on data which they may have or wish to study with simulation on specific patterns of missing data.