5.2 Variable Star Classification
5.2.1 Fourier Decomposition
Fourier decomposition makes use of the property of Fourier series. A Fourier series is a method of approximating any continuous univariate function as a sum of an infinite number of sine and cosine functions (Fourier, 1878). The amplitudes and phases of these sine and cosine components define the shape of the resulting Fourier series. A truncated Fourier series is a finite number of these infinite components which will result in an imperfect approximation with an accuracy depending on how many components are retained (Debosscher et al.,2007;Deb and Singh,2009;Richards et al.,2011b,2012;
Kim and Bailer-Jones,2016). For light curve modelling, the univariate time-series can be fit using regression by a truncated Fourier series with properties determined by the developer based on the quality and complexity of the light curves. The amplitudes and phases can be extracted from the regressed model and form a set of features that describe the shape of the light curve (Debosscher et al.,2007).
The method we utilise on the SkycamT light curves is a variant of the approach in- troduced into Machine-Learned periodic light curve classification by Debosscher et al (Debosscher et al., 2007) and improved with a regularisation technique by Richards et al. (Richards et al.,2011b,2012). To model periodic and multi-periodic light curves, a period estimation algorithm is applied to the light curve time-series data. Any efficient and reliable period estimation method may be chosen for Fourier decomposition as it only requires the real-valued period. Upon the determination of the candidate period, a harmonic model with a linear trend is fitted with this period across the time-series data. In order for the model to have the degree of freedom required to accurately fit the data, artificial data points were bound into the time-series. These artificial data points have a uniform distribution in time and a magnitude equal to the mean magnitude of the time-series. The artificial data points must be assigned zero weight as to not contribute
to the fitted model. As the weights are defined as the reciprocal of the magnitude error, the artificial data points are assigned a magnitude error of positive infinity. Weighted linear regression is performed on this new time-series to fit an m-harmonic sinusoid model using the period detected by the periodogram and is demonstrated by Equation
5.21. This model has ten coefficients in our method as we use m = 4 to sufficiently model the light curve.
y(t) = ct +
m
X
j=1
ajsin(2πjf1t) + bjcos(2πjf1t) + b0 (5.21)
Where b0 is the mean magnitude of the light curve and c is the linear trend of the time-
series. aj and bj are Fourier coefficients for the fitted model. f1 is the frequency from
the periodogram. This model is subtracted from the time-series in a process called pre- whitening to eliminate any periodic activity within the time-series based on the dominant period detected by the periodogram. This pre-whitened time-series is processed with another period estimation method to identify a second period independent of the first dominant period. A harmonic model is fitted for this period and subtracted off in a second pre-whitening phase. Finally, a third period is identified independent to the first two periods. The time-series is then restored to its original, archived prior to the pre-whitening operations. A harmonic best-fit is computed using weighted linear regression and a model with twenty six coefficients. This is shown in Equation 5.22
when n = 3. It is possible to perform additional pre-whitening operations for additional model complexity although overfitting is a concern.
y(t) = ct + n X i=1 m X j=1 aijsin(2πjfit) + bijcos(2πjfit) + b0 (5.22)
The linear trend of the time-series, calculated alongside the sinusoidal model is retained as a feature of the object. By calculating this linear trend in the same regression oper- ation as the sinusoidal models, a time-series with a non-integer number of wavelengths within the sampling period (with a corresponding trend) will not interfere with the linear trend caused by a gradual brightening or dimming of the object, an important feature. The frequencies fiand the coefficients aij and bij are retained to provide a good descrip-
tion of the light curve as long as it is periodic and accurately reproducible as a sum of sinusoids. These coefficients are not time-translation invariant and are transformed into better descriptors of the light curve. This was accomplished by transforming Fourier coefficients into a set of amplitudes Aij and phases P Hij and are determined according
to Equations5.23 and5.24 respectively (Debosscher et al.,2007). Aij =
q a2
Feature Extraction 160 P Hij = arctan bij aij (5.24) The phases are not time-translation invariant and are defined relative to P H11, the
phase of the first harmonic of the dominant period using Equation5.25. P Hij0 = arctan bij aij − jfi f1 arctan b11 a11 (5.25)
The phases were then constrained between −π to +π by the transformation as indicated in Equation 5.26. Please note that for simplicity, the double dash is dropped through the rest of the paper.
P Hij00 = arctan sinP Hij0 cosP Hij0 (5.26)
For light curves that are primarily monoperiodic, this results in the production of 28 fea- tures that are time-translation invariant allowing direct comparison between light curves measured at different phases (Debosscher et al., 2007). Monoperiodic light curves are those that oscillate with one dominant period (Aerts et al.,2006). This assumption does not hold for all potential variable stars but as the primary period is usually highly dom- inant in multi-period variables, this assumption is a good approximation (Debosscher et al.,2007). These features include the slope of the linear trend, the three frequencies used in the final harmonic model, the twelve amplitude coefficients and eleven phase coefficients (as P H11is always zero it is discarded) and the ratio of data variance (called
variance ratio) between the variance before the pre-whitening of the harmonic model of the primary period and after. This statistic is a strong indicator of the importance of the primary period to the light curve relative to the other periods.
The Fourier decomposition method is powerful but is sensitive to noise and outliers. Re- ducing the number of model parameters is a solution to preventing overfitting on noise but can also result in a loss of information on complicated astrophysical signals. An alternative method is the use of L2 regularisation (Richards et al., 2012). L2 regulari- sation is a technique applied to optimisation methods such as the harmonic regression used in fitting the Fourier models. It is a second term to the least squares minimisation used in the harmonic regression which places a penalisation on the higher harmonic amplitudes and phases. If the higher harmonic components attempt to fit on noise, the penalty will cause their parameters to asymptotically approach zero eliminating their effect. However, in the case of fitting a complicated signal with sufficient supporting evidence in the light curve, the higher harmonic parameters will resist the regularisation and maintain their contribution. Equation5.27demonstrates the optimisation function
Figure 5.2: The light curve of the star Mira with a regularised harmonic fit with a primary period of 332.57 days determined by the GRAPE method.
to be minimised in this regularised regression fitting.
R (θ, λ) = N X i=1 (di− mi)2 σi2 + N λ 4 X n=1 n4 A2n+ Bn2 (5.27)
Where θ is the model parameters, λ is the regularisation parameter, N is the number of light curve points, diare the photometric data points, mi are the model data points and
pA2
n+ B2nis the amplitude of the nth Fourier harmonic. The value of the regularisation
parameter allows the control of the smoothing of the model with small values allowing the modelling of high frequency structure and large values smooth this structure out. For the SkycamT light curves the regularised fit is applied with a regularisation parameter of 0.01 determined by manual inspection of the resulting models across a set of known variable light curves. In the event that the regularisation procedure fails, the light curves are fit with a non-regularised procedure at the risk of overfitting. The regularisation is performed using the normal equation shown in equation5.28.
θ =X>X + λW
−1
X>y (5.28)
Where X is the ‘design matrix’ of the problem, a matrix with the Fourier components in columns and the data points in rows, λ is the regularisation parameter, W is the regularisation weights determined from equation 5.27, y is the vector of data point
Feature Extraction 162
values and > is the transpose operator. This also provides an additional benefit of making the X>X + λW invertible as the normal equation cannot be solved if X>X is non-invertible, also known as a singular matrix.
Figure 5.2demonstrates the model produced using described method for the SkycamT data collected on the star Mira, the prototype of the Mira class variables showing a clear sinusoidal oscillation. The period of Mira has been widely reported as 332 days, verified by surveys such as Hipparcos (Bedding and Zijlstra,1998). In the event of the periodogram returning a result similar to the stars correct period, linear regression can produce an accurate model despite the prevalence of noise within the time-series data.