Multiscale Complexity and Irreversibility of Non-Stationary Time Series in Commodity Futures Markets

1. Introduction

Commodity futures markets exemplify complex systems, yet the ways in which their price dynamics depend on observation scale have not been systematically characterized. Prices in these markets reflect a confluence of factors: supply-demand fundamentals, macroeconomic conditions, geopolitical risks, inventory fluctuations, and market microstructure. These forces operate across markedly different time horizons—intraday swings driven by algorithmic trading, seasonal agricultural cycles and OPEC adjustments spanning months, and multi-year shifts in infrastructure investment and global trade architecture. The variation across scales, combined with trend persistence and structural heterogeneity across commodities, tends to obscure the actual relationships and causal pathways among variables. Complex systems, as Waldrop observed, are defined less by their components than by the interactions among them and the emergent properties that follow—properties not inferable from isolated observations [1]. Commodity futures markets fit this description. A diverse set of participants—producers, consumers, speculators, arbitrageurs, hedgers—operate under different information sets, risk preferences, and time horizons. The resulting price dynamics are distinctly nonlinear: long-range correlations, volatility clustering, intermittent bursts, and asymmetric responses to information shocks all appear. These markets are fundamentally non-equilibrium systems. Prices are repeatedly pushed away from equilibrium by external shocks and internal feedback, then gradually return through intricate, multi-stage adjustment processes. How these adjustments vary with the chosen observation scale and with the structural attributes of different commodities is the central question this study addresses.

Two broad frameworks underlie much of the analysis in this area: fractal theory and information theory. Mandelbrot’s fractal theory [2] suggests that many natural and social phenomena exhibit self-similarity under scale changes—statistical regularities persist, aside from a scaling exponent, when data are coarsened or magnified. This relationship is often expressed as $F\left(s\right)\sim s^{\alpha }$, where $\alpha$ denotes the Hurst exponent, with values above 0.5 indicating persistence, below 0.5 anti-persistence, and near 0.5 uncorrelated behavior. A single exponent, however, seldom suffices for real-world data. Ecological, economic, and financial signals typically show different scaling behavior across small versus large fluctuations, a property known as multifractality. The multifractal spectrum, summarized by the generalized Hurst exponent $h\left(q\right)$ and the singularity spectrum $f\left(\alpha \right)$, captures the range of local scaling exponents present in the data [3,4,5]. One empirical pattern we explore below is that multifractality itself shifts with commodity type and with the time scale examined.

The introduction of detrended fluctuation analysis (DFA) marked a significant advance [6,7]. At each scale, a polynomial is fitted to the local trend and the residuals examined, thereby removing non-stationary trends that might otherwise distort correlation estimates. Kantelhardt and colleagues later generalized this to multifractal DFA (MFDFA) [8], opening the door to multifractal analysis of non-stationary series. Given that complex systems are typically composed of interacting components with correlated outputs, Podobnik and Stanley [9] introduced detrended cross-correlation analysis (DCCA) to quantify long-range cross-correlations between two time series. Zhou [10] then extended this to MF-DCCA, allowing for scale-dependent cross-multifractal analysis between signal pairs. Alternative formulations have also been proposed. Wang, Shang, and Ge [11] developed a moment-based approach that performs well with non-stationary signals. Recent work has pushed further: Wu et al. [12] incorporated deep learning into MF-DCCA estimation, improving robustness on short or noisy records; Stosic and Stosic [13] conducted a careful methodological assessment, clarifying the conditions under which MF-DCCA produces reliable output and highlighting common pitfalls. Other work has examined what affects DFA-type estimates: trends [14], non-stationarities, and nonlinear filters [15,16] have all been scrutinized. Wang, Shang, and Dong [17] showed that linear filters have little effect whereas nonlinear polynomial filters can shift results depending on their order. Extensions to moving-average and time-delayed formulations have also been developed [18,19,20]. Yet fractal methods, while powerful for correlation structure and scale invariance, are less suited to capturing other facets of complexity—ordinal patterns, informational hierarchies, or directional asymmetries. Information-theoretic tools fill some of these gaps.

Costa and co-authors [21,22] observed that conventional entropy measures like approximate entropy or sample entropy assign their highest values to white noise and their lowest to perfectly ordered sequences. Yet real complexity, intuitively, sits in between—order across multiple scales, fragile under randomization but absent in purely deterministic signals. This motivated the development of multiscale entropy (MSE), which computes sample entropy at progressively coarse-grained scales and draws out structural contributions at different temporal resolutions [23]. Humeau-Heurtier [24] provided a comprehensive review of the MSE algorithm and its numerous variants, systematically cataloging the methodological developments and their applications across diverse fields. Jamin and Humeau-Heurtier [25] further extended this line of inquiry by reviewing (multiscale) cross-entropy methods, which quantify the similarity or dissimilarity between two time series across multiple scales, offering complementary tools for analyzing coupled dynamical systems. Refinements followed: composite MSE (CMSE) and refined composite MSE (RCMSE) addressed biases and undefined values in short records. Zhou et al. [26] applied MSE-based tools to assess information dominance between innovative and traditional financial assets; Fu, Sun, and Liu [27] extended similar logic to carbon and new energy markets. Zhao et al. [28] employed transfer entropy networks to construct early warning systems for systemic risk in Chinese commodity markets, demonstrating that entropy-based information flow measures can effectively capture the propagation of risk across interconnected commodity sectors. Xue et al. [29] further applied transfer entropy causal networks to examine the nonlinear information transmission between global banking sectors and green markets, revealing that entropy-based causality measures provide valuable insights into cross-market interconnectedness during periods of financial stress. Lin and Liu [30] proposed a framework combining return-adaptive piecewise linear representation with attention-based neural networks to enhance stock investment performance, leveraging entropy-based pattern recognition to identify important trading points, illustrating the broader applicability of entropy methods beyond traditional time series analysis. A related development is permutation entropy, due to Bandt and Pompe [31], which relies on the relative ordering of successive observations rather than actual magnitudes. This makes the measure robust against noise, computationally light, and nearly preprocessing-free—properties well-suited to financial data. Zhao, Shang, and Wang [32] later adapted the framework to quantify information interactions on ordinal patterns of stock returns. Combining permutation entropy with multiscale coarse-graining gives multiscale permutation entropy (MSPE), which tracks how ordinal structure changes across scales—a dimension that amplitude-based entropy measures do not capture. Masoudi, Shahbazi, and Sharifi [33] recently compared multifractal and multiscale entropy approaches side by side, concluding that the two together characterize market complexity more fully than either alone. A complication is trend interference: trends can distort correlation estimates, introduce spurious dependencies, and amplify random noise in entropy computations, especially when series contain pronounced deterministic movements.

Several approaches have been proposed to address this issue. Among them, empirical mode decomposition (EMD) and Fourier-based filtering are widely used. Huang et al. [34] introduced EMD, an adaptive, data-driven procedure that separates a signal into a set of intrinsic mode functions (IMFs) and a residual trend. Wu and Huang [35] and Flandrin et al. [36] later elaborated on EMD’s filter-bank properties and its usefulness for denoising and detrending. Fourier-based alternatives also remain in wide use [37]. Integrated with multiscale entropy, these techniques yield EMD-RCMSE and FT-RCMSE frameworks that remove deterministic trends and uncover intrinsic complexity. Wang, Shang, Xia, and Shi [38] showed that EMD-based RCMSE substantially improves robustness and reliability. This point is particularly relevant here: commodity prices tend to carry strong long-term trends that would otherwise mask the complexity structure along the soybean processing chain (soybean → meal → oil). In a related development, Zhang, Wang, and Li [39] introduced a multiscale detrended cross-correlation coefficient that provides a standardized measure of coupling across scales, complementing our analytical approach. Dolfin et al. [40], meanwhile, examined investor behavior through the lens of multiscale cross-correlations and identified regime shifts in global financial markets.

Time series irreversibility introduces a further dimension. The question is whether reversing a time series leaves its statistical properties unchanged. If not, the series exhibits directional asymmetry. Irreversibility typically signals non-equilibrium dynamics—systems driven externally or subject to internal dissipation. Porat and Friedland showed that irreversibility can be captured through asymmetry in the distribution of increments, offering a way to detect directional information flow and non-conservative forces. In financial settings, irreversibility has been linked to leverage effects, volatility asymmetries, and the uneven impact of positive versus negative news—all features that symmetric equilibrium models cannot reproduce. Zhao, Shang, and Wang [41] developed a framework to measure the asymmetric contributions of subsystems, finding that uneven contribution patterns reveal something about the system’s internal organization. The degree of irreversibility is not static; it shifts with market conditions, the chosen investment horizon, and the type of shocks involved. Major exogenous shocks, in particular, tend to leave structural breaks that push the system into new, non-stationary regimes. Zanin and Papo [42] recently surveyed methods for assessing multiscale irreversibility, while Zhang, Li, and Wang [43] proposed a trend-pattern-length approach that complements amplitude-based measures. The key question for our purposes is whether directional asymmetry depends on scale—and if so, whether a characteristic time scale exists.

Structural break detection adds yet another layer. Abrupt shifts in the data-generating process often point to regime changes driven by exogenous shocks or internal transitions. One segmentation method, based on Jensen–Shannon (JS) divergence, offers a non-parametric way to locate change points by maximizing distributional divergence across adjacent windows. Wang, Shang, Lin, and Chen [44] introduced segmented inner composition alignment to detect coupling patterns in complex systems, even when these patterns vary over time. For commodity markets, this method is particularly relevant: geopolitical events—wars, sanctions, supply disruptions, trade disputes—can trigger sudden and long-lasting changes in price behavior and cross-market linkages. Mapping detected breaks to external events builds an empirical bridge between geopolitical risk and market dynamics; this mapping is validated later by aligning breakpoints with the GPR index.

While each of these four methodological streams has been applied individually to financial data, their integration within a single analytical framework—and, more importantly, whether they converge on a common set of empirical regularities—has received little attention. We bring these threads—multifractal cross-correlation analysis, entropy-based complexity measures, irreversibility diagnostics, and structural break detection—together in a single framework. The aim is to move beyond piecewise characterizations and toward a more integrated view of how commodity futures markets behave as multi-scale systems. Three questions structure the empirical work: (i) How does cross-market coupling change across time scales, and what explains the differences across commodities? (ii) Does directional asymmetry shift systematically with the investment horizon, and if so, at what characteristic time scale? (iii) Do geopolitical shocks induce persistent structural changes beyond short-lived volatility spikes, and does their persistence depend on the depth of systemic penetration—that is, the extent to which a shock propagates beyond the market in which it originates?

The findings suggest a tripartite pattern: supply elasticity shapes the basic form of scale dependence; physical logistics rhythms anchor directional reversal at roughly 20 days; and systemic penetration depth determines how long structural reconstruction lasts. These findings bear directly on cross-commodity allocation, multi-horizon risk management, and geopolitical scenario planning.

To this end, we use MF-DCCA to measure scale-dependent cross-correlations between WTI crude oil, the US dollar index, and US–China soybean prices—to examine whether correlation strength and multifractal breadth vary with horizon and whether supply-side attributes (concentration versus substitutability) drive these variations. We apply the PG irreversibility index across multiple scales to test whether sign reversal occurs, which would signal a shift in the dominant direction of price adjustment. We detect structural breaks in WTI prices via JS divergence and align them with GPR-indexed events to test the connection between geopolitical risk and structural market change. We also apply MSWPE to the soybean crushing chain to test whether downstream products show lower complexity than upstream raw materials—a pattern that would clarify how shocks propagate along production chains and offer indirect evidence on systemic penetration depth.

The rest of this paper is structured as follows. Section 2 lays out the methodological framework. Section 3 describes the data and preprocessing steps. Section 4 presents the empirical findings. Section 5 discusses their implications and limitations. Section 6 concludes.

2. Data and Methods

2.1. Data and Preprocessing

The daily data cover the period from January 2, 2018 to October 1, 2025, with a total of 2,022 trading days. Table 1 lists the nine variables used in our empirical analysis, grouped into three categories: macro-financial indicators, commodity futures prices, and the geopolitical risk index employed for event interpretation. The GPR Index, constructed by Caldara and Iacoviello [45], measures geopolitical tensions by counting the number of newspaper articles covering adverse geopolitical events. This index is commonly used in empirical finance to capture exogenous political shocks to commodity markets.

All price series are converted into log-returns:

$$r_t=ln(P_t/P_{t-1}),$$

where $P_t$ is the closing price on day t. For structural break analysis, in addition to log-returns, raw prices and log-prices are also used directly to capture level shifts. Missing values are forward-filled, yielding a final sample of 2,022 daily return observations. Augmented Dickey–Fuller (ADF) tests confirm that all log-return series are stationary at the 1% significance level, confirming the suitability of the subsequent time-series methodologies.

2.2. Multifractal Detrended Cross-Correlation Analysis (MF-DCCA)

We employ MF-DCCA, an extension of detrended fluctuation analysis (DFA), to capture scale-dependent cross-correlations between energy and agricultural markets and to quantify the multifractal properties of cross-correlations between two non-stationary series [10]. Unlike conventional correlation measures that assume linearity and stationarity, MF-DCCA accommodates non-stationary trends and distinguishes between cross-correlations of small versus large fluctuations via the generalized Hurst exponent $h_{xy}\left(q\right)$. This enables us to test whether cross-market linkages differ between turbulent periods (captured by $q>0$) and relatively quiet periods ($q<0$).

The computational procedure is as follows. Given two series $\left\{x_i\right\}$ and $\left\{y_i\right\}$, each of length N, we first construct their integrated profiles:

$$X\left(i\right)={\displaystyle \sum _{k=1}^i}(x_k-\overline{x}),Y\left(i\right)={\displaystyle \sum _{k=1}^i}(y_k-\overline{y}),i=1,2,\dots ,N$$

Each profile is divided into $N_s=\lfloor N/s\rfloor$ non-overlapping segments of length s. For each segment v, a polynomial of order m is fitted to the local trend and subtracted. Following Kantelhardt et al. [8] and standard practice in the literature, we set $m=2$. A second-order polynomial removes both linear and quadratic trends without overfitting local fluctuations. The detrended covariance is computed as:

$$F^2(s,v)={\displaystyle \frac{1}{s}}{\displaystyle \sum _{j=1}^s}[X_v\left(j\right)-{\tilde{X}}_v\left(j\right)][Y_v\left(j\right)-{\tilde{Y}}_v\left(j\right)]$$

where ${\tilde{X}}_v\left(j\right)$ and ${\tilde{Y}}_v\left(j\right)$ are the fitted polynomial trends. The q-th order fluctuation function is then:

$$F_q\left(s\right)={\left[{\displaystyle \frac{1}{2N_s}}{\displaystyle \sum _{v=1}^{2N_s}}{\left(F^2(s,v)\right)}^{q/2}\right]}^{1/q},q\ne 0$$

If $F_q\left(s\right)$ and s satisfy the power-law relation $F_q\left(s\right)\sim s^{h_{xy}\left(q\right)}$, then $h_{xy}\left(q\right)$ is the generalized cross-Hurst exponent. When $q=2$, this reduces to the standard Hurst exponent H: 0.5 indicates uncorrelated behavior, values above 0.5 indicate persistence, and values below 0.5 indicate anti-persistence. The multifractal spectrum width $\Delta h=h_{xy}\left(q_{min}\right)-h_{xy}\left(q_{max}\right)$ measures the degree of multifractality—larger values indicate richer multifractal structure. The singularity spectrum $f\left(\alpha \right)$ is obtained via the Legendre transform:

$$\alpha =h\left(q\right)+q{h}^{\prime }\left(q\right),f\left(\alpha \right)=q[\alpha -h\left(q\right)]+1$$

We compute MF-DCCA over six scales: $s\in \{2,5,10,20,40,60\}$ days, covering intra-week, monthly, and quarterly investment horizons. The fluctuation order is set to $q\in [-5,-4,\dots ,4,5]$, a range commonly adopted in the multifractal literature [8,10,20]. This range captures the asymptotic scaling behavior without entering the unstable region associated with extreme q values.

2.3. Porat–Friedland (PG) Irreversibility Index

Time irreversibility refers to whether the statistical properties of a time series remain invariant under time reversal. When they do not, the system is out of equilibrium, typically indicating dissipative dynamics [46]. In financial markets, time irreversibility manifests as asymmetric price responses to upward versus downward movements. This asymmetry is commonly attributed to the leverage effect—the negative correlation between volatility and returns—and to the differential impact of positive versus negative news on volatility.

We use the PG index to quantify this asymmetry because it is structurally simple, interpretable, and captures both directional bias and magnitude asymmetry in a single scalar.

Given a sequence $\left\{z_t\right\}$, define the difference ${\Delta }_t=z_t-z_{t-1}$. Let $p_{+}$ and $p_{-}$ denote the proportions of positive and negative differences, and ${\mu }_{+}$ and ${\mu }_{-}$ their respective means. The PG index is defined as:

$$PG={\displaystyle \frac{1}{2}}\left[p_{+}log\left({\displaystyle \frac{p_{+}}{p_{-}}}\right)+{\displaystyle \frac{{\mu }_{+}-{\mu }_{-}}{{\mu }_{+}+{\mu }_{-}}}\right]$$

The first term captures directional bias; the second captures magnitude asymmetry. Positive PG values indicate that upward movements are faster or more frequent than downward movements; negative values indicate the opposite.

To examine whether irreversibility varies with scale, we compute the PG index separately at each scale $s\in \{2,5,10,20,40,60\}$. The coarse-graining procedure follows the same logic as in multiscale entropy analysis [21]:

$$z_j^{\left(s\right)}={\displaystyle \frac{1}{s}}{\displaystyle \sum _{i=(j-1)s+1}^{js}}{z}_i$$

where $z^{\left(s\right)}$ is the coarse-grained series at scale s. We examine whether the direction of irreversibility—and hence the dominant forces driving price dynamics—changes with the observation horizon, a question that has not been systematically examined in commodity markets.

2.4. Multiscale Weighted Permutation Entropy (MSWPE)

Permutation entropy measures complexity by mapping consecutive values to ordinal patterns. The method is robust, computationally efficient, and insensitive to amplitude noise [31]. Weighted permutation entropy extends the standard version by incorporating amplitude information through a variance-based weighting mechanism. This reduces sensitivity to observational noise and enhances robustness, making it particularly suitable for futures return series, which are known to exhibit heavy-tailed features [47].

For an embedding dimension d and delay $\tau =1$, each vector $[z_t,z_{t+\tau },\dots ,z_{t+(d-1)\tau }]$ is mapped to a permutation $\pi$ of order d based on the rank order of its elements. For each pattern ${\pi }_i$, the weighted relative frequency is:

$$p\left({\pi }_i\right)={\displaystyle \frac{{\sum }_t{w}_t·{\mathbf{1}}_{patter{n}_t={\pi }_i}}{{\sum }_t{w}_t}}$$

where the weight $w_t$ is the variance of the embedded vector:

$$w_t={\displaystyle \frac{1}{d}}{\displaystyle \sum _{j=1}^d}{(z_{t+(j-1)\tau }-{\overline{z}}_t)}^2,$$

with ${\overline{z}}_t$ denoting the mean of the embedded vector. The weighted permutation entropy is then:

$$H_w\left(d\right)=-{\displaystyle \sum _{i=1}^{d!}}p\left({\pi }_i\right){log}_{2}p\left({\pi }_i\right)$$

We set $d=3$, which yields $d!=6$ possible ordinal patterns. This choice is motivated by two considerations. First, the sample size $N=2022$ is far larger than $d!$, satisfying the statistical requirement $N\gg d!$ [31]. Second, this value provides a reasonable balance between pattern richness and estimation reliability. Higher entropy values indicate greater irregularity and lower predictability. We compute MSWPE across the same set of scales $s\in \{2,5,10,20,40,60\}$, ensuring consistency with the MF-DCCA and PG analyses.

2.5. Jensen–Shannon Divergence for Structural Break Detection

The Jensen–Shannon divergence measures the similarity between two probability distributions P and Q [48]:

$$JS\left(P\right|\left|Q\right)={\displaystyle \frac{1}{2}}{D}_{KL}\left(P\right|\left|M\right)+{\displaystyle \frac{1}{2}}{D}_{KL}\left(Q\right|\left|M\right)$$

where $M={\displaystyle \frac{1}{2}}(P+Q)$ and $D_{KL}$ is the Kullback–Leibler divergence. $JS\in [0,ln2]$, with $JS=0$ indicating identical distributions.

For structural break detection, we employ a sliding window of width $w=120$ trading days—approximately six months. This window length balances two requirements: sufficient observations for reliable distribution estimation and reasonable timeliness in break detection. For each position t, we compare the empirical distributions of the left segment $[t-w,t]$ and the right segment $[t,t+w]$. The distributions are estimated via histograms with 10 bins, with boundaries set at the 5th and 95th percentiles of the pooled data. A breakpoint is identified when the JS divergence exceeds the threshold $\theta =0.3$:

$$\mathrm{Break}\mathrm{at}t\mathrm{if}JS\left(P_{[t-w,t]}\right|\left|P_{[t,t+w]}\right)>\theta .$$

The threshold $\theta =0.3$ is chosen based on the empirical distribution of JS values across the sample, corresponding to the 95th percentile of the null distribution estimated from 1,000 simulated random walks of the same length. This calibration is consistent with the non-parametric nature of the method and imposes no distributional assumptions on the underlying data.

2.6. Rolling-Window Time-Varying Analysis

To examine whether the relationships captured by MF-DCCA and PG change over time, we implement a rolling-window framework with window width $W=252$ trading days—approximately one year—and step size $L=21$ trading days—approximately one month. In each window, we compute the MF-DCCA spectrum width $\Delta h_s\left(t\right)$ at scale $s=5$ days, and the PG index $P{G}_s\left(t\right)$ at scales $s\in \{2,5,10\}$ days.

This approach captures the temporal heterogeneity of market dynamics and enables systematic mapping to external shocks. We align the resulting trajectories with the timeline of major events—the 2018 US–China trade war, the 2020 negative oil price event, the 2022 Russia–Ukraine conflict, and the 2023 Red Sea crisis—enabling direct observation of how exogenous geopolitical events reshape cross-market correlations and directional asymmetry. All computations are implemented in Python, with NumPy, SciPy, and statsmodels handling the numerical and statistical work.

The key parameters and interpretation guidelines of the four methods described above are summarized in Table 2.

3. Results

3.1. Scale-Dependent Multifractal Cross-Correlations

We begin by examining how the cross-correlation structure between energy and financial markets evolves across different observation scales. The spectrum width $\Delta h$ expands steadily from 0.139 at the 2-day scale to 0.602 at the 20-day scale—an increase of more than fourfold.

At short horizons (2–5 days), the spectrum width remains relatively narrow ($\Delta h\approx 0.14$–0.16), indicating a pronounced monofractal signature. Short-term markets are predominantly driven by relatively homogeneous factors—speculative flows, algorithmic trading, and microstructure frictions—that respond to small and large fluctuations in a broadly similar manner. At the 10-day and 20-day scales, however, the spectrum width jumps to 0.530 and 0.602, more than tripling from the short-horizon level. This sharp increase signals a transition to a multifractal regime: the WTI–dollar correlation is no longer independent of fluctuation amplitude, but varies systematically with the scale of price movements.

The source of this widening lies in the marked divergence of the generalized Hurst exponents across small and large fluctuation regimes. For small fluctuations ($q=-5$), $h_{xy}(-5)=0.852$, close to 1, indicating strong trend persistence. For large fluctuations ($q=5$), $h_{xy}\left(5\right)=0.250$, well below 0.5 and approaching anti-persistence, implying rapid reversal after large shocks. This differentiation points to a consistent pattern: at the monthly scale, WTI–dollar co-movements exhibit significant fluctuation-size asymmetry—small fluctuations drive gradual trend accumulation, while large fluctuations trigger mean-reverting corrections. This asymmetric behavior constitutes the nonlinear foundation of medium- to long-term cross-market risk transmission.

This scale-dependent amplification reflects the differential penetration of macro-financial factors across investment horizons. At the monthly scale, monetary policy expectations, global liquidity cycles, and long-term supply–demand trends become dominant. The asymmetric response of the dollar to oil price shocks—the "commodity currency" effect—further enriches the multifractal structure. In contrast, the near-monofractal state at short horizons persists because price fluctuations are predominantly governed by inventory surprises, technical patterns, and daily positioning adjustments—factors that respond uniformly across fluctuation sizes.

The contrasting results for the cross-market soybean pair (US soybeans vs. Chinese soybeans) provide a powerful counterpoint. The spectrum width remains consistently below 0.015 across all scales, effectively indistinguishable from zero. Moreover, $h_{xy}(-5)$ and $h_{xy}\left(5\right)$ are nearly identical at each scale, indicating that small and large price fluctuations follow the same scaling relationship—reflecting efficient cross-market arbitrage. This robustness persisted throughout the 2018–2020 US–China trade tensions, demonstrating that the fungible nature of soybeans—with alternative supply sources in Brazil and Argentina—preserved price equilibrium despite bilateral friction. Substitutability determines the depth of shocks: agricultural commodities with diversified supply chains absorb trade frictions as transient noise, while energy commodities with concentrated supply embed geopolitics into long-term pricing.

Table 3. Multifractal spectrum parameters for WTI crude oil vs. US dollar index.

Scale	${\mathit{h}}_{\mathbf{xy}}(-5)$	${\mathit{h}}_{\mathbf{xy}}\left(5\right)$	Spectrum Width Δ$\mathit{h}$
2-day	0.483	0.344	0.139
5-day	0.551	0.388	0.163
10-day	0.794	0.264	0.530
20-day	0.852	0.250	0.602
40-day	–	–	–
60-day	–	–	–

Note: Results at 40-day and 60-day scales are not reported due to insufficient data segments for reliable estimation ($N_s<30$).

Table 3. Multifractal spectrum parameters for WTI crude oil vs. US dollar index.

Scale	${\mathit{h}}_{\mathbf{xy}}(-5)$	${\mathit{h}}_{\mathbf{xy}}\left(5\right)$	Spectrum Width Δ$\mathit{h}$
2-day	0.483	0.344	0.139
5-day	0.551	0.388	0.163
10-day	0.794	0.264	0.530
20-day	0.852	0.250	0.602
40-day	–	–	–
60-day	–	–	–

Note: Results at 40-day and 60-day scales are not reported due to insufficient data segments for reliable estimation ($N_s<30$).

Table 4. Multifractal spectrum parameters for US soybean vs. Chinese soybean prices.

Scale	${\mathit{h}}_{\mathbf{xy}}(-5)$	${\mathit{h}}_{\mathbf{xy}}\left(5\right)$	Spectrum Width Δ$\mathit{h}$
2-day	0.395	0.390	0.005
5-day	0.421	0.411	0.010
10-day	0.449	0.444	0.005
20-day	0.473	0.461	0.012
40-day	–	–	–
60-day	–	–	–

Table 4. Multifractal spectrum parameters for US soybean vs. Chinese soybean prices.

Scale	${\mathit{h}}_{\mathbf{xy}}(-5)$	${\mathit{h}}_{\mathbf{xy}}\left(5\right)$	Spectrum Width Δ$\mathit{h}$
2-day	0.395	0.390	0.005
5-day	0.421	0.411	0.010
10-day	0.449	0.444	0.005
20-day	0.473	0.461	0.012
40-day	–	–	–
60-day	–	–	–

3.2. Scale-Dependent Time Irreversibility

Time irreversibility refers to the statistical asymmetry of a time series under time reversal, which is a signature of dissipative dynamical systems operating away from equilibrium. In commodity futures markets, this manifests as asymmetric price responses to positive versus negative shocks: adverse news triggers more violent fluctuations than favorable news, declines are more abrupt than advances, and recoveries are more gradual. The central finding of this subsection is that this asymmetry undergoes a fundamental shift with the observation horizon.

For WTI crude oil, the PG irreversibility index is positive at short horizons (2–10 days), with the 5-day value reaching +0.0177. This positive irreversibility corresponds to the well-documented "slow grind up, fast crash down" phenomenon: positive shocks from supply disruptions or demand surges unfold over days to weeks, while negative shocks manifest as sharp sell-offs triggered by panic selling or margin calls. At the 20-day scale, however, the PG reverses to $-0.0267$—a shift of approximately 0.044 relative to the 5-day value. Over monthly horizons, downward movements become relatively faster or more frequent than upward movements. At the 40-day and 60-day scales, the index remains negative but with diminishing magnitude ($-0.0032$, $-0.0011$), suggesting gradual dissipation toward equilibrium.

This cross-scale sign reversal reflects the dominance of different investor populations across time scales. Short-term speculators and momentum traders amplify upward momentum over daily to weekly intervals, generating positive PG values. At monthly horizons, commercial hedgers, producers, and institutional investors adjust positions in response to fundamental supply–demand assessments, exhibiting greater sensitivity to downside risks and producing sharper downward corrections. The near-zero PG at 60 days (approximately one quarter) is consistent with finite memory of directional bias, dissipating over a timescale comparable to a typical commodity cycle. Notably, the 20-day reversal window coincides with the one-way shipping time from the Middle East to East Asia and the global refinery inventory turnover cycle, suggesting that physical logistics rhythms have been internalized into WTI futures pricing through the hedging behaviors of East Asian market participants.

The corresponding results for US wheat exhibit an even more pronounced but structurally different pattern. The 5-day PG reaches 0.0229, and the 10-day PG peaks at 0.0363—the highest short-horizon values in this study. This strong positive irreversibility reflects rapid price responses to weather reports, crop condition updates, and USDA announcements, with adverse reports triggering immediate upward jumps followed by gradual adjustment. At the 40-day scale, the PG reverses to $-0.0073$, consistent with the medium-term downward bias observed in WTI. However, at the 60-day scale, a remarkable re-reversal occurs: the PG reaches $+0.1142$, the highest value across all commodities and scales. This exceptionally strong positive irreversibility at the quarterly horizon is attributable to the asymmetric nature of agricultural supply shocks: adverse weather or geopolitical disruptions generate rapid price spikes at the onset of the growing season, while the subsequent supply–demand rebalancing—through planting adjustments, import substitutions, and strategic reserve drawdowns—unfolds gradually over months.

The "dual-reversal" pattern of US wheat demonstrates that directional asymmetry in agricultural commodities is not a monotonic decay process, but rather a repetitive structural alternation anchored by the growing season cycle. This contrasts with the monotonic reversal pattern observed in crude oil. Overall, time irreversibility in commodity markets is jointly determined by supply adjustment elasticity and shock accumulation patterns: energy commodities exhibit finite memory (dissipating over approximately one quarter), while agricultural commodities display cycle-anchored asymmetry (accumulating over the growing season and releasing at season-end).

Table 5. Multiscale PG irreversibility index for WTI crude oil.

Scale	PG Index	Directional Interpretation
2-day	+0.0087	Weak upward bias
5-day	+0.0177	Moderate upward bias
10-day	+0.0070	Weak upward bias
20-day	-0.0267	Moderate downward bias
40-day	-0.0032	Weak downward bias
60-day	-0.0011	Near-symmetric

Table 5. Multiscale PG irreversibility index for WTI crude oil.

Scale	PG Index	Directional Interpretation
2-day	+0.0087	Weak upward bias
5-day	+0.0177	Moderate upward bias
10-day	+0.0070	Weak upward bias
20-day	-0.0267	Moderate downward bias
40-day	-0.0032	Weak downward bias
60-day	-0.0011	Near-symmetric

Table 6. Multiscale PG irreversibility index for US wheat.

Scale	PG Index	Directional Interpretation
2-day	+0.0084	Weak upward bias
5-day	+0.0229	Moderate upward bias
10-day	+0.0363	Moderate upward bias
20-day	+0.0148	Weak upward bias
40-day	-0.0073	Weak downward bias
60-day	+0.1142	Strong upward bias

Table 6. Multiscale PG irreversibility index for US wheat.

Scale	PG Index	Directional Interpretation
2-day	+0.0084	Weak upward bias
5-day	+0.0229	Moderate upward bias
10-day	+0.0363	Moderate upward bias
20-day	+0.0148	Weak upward bias
40-day	-0.0073	Weak downward bias
60-day	+0.1142	Strong upward bias

3.3. Complexity Hierarchy Along the Soybean Crushing Chain

The soybean crushing value chain—soybeans, soybean meal, and soybean oil—provides a setting for examining how market complexity propagates along a production chain. Each processing stage alters the commodity’s physical form and economic use, generating differentiated price dynamics. Soybeans are easily storable and globally traded, with prices determined by international supply–demand; soybean meal, less storable and tied to livestock cycles, reflects regional demand; soybean oil, with dual uses as edible oil and biofuel feedstock, directly links the agricultural chain to energy markets.

Entropy declines monotonically with scale for all three products, indicating that price dynamics become more predictable and less complex as the horizon lengthens. At short scales, microstructure noise and stochastic trading shocks dominate price fluctuations, resulting in higher entropy and price paths that resemble random walks. As the scale extends, these transitory disturbances are gradually smoothed out, and fundamental trend information emerges, making price behavior more orderly and predictable. The entropy values across all products fall roughly between 2.35 and 2.58. For $d=3$, the maximum possible entropy is ${log}_2\left(6\right)\approx 2.585$ (with six possible ordinal patterns), so this range indicates a fairly high level of complexity. The distribution of the six ordinal patterns is nearly uniform, with no single pattern dominating. This near-uniform distribution is a hallmark of high informational efficiency—prices have largely absorbed available information, making predictable regular structures difficult to form.

A notable exception appears at the 40-day scale: the entropy of soybean oil (2.348) drops significantly below that of soybeans (2.562) and soybean meal (2.551), a gap of approximately 0.21 bits. This divergence emerges specifically at 40 days, likely related to the institutional cycles soybean oil faces as a biofuel feedstock—the US EPA’s Renewable Fuel Standard (RFS) compliance cycles and the monthly-to-bimonthly adjustment frequency of biofuel production margins both fall within this window. Energy market trend signals become embedded in soybean oil pricing at this horizon, whereas soybeans and soybean meal lack this cross-energy-market transmission channel. At shorter scales, all three products are driven by common agricultural news and weather trading; at 60 days, all segments have synchronized to longer-term supply–demand equilibrium, and the divergence dissipates.

Table 7. Multiscale weighted permutation entropy for the soybean complex.

Scale	Soybean	Soybean Meal	Soybean Oil
2-day	2.580	2.584	2.582
5-day	2.575	2.581	2.573
10-day	2.563	2.581	2.567
20-day	2.574	2.558	2.574
40-day	2.562	2.551	2.348
60-day	2.502	2.510	2.476

Table 7. Multiscale weighted permutation entropy for the soybean complex.

Scale	Soybean	Soybean Meal	Soybean Oil
2-day	2.580	2.584	2.582
5-day	2.575	2.581	2.573
10-day	2.563	2.581	2.567
20-day	2.574	2.558	2.574
40-day	2.562	2.551	2.348
60-day	2.502	2.510	2.476

3.4. Structural Break Detection and Geopolitical Linkages

The Jensen–Shannon divergence segmentation algorithm identifies 12 structural breakpoints in the WTI price series over the sample period. Over 90% coincide closely with major geopolitical events—the 2018 trade war, the 2020 negative oil price event, the 2022 Russia–Ukraine conflict, and the 2023 Red Sea crisis—with the corresponding GPR Index reaching elevated levels within the event windows. Classified by source, the 12 breakpoints fall into two main categories: geopolitical conflicts (approximately 60%) and supply–demand policy interventions (approximately 40%). Conventional macro-financial variables such as interest rates and exchange rates exhibit markedly weaker explanatory power at these breakpoints.

The JS divergence values range from 0.312 to 0.516. The highest values cluster around OPEC+ events—0.516 for the December 2020 agreement and 0.503 for the April 2023 production cuts—underscoring the organization’s policy decisions as the primary driver of crude oil market breaks. COVID-19 (0.483) and US election uncertainty (0.479) register comparable magnitudes, indicating that macro shocks exert impacts no less significant than supply interventions. The negative oil price event (0.381), while a landmark that overturned the futures term structure, registers a lower divergence than the pandemic and election shocks. This contrast indicates that exogenous macro uncertainties exert influence through multiple channels—demand expectations, policy pathways, and risk premia—with impact scope and persistence far exceeding those of market-internal structural adjustments.

All identified breakpoints correspond one-to-one with documented events, with the GPR Index at an elevated level at each breakpoint, establishing a direct empirical link between geopolitical risk and structural market shifts. These breaks cover three distinct shock categories: demand-side shocks (COVID-19, China’s reopening), supply-side shocks (EU sanctions, Ukrainian attacks, Red Sea blockage), and policy expectation shifts (OPEC+ agreements, US elections). The JS divergence, as a non-parametric detection tool, captures all these shifts uniformly without requiring pre-specification of shock categories. These findings indicate that geopolitical shocks induce persistent changes in the statistical structure of commodity prices, rather than merely transient volatility spikes. Structural breaks imply a permanent shift in the data-generating process—the mean, variance, or autocorrelation structure has been fundamentally altered. The successful alignment of JS-identified breakpoints with geopolitical events indicates that geopolitical risk is not merely short-term noise, but a core driver of long-term market evolution—with clear implications for cross-horizon risk management and policy timing.

Table 8. Structural breakpoints in WTI price series identified by JS-divergence segmentation.

No.	Date	JS Divergence	Associated Event
1	2018-06-19	0.338	US–China trade war escalation
2	2020-01-28	0.483	COVID-19 pandemic outbreak
3	2020-04-21	0.381	WTI negative oil price event
4	2020-12-29	0.516	OPEC+ production agreement
5	2021-03-23	0.439	Suez Canal blockage
6	2022-05-17	0.339	EU sanctions on Russian oil
7	2022-08-09	0.322	Biden visit to the Middle East
8	2023-01-24	0.312	China’s post-COVID reopening
9	2023-04-18	0.503	OPEC+ surprise production cuts
10	2023-12-26	0.468	Red Sea crisis
11	2024-03-19	0.451	Ukrainian attacks on Russian refineries
12	2024-11-26	0.479	US post-election policy uncertainty

Table 8. Structural breakpoints in WTI price series identified by JS-divergence segmentation.

No.	Date	JS Divergence	Associated Event
1	2018-06-19	0.338	US–China trade war escalation
2	2020-01-28	0.483	COVID-19 pandemic outbreak
3	2020-04-21	0.381	WTI negative oil price event
4	2020-12-29	0.516	OPEC+ production agreement
5	2021-03-23	0.439	Suez Canal blockage
6	2022-05-17	0.339	EU sanctions on Russian oil
7	2022-08-09	0.322	Biden visit to the Middle East
8	2023-01-24	0.312	China’s post-COVID reopening
9	2023-04-18	0.503	OPEC+ surprise production cuts
10	2023-12-26	0.468	Red Sea crisis
11	2024-03-19	0.451	Ukrainian attacks on Russian refineries
12	2024-11-26	0.479	US post-election policy uncertainty

3.5. Time-Varying Dynamics of Irreversibility

To examine whether market directionality changes over time, we perform a rolling-window estimation of the PG index for WTI crude oil with window length 252 trading days (approximately one year) and step size 21 trading days (approximately one month).

In the earliest reported window (January 2018), the PG index is negative across all scales, with the 10-day PG reaching $-0.0553$, reflecting downward pressure during the initial phase of US–China trade tensions. In the subsequent window (February 2018), however, the PG turns positive across all scales, with the 10-day PG rising to $+0.0768$, suggesting the market gradually digested the shock after initial panic subsided. The COVID-19 window (October 2019 to September 2020) exhibits strongly negative PG values across all scales, with the 10-day PG reaching $-0.0579$, capturing the extreme bearishness of lockdowns, demand destruction, and the unprecedented negative price event. In contrast, the post-Red Sea window (December 2023 to November 2024) exhibits the strongest positive PG values across all three scales, with the 10-day PG reaching $+0.0644$, reflecting sustained upward pressure from shipping route disruptions.

These time-varying patterns indicate that directional asymmetry of crude oil evolves with changing macroeconomic and geopolitical conditions—demand-side shocks push PG into negative territory, while supply-side shocks push it positive. A closer comparison of PG values within each window reveals that during extreme stress, the magnitude of asymmetry at the 10-day scale exceeds that at shorter scales (e.g., $-0.0579$ at 10 days versus $-0.0187$ at 2 days during COVID-19). This suggests directional bias is most pronounced at medium-term horizons during crises, possibly reflecting the time required for market participants to re-evaluate fundamentals and adjust medium-term positions. In quieter periods, asymmetry across scales shows smaller differences or is overall weaker.

Time irreversibility arises from the nonlinear superposition of external shocks and market response lags at specific time scales. Following a shock, directional bias amplifies to its peak within approximately 10 days, then gradually decays. The cross-scale structure of directional bias thus carries information about shock type, transmission speed, and participant behavior patterns. Different shock types and market states produce differentiated scale-response patterns through distinct coupling pathways. For risk management, this implies that during crises, in addition to monitoring short-term volatility elevation, investors should pay particular attention to the asymmetric risk exposure of medium-term positions.

Table 9. Time-varying PG irreversibility indices for WTI crude oil (selected windows).

Window Start	Window End	PG (2-day)	PG (5-day)	PG (10-day)
2018-01-03	2018-12-20	-0.0078	-0.0281	-0.0553
2018-02-01	2019-01-18	+0.0111	+0.0215	+0.0768
2018-05-01	2019-04-17	-0.0162	-0.0097	-0.0322
2018-07-27	2019-07-15	-0.0087	+0.0147	+0.0148
2019-10-02	2020-09-18	-0.0187	-0.0254	-0.0579
2020-01-30	2021-01-15	+0.0020	-0.0178	-0.0105
2020-05-06	2021-04-23	+0.0086	+0.0306	-0.0162
2021-08-31	2022-08-19	+0.0229	+0.0068	-0.0053
2022-03-16	2023-03-03	-0.0177	-0.0043	+0.0053
2023-12-12	2024-11-29	+0.0528	+0.0559	+0.0644

Table 9. Time-varying PG irreversibility indices for WTI crude oil (selected windows).

Window Start	Window End	PG (2-day)	PG (5-day)	PG (10-day)
2018-01-03	2018-12-20	-0.0078	-0.0281	-0.0553
2018-02-01	2019-01-18	+0.0111	+0.0215	+0.0768
2018-05-01	2019-04-17	-0.0162	-0.0097	-0.0322
2018-07-27	2019-07-15	-0.0087	+0.0147	+0.0148
2019-10-02	2020-09-18	-0.0187	-0.0254	-0.0579
2020-01-30	2021-01-15	+0.0020	-0.0178	-0.0105
2020-05-06	2021-04-23	+0.0086	+0.0306	-0.0162
2021-08-31	2022-08-19	+0.0229	+0.0068	-0.0053
2022-03-16	2023-03-03	-0.0177	-0.0043	+0.0053
2023-12-12	2024-11-29	+0.0528	+0.0559	+0.0644

4. Discussion

The preceding subsections indicate that the nonlinear dynamics of commodity futures markets are shaped by three factors together: commodity supply elasticity, physical logistics rhythms, and shock penetration depth. Supply elasticity determines the fundamental pattern of scale dependence; physical logistics rhythms provide the characteristic time anchor for directional reversals; and shock penetration depth governs the persistence of structural breaks. Together, these three dimensions provide a coherent framework for interpreting the complex system dynamics of commodity markets.

First, supply elasticity shapes the evolutionary path of cross-market linkages and directional asymmetry. The time-scale dependence of commodity markets is not homogeneous across commodities, but varies with their supply adjustment elasticity. Energy commodities such as WTI crude oil, whose supply is constrained by geographic concentration and physical logistics, exhibit cross-market linkages that become sharply more complex with expanding scales, while directional asymmetry undergoes a fundamental sign reversal at the medium-term horizon. By contrast, agricultural commodities such as soybeans, with global supply substitutability, maintain a near-monofractal cross-market structure across all scales, with directional bias close to zero. Wheat, meanwhile, displays a distinctive "positive → negative → strong positive" dual-reversal pattern, reflecting the asymmetric accumulation of shocks arising from rigidities in agricultural supply. This finding links multifractal cross-correlation analysis and time irreversibility analysis through a consistent economic logic: a commodity’s supply adjustment elasticity determines the sensitivity of its price dynamics to scale—the higher the elasticity, the lower the heterogeneity of cross-scale structure; the lower the elasticity, the more pronounced the scale-dependent nonlinearity.

Second, a characteristic time scale of approximately 20 days is identified consistently across three independent methodological approaches—representing the coupling point between the physical logistics rhythm of the global crude oil market and financial pricing. The MF-DCCA spectrum width peaks at this scale (0.602); the PG irreversibility index undergoes sign reversal at this scale (from +0.018 to −0.027); and the majority of OPEC+ events identified by JS divergence influence the market on approximately a monthly cycle. This cross-methodological convergence suggests that maritime transport lags and inventory buffers jointly determine the inherent frequency at which the market absorbs shocks and completes directional turns. The convergence across methods indicates that linkage complexity, directional asymmetry, and structural breaks—three seemingly distinct nonlinear dimensions—share the same underlying time constant, providing a consistent empirical foundation for understanding the multiscale dynamics of energy markets.

Third, the intensity of a shock’s reconstruction of the price statistical structure depends on its "systemic penetration depth"—the extent to which the shock transcends single-market mechanisms and diffuses through broader channels such as demand expectations, policy pathways, and risk premia into the wider macro-financial network, thereby altering the price-generating process. The JS divergence analysis reveals a critical distinction. Structural events internal to the futures market (such as the 2020 negative oil price event, JS = 0.381), despite their high news salience, exhibit lower reconstruction intensity on the price statistical structure than exogenous macro uncertainty shocks (COVID-19 at 0.483, the US election at 0.479). The negative oil price event represents an extreme failure of market mechanisms, but its impact remains relatively confined to structural repairs within the futures market itself. The COVID-19 and election shocks, by contrast, exert more comprehensive and more persistent systemic reconstruction pressures through global demand expectations, policy uncertainty, and risk appetite channels. This conclusion also links the JS break analysis with the time-varying patterns of irreversibility: demand-side shocks drive PG toward negative values, while supply-side shocks push it toward positive values. The deeper the systemic penetration depth of a shock, the more persistent and comprehensive its reconstruction of the price statistical structure.

5. Conclusions

This paper integrates four methodologies—multifractal analysis, irreversibility measurement, complexity quantification, and structural break detection—into a multiscale analytical framework for examining the nonlinear dynamics of commodity futures markets. Three main conclusions emerge.

First, the physical and biological constraints inherent in commodity supply are the primary source of scale-dependent heterogeneity. Energy and agricultural commodities exhibit systematic differences in the scale evolution of cross-market linkages and in their patterns of directional asymmetry—differences that are endogenously determined by their respective supply adjustment elasticities. Second, three categories of nonlinear characteristics—linkage complexity, directional asymmetry, and structural breaks—converge around a characteristic time scale of approximately 20 days. Observed across three independent approaches, this convergence indicates that the scale reflects an intrinsic coupling frequency between physical logistics rhythms and financial pricing. Third, whether a shock induces persistent reconstruction of the price statistical structure depends not on its news salience, but on its systemic penetration depth—the extent to which it can transcend single-market mechanisms, diffuse through multiple channels within the broader macro-financial network, and thereby alter the price-generating process.

These findings suggest that scale-dependent heterogeneity in commodity markets is not incidental but systematically linked to supply-side fundamentals. The framework developed here—anchored in supply elasticity, physical logistics rhythms, and systemic penetration depth—offers a coherent basis for cross-commodity allocation, multi-horizon risk management, and geopolitical scenario analysis. At the same time, the generalizability of these results to other commodity classes, such as metals and livestock, and to alternative market conditions warrants further investigation.