ARIMA Model

ARIMA decomposes a time series into three distinct structural elements:

• AR (Autoregressive): The current channel state is regressed on its own prior values. The parameter p defines the lag window, capturing the persistence of spectrum occupancy.
• I (Integrated): Differencing is applied to the raw signal data to achieve stationarity, removing trends in channel utilization to stabilize the mean.
• MA (Moving Average): The model accounts for dependency between an observation and the residual errors from a moving average applied to lagged observations, parameterized by q.

A fundamental assumption of the ARIMA framework is that the input time series must be stationary—its statistical properties like mean and variance must remain constant over time.

• Raw spectrum data often exhibits trends (e.g., diurnal usage patterns) and is non-stationary.
• The Integrated (I) order d applies successive differencing (e.g., subtracting the previous observation) to remove these trends.
• The Augmented Dickey-Fuller (ADF) test is typically used to statistically verify that stationarity has been achieved before fitting the model.

The optimal orders p and q are identified by analyzing the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots of the stationary time series:

• ACF: Measures the correlation between observations at different time lags. A sharp cut-off after lag q suggests the order of the MA component.
• PACF: Measures the direct correlation at a specific lag, removing the influence of intermediate lags. A cut-off after lag p indicates the AR order.
• This visual inspection method provides a principled way to initialize the model before fine-tuning with information criteria like AIC or BIC.

Spectrum occupancy often exhibits strong seasonal patterns tied to human activity cycles (e.g., rush hour vs. midnight). The standard ARIMA model cannot capture these repeating cycles.

• SARIMA (Seasonal ARIMA) extends the model with additional seasonal hyperparameters: (P, D, Q, s).
• s defines the seasonal period (e.g., 24 for hourly data with a daily cycle).
• P, D, and Q mirror the non-seasonal parameters but operate on the seasonal lag, allowing the model to learn that occupancy at 9 AM today is highly correlated with occupancy at 9 AM yesterday.

Unlike some black-box deep learning models, ARIMA provides a prediction interval alongside its point forecast, quantifying the uncertainty of the channel state prediction.

• The model outputs a Gaussian predictive distribution, allowing a cognitive radio to make risk-aware decisions.
• A wide prediction interval signals high uncertainty in the forecast, which might trigger a conservative spectrum handoff strategy.
• This inherent uncertainty quantification is critical for proactive spectrum mobility, where overconfidence in a false "idle" prediction leads to a collision with the primary user.

While computationally efficient, ARIMA models have strict limitations in complex electromagnetic environments:

• Linearity Assumption: ARIMA captures linear relationships but fails to model non-linear dynamics or abrupt regime changes common in bursty data traffic.
• Univariate Focus: Standard ARIMA models a single frequency channel in isolation, ignoring spatial correlations with adjacent channels.
• Concept Drift: The model parameters are static after training. If a primary user's traffic pattern changes permanently, the model's accuracy degrades until it is manually retrained, unlike adaptive online learning methods.