Fractal Feature Corrector (FFC)

• FFC is a high-fidelity complexity extractor using overlapping multifractal analysis to generate stable, directional Hurst exponent features essential for risk-aware forecasting.
• It employs an OSW-MF-ADCCA algorithm that overcomes traditional single-series limitations and estimator instability via overlapping segmentation and local detrending.
• Integrating with models like LSTM and Transformer, FFC enhances forecasting accuracy by dynamically combining raw financial data with fractal features in the FCOC framework.

The Fractal Feature Corrector (FFC) is a high-fidelity complexity extractor designed for financial volatility forecasting, introduced within the Fractal-Chaotic Oscillation Co-driven (FCOC) framework. FFC synthesizes advances in multifractal analysis—specifically, an asymmetric, cross-correlation-based generalization of multifractal detrended cross-correlation analysis (MF-ADCCA) with overlapping windowing—to generate time-varying, directionally aware features suitable for downstream machine learning models. Unlike classical approaches, FFC systematically resolves both the single-series limitation and the instability of non-overlapping segmentation, making it a central component for risk-aware forecasting of financial time series (Zenga et al., 13 Nov 2025).

1. Motivation and Theoretical Underpinnings

Financial time series are characterized by long-range memory, heavy tails, and multifractal cross-correlations between assets. Traditional fractal methods such as MF-DFA or vanilla MF-ADCCA face two core issues:

• Single-series limitation: Inability to capture asymmetric, cross-asset dynamics responsible for systemic risk.
• Estimator instability: Non-overlapping segmentations create spurious fluctuations, leading to degraded feature fidelity.

FFC addresses these problems by employing an asymmetric cross-correlation-based multifractal measure (MF-ADCCA), augmented via an Overlapping Sliding Window (OSW) scheme. The output consists of directional (positive/negative trend) generalized Hurst exponents, which are empirically shown to encode fine-grained, time-localized multifractal structure essential for robust volatility prediction.

2. Mathematical Construction

Given two synchronized return series $r_x(t), r_y(t)$ of length $N$ (with $r_x$ typically representing a market proxy, and $r_y$ volatility increments), the OSW-MF-ADCCA proceeds in six principal steps:

1. Profile Construction:

$$P_x(k) = \sum_{t=1}^k (r_x(t) - \mu_x), \quad P_y(k) = \sum_{t=1}^k (r_y(t) - \mu_y)$$

where $\mu_x$, $\mu_y$ are sample means.

1. Overlapping Segmentation:
   • Segment length $s$ and overlap ratio $\rho \in [0,1)$. Effective step: $$s_{\text{step}} = \left\lfloor s \cdot (1-\rho) + 0.5 \right\rfloor$$.
   • Number of segments: $$N_{\text{seg}} = \left\lfloor (N-s)/s_{\text{step}} \right\rfloor + 1$$.
2. Local Detrending:
   • For each segment $j$, fit polynomials $p_x^{(j)}(i), p_y^{(j)}(i)$ of order $m=2$.
   • Compute local detrended cross-covariance:

   $$F^2_{\text{seg}}(s,j) = \frac{1}{s}\sum_{i=1}^s\left|[P_x(i)-p_x^{(j)}(i)]\cdot[P_y(i)-p_y^{(j)}(i)]\right|$$

• Trend indicators from the slope $\beta_j$ of a linear fit to $P_x(i)$:

  $$I^+(\beta_j) = \frac{1+\text{sgn}(\beta_j)}{2}, \quad I^-(\beta_j) = \frac{1 - \text{sgn}(\beta_j)}{2}$$

1. Directional q-order Fluctuations: For real $q\neq 0$:

$$F_q^+(s) = \left[\frac{1}{N_{\text{pos}}} \sum_{j=1}^{N_{\text{seg}}} I^+(\beta_j) [F^2_{\text{seg}}(s,j)]^{q/2}\right]^{1/q}$$

$$F_q^-(s) = \left[\frac{1}{N_{\text{neg}}} \sum_{j=1}^{N_{\text{seg}}} I^-(\beta_j) [F^2_{\text{seg}}(s,j)]^{q/2}\right]^{1/q}$$

$$F_q(s) = \left[\frac{1}{N_{\text{seg}}} \sum_{j=1}^{N_{\text{seg}}} [F^2_{\text{seg}}(s,j)]^{q/2}\right]^{1/q}$$

For $q=0$, logarithmic averaging is used.

2. Scaling and Hurst Exponents: The fluctuation functions scale as:

$$F_q(s) \propto s^{H(q)}, \quad F_q^+(s) \propto s^{H^+(q)}, \quad F_q^-(s) \propto s^{H^-(q)}$$

The slopes $H(2), H^+(2), H^-(2)$ from regressing $\log F_2$ vs. $\log s$ yield the generalized Hurst exponents for overall, positive, and negative trend directions.

3. Rolling-Window Feature Generation: On a rolling window basis (e.g., length $T=252$), the procedure is repeated for each window, sliding forward by step $k$. This produces time series $$\mathcal{H}_{all}(t), \mathcal{H}_{pos}(t), \mathcal{H}_{neg}(t)$$ of length approximately $(N-T)/k$.

A full schematic is encapsulated in Algorithm 1 in (Zenga et al., 13 Nov 2025).

3. Algorithmic Workflow

The following table summarizes the workflow of the FFC:

Step	Description	Parameterization
Input	Synchronized series $r_x$, $r_y$	$N$, $T$, $k$, $\rho$, $s$, $m$
Windowing	Rolling windows of $T$ steps, step size $k$	$T=252$, $k=1$ (empirical)
Segmentation	Overlap ratio for window segments	$\rho=1/3$
Polynomial Detrending	Quadratic fit in each segment	$m=2$
Feature Extraction	Compute $H$, $H^+$, $H^-$ for each window	Fitted over power-law grid for $s$
Output	3 time series of Hurst exponents per window	$\sim (N-T)/k$ points

Performance is insensitive to the polynomial order beyond $m=2$ or modest variations in $\rho$ and $s$, but optimality is dataset-dependent.

4. Empirical Validation and Benchmarking

Empirical evaluation utilizes S&P 500 (2005–2025) and DJI (2009–2025) 5-minute intraday returns, aggregated to daily realized volatility (RV) targets. The experimental protocol includes:

• Features: Raw (lagged RV, log-returns, BPV-based volatility increments); Fractal (OSW-MF-ADCCA exponents $H, H^+, H^-$).
• Baselines: LSTM, GRU, Transformer, Mamba architectures without fractal features; with standard MF-ADCCA; with OSW-MF-ADCCA (FFC).
• Metrics: $R^2$ (higher is better), QLIKE (lower is better), MAE, MSE. Model Confidence Set (MCS) at 25% over 10,000 block-bootstrap resamples.

Key results:

• Enriching any backbone with standard MF-ADCCA yields mild gains; OSW-MF-ADCCA yields substantial gains (e.g., Transformer $R^2$ on S&P 500 increases from 0.123 to 0.383).
• In MCS tests, only OSW-MF-ADCCA variants appear in the superior set ($p=1.0000$); no standard baseline survives.
• In ablation, FFC-only improves $R^2$ and QLIKE uniformly; adding COC (“COC-only”) also helps; full FFC+COC (FCOC) yields the highest performance (e.g., Mamba on DJI with $R^2$ moving from 0.263 $\to$ 0.437 (FFC) $\to$ 0.491 (COC) $\to$ 0.507 (full)).

These findings demonstrate that the FFC is a high-fidelity complexity extractor whose impact is magnified when paired with dynamically adaptive processing (COC).

5. Integration with Downstream Architectures

Within the FCOC framework, the FFC output—the triplet $$(\mathcal{H}_{all}(t), \mathcal{H}_{pos}(t), \mathcal{H}_{neg}(t))$$—is concatenated with raw inputs at each time $t$ for any volatility-forecasting model such as LSTM, GRU, Transformer, or Mamba.

• FFC-only: The fractal features are fed directly into backbone models utilizing standard static activations (e.g., ReLU, GELU).
• Full FCOC: All static activations are replaced by the Chaotic Oscillation Component (COC), a dynamic system that processes the fractal features for enhanced model responsiveness.

OSW-MF-ADCCA’s stability (from overlapping windows) and its directional decomposition (positive/negative trends) are essential to this integration, conferring both statistical robustness and sensitivity to market regime shifts.

6. Theoretical Properties and Limitations

FFC offers two main theoretical properties:

• Estimation stability: Overlapping windowing reduces variance compared to non-overlapping MF-DCCA and mitigates spurious log-periodic fluctuations, as documented by Bashan et al. (2008).
• Directional multifractality: By separately modeling positive and negative trends, FFC captures the asymmetric, regime-dependent multifractality that is intrinsic to financial markets.

No closed-form convergence result is provided for the estimator. Reliability is established empirically via stationarity tests (ADF/KPSS) and statistical significance using the Model Confidence Set framework.

A plausible implication is that while FFC delivers empirical robustness, its theoretical behavior—especially in nonstationary or short-length settings—remains an open area for further investigation.

7. Practical Implementation Considerations

FFC is straightforward to implement, requiring only Algorithm 1 of (Zenga et al., 13 Nov 2025). Parameter tuning for segment length $s$, overlap ratio $\rho$, and polynomial degree $m$ adapts the method to various asset classes and forecast horizons. Adoption entails the following:

• Compute the three rolling Hurst exponent time series from synchronized input series.
• Concatenate with additional features and input to any downstream model.
• For full FCOC integration, substitute standard activation functions with COC modules, fostering dynamic, context-sensitive processing of fractal features.

Resource and computational requirements are dominated by the repeated detrending and linear regression steps for each window, which scale linearly with data length and number of windows. Empirical results indicate the approach scales well to years of minute-resolution data without recourse to specialized hardware.

FFC’s modularity allows it to serve as an independent feature extractor or as a component in hybrid frameworks, with demonstrated transferability across model classes and regimes.