statsmodels.tsa.forecasting.theta.ThetaModel¶

class statsmodels.tsa.forecasting.theta.ThetaModel(endog, *, period=None, deseasonalize=True, use_test=True, method='auto', difference=False)[source]¶

The Theta forecasting model of Assimakopoulos and Nikolopoulos (2000)

Parameters:

endogarray_like, 1d: The data to forecast.
periodint, default None: The period of the data that is used in the seasonality test and adjustment. If None then the period is determined from y’s index, if available.
deseasonalizebool, default True: A flag indicating whether the deseasonalize the data. If True and use_test is True, the data is only deseasonalized if the null of no seasonal component is rejected.
use_testbool, default True: A flag indicating whether test the period-th autocorrelation. If this test rejects using a size of 10%, then decomposition is used. Set to False to skip the test.
method{“auto”, “additive”, “multiplicative”}, default “auto”: The model used for the seasonal decomposition. “auto” uses a multiplicative if y is non-negative and all estimated seasonal components are positive. If either of these conditions is False, then it uses an additive decomposition.
differencebool, default False: A flag indicating to difference the data before testing for seasonality.

See also

statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothing: Exponential smoothing parameter estimation and forecasting
statsmodels.tsa.statespace.sarimax.SARIMAX: Seasonal ARIMA parameter estimation and forecasting

Notes

The Theta model forecasts the future as a weighted combination of two Theta lines. This class supports combinations of models with two thetas: 0 and a user-specified choice (default 2). The forecasts are then

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

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.

The model is estimated in steps:

Test for seasonality
Deseasonalize if seasonality detected
Estimate \(\alpha\) by fitting a SES model to the data and \(b_0\) by OLS.
Forecast the series
Reseasonalize if the data was deseasonalized.

The seasonality test examines where the autocorrelation at the seasonal period is different from zero. The seasonality is then removed using a seasonal decomposition with a multiplicative trend. If the seasonality estimate is non-positive then an additive trend is used instead. The default deseasonalizing method can be changed using the options.

References

[1]

Assimakopoulos, V., & Nikolopoulos, K. (2000). The theta model: a decomposition approach to forecasting. International Journal of Forecasting, 16(4), 521-530.

[2]

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

[3]

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

Attributes:

deseasonalize: Whether to deseasonalize the data
difference: Whether the data is differenced in the seasonality test
method: The method used to deseasonalize the data
period: The period of the seasonality
use_test: Whether to test the data for seasonality

Methods

fit([use_mle, disp])

Estimate model parameters.

Properties

`deseasonalize`	Whether to deseasonalize the data
`difference`	Whether the data is differenced in the seasonality test
`method`	The method used to deseasonalize the data
`period`	The period of the seasonality
`use_test`	Whether to test the data for seasonality