SARIMA (Seasonal ARIMA) › Implementation in Statsmodels

Now let's see how the SARIMA formula we've discussed translates into actual code. We'll implement SARIMA using the statsmodels library, which provides time series analysis tools. As we work through the implementation, you'll see how each component of the formula.differencing, autoregressive terms, and moving average terms.is handled by the library, making the connection between theory and practice concrete.

Statsmodels is a good choice here because it offers robust statistical inference, diagnostic tools, and detailed model summaries that are important for understanding SARIMA model performance.

The generated time series shows a clear upward trend from approximately $100K to $150K over 5 years, with seasonal fluctuations. The mean and standard deviation provide a baseline for understanding the data's central tendency and variability before modeling.

The model summary provides diagnostic information including parameter estimates, standard errors, confidence intervals, and information criteria. The coefficients indicate how strongly current values depend on past values and past forecast errors, both for non-seasonal and seasonal components.

The parameter values indicate the strength of relationships in the model. Positive AR coefficients suggest that high values tend to follow high values, while MA coefficients capture how the model adjusts based on past forecast errors. The AIC and BIC values allow comparison with alternative model specifications, with lower values typically indicating better model fit adjusted for complexity.

The forecasts show expected values for the next 12 months along with 95% confidence intervals. The confidence intervals widen as we forecast further into the future. The forecasts capture both the upward trend and seasonal patterns observed in the historical data.

The implementation demonstrates the SARIMA modeling process. We generate realistic sales data with both trend and seasonal components, fit a SARIMA(1,1,1)(1,1,1)12 model, and generate 12-month forecasts with confidence intervals. The model captures both the upward trend and the seasonal patterns.

Key ParametersLink Copied

Below are some of the main parameters that affect how the SARIMA model works and performs.

• order: A tuple (p, d, q) specifying the non-seasonal ARIMA parameters. p is the order of the autoregressive part, d is the degree of differencing, and q is the order of the moving average part. Common starting values are (1,1,1) for many time series.

• seasonal_order: A tuple (P, D, Q, s) specifying the seasonal ARIMA parameters. P is the seasonal autoregressive order, D is the seasonal differencing order, Q is the seasonal moving average order, and s is the seasonal period (e.g., 12 for monthly data with annual seasonality). Start with (1,1,1,12) for monthly data.

• trend: Optional parameter to include a constant term in the model. Can be 'c' for constant, 't' for linear trend, 'ct' for both, or 'n' for neither. Default is 'c' which includes a constant term.

• enforce_stationarity: Boolean (default: True) that determines whether to enforce stationarity constraints on autoregressive parameters. Setting to False allows non-stationary models but may lead to unreliable forecasts.

• enforce_invertibility: Boolean (default: True) that determines whether to enforce invertibility constraints on moving average parameters. Invertibility ensures the model can be expressed in terms of past observations.

• method: Optimization method for parameter estimation. Options include 'css-ml' (conditional sum of squares then maximum likelihood, default), 'ml' (maximum likelihood), or 'css' (conditional sum of squares). 'css-ml' typically provides good balance between speed and accuracy.

Key MethodsLink Copied

The following are the most commonly used methods for interacting with SARIMA models.

• fit(): Estimates model parameters using maximum likelihood or conditional sum of squares. Returns a fitted model object that supports parameter inspection, diagnostic checks, and forecasting.

• forecast(steps): Generates point forecasts for the specified number of steps ahead. Returns a pandas Series with forecasted values.

• get_forecast(steps): Generates forecasts with confidence intervals. Returns a forecast object with .predicted_mean for point forecasts and .conf_int() for confidence intervals.

• predict(start, end): Generates in-sample or out-of-sample predictions for specified time periods. Useful for generating predictions on historical data or specific future dates.

• summary(): Displays a model summary including parameter estimates and their uncertainty, information criteria (AIC, BIC), and diagnostic statistics.