Delta-Chi2 Model Comparison

What it is

Several PQC event detectors (transient, step, DM-step, dip, bump, glitch) accept candidates using improvement in weighted fit:

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

Why PQC uses it

It provides a simple, fast statistic for ranking candidate event templates and applying practical acceptance thresholds.

Weighted least-squares context

With residuals \(y_i\), uncertainties \(\sigma_i\), and model \(m_i\), weights are \(w_i=1/\sigma_i^2\):

\[\chi^2_{\mathrm{model}} = \sum_i w_i (y_i - m_i)^2,\quad \chi^2_{\mathrm{null}} = \sum_i w_i y_i^2\]

Interpretation

  • larger \(\Delta\chi^2\) means stronger evidence for event model

  • threshold choice controls detector aggressiveness

Assumptions and caveats

  • assumes sigma values are meaningful relative weights

  • not a full Bayesian model comparison

  • scanning many candidate epochs/windows introduces a look-elsewhere effect, so absolute statistical significance is approximate

Small worked example

If \(\chi^2_{\mathrm{null}}=120\) and \(\chi^2_{\mathrm{model}}=92\), then \(\Delta\chi^2=28\). If detector threshold is 25, this candidate is accepted.

References

[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.

[LKH2005]

Lorimer, D. R., & Kramer, M. (2005). Handbook of Pulsar Astronomy. Cambridge University Press.