Seasonal time series forecasting by the Walsh-transformation based technique

It is relatively well known that Walsh–Fourier analysis is capable of approximating functions by decomposing them into simple values: 1 and − 1. This method and its valuable characteristics however, are seldom applied in time series forecasting. Moreover, despite the promising applicability of the technique, there is gap in the scientific literature for a comprehensive and detailed introduction and application. Inspired by the results Stoffer (J Time Ser Anal 6:261–267, 1985, J Time Ser Anal 8:449–167, 1987, J Time Ser Anal 11:57–73, 1990, J Am Stat Assoc 86(414):461–479, 1991), Stoffer et al. (J Am Stat Assoc 83:954–963, 1988) and Basu et al. (Electr Power Energy Syst 13(4):193–200, 1991) in the current paper, we aim to describe the theory of Walsh–Fourier analysis, to introduce the so-called Walsh transformation based “row-wise” forecasting process—using the characteristics of the Walsh-matrix—for its application and to compare numerically in R-programming environment (RStudio) using the MComp test data set and the state-of-the-art methods (generic function forecast()) the following two approaches: (1) the 1st of them may be referred to as “direct” since we apply function forecast() to time-series directly in “conventional” mode, (2) and the 2nd one may be called “Walsh-based row-wise” since we apply the same function forecast() to transformed time-series row-wisely. As can be seen—based on our intentions—a significant advantage of the proposed forecasting process is that when forecasting more than one-step ahead it does not use previously forecasted values and, hence, does not accumulate inaccuracy, thus the forecasted results can be more proper compared to other mainstream techniques. Moreover, as show our numeric results the longer the horizon of forecasting, the more attractive relative accuracy achieved.