Approaches to modelling functional time series with an application to electricity generation data

We study the half-hourly electricity generation by coal and by gas in the UK over a period of three years from 2012 to 2014. As a highly frequent time series, daily cycles along with seasonality and trend across days can be seen in the data for each fuel. Taylor (2003), Taylor et al. (2006), and Taylor (2008) studied time series of the similar features by introducing double seasonality into the methods for a single univariate time series. As we are interested in the continuous variation in the generation within a day, the half-hourly observations within a day are considered as a continuous function. In this way, a time series of half-hourly discrete observations is transformed into a time series of daily functions. The idea of a time series of functions can also seen in Shang (2013), Shang and Hyndman (2011) and Hyndman and Ullah (2007). We improve their methods in a few ways. Firstly, we identify the systematic effect due to the factors that take effect in a long term, such as weather and prices of fuels, and the intrinsic differences between the days of the week. The systematic effect is modeled and removed before we study the day-by-day random variation in the functions. Secondly, we extend functional principal component analysis (PCA), which was applied on one group of functions in Shang (2013), Shang and Hyndman (2011) and Hyndman and Ullah (2007), into partial common PCA, in order to consider the covariance structures of two groups of functions and their similarities. A test on the goodness of the approximation to the functions given by the common eigenfunctions is also proposed. The idea of bootstrapping residuals from the approximation seen in Shang (2014) is employed but is improved with non-overlapping blocks and moving blocks of residuals. Thirdly, we use a vector autoregressive (VAR) model, which is a multivariate approach, to model the scores on common eigenfunctions of a group such that the cross-correlation between the scores can be considered. We include Lasso penalties in the VAR model to select the significant covariates and refit the selection with ordinary least squares to reduce the bias. Our method is compared with the stepwise procedure by Pfaff (2007), and is proved to be less variable and more accurate on estimation and prediction. Finally, we propose the method to give the point forecasts of the daily functions. It is more complicated than the methods of Shang (2013), Shang and Hyndman (2011) and Hyndman and Ullah (2007) as the systematic effect needs to be included. An adjustment interval is also given along with a point forecast, which represents the range within which the true function might vary. Our methods to give the point forecast and the adjustment interval include the information updating after the training period, which is not considered in the classical predicting equations of VAR and GARCH seen in Tsay (2013) and Engle and Bollerslev (1986).