Detector-to-Statistic Mapping

This page maps each PQC detector to its core statistical test(s), scoring rules, and key assumptions.

Summary table

Detailed mapping notes

Bad measurements (OU + FDR)

  1. Build innovation z-scores under OU correlation.

  2. Aggregate to day-level maxima of \(|z|\).

  3. Convert to p-values and apply BH-FDR.

This controls multiplicity across many tested days while accounting for short time correlation.

Robust outliers (MAD)

Uses robust standardized residuals:

\[z_i = 0.6745\frac{y_i-\mathrm{median}(y)}{\mathrm{MAD}}\]

Flags points with \(|z_i| \ge z_\mathrm{thresh}\).

Event detectors (\(\Delta\chi^2\) family)

Most event detectors compare a null and event model in weighted least squares:

\[\Delta\chi^2 = \chi^2_{\mathrm{null}} - \chi^2_{\mathrm{model}}\]

Accepted events exceed configured delta_chi2_thresh and then apply membership rules based on per-point model SNR (member_eta).

Frequency-dependent detectors

Several detectors support chromatic scaling:

\[m(f) \propto \frac{1}{f^\alpha}\]

where \(\alpha\) may be fixed or fitted in configured bounds. DM-step uses the physically motivated \(\alpha=2\).

Common caveats across detectors

  • sigma quality strongly impacts weighted statistics.

  • scanning many candidate epochs/windows induces look-elsewhere effects.

  • model mismatch can turn real structure into apparent outliers (or vice versa).

  • event precedence and overlap suppression settings affect final labels.

References

[BH1995]

Benjamini, Y., & Hochberg, Y. (1995). “Controlling the false discovery rate: a practical and powerful approach to multiple testing.” Journal of the Royal Statistical Society Series B, 57(1), 289-300.

[UO1930]

Uhlenbeck, G. E., & Ornstein, L. S. (1930). “On the theory of the Brownian motion.” Physical Review, 36, 823-841.

[Hampel1974]

Hampel, F. R. (1974). “The influence curve and its role in robust estimation.” Journal of the American Statistical Association, 69(346), 383-393.

[Rousseeuw1993]

Rousseeuw, P. J., & Croux, C. (1993). “Alternatives to the median absolute deviation.” Journal of the American Statistical Association, 88(424), 1273-1283.

[Edwards2006]

Edwards, R. T., Hobbs, G. B., & Manchester, R. N. (2006). “tempo2, a new pulsar timing package - II. The timing model and precision estimates.” MNRAS, 372(4), 1549-1574.