Spectrum Occupancy Seasonality Decomposition

The choice between additive and multiplicative models depends on whether the amplitude of the seasonal cycle remains constant or varies with the trend.

• Additive Model: Assumes the seasonal fluctuations are constant over time. Best when the trend is flat or linear.
  • Formula: Y(t) = Trend(t) + Seasonality(t) + Residual(t)
• Multiplicative Model: Assumes the seasonal amplitude scales proportionally with the trend level. Ideal when traffic volume grows over months.
  • Formula: Y(t) = Trend(t) × Seasonality(t) × Residual(t)

Selecting the wrong model introduces systematic bias into the residual component, degrading downstream forecast accuracy.

Human activity creates highly predictable cyclical patterns in spectrum usage that decomposition explicitly isolates.

• Diurnal (24-hour) Cycle: Captures the day/night rhythm. Commercial bands peak during business hours; residential bands peak in the evening.
• Weekly Cycle: Distinguishes weekday traffic from weekend lulls. Industrial and enterprise bands show a sharp drop on Saturdays and Sundays.
• Harmonic Modeling: Fourier series can be used to extract these cycles, representing the seasonality as a sum of sine and cosine waves with 24-hour and 168-hour periods.

By removing these known cycles, the residual component reveals anomalies like unexpected interference or equipment failures.

STL is a robust, non-parametric method for decomposing a time series that handles complex seasonality and outliers gracefully.

• LOESS Smoothing: Uses locally weighted regression to estimate the trend, making it resistant to short-term anomalies.
• Iterative Detrending: The algorithm iteratively extracts and refines the seasonal component, allowing the seasonal pattern to evolve slowly over time.
• Robustness Weights: Automatically down-weights outliers during the fitting process, preventing a single jamming event from distorting the estimated trend.

STL is particularly effective for spectrum data because it does not assume a fixed-length seasonality, making it adaptable to the slightly irregular patterns of human behavior.

The residual component—what remains after removing trend and seasonality—should ideally be stationary white noise. Systematic structure in the residuals indicates a modeling failure.

• Anomaly Thresholding: A transmission that causes a residual spike exceeding 3 standard deviations from the mean is a statistical outlier, likely an unauthorized or emergency signal.
• Autocorrelation Check: Plotting the autocorrelation function (ACF) of the residuals validates the decomposition. Significant autocorrelation means the model failed to capture all predictable structure.
• Heteroskedasticity Detection: If the variance of the residuals changes over time, a multiplicative model or a variance-stabilizing transformation is required.

This process transforms a forecasting problem into a statistical process control problem for spectrum enforcement.

Many predictive models, including ARIMA and LSTM networks, perform better on stationary data. Seasonal decomposition is a critical pre-processing step.

• Feature Engineering: The extracted seasonal component can be fed as an explicit exogenous feature into a multivariate forecasting model, rather than forcing the model to learn the cycle from raw data.
• Seasonal Differencing: For ARIMA models, a seasonal difference (e.g., Y(t) - Y(t-24)) removes the diurnal cycle, making the series stationary.
• Hybrid Pipelines: A common architecture uses STL to decompose the signal, an LSTM to forecast the deseasonalized trend, and then re-applies the seasonal component to generate the final prediction.

This separation of concerns allows the neural network to focus on the complex, non-linear trend dynamics without wasting capacity on memorizing a fixed daily cycle.

Spectrum occupancy often exhibits nested seasonality—a daily pattern within a weekly pattern within an annual pattern. Standard decomposition must be extended.

• Double Seasonal Holt-Winters: An extension of exponential smoothing that models two simultaneous seasonal cycles (e.g., 24-hour and 168-hour).
• TBATS (Trigonometric, Box-Cox, ARMA, Trend, Seasonal): A fully automated framework that handles complex, non-integer, and multiple seasonal periods. It uses trigonometric terms to reduce the parameter load of high-frequency seasonality.
• Sequential Decomposition: A practical approach first removes the weekly cycle, then decomposes the resulting series to extract the daily cycle from the weekly-adjusted data.

Ignoring nested seasonality causes the model to misinterpret a weekend lull as a sudden trend downturn, leading to false predictions of spectrum availability.