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.. [Lomb1976] `Least-squares frequency analysis of unequally spaced data <http://adsabs.harvard.edu/abs/1976Ap%26SS..39..447L>`_
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.. [Scargle1982] `Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data <http://adsabs.harvard.edu/abs/1982ApJ...263..835S>`_
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.. [Barning1963] `The numerical analysis of the light-curve of 12 Lacertae <http://adsabs.harvard.edu/abs/1963BAN....17...22B>`_
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.. [Vanicek1971] `Further Development and Properties of the Spectral Analysis by Least-Squares <http://adsabs.harvard.edu/abs/1971Ap%26SS..12...10V>`_
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.. [VanderPlas2017] `'Understanding the Lomb-Scargle Periodogram <https://arxiv.org/abs/1703.09824>`_
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Installing
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----------
@@ -61,7 +48,7 @@ As long as you have ``scipy`` and ``numpy`` installed, you should be able to run
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``pip install ftperiodogram`` and everything should work fine.
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If this doesn't work, consult the instructions in ``CONDA_INSTALL.md`` for installing ``ftperiodogram`` and its dependencies with with
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Updates
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-------
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* See the `issues <https://github.com/PrincetonUniversity/FastTemplatePeriodogram/issues>`_
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section for known bugs! You can also submit bugs through this interface.
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* See the `issues <https://github.com/PrincetonUniversity/FastTemplatePeriodogram/issues>`_section for known bugs! You can also submit bugs through this interface.
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More information
@@ -101,8 +87,6 @@ multiband model that locked the phases, amplitudes and
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offsets of all bands together. They found that template fitting was significantly more accurate for estimating periods of RR Lyrae stars, but the computational resources
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needed for these fits were enormous (~30 minutes per object per CPU core).
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.. [Sesar2017] https://arxiv.org/abs/1611.08596
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How does the fast template periodogram improve things?
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(Hoffman *et al.* 2017, *in prep*), but the important part is you can reduce
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the non-linearity of the problem to the following:
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* Finding the zeros of an order ``6H-1`` complex polynomial at each trial frequency
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* This is done via the ``numpy.polynomial`` library, which performs singular-value decomposition on the polynomial "companion matrix", and scales as ``O(H^3)``.
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* Computing the coefficients of these polynomials for all trial frequencies simultaneously by leveraging the non-equispaced fast Fourier transform, a process that scales as ``O(HN_f log(HN_f))``.
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- Finding the zeros of an order ``6H-1`` complex polynomial at each trial frequency
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- This is done via the ``numpy.polynomial`` library, which performs singular-value decomposition on the polynomial "companion matrix", and scales as ``O(H^3)``.
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- Computing the coefficients of these polynomials for all trial frequencies simultaneously by leveraging the non-equispaced fast Fourier transform, a process that scales as ``O(HN_f log(HN_f))``.
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This provides two advantages:
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:Computational speed and scaling:
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:Improved computational speed and scaling:
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.. image:: plots/timing_vs_ndata_const_freq.png
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Speed comparison for a test case using a constant
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number of trial frequencies but varying the number
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of observations.
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:Numerically stable and accurate:
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.. image:: plots/correlation_with_nonlinopt.png
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Accuracy comparison between the fast template periodogram
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and a ``gatspy``-like method that uses the ``scipy.optimize.minimize``
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function to find the optimal phase shift parameter. The minimization
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method is given 10 random starting values and the best result is kept.
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Though in most cases the truly optimal solution is found, in many cases
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a sub-optimal solution is chosen instead (i.e. only a locally optimal
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solution was chosen).
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* The non-equispaced fast Fourier transform (NFFT)
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* Polynomial zero-finding
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The FTP is a non-linear extension of the GLS. The nonlinearity
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of the problem can be reduced to finding the zeros of
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a complex, order `6H-1` polynomial at each trial frequency.
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Templates are assumed to be well-approximated by a short truncated Fourier series
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of length `H`. Using this representation, the optimal parameters
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(amplitude, phase, offset) of the template fit at a given trial frequency
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can then be found *exactly* after finding the roots of
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a polynomial at each trial frequency.
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The coefficients of these polynomials involve sums that can be efficiently
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evaluated with non-equispaced fast Fourier transforms. These sums
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can be computed in `O(HN_f log(HN_f))` time.
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In its current state, the root-finding procedure is the rate limiting step.
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This unfortunately means that for now the fast template periodogram scales as
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`N_f*(H^3)`. We are working to reduce the computation time so that the entire
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procedure scales as `HN_f log(HN_f)` for reasonable values of `H` (`< 10`).
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However, even for small cases where `H=6` and `N_obs=10`, this procedure is
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about an order of magnitude faster than the `gatspy` template modeler.
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How is this different than the multi-harmonic periodogram?
### How is this different than the multi-harmonic periodogram?
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The multi-harmonic periodogram ([Bretthorst 1988](https://link.springer.com/book/10.1007%2F978-1-4684-9399-3), [Schwarzenberg-Czerny (1996)](http://iopscience.iop.org/article/10.1086/309985/meta)) is another
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The multi-harmonic periodogram ([Bretthorst1988]_,[SchwarzenbergCzerny1996]_) is another
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extension of Lomb-Scargle that fits a truncated Fourier series to the data
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at each trial frequency. This is nice if you have a strong non-sinusoidal signal
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and a large dataset. This algorithm can also be made to scale as
The FTP scales sub-linearly to linearly with the number of harmonics `H`
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for `H < 10`, and for larger number of harmonics scales as `H^3` (since
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zeros are found via singular value decomposition of the polynomial companion matrix).
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This limits the set of templates to those that are sufficiently approximated by a small
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number of Fourier terms.
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.. [Lomb1976] `Least-squares frequency analysis of unequally spaced data <http://adsabs.harvard.edu/abs/1976Ap%26SS..39..447L>`_
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### Accuracy
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.. [Scargle1982] `Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data <http://adsabs.harvard.edu/abs/1982ApJ...263..835S>`_
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Compared with the Gatspy template modeler, the FTP provides improved accuracy as well as speed.
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For many values of `p(freq)`, the FTP correlates strongly with results obtained from
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non-linear optimization. However, since the problem is not convex, the solution recovered from
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non-linear optimization techniques may only represent a *local* minima. FTP, on the other
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hand, solves for all local minima simultaneously, from which the globally optimal solution can be
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found easily.
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.. [Barning1963] `The numerical analysis of the light-curve of 12 Lacertae <http://adsabs.harvard.edu/abs/1963BAN....17...22B>`_
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.. [Vanicek1971] `Further Development and Properties of the Spectral Analysis by Least-Squares <http://adsabs.harvard.edu/abs/1971Ap%26SS..12...10V>`_
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.. [VanderPlas2017] `Understanding the Lomb-Scargle Periodogram <https://arxiv.org/abs/1703.09824>`_
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The FTP requires that templates be *approximated* by a truncated Fourier expansion. The figure
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below compares the template periodograms for a single template approximated by different numbers
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of harmonics:
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.. [Sesar2017] https://arxiv.org/abs/1611.08596
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