Fractional lower-order covariance-based measures for cyclostationary time series with heavy-tailed distributions: application to dependence testing and model order identification

1. Problem Statement

Traditional analysis of cyclostationary time series (processes with statistical characteristics that vary periodically) relies on the autocovariance function (ACVF) and the concept of Periodically Correlated (PC) processes. However, the ACVF is undefined for processes with infinite variance, which are common in real-world data exhibiting heavy-tailed distributions (e.g., $\alpha$-stable distributions with stability index $\alpha < 2$).

When data contains outliers or follows heavy-tailed laws, classical second-order methods fail. Existing alternatives for infinite-variance processes (such as covariation or codifference) have been explored, but there is a lack of robust, time-domain tools specifically designed for cyclostationary processes with infinite variance. The paper addresses the need for:

1. A rigorous definition of cyclostationarity for infinite-variance processes.
2. New dependence measures analogous to the Periodic Autocorrelation Function (peACF) and Periodic Partial Autocorrelation Function (pePACF) but valid for heavy-tailed data.
3. Practical methods for dependence testing and model order identification (for PAR and PMA models) using these new measures.

2. Methodology

2.1. Theoretical Framework: FLOC-Cyclostationarity

The authors introduce FLOC-cyclostationarity, a generalization of PC processes. Instead of requiring periodic mean and ACVF, this framework requires the periodicity of the mean and the Fractional Lower-Order Covariance (FLOC).

• FLOC Definition: For random variables $Y_1, Y_2$ with finite fractional moments up to order $a$ ($1 < a \le 2$), FLOC is defined as:
  $$\text{FLOC}(Y_1, Y_2; A, B) = E[Y_1^{\langle A \rangle} Y_2^{\langle B \rangle}]$$
  where $x^{\langle c \rangle} = |x|^c \text{sgn}(x)$ and $A+B < a$.
• FLOC-Cyclostationary Process: A process $\{X_t\}$ is FLOC-cyclostationary with period $T$ if $E[X_t]$ and $\text{FLOC}(X_t, X_{t+h})$ are periodic in $t$ with period $T$.

2.2. New Autodependence Measures

The paper proposes two novel measures derived from FLOC:

1. Periodic Fractional Lower-Order Autocorrelation Function (peFLOACF):
   • A normalized version of the Periodic FLOC (peFLOACVF).
   • Defined by standardizing the time series using the Fractional Lower-Order Moment (FLOM) (the FLOC of a variable with itself) rather than variance.
   • Key Property: For Periodic Moving Average (PMA) models, the peFLOACF exhibits a "cut-off" property, becoming zero for lags $|h| > q$ (the model order). This mirrors the behavior of classical ACF in MA models.
2. Periodic Fractional Lower-Order Partial Autocorrelation Function (peFLOPACF):
   • Derived using FLOC-based Yule-Walker equations.
   • Key Property: For Periodic Autoregressive (PAR) models, the peFLOPACF exhibits a "cut-off" property, becoming zero for lags $h > p$ (the model order). This mirrors the behavior of classical PACF in AR models.

2.3. Proposed Applications

Based on these measures, the authors develop two algorithms:

• Portmanteau Test for Dependence: A statistical test to determine if a cyclostationary series is "white noise" (peFLOWN) or possesses autodependence. It aggregates squared peFLOACF values across lags and seasons, using Monte Carlo simulations to establish critical regions (since analytical distributions are complex).
• Model Order Identification:
  • PAR Order ($p$): Identified by finding the lag $h$ where the sample peFLOPACF $\hat{\zeta}_v(h)$ first falls within the confidence interval (becomes statistically insignificant).
  • PMA Order ($q$): Identified by finding the lag $h$ where the sample peFLOACF $\hat{\eta}_v(h)$ cuts off.
  • The method determines "seasonal orders" $p(v)$ and $q(v)$ for each phase $v=1,\dots,T$, and the global order is the maximum of these.

3. Key Contributions

1. Generalization of Cyclostationarity: Successfully extends the concept of cyclostationarity to infinite-variance processes using FLOC, providing a mathematically sound alternative to PC processes.
2. Novel Measures: Introduces peFLOACF and peFLOPACF, which are the first robust, time-domain tools specifically for identifying dependence structures in heavy-tailed cyclostationary data.
3. Practical Algorithms: Develops complete statistical procedures for:
   • Testing dependence in heavy-tailed cyclostationary series.
   • Identifying the order of PAR and PMA models with infinite variance.
4. Validation: Provides extensive Monte Carlo simulations and a real-world case study to validate the efficiency and robustness of the proposed methods.

4. Results

4.1. Simulation Study

• Dependence Testing: The portmanteau test demonstrated high empirical power. For PAR and PMA models with coefficients far from zero, the test achieved near-perfect efficiency (100% power). Efficiency decreased for coefficients close to zero but improved significantly with larger sample sizes ($N_T=1000$ vs $N_T=100$).
• Order Identification:
  • The methods correctly identified seasonal orders $p(v)$ and $q(v)$ in the majority of cases.
  • Performance was strongly correlated with the magnitude of the model coefficients; larger coefficients led to higher identification accuracy.
  • The "cut-off" property was clearly observed in the simulated plots of peFLOACF and peFLOPACF, confirming theoretical expectations.

4.2. Real Data Application

• Dataset: Daily average PM10 (particulate matter) air pollution data from Vitória, Brazil (546 observations).
• Preprocessing: Log-transformation and centering. The data exhibited heavy tails and weekly seasonality ($T=7$).
• Analysis:
  • The data was modeled as a PAR process with symmetric $\alpha$-stable innovations ($\alpha \approx 1.9$).
  • Dependence Test: The portmanteau test rejected the null hypothesis of white noise for most days of the week, confirming autodependence.
  • Order Identification: The peFLOPACF plots indicated a global order of $p=3$ (specifically for Thursday, $v=4$), while other days showed lower orders.
  • Model Fitting: A PAR(3) model was fitted. Residual analysis confirmed that the residuals were free of significant autodependence (passed the portmanteau test) and retained the heavy-tailed nature ($\alpha \approx 1.89$), validating the model fit.

5. Significance

This paper bridges a critical gap in time series analysis by providing tools for non-Gaussian, heavy-tailed cyclostationary data.

• Robustness: It offers a solution for datasets where classical variance-based methods fail due to outliers or infinite variance, a common issue in fields like finance, environmental engineering, and signal processing.
• Extensibility: The proposed measures generalize classical concepts (ACF/PACF) to the infinite-variance domain, allowing practitioners to use familiar model selection logic (cut-off properties) even when data is non-Gaussian.
• Practical Utility: The successful application to air pollution data demonstrates that these theoretical advances can be directly applied to real-world problems involving cyclic patterns and extreme events.

In conclusion, the authors have established a robust framework for analyzing cyclostationary processes with heavy tails, introducing reliable measures for dependence testing and model order identification that outperform or complement existing methods in non-Gaussian contexts.