Fractal Time Series Analysis: Mathematical Foundations¶

Research Documentation for FracTime Framework

Table of Contents¶

Introduction
Fractal Geometry in Financial Markets
The Hurst Exponent
Fractal Dimension
Trading Time vs. Clock Time
Cross-Dimensional Fractal Analysis
Pattern Recognition in Self-Similar Systems
Regime-Matched Path Simulation
Statistical Testing Framework
References

Introduction¶

Traditional financial models assume that market returns follow a Gaussian (normal) distribution with independent increments. However, empirical evidence consistently shows that financial markets exhibit:

Fat tails: Extreme events occur more frequently than predicted by normal distributions
Volatility clustering: High volatility periods cluster together
Long-range dependence: Returns show persistent correlations over long time horizons
Self-similarity: Market patterns repeat across different time scales

Fractal geometry provides a mathematical framework for modeling these empirically observed phenomena. Unlike traditional models based on Brownian motion, fractal models can capture the complex, scale-invariant structures inherent in financial time series.

Fractal Geometry in Financial Markets¶

Self-Similarity and Scale Invariance¶

A fractal object exhibits self-similarity if it appears similar at different scales of magnification. Mathematically, a time series X(t) is self-similar with Hurst parameter H if:

X(ct) =_d c^H X(t)

where =_d denotes equality in distribution, and c is a scaling constant.

For financial time series, this implies that the statistical properties of price movements over different time horizons (daily, weekly, monthly) maintain similar characteristics when appropriately rescaled.

Mandelbrot's Multifractal Model¶

Mandelbrot proposed that price changes follow a multifractal process rather than simple Brownian motion. In a multifractal model, volatility itself varies according to a fractal process, creating:

Heterogeneous scaling: Different parts of the time series scale differently
Intermittency: Bursts of high activity followed by calm periods
Fat-tailed distributions: Power-law tails rather than exponential decay

The log-price process can be modeled as:

log P(t) = log P(0) + ∫₀ᵗ σ(s) dB_H(s)

where B_H(s) is fractional Brownian motion with Hurst exponent H, and σ(s) is a multifractal volatility process.

The Hurst Exponent¶

Definition and Interpretation¶

The Hurst exponent (H) quantifies the long-term memory and persistence of a time series. It ranges from 0 to 1:

H = 0.5: Random walk (no memory, Brownian motion)
0.5 < H < 1: Persistent series (trending behavior, positive autocorrelation)
0 < H < 0.5: Anti-persistent series (mean-reverting, negative autocorrelation)

R/S Analysis Method¶

The rescaled range (R/S) analysis method estimates H by examining how the range of cumulative deviations scales with time:

Step 1: For a time series of returns {r_i}, divide into subseries of length τ

Step 2: For each subseries, compute the mean:

m_τ = (1/τ) Σᵢ₌₁ᵗ rᵢ

Step 3: Create mean-adjusted cumulative deviations:

Y(t,τ) = Σᵢ₌₁ᵗ (rᵢ - m_τ)

Step 4: Calculate the range:

R(τ) = max Y(t,τ) - min Y(t,τ)

Step 5: Calculate the standard deviation:

S(τ) = √[(1/τ) Σᵢ₌₁ᵗ (rᵢ - m_τ)²]

Step 6: The rescaled range:

(R/S)(τ) = R(τ)/S(τ)

Step 7: The Hurst exponent is the slope of:

log(R/S) = H log(τ) + constant

Fractional Brownian Motion¶

A process with Hurst exponent H can be modeled as fractional Brownian motion (fBm):

B_H(t) ~ N(0, t^(2H))

The covariance structure is:

Cov[B_H(s), B_H(t)] = (1/2)[|s|^(2H) + |t|^(2H) - |t-s|^(2H)]

This captures long-range dependence when H ≠ 0.5.

Trading Implications¶

H > 0.7: Strong trending behavior → momentum strategies may be effective
H < 0.3: Strong mean reversion → contrarian strategies may work
0.4 < H < 0.6: Near-random walk → difficult to predict, market efficiency

Fractal Dimension¶

Box-Counting Dimension¶

The box-counting dimension D_B measures the complexity and roughness of a time series curve. For a curve embedded in 2D space:

Algorithm: 1. Cover the curve with boxes of size ε 2. Count the number N(ε) of boxes needed 3. The dimension is:

D_B = lim_(ε→0) log N(ε) / log(1/ε)

Practical Estimation:

D_B ≈ slope of log N(ε) vs log(1/ε) plot

Relationship to Hurst Exponent¶

For self-affine fractals (like financial time series), the relationship is:

D_B = 2 - H

This means: - Higher H → Lower dimension → Smoother, more persistent - Lower H → Higher dimension → Rougher, more erratic

Volatility Estimation¶

Fractal dimension can estimate volatility without assuming normal distributions:

σ_fractal ∝ (D_B - 1)

This provides a robust volatility measure that adapts to fat tails and clustering.

Trading Time vs. Clock Time¶

Mandelbrot's Subordinated Process¶

Mandelbrot proposed that financial markets operate on trading time rather than clock time. Trading time flows: - Faster during high volatility (more "information" per unit time) - Slower during quiet periods (less market activity)

Mathematically, price returns in trading time follow:

dP/P = σ dB(τ(t))

where τ(t) is a stochastic time change process, and B is Brownian motion in trading time.

Time Dilation Model¶

The time dilation factor λ(t) transforms clock time t to trading time τ:

dτ/dt = λ(t) = f(σ(t), V(t))

where: - σ(t) is local volatility - V(t) is trading volume - f is a combining function (e.g., geometric mean)

Power-law transformation:

λ(t) = [σ(t)/σ̄]^α · [V(t)/V̄]^β

where α and β control the sensitivity to volatility and volume.

Implications for Forecasting¶

Equal trading time intervals have more similar statistical properties than equal clock time intervals
Volatility forecasts should account for time dilation
Regime detection is more accurate in trading time
Pattern matching works better when patterns are aligned in trading time

Cross-Dimensional Fractal Analysis¶

Multidimensional Fractals¶

Financial systems are inherently multidimensional, with interdependencies between: - Price and volume - Multiple securities - Different market indicators

Cross-dimensional fractal coherence measures how fractal properties align across dimensions:

C = Σᵢⱼ w_ij · corr(Dᵢ, Dⱼ)

where Dᵢ and Dⱼ are fractal dimensions of different variables.

Joint Hurst Estimation¶

For bivariate series (X, Y), the joint Hurst exponent H_XY characterizes co-movement:

H_XY = [H_X + H_Y + H_(X+Y)]/2

Interpretation: - H_XY > (H_X + H_Y)/2: Positive co-persistence - H_XY < (H_X + H_Y)/2: Negative co-persistence (divergence)

Regime Classification¶

Cross-dimensional analysis enables regime classification based on fractal properties:

Regime Features:

Feature vector: [H_price, D_price, H_volume, D_volume, H_XY, C]

Clustering: K-means or Gaussian mixture models on feature space identify: 1. Trending regimes: High H, low D across dimensions 2. Mean-reverting regimes: Low H, high D 3. Transition regimes: Mixed properties, low coherence

Pattern Recognition in Self-Similar Systems¶

Fractal Pattern Matching¶

Traditional pattern recognition fails in self-similar systems because patterns appear at multiple scales. Fractal pattern matching accounts for scale invariance:

Normalized Distance Metric:

d(P₁, P₂) = min_s ||P₁ - s·P₂||_H

where s is a scaling factor and ||·||_H is the Hurst-weighted norm:

||X||_H = √[Σᵢ (xᵢ - x̄)² · i^(-2H)]

This metric: 1. Rescales patterns to find best match 2. Weights by persistence: More weight on recent values if H > 0.5 3. Accounts for heteroscedasticity: Adapts to varying volatility

Wavelet-Based Pattern Detection¶

Continuous wavelet transform decomposes signals into scale and position:

W(a,b) = ∫ f(t) ψ*((t-b)/a) dt

where: - a is the scale parameter - b is the translation parameter - ψ is the mother wavelet

Pattern identification: 1. Transform both historical and forecast patterns 2. Compare wavelet coefficients across scales 3. Match patterns with similar multi-scale structure

Self-Similar Pattern Synthesis¶

Recursive subdivision: Generate new patterns by combining historical sub-patterns:

P_new(t) = Σₖ wₖ · s_k · P_hist(φₖ(t))

where: - wₖ are weights based on pattern similarity - sₖ are scaling factors - φₖ are time warping functions

Regime-Matched Path Simulation¶

Volatility Regime Detection¶

Multi-scale volatility clustering: Analyze volatility at different timeframes (daily, weekly, monthly):

σ_τ(t) = √[252/τ · Σᵢ₌₀^(τ-1) r²(t-i)]

Regime features: - Mean volatility: σ̄_τ - Volatility of volatility: σ(σ_τ) - Hurst exponent: H_τ - Fractal dimension: D_τ

Similarity metric between current and historical regimes:

S = Σ_τ w_τ [|σ̄_τ^now - σ̄_τ^hist|/σ̄_τ^now + |H_τ^now - H_τ^hist|]

Path Generation Algorithm¶

Step 1: Regime Identification - Compute current regime features R_current - Find N most similar historical regimes: {R_1, ..., R_N}

Step 2: Return Sampling - For each historical regime Rᵢ, sample n_steps returns - Apply trading time transformation if enabled - Preserve empirical distribution (no parametric assumptions)

Step 3: Path Assembly - Construct cumulative returns:

Path(t) = P₀ · exp(Σᵢ₌₁ᵗ rᵢ)

Step 4: Probability Weighting - Assign probability to each path based on regime similarity:

p_i = exp(-λ · S_i) / Σⱼ exp(-λ · S_j)

where λ controls concentration on most similar regimes.

Volatility Preservation¶

To ensure forecasts maintain realistic volatility:

Step 1: Measure historical volatility

σ_hist = std(log-returns)

Step 2: Measure forecast volatility

σ_forecast = mean over paths [std(log-returns per path)]

Step 3: If σ_forecast < threshold · σ_hist, add scaled noise:

Path'(t) = Path(t) · exp(ε(t))
where ε(t) ~ Empirical(historical returns) · scaling_factor

Statistical Testing Framework¶

Diebold-Mariano Test¶

Tests whether two forecasts have significantly different accuracy:

Null hypothesis: E[L(e₁) - L(e₂)] = 0

where L is a loss function (e.g., squared error) and e₁, e₂ are forecast errors.

Test statistic:

DM = d̄ / √[Var(d)/T]

where d̄ is mean loss differential, T is sample size.

Distribution: DM ~ N(0,1) under H₀ for large T

Interpretation: - |DM| > 1.96 → Reject H₀ at 5% level → Forecasts differ significantly

Model Confidence Set (MCS)¶

Identifies the set of models that are not significantly outperformed:

Algorithm: 1. Start with all M models 2. Compute pairwise loss differentials 3. Eliminate model with worst performance 4. Test if remaining models are equivalent 5. Repeat until no significant differences

Equivalence test statistic:

T_max = max_i,j |t_ij|

where t_ij is the t-statistic for loss differential between models i and j.

Result: Superior Set of Models (SSM) with confidence level α

Continuous Ranked Probability Score (CRPS)¶

Evaluates probabilistic forecasts using the entire predictive distribution:

CRPS(F, x) = ∫_{-∞}^{∞} [F(y) - 1{y ≥ x}]² dy

where F is the forecast CDF and x is the realized value.

Interpretation: - Lower CRPS → Better calibrated probabilistic forecast - CRPS = 0 → Perfect forecast - Generalizes MAE for probabilistic forecasts

Coverage Tests¶

Assess whether prediction intervals have correct coverage:

Unconditional coverage:

H₀: P(y_t ∈ CI_t) = 1 - α

Test using binomial test on hit rate.

Conditional coverage (Christoffersen test): - Tests both correct coverage and independence of violations - Uses likelihood ratio test on Markov chain of hits/misses

References¶

Foundational Papers¶

Mandelbrot, B. B. (1963). "The Variation of Certain Speculative Prices." Journal of Business, 36(4), 394-419.
Introduced fractal concepts to finance
Showed fat tails and infinite variance in cotton prices
Hurst, H. E. (1951). "Long-Term Storage Capacity of Reservoirs." Transactions of the American Society of Civil Engineers, 116, 770-808.
Original R/S analysis method
Long-range dependence in Nile River data
Peters, E. E. (1994). Fractal Market Analysis: Applying Chaos Theory to Investment and Economics. Wiley.
Comprehensive treatment of fractals in finance
Market microstructure and fractal time
Mandelbrot, B. B., & Van Ness, J. W. (1968). "Fractional Brownian Motions, Fractional Noises and Applications." SIAM Review, 10(4), 422-437.
Mathematical foundations of fBm
Connection to Hurst exponent

Fractal Time Series Methods¶

Granger, C. W., & Joyeux, R. (1980). "An Introduction to Long-Memory Time Series Models and Fractional Differencing." Journal of Time Series Analysis, 1(1), 15-29.
ARFIMA models for long memory
Fractional integration
Peng, C. K., et al. (1994). "Mosaic Organization of DNA Nucleotides." Physical Review E, 49(2), 1685.
Detrended fluctuation analysis (DFA)
Alternative to R/S analysis
Kantelhardt, J. W., et al. (2002). "Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series." Physica A, 316(1-4), 87-114.
MFDFA method
Local Hurst exponents

Trading Time and Subordinated Processes¶

Clark, P. K. (1973). "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices." Econometrica, 41(1), 135-155.
Subordinated process framework
Volume as directing process
Ané, T., & Geman, H. (2000). "Order Flow, Transaction Clock, and Normality of Asset Returns." Journal of Finance, 55(5), 2259-2284.
Transaction time vs. calendar time
Conditional normality in transaction time

Statistical Testing¶

Diebold, F. X., & Mariano, R. S. (1995). "Comparing Predictive Accuracy." Journal of Business & Economic Statistics, 13(3), 253-263.
- DM test for forecast comparison
- Asymptotic theory
Hansen, P. R., Lunde, A., & Nason, J. M. (2011). "The Model Confidence Set." Econometrica, 79(2), 453-497.
- MCS procedure
- Multiple testing corrections
Gneiting, T., & Raftery, A. E. (2007). "Strictly Proper Scoring Rules, Prediction, and Estimation." Journal of the American Statistical Association, 102(477), 359-378.
- Proper scoring rules
- CRPS and other metrics

Recent Advances¶

Cont, R. (2007). "Volatility Clustering in Financial Markets: Empirical Facts and Agent-Based Models." In Long Memory in Economics (pp. 289-309). Springer.
- Stylized facts of volatility
- Agent-based explanations
Tsay, R. S. (2010). Analysis of Financial Time Series (3rd ed.). Wiley.
- Modern time series methods
- GARCH and stochastic volatility
Taylor, S. J. (2008). Modelling Financial Time Series (2nd ed.). World Scientific.
- Comprehensive treatment
- High-frequency data analysis

Appendix: Mathematical Notation¶

Symbol	Meaning
H	Hurst exponent
D_B	Box-counting dimension
B_H(t)	Fractional Brownian motion
τ	Time lag or trading time
σ(t)	Volatility at time t
λ(t)	Time dilation factor
P(t)	Price at time t
r_t	Log return at time t
=_d	Equality in distribution
~	Distributed as
∫	Integral
Σ	Summation

E[·]	Expected value
Var(·)	Variance
Cov(·, ·)	Covariance
corr(·, ·)	Correlation

This research document provides the theoretical foundations for the FracTime forecasting framework. For implementation details, see the codebase documentation.