Comments on John Rice’s paper

“On Detecting Periodicity in Astronomical Point Processes”

Jeff Scargle

This paper should be quite accessible to astronomers:

• Clear description of a well-defined, practical problem

• Does not go off into asymptotia

• Choices for the analyst not overwhelming

• Nice description of problem of unspecified alternative -- geometrical view: tests good in restricted sets of directions in parameter space

• Ideas have direct parallels with Bayesian methods [Bretthorst periodogram is similar to Rice’s power formula for the case of fundamental only (no harnomics; Rayleigh test)]

Sampling in frequency:

In periodogram/spectral analysis with evenly spaced data, one normally evaluates the power spectrum at the Fourier frequencies

fn = n / T n = 0, 1, 2, … (N/2) - 1

But often one is tempted to oversample, to avoid “missing a real peak”

For uneven sampling or for point data as Rice considers, the Nyquist frequency is not well defined and it is not obvious how many and which frequencies should be sampled. He proposes a novel and potentially extremely useful procedure to integrate over finite frequency bands.

One needs to carefully assess the statistical significance of any peak found this way. Rice’s analysis of this is a good beginning, but needs more development … examples, etc., the usual stuff astronomers need to feel good about a method.

Incorporating Frequency Drift

Simple extension of model to include frequency derivative:Power = P( f, df/dt )

Perhaps this can be interpreted as a time-frequency plot:change of variables, such as

( f , df/dt ) ? (f true , t ) where t = ( f true – f ) / df/dt

This would be an excellent tool for astronomers, for not just pulsars, but for tracking QPOs in accreting black hole/neutron star systems, etc.!:

• Photons in• P( f, t ) out• knob tunes time-frequency resolution trade-off

Alice Harding

Alice Harding

Bayesian Estimation of Time Series Lags and Structure , J. Scargle, in Bayesian Inference and Maximum Entropy in Science and Engineering, 2001. AIP Conference Proceedings, ed. Robert L. Fry. http://proceedings.aip.org/ (volume617),

Bayesian Time Series Tools

Periodogram sin( ωt + a ) Pr(ω) ~ exp( - P(ω) / σ2 ) Bretthorstperiod/phase

Cross-corr. f1(t), f

2( t – τ ) Pr(τ) ~ exp( - ρ(τ) / const ) Scargle

Time Lag

Scalegram s = scale Pr(s) ~ exp( - W(s) / const ) Scargle, Loredo

Time-Freq. Bayesianization of Rice’s frequency drift tracker?Distribution

Others ?