statsmodels.tsa.forecasting.theta.ThetaModelResults.forecast

ThetaModelResults.forecast(steps=1, theta=2)[source]

Forecast the model for a given theta

Parameters:
stepsint

The number of steps ahead to compute the forecast components.

thetafloat

The theta value to use when computing the weight to combine the trend and the SES forecasts.

Returns:
Series

A Series containing the forecasts

Notes

The forecast is computed as

\[\hat{X}_{T+h|T} = \frac{\theta-1}{\theta} b_0 \left[h - 1 + \frac{1}{\alpha} - \frac{(1-\alpha)^T}{\alpha} \right] + \tilde{X}_{T+h|T}\]

where \(\tilde{X}_{T+h|T}\) is the SES forecast of the endogenous variable using the parameter \(\alpha\). \(b_0\) is the slope of a time trend line fitted to X using the terms 0, 1, …, T-1.

This expression follows from [1] and [2] when the combination weights are restricted to be (theta-1)/theta and 1/theta. This nests the original implementation when theta=2 and the two weights are both 1/2.

References

[1]

Hyndman, R. J., & Billah, B. (2003). Unmasking the Theta method. International Journal of Forecasting, 19(2), 287-290.

[2]

Fioruci, J. A., Pellegrini, T. R., Louzada, F., & Petropoulos, F. (2015). The optimized theta method. arXiv preprint arXiv:1503.03529.


Last update: Jun 14, 2024