Create plot routine for extractor feets.extractors.ext_fourier_components.FourierComponents

[leliel12]

Create plot routine for extractor FourierComponents.

Path: feets.extractors.ext_fourier_components.py

Features

• Freq2_harmonics

• Freq3_harmonics

• Freq1_harmonics

Extractor Documentation

Periodic features extracted from light-curves using Lomb-Scargle (Richards et al., 2011)

Here, we adopt a model where the time series of the photometric magnitudes of variable stars is modeled as a superposition of sines and cosines:

y_i(t|f_i)=a_isin(2π**f_it)+b_icos(2π**f_it)+b_i, ∘

where a and b are normalization constants for the sinusoids of frequency f_i and b_i, ∘ is the magnitude offset.

To find periodic variations in the data, we fit the equation above by minimizing the sum of squares, which we denote χ²:

$$\chi^2 = \sum_k \frac{(d_k - y_i(t_k))^2}{\sigma_k^2}$$

where σ_k is the measurement uncertainty in data point d_k. We allow the mean to float, leading to more robust period estimates in the case where the periodic phase is not uniformly sampled; in these cases, the model light curve has a non-zero mean. This can be important when searching for periods on the order of the data span T_tot. Now, define

$$\chi^2_{\circ} = \sum_k \frac{(d_k - \mu)^2}{\sigma_k^2}$$

where μ is the weighted mean

$$\mu = \frac{\sum_k d_k / \sigma_k^2}{\sum_k 1/\sigma_k^2}$$

Then, the generalized Lomb-Scargle periodogram is:

$$P_f(f) = \frac{(N-1)}{2} \frac{\chi_{\circ}^2 - \chi_m^2(f)} {\chi_{\circ}^2}$$

where χ_m²(f) is χ² minimized with respect to a, b and b_∘.

Following Debosscher et al. (2007), we fit each light curve with a linear term plus a harmonic sum of sinusoids:

$$y(t) = ct + \sum_{i=1}^{3}\sum_{j=1}^{4} y_i(t|jf_i)$$

where each of the three test frequencies f_i is allowed to have four harmonics at frequencies f_i, j = j**f_i. The three test frequencies f_i are found iteratively, by successfully finding and removing periodic signal producing a peak in P_f(f) , where P_f(f) is the Lomb-Scargle periodogram as defined above.

Given a peak in P_f(f), we whiten the data with respect to that frequency by fitting away a model containing that frequency as well as components with frequencies at 2, 3, and 4 times that fundamental frequency (harmonics). Then, we subtract that model from the data, update χ_∘², and recalculate P_f(f) to find more periodic components.

Algorithm:

1. For i = 1, 2, 3
2. Calculate Lomb-Scargle periodogram P_f(f) for light curve.
3. Find peak in P_f(f), subtract that model from data.
4. Update χ_∘², return to Step 1.

Then, the features extracted are given as an amplitude and a phase:

$$A_{i,j} = \sqrt{a_{i,j}^2 + b_{i,j}^2}\\\ \textrm{PH}_{i,j} = \arctan(\frac{b_{i,j}}{a_{i,j}})$$

where A_i, j is the amplitude of the j − t**h harmonic of the i − t**h frequency component and PH_i, j is the phase component, which we then correct to a relative phase with respect to the phase of the first component:

PH′_i, j = PH_i, j − PH₀₀

and remapped to | − π, +π|

References