Many astrophysical phenomena are time-varying, in the sense that their brightness change over time. In the case of periodic stars, previous approaches assumed that changes in period, amplitude, and phase are well described by either parametric or piecewise-constant functions. With this paper, we introduce a new mathematical model for the description of the so-called modulated light curves, as found in periodic variable stars that exhibit smoothly time-varying parameters such as amplitude, frequency, and/or phase. Our model accounts for a smoothly time-varying trend, and a harmonic sum with smoothly time-varying weights. In this sense, our approach is flexible because it avoids restrictive assumptions (parametric or piecewise-constant) about the functional form of trend and amplitudes. We apply our methodology to the light curve of a pulsating RR Lyrae star characterised by the Blazhko effect. To estimate the time-varying parameters of our model, we develop a semi-parametric method for unequally spaced time series. The estimation of our time-varying curves translates into the estimation of time-invariant parameters that can be performed by ordinary least-squares, with the following two advantages: modeling and forecasting can be implemented in a parametric fashion, and we are able to cope with missing observations. To detect serial correlation in the residuals of our fitted model, we derive the mathematical definition of the spectral density for unequally spaced time series. The proposed method is designed to estimate smoothly time-varying trend and amplitudes, as well as the spectral density function of the errors. We provide simulation results and applications to real data.
翻译:许多天体物理现象在时间上变化不定,也就是说,它们的亮度会随时间变化。对于周期性恒星,先前的方法假设,周期性、振幅和阶段的变化是用参数性函数或片断式函数来描述的。有了本文件,我们引入了一个新的数学模型来描述所谓的调制光曲线,如在周期性变异恒星中发现的,这些恒星显示的时间变化参数,如振动、频率和/或阶段。我们模型计算了一个平稳时间变化的模拟趋势,以及一个调和数,并具有平稳时间变化的比重。在这个意义上,我们采用的方法是灵活的,因为它避免了对趋势和振动的功能形式的限制性假设(参数性或片断式)或片断式函数。我们采用了一种新的数学模型性模型性曲线来描述所谓的调调色光曲线,这些恒星显示的是振动时间变化的参数,我们为不均匀的时空时间序列制定了一种半偏移的测算方法。我们用时间变化的轨变数计算曲线来估算我们的时间变化最慢的时序变曲线,可以按照时间性变的轨变的变的变的轨变的曲线来计算。