(c) 2016; John Hoffman, Jake Vanderplas

The Fast Template Periodogram extends the Generalized Lomb-Scargle periodogram (Zechmeister and Kurster 2009) for arbitrary (periodic) signal shapes. A template is first approximated by a truncated Fourier series of length H. The Non-equispaced Fast Fourier Transform NFFT is used to efficiently compute frequency-dependent sums.

The tempfit library is complete with API documentation and consistency checks using py.test.

See the CONDA_INSTALL.md for installing tempfit with conda) (recommended).

See the Examples.ipynb located in the notebooks/ directory.

To run this notebook, use the jupyter notebook command from inside the notebooks/ directory

$ cd notebooks/
$ jupyter notebook

• See the issues section for known bugs! You can also submit bugs through this interface.

The gatspy library has an implementation of both single and multiband template fitting, however this implementation uses non-linear least-squares fitting to compute the optimal parameters (amplitude, phase, constant offset) of the template fit at each frequency. That process scales as N_obs*N_f, where N is the number of observations and N_f is the number of frequencies at which to calculate the periodogram.

This process is extremely slow. Sesar et al. (2016) applied a similar template fitting procedure to multiband Pan-STARRS photometry and found that (1) template fitting was significantly more accurate for estimating periods of RR Lyrae stars, but that (2) it required a substantial amount of computational resources to perform these fits.

• The non-equispaced fast Fourier transform (NFFT)
• Polynomial zero-finding

The FTP is a non-linear extension of the GLS. The nonlinearity of the problem can be reduced to finding the zeros of an order ~6H polynomial at each trial frequency.

Templates must be well-approximated by a short truncated Fourier series of length H. Using this representation, the optimal parameters (amplitude, phase, offset) of the template fit at a given trial frequency can then be found exactly after finding the roots of a polynomial at each trial frequency.

The coefficients of these polynomials involve sums that can be efficiently computed with (non-equispaced) fast Fourier transforms. These sums can be computed in HN_f log(HN_f) time.

In its current state, the root-finding procedure is the rate limiting step. This unfortunately means that for now the fast template periodogram scales as N_f*(H^4). We are working to reduce the computation time so that the entire procedure scales as HN_f log(HN_f) for reasonable values of H (< 10).

However, even for small cases where H=6 and N_obs=10, this procedure is about twice as fast as the gatspy template modeler. And, the speedup over gatspy grows linearly with N_obs!

The multi-harmonic periodogram (Schwarzenberg-Czerny (1996)) is another extension of Lomb-Scargle that fits a truncated Fourier series to the data at each trial frequency. This is nice if you have a strong non-sinusoidal signal and a large dataset. This algorithm can also be made to scale as HN_f logHN_f (Palmer 2009).

However, the multi-harmonic periodogram is fundamentally different than template fitting. In template fitting, the relative amplitudes and phases of the Fourier series are fixed. In a multi-harmonic periodogram, the relative amplitudes and phases of the Fourier series are free parameters. These extra free parameters mean that (1) you need a larger number of observations N_obs to reach the same signal to noise, and (2) you are more likely to detect a multiple of the true frequency. For a discussion of this effect, possible remedies with Tikhonov regularization, and an illuminating review of periodograms in general, see Vanderplas et al. (2015).

The Fast Template Periodogram seems to do better than Gatspy for virtually all reasonable cases (reasonable meaning a small-ish number of harmonics are needed to accurately approximate the template, small-ish meaning less than about 10).

It may be surprising that FTP appears to scale as NH, instead of NH log NH, but that's because the NFFT is not the limiting factor (yet). Most of the computation time is spent calculating polynomial coefficients, and this computation scales as roughly NH^4.

The FTP scales sub-linearly to linearly with the number of harmonics H for H < 10, and for larger number of harmonics scales as H^4. This is the main limitation of FTP.

Compared with the Gatspy template modeler, the FTP provides improved accuracy as well as speed. For large values of p(freq), the FTP correlates strongly with the Gatspy template algorithm; however, since Gatspy uses non-linear function fitting (Levenberg-Marquardt), the predicted value for p(freq) may not be optimal if the data is poorly modeled by the template. FTP, on the other hand, solves for the optimal solution directly, and thus tends to find equally good or better solutions when p(freq) is small.

For some frequencies, the Gatspy modeler finds no improvement over a constant fit (p_gatspy(freq) = 0). However, for these frequencies, the FTP consistently finds better solutions.

At frequencies where the template models the data at least moderately well (p(freq) ~> 0.01), the Gatspy modeler and the FTP are in good agreement.

Assuming, then, that the FTP is indeed producing the "correct" periodogram, we can then ask how many harmonics we must use in order to achieve an estimate of the periodogram to a given accuracy.

• Multi-band extensions
• Speed improvements