Statistics

All estimators take phase data (offsets, seconds) on a uniform grid. Irregular logs are resampled at their median interval τ₀. Grid points inside gaps longer than --max-gap·τ₀ are NaN, and every term touching them is dropped, so gaps are never bridged.

Kind	Name	Use
adev, oadev	Allan deviation (non-/overlapping)	frequency stability; OADEV preferred
mdev	modified Allan deviation	separates white and flicker PM
tdev	time deviation, τ·MDEV/√3	telecom/PTP time stability (ITU-T G.810)
hdev	overlapping Hadamard deviation	insensitive to linear frequency drift
totdev, mtot, ttot	total, modified total and time total deviation	reflection-extended; tighter at long τ; MTOT/TTOT bias-corrected per noise type as in Stable32 (--raw-mtot to disable)
htot	Hadamard total deviation	drift-insensitive like HDEV, tighter at long τ; bias-corrected per noise type (SP 1065 tables within 0.3 %)
theo1, theobr, theoh	Theo1, bias-removed TheoBR, hybrid TheoH	reach τ = 0.75 × record length
mtie, tierms	maximum and RMS time interval error	network time-error limits

Each estimator is tested against a literal implementation of its published formula (NIST SP 1065) and, where one exists, against analytic power-law results.

Long records: MTOT and the Theo family

MTOT and Theo1 cost O(N·m) per τ, and TheoBR's bias ratio averages about N/6 Theo1 values. They are computed with vectorised block operations, and when one τ would exceed a work limit (stability.MAX_WORK element operations) ntpstats averages every k-th subsequence, with k never larger than m. Adjacent subsequences share all but one sample, so this changes the estimate by well under 1 % in the tests. The TheoBR ratio uses 64 evenly spaced terms instead of all of them. A million-sample log takes a few seconds per statistic.

What was sampled is reported in the result: meta["stride"] per τ (1 = every subsequence) and meta["theobr_ratio_terms"] (used, total). compute(..., max_work=0) or ntpstats stability --exact always computes the full definitions.

Confidence intervals

Intervals are χ² intervals based on the equivalent degrees of freedom (EDF). For ADEV/OADEV/MDEV/TDEV/HDEV, ntpstats computes the EDF exactly for the discrete power-law noise model, from the estimator filter convolved with the Kasdin noise filter. This avoids the continuous-time approximation of Greenhall & Riley (2003), which overestimates the EDF by up to ~20 % at small averaging factors for sampled offsets. Monte Carlo tests check it for every noise type. TOTDEV uses the NIST table for FM noise.

Noise identification

The dominant power-law noise at each τ is identified with the lag-1 autocorrelation method of Riley & Greenhall (2004): white PM (α = 2), flicker PM (1), white FM (0), flicker FM (−1) and random-walk FM (−2). It selects the EDF model and is shown in every table.

Masks

--mask FILE checks one statistic against a limit curve: a CSV of tau,<kind> (e.g. tau,tdev), interpolated in log-log. It uses the upper confidence bound, prints the margin per τ and exits with code 3 on failure. No standards text is bundled; bring your own limits.

Dynamic stability

ntpstats dynamic (and the UI heat-map) computes a statistic over sliding windows, which exposes route changes, load cycles and temperature effects that a single curve averages away.

Noise model, spectra, hat and holdover

Fitting the power-law coefficients h_α, spectra, the N-cornered hat and holdover prediction are described in Metrology.