Testing Intrinsic Mode Functions for Analising Red Noise

Testing Intrinsic Mode Functions for Analising Red Noise Julieta Paz S´ anchez Arias1, Suryani Guha1,2, Alejandra Christen3 1Astronomical Institute of the Czech Academy of Sciences, Friˇ cova 298, CZ-251 65 Ondˇ rejov, Czech Republic 2Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic. 3Instituto de Estad´ ıstica, Universidad de Valpara´ ıso, Valpara´ ıso, Chile. Motivation The frequency spectra of massive OB stars are known to be affected by red noise, which manifests as an excess of power at low frequencies. Proposed origins include surface convection and granulation, inhomogeneities in stellar winds, and stochastically excited internal gravity waves at subsurface convective zones [1],though the underlying physics remains poorly understood. The presence of coherent pulsation modes in the same low-frequency range complicates efforts to disentangle these phenomena. In this work, we introduce a novel mathematical approach for frequency analysis in astronomy: the Empirical Mode Decomposition (EMD). This method decomposes signals into Intrinsic Mode Functions (IMFs), which isolate oscillations across distinct frequency scales without assuming linearity or stationarity. We assess the ability of IMFs to separate noise from coherent oscillations in a controlled experiment with high and low frequencies and noise generated with an AutoRegressive Fractionally Integrated Moving Average (ARFIMA) model. We study the frequency spectra of the IMFs and the corresponding Autocorrelation Functions (ACF) to gain insights into their nature. We complement our study with the Weighted Wavelet Z-transform (WWZ) scalograms to examine further our selected IMFs to reproduce the signals and the ARFIMA noise. Synthetic light curve We designed an experiment with two frequencies at high range (5 and 5.5 d−1) and two at low range (0.16 and 0.3 d−1), mimicking coherent pulsations. 0 5 10 15 20 25 Time [d] 15 10 5 0 5 10 15 20 Normalized Flux signal+noise noise signal Figure 1: Synthetic light curve studied (in green), along with the ARFIMA noise, depicted in red and the four-frequencies signal is indicated in black. The noise component was created with an ARFIMA model, which corresponds to the Fourier behaviours known as 1/fα‘red noise’ where f is the frequency and αis the slope of a power law fit to the low-frequency Fourier power spectrum [2]. The general form of an ARFIMA model is: Φ(B)(1 −B)dXt= Θ(B)ϵt(1) where Xtis the time serie at time t;Bis the backward shift operator BXt=Xt−1.Φis the autoregressive (AR) polynomial of degree p;(1 −B)drepresents the fractional differencing operator and Θ(B)is the moving average (MA) polynomial of degree q. We adopted ARFIMA(1,0.4,1), with ϕ= 0.8and θ= 0.1, which represents a stationary, longmemory process with a strong dependency of the previous measurement and a slight correlation with the previous white noise value. The synthetic light curve, indicating a strong ARFIMA noise component, is shown in Figure 1. Intrinsic mode functions IMFs are the orthogonal functions into which a signal is decomposed using EMD, a data-driven technique capable of analysing non-linear and non-stationary signals. 4 3 2 1 0 1 2 3 4 Normalized Flux IMF 7 IMF 6 IMF 5 IMF 4 IMF 3 IMF 2 IMF 1 0 5 10 15 20 25 Time 4 2 0 2 4 Normalized Flux IMF 8 IMF 9 IMF 10 IMF 11 IMF 12 IMF 13 IMF 14 Figure 2: IMFs obtained for the synthetic light curve separated into high (upper panel) and low (lower panel) frequency oscillations. They are extracted through an iterative process known as ”sifting,” which progressively isolates these modes by identifying local extrema, constructing upper and lower envelopes, computing their mean, and subtracting it from the original signal until the result meets the criteria for an IMF. These criteria are 1. IMFs have exactly one zero between any two consecutive local extrema, and 2. have zero ”local mean”. This process is repeated on the residual signal to obtain subsequent IMFs, with the final residual often representing a long-term trend. We employed the Complete Ensemble EMD with Adaptive Noise (CEEMDAN) within the R package RLIBEEMD. This variant of the EMD enhances the stability and accuracy of the decomposition by adding white noise and averaging results across multiple realisations. Figure 2 shows the obtained IMFs for the synthetic light curve. Fourier Transform and Autocorrelation Functions We compared the First Fourier Transform (FFT) of the coherent signal and the ARFIMA noise with the one corresponding to the sum of different IMFs, with the aim of separating these components in the light curve. Note that the red-like noise introduces additional frequencies in the full signal. Our best result is shown in Figure 3. The sum of IMF 8, 12 and 13 most accurately reproduces the coherent signal, while the remaining components correspond to the ARFIMA noise. This configuration allows for the recovery of the coherent signals; however, it still contains information of ARFIMA noise frequencies below 0.1 d−1. The sum of the selected IMF for mimicking the ARFIMA noise, reproduces these low-frequencies, however the content of the frequency at 0.16 d−1is absorbed by the sum of the IMF representing the coherent signal. Studying the ACFs is a powerful tool for identifying the nature of a signal because they reveal how the signal correlates with itself at different time lags. The ACFs corresponding to ARFIMA noise (middle panel, in red) show a typical hyperbolic decay associated with this process. The ACF for the case of the ARFIMA-related IMFs, shows a substantial decrease in autocorrelation around lag 20, after which it resumes the typical ARFIMA trend near lag 40, suggesting that the IMFs do not fully capture the ARFIMA signal. The ACF of the full signal (in green) shows a strong influence of the ARFIMA noise component in the initial decay. However, from lag 20 onwards, the decay slope becomes steeper than in the case of ARFIMA noise alone, due to the presence of the periodic component. Both ACFs corresponding to the coherent signals (in black) are similar; however, the slower decay for high lags in the signal recovered with the IMFs suggest the absortion of a component from the ARFIMA noise. 012345 Frequency [days 1] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Power Signal only ARFIMA only Signal + ARFIMA Signal IMF ARFIMA IMF 0.0 0.1 0.2 0.3 0.4 Frequency [days 1] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Power 4.8 5.0 5.2 5.4 5.6 Frequency [days 1] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Power 0 20 40 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 0.0 0.2 0.4 0.6 0.8 1.0 ACF ACF IMFs Figure 3: Left panels: FFT corresponding to the full signal (in green), the coherent frequencies (in black solid line), the sum of the IMF representing the coherent signal (in black dashed line), the ARFIMA noise (in red solid line) and the sum of the IMF representing the ARFIMA noise (in red dashed line). Middle panels: the ACF vs. Lag for the coherent signal, ARFIMA noise and full signal in black, red and green, respectively. Right panels: the ACF corresponding to the sum of the IMF representing the coherent signal and the ARFIMA noise in black and red, respectively WWZ analysis We conducted an additional analysis of the WWZ scalograms corresponding to the full signal and its coherent and ARFIMA-noise components. We observe, in Figure 4, that the ARFIMA-noise washes the frequency at 0.3 d−1from the full signal. The selected IMFs to reproduce the coherent signal, recover the signal at 0.16 d−1and one at 0.3 d−1, although the latter exhibits very low intensity. The lowest frequencies present in the ARFIMA noise are also recovered by our selection of the IMFs. At higher frequencies, we observe a beating phenomenon pattern between the 5 and 5.5 d−1in the coherent component, which is washed by the ARFIMA noise in the full light curve. Our selection of the IMFs, partially poorly recovers this pattern; however, it keeps the strongest intensities at approximately these high frequencies. The scalogram of selected IMFs corresponding to the ARFIMA-noise component, has structures with strong intensity at 5 and 5.5 d−1, indicating periodic components still present in our selection. 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 Frequency [d 1] Original 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 Only Signals 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 Signals IMFs 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 ARFIMA Noise 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 ARFIMA Noise IMFs 0 5 10 15 20 25 Time [d] 0.1 0.2 0.3 0.4 0.5 Frequency [d 1] 0 5 10 15 20 25 Time [d] 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 Time [d] 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 Time [d] 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 Time [d] 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Power Figure 4: WWZ scalograms for different components of the signal. From left to right: the full signal, the coherent component, the IMF representing the coherent signal, the ARFIMA noise, and the IMF associated with the ARFIMA component.Upper pannels: frequency range from 4.5 to 6 d−1.Lower pannels: frequency range from 0 to 0.5 d−1. Conclusions The presence of ’red-like’ noise induces the appearance of new frequency components at lowfrequency ranges, which can overlap coherent pulsations in this range We could partially recover the signals from both the coherent signal and ARFIMA noise components at high and low frequencies, highlighting the potential of using the IMFs to distinguish between these components. Applications to real light curves should be based on the study of ACFs or more sophisticated statistical tests to distinguish the coherent signals from red-like noise. References [1] Dominic M. Bowman and Trevor Z. Dorn-Wallenstein. . A&A, 668:A134, December 2022. [2] Eric D. Feigelson, G. Jogesh Babu, and Gabriel A. Caceres. Frontiers in Physics, 6:80, 2018. This research was supported by the Grant Agency of the Czech Republic (GA ˇ CR (GA ˇ CR 25-17532S) and the European Union (Project 101183150-OCEANS) TASC9/KASC16 Workshop — from July 7 to 11, 2025 — ISTA, Austria