Energy-Spectrum Analysis of Bitcoin

• The paper demonstrates that energy-spectrum analysis using ACE-EMD/Hilbert and Fourier PSD methods quantifies Bitcoin’s multiscale dynamics and regime-dependent features.
• It shows that adaptive decomposition and stringent stationarity conditioning reveal a power-law decay in energy distribution, highlighting a dominant trend-following component.
• The analysis recommends best practices in noise-assisted preprocessing and segmentation protocols to effectively address Bitcoin's nonstationary market behavior.

Energy-spectrum analysis of Bitcoin concerns the empirical quantification and interpretation of how variance or "energy" in the Bitcoin price or return series is distributed across temporal frequencies and scales. This approach leverages advanced time-frequency methodologies to dissect the highly nonstationary, regime-switching nature of Bitcoin markets and to reveal both transient and persistent dynamical features. Two leading frameworks—Adaptive Complementary Ensemble Empirical Mode Decomposition with Hilbert spectral analysis and Fourier-based power spectral density estimation under rigorous stationarity control—provide mutually illuminating yet methodologically distinct portraits of Bitcoin's spectral landscape (Leung et al., 2021, Tang et al., 6 Aug 2024).

1. Mathematical Foundations and Decomposition Methods

The energy-spectrum structure of Bitcoin time series is frequently probed through two complementary mathematical paradigms:

Multiscale Adaptive Decomposition. The ACE-EMD (Adaptive Complementary Ensemble Empirical Mode Decomposition) scheme decomposes the log-price series $x(t)$ into a sum of $n$ intrinsic mode functions (IMFs) $c_j(t)$, each representing price oscillations at a characteristic scale, and a residual $r_n(t)$: $x(t) = \sum_{j=1}^n c_j(t) + r_n(t)$ This fully data-driven procedure tailors to the time-varying volatility of Bitcoin by adaptively modulating additive noise injected at each sifting trial, with the magnitude controlled via a local amplitude spline derived from the signal's fastest IMF.

Stationarity and Power Spectrum. The Fourier-based approach hinges on the wide-sense stationarity (WSS) criterion. For a process $X_t$ to be WSS, mean and autocovariance must be invariant under time shifts. The Wiener–Khinchin theorem then guarantees that the (two-sided) power spectral density $S(f)$ and the autocorrelation $R(\tau)$ are Fourier pairs: $$S(f) = \int_{-\infty}^{\infty} R(\tau) e^{-i2\pi f\tau} \, d\tau$$ Empirical enforcement of WSS is achieved by local detrending and normalization—quantified via moving-mean and moving-standard-deviation windows (denoted $\Delta_1 t$ and $\Delta_2 t$)—before computation of the sample spectral density via FFT (Tang et al., 6 Aug 2024).

2. Data Preparation and Segmentation Protocols

High-fidelity energy-spectrum analysis mandates stationarity conditioning and careful data partitioning:

• Source Data: One-minute price-index data spanning from 2019–2023 for Bitcoin (USD) (Tang et al., 6 Aug 2024), and multi-year daily price series for decomposition studies (Leung et al., 2021).
• Volatility Regime Segmentation: Empirical regimes are identified based on market volatility, e.g., dividing the timeline into (1) pre-2021, (2) the 2021–mid-2022 high-volatility window, and (3) post-May 2022 (Tang et al., 6 Aug 2024).
• Detrending and Normalization: For returns, the mean is removed using a centered moving average over a calendar week ($\Delta_1 t = 10\,080$ min). The local variance is estimated over a secondary window ($\Delta_2 t$) varied between 10–60 min to optimize for local homogeneity (stationarity).
• Noise-Assisted Decomposition: In ACE-EMD, Gaussian white noise of time-varying amplitude is injected in positive/negative pairs per trial to achieve mode orthogonality and maximize scale separation.

3. Frequency-Energy Spectrum Extraction and Estimation

The extraction of an energy-frequency map proceeds differently in each framework:

IMF-wise Instantaneous Analysis. Each IMF $c_j(t)$ is subjected to the Hilbert transform to obtain its analytic signal and thus instantaneous amplitude $a_j(t)$ and phase $\theta_j(t)$: $$a_j(t) = \sqrt{c_j^2(t) + \mathcal{H}(c_j)^2(t)}, \quad \omega_j(t) = \frac{d\theta_j(t)}{dt}$$ The instantaneous energy is $E_j(t) = a_j^2(t)$, and the instantaneous spectrum is defined as: $$H(\omega, t) = \sum_{j=1}^n a_j(t)\, \delta(\omega - \omega_j(t))$$ Integration over time yields the marginal spectrum $h(\omega)$, which quantifies the cumulative energy at each frequency. Clustering in $(\log\omega, \log E)$ space sharply distinguishes the timescales inherent to the Bitcoin price process (Leung et al., 2021).

Fourier-based Power Spectral Density. After stationarity conditioning, the sample return series is transformed into the frequency domain, with energy at frequency $f$ proportional to $| \hat X(f) |^2$, and compared directly—via the Wiener–Khinchin theorem—to the Fourier transform of the sample autocorrelation function after smoothing (e.g., via Savitzky–Golay) to reduce high-frequency noise (Tang et al., 6 Aug 2024).

4. Empirical Results for Bitcoin Energy Spectra

Key spectral features for Bitcoin are as follows:

ACE-EMD/Hilbert Spectra (2016–2021, log-price):

• Five well-separated IMFs dominate, with central frequencies ($\bar f_j$) decreasing and associated energies ($\bar E_j$) increasing with mode index $j$.
• Quantitatively: $$\begin{array}{c|ccccc} j & 1 & 2 & 3 & 4 & 5 \\hline \bar f_j\ (\text{cycles/day}) & 0.33 & 0.10 & 0.028 & 0.0074 & 0.0019 \ \bar E_j & 0.00012 & 0.00095 & 0.0051 & 0.032 & 0.21 \ \end{array}$$
• A log-log regression reveals power-law decay: $\bar E \propto \bar f^{-\alpha}$, with spectral exponent $\alpha_{BTC}\approx1.207$ ($R^2\approx0.99$), slightly faster than canonical pink noise.
• Slowest modes (IMFs 3–5, multi-week to secular scale) retain $\sim70\%$ of total energy, reflecting a pronounced trend-following component.

Fourier PSD (2019–2023, 1-min returns):

• In high-volatility periods, stationarity is satisfied for normalization windows up to 60 min, with the power spectrum flat (exponent $\approx0$) at frequencies $f\lesssim 1/(30\, \text{min})$. Lower-volatility segments required much shorter normalization windows (10 min) for stationarity compliance.
• Across all segments, spectra lack pronounced line features; energy is concentrated at lowest frequencies once nonstationarities are removed (Tang et al., 6 Aug 2024).
• No quantitative $\alpha$ exponents or 1/$f^\beta$ scaling have been fitted for returns, but after standardization and smoothing, the flat spectrum implies negligible autocorrelation at timescales above a few minutes, consistent with a memoryless process.

5. Interpretations and Regime Dependencies

Multiple studies highlight regime-dependent dynamical structure in Bitcoin's energy spectrum:

• Regime Transitions: Bull-to-bear events are reflected in energy transfer from low-frequency (trend) modes into mid-frequency (oscillatory correction) modes; recovery phases display the reverse, with mid- and high-frequency modes (IMFs 3 and 1) leading (Leung et al., 2021).
• Spectral Similarity to Equities: Comparisons to the S&P 500 show that Bitcoin, while historically exhibiting higher $\alpha$ (steeper falloff), has seen its exponent trend downward in rolling windows, approaching SP500's $\alpha\approx0.89$ (Leung et al., 2021). In both markets, flat spectra at the return level dominate once stationarity is enforced (Tang et al., 6 Aug 2024).
• Temporal Inhomogeneity: Counter-intuitively, high-volatility periods for Bitcoin produce more homogeneous (i.e., WSS) return statistics over broader temporal windows than quieter regimes.

6. Recommendations and Future Directions for Spectral Analysis

Current evidence suggests several methodological best practices:

• Verify stationarity using explicit Wiener–Khinchin criteria for each segment, and report detrending/normalization window choices transparently (Tang et al., 6 Aug 2024).
• Apply adaptive, noise-assisted decompositions (ACE-EMD) for resolving nonstationary multiscale phenomena, and report instantaneous amplitude/frequency statistics via Hilbert methods (Leung et al., 2021).
• Use smoothing filters such as Savitzky–Golay to suppress artifacts in empirical PSDs while avoiding bias in estimated spectral slopes (Tang et al., 6 Aug 2024).
• Examine persistence of flat-spectrum behavior at ultra-high frequency (sub-minute) or multi-day aggregations; investigate the emergence of $1/f^\beta$ scaling at coarser temporal resolutions.
• Account for possible market microstructure effects—such as opening-hour spikes or lunch breaks—through targeted spectral line analysis after rigorous stationarity enforcement.

A plausible implication is that the Bitcoin price process, when locally detrended and normalized, approximates an efficient-market–type structure at high frequencies, but long-memory and trend phenomena are prominent at coarser scales, as revealed by the energy distribution across IMFs.

7. Comparative Summary

Method	Data Domain / Variable	Key Spectral Summary
ACE-EMD + Hilbert (Leung et al., 2021)	log-price (2016–2021)	Quantized energy clustering; power-law decay ($\alpha\approx1.2$); trend energy dominance
PSD under WSS (Tang et al., 6 Aug 2024)	normalized 1-min returns (2019–2023)	Flat low-frequency spectrum in all stationary segments; no line features; rapid decay of autocorrelation

In summary, energy-spectrum analysis of Bitcoin reveals rich, scale-dependent dynamical structure, punctuated by regime-dependent spectral signatures and dual timescale behavior—efficient-style randomness at high frequency, coexisting with nonlocal, trend-dominated energy at low frequencies.