# Stability of Multiple Seasonal Holt-Winters Models Applied to Hourly Electricity Demand in Spain
## 2 Related Work  
Short and medium term electricity demand time series forecasting were extensively studied in the literature applying both statistical methods and machine learning techniques. Recently, the machine learning methods have focused on big data [25], especially deep learning [26; 27], and ensemble methodologies [28; 29].  
Within classical methods, the accuracy of the forecasts in the Holt-Winters models has been widely studied, especially how to select the best method and adjust the parameters [30]. As the Holt-Winters models are recursive, an initialization value is needed to feed the model. Thus, the initialization methods for Holt-Winters were also another intense field of research especially with the emergence of double and triple seasonal Holt-Winters models [31; 32]. In fact, new methods to initialize the level, trend and seasonality in multiple seasonal Holt-Winters models were recently developed in [14]. These seed values have influence on the smoothing parameters' values as well as the forecasting accuracy [33]. However, when the series adjustment is made, the initial values lose influence [15]. After fitting the Holt-Winters model, the obtained smoothing parameters are assumed to be optimal for the data set. Thus, the study of their values or their stability has not been worked in depth in the literature. This is due, in large part, to the fact that the models do not allow a theoretical mathematical analysis.  
[PERSON] [34] used 406 monthly series from the M competitions. He analyzed the models with additive seasonality, and demonstrated that the values of the parameters within the range [0,1] are not always invertible. Thus, it is necessary to use only a set range within the region of invertibility.  
[PERSON] [35] used state spaces to analyze Holt-Winters models, and although his work focused on the normalization of the seasonal component, he also analyzed the stability of the parameters. From previous work on filters [36; 37], it determined that Holt-Winters models are not asymptotically stable. The values of the eigenvectors of the stability matrices depend on the \(\alpha\), \(\gamma\) and \(\delta\) parameters.  
Some authors [38] worked on obtaining the limits of the smoothing parameters. They analyzed state space parameters, and stated the parameters ought to meet \(0<\alpha<\gamma\). Finally, the authors in [39] established a series of criteria in the parameters so that the models can be \"predictable\", a term that mints whether a series will be able to make forecasts with constant mean and variance. [PERSON] et al. [40] obtained the values of the main vectors and confirmed that the models proposed in [39] produce stable predictions, but if regressors are used, they must be invariant over time.  
Another interesting study related to this paper is the study of the minimum size required for the sample. [PERSON] et al. [41] analyzed this situation and concluded that there is no clear answer, but that there should be many more observations than parameters. In this regard, [PERSON] and [PERSON] [13] verified that for a double seasonal model, a data set length of one year produces more stable parameters than 8-10 weeks.