Hurst Exponent

The Hurst exponent directly measures the degree of self-similarity in spectrum occupancy data. A process is self-similar if its statistical properties remain invariant across different time scales. In cognitive radio, this means busy/idle patterns observed over milliseconds exhibit the same burstiness characteristics as patterns observed over hours. This fractal-like behavior invalidates traditional Poisson traffic models, which assume independence between arrivals. Accurately estimating H is critical for selecting the correct stochastic model for primary user activity.

The value of H strictly defines three distinct statistical regimes for a time series:

• 0 < H < 0.5: Mean-Reverting (Anti-Persistent). A busy period is statistically more likely to be followed by an idle period. High values tend to be followed by low values. The process is choppier than random noise.
• H = 0.5: Random Walk (Short-Range Dependence). Observations are uncorrelated. This corresponds to standard Brownian motion and validates the use of Markovian models for spectrum occupancy.
• 0.5 < H < 1: Trending (Persistent). A busy period is likely to be followed by another busy period. Long runs of channel occupancy are common, indicating strong positive autocorrelation and long memory.

The classical Rescaled Range (R/S) analysis, developed by Harold Hurst, is the foundational estimation technique. It calculates the ratio of the range of cumulative deviations from the mean (R) to the standard deviation (S) over multiple sub-periods of the time series. The Hurst exponent is the slope of the log-log plot of R/S versus the sub-period length (n). While intuitive, R/S is sensitive to short-range dependence, which can bias the estimate. Modern alternatives like Detrended Fluctuation Analysis (DFA) are often preferred for non-stationary spectrum data.

The estimated Hurst exponent directly dictates the choice of prediction architecture. For H ≈ 0.5, a standard Hidden Markov Model (HMM) or ARIMA model is sufficient. For H > 0.5, the long memory requires architectures capable of capturing distant temporal dependencies, such as LSTM Spectrum Predictors or fractional ARIMA (FARIMA) models. Ignoring a high H value and using a memoryless predictor leads to a high forced termination probability, as the model will fail to anticipate the extended busy periods characteristic of persistent traffic.

A single Hurst exponent assumes a monofractal process, which may be insufficient for complex spectrum dynamics. MF-DFA generalizes the concept to a spectrum of generalized Hurst exponents, h(q), which vary with the statistical moment q. This reveals whether small fluctuations (negative q) scale differently from large fluctuations (positive q). In spectrum occupancy, a multifractal signature indicates that rare, long-duration busy events are governed by a different scaling law than the frequent, short idle gaps, requiring a Copula Model or extreme value theory for accurate tail risk assessment.

A visual diagnostic for long-range dependence is the variance-time plot. For an independent process (H=0.5), the variance of the aggregated time series decreases proportionally to the aggregation level m (slope of -1 on a log-log plot). For a self-similar process with H > 0.5, the variance decays more slowly (slope > -1). This slow decay is a hallmark of long-range dependence, confirming that aggregating spectrum occupancy samples does not smooth out the burstiness as quickly as expected under Poisson assumptions.

A property where the statistical structure of a time series remains consistent across different time scales. In spectrum occupancy, self-similar traffic exhibits burstiness that looks identical whether observed over milliseconds or hours. The Hurst Exponent (H) quantifies this: H > 0.5 indicates persistence and long-range dependence, while H < 0.5 signals mean-reverting behavior. This directly impacts the design of prediction horizons for proactive handoff.

A phenomenon where autocorrelation in a time series decays hyperbolically rather than exponentially, meaning distant observations retain significant influence. In cognitive radio, LRD implies that a channel's state minutes ago can still inform current predictions. Key characteristics:

• Slow decay of the autocorrelation function
• Contrasts with short-range dependent Markov models
• Requires models like fractional ARIMA or LSTM to capture

A stochastic process parameterized directly by the Hurst Exponent, serving as the canonical model for self-similar traffic. fGn generalizes white Gaussian noise to exhibit long-range dependence. In spectrum modeling, fGn is used to:

• Generate synthetic primary user activity traces for simulation
• Validate whether observed occupancy data deviates from pure randomness
• Calibrate prediction algorithms against known LRD benchmarks

The classical statistical method for estimating the Hurst Exponent from empirical data. The algorithm:

1. Divides the time series into sub-windows of varying length n
2. Computes the rescaled range (range divided by standard deviation) for each window
3. Fits a power-law: R/S ∝ n^H The slope of the log-log plot yields the Hurst estimate. Sensitive to short-term biases, it is often paired with Detrended Fluctuation Analysis (DFA) for validation.

A robust alternative to R/S analysis for calculating the Hurst Exponent, designed to filter out spurious long-range correlations caused by non-stationary trends. DFA works by:

• Integrating the time series and segmenting it into windows
• Fitting and subtracting a local polynomial trend in each window
• Computing the fluctuation function F(n) ∝ n^α The scaling exponent α is equivalent to H for stationary signals. Preferred for real-world spectrum occupancy data with drift.

An extension beyond the single Hurst Exponent to characterize signals with intermittent, bursty scaling behavior. While a monofractal process has one H, a multifractal requires a spectrum of Hölder exponents. In dynamic spectrum access, multifractal analysis reveals:

• Intermittency in primary user arrivals
• Time-varying scaling properties that a single H cannot capture
• The need for adaptive concept drift adaptation in predictors