Complex exponential smoothing

Exponential smoothing is one of the most popular forecasting methods in practice. It has been used and researched for more than half a century. It started as an ad-hoc forecasting method and developed to a family of state-space models. Still all exponential smoothing methods are based on time series decomposition and the usage of such components as "level", "trend", "seasonality" and "error". It is assumed that these components may vary from one time series to another and take different forms depending on data characteristics. This makes their definition arbitrary and in fact there is no single way of identifying these components. At the same time the introduction of different types of exponential smoothing components implies that a model selection procedure is needed. This means that a researcher needs to select an appropriate type of model out of 30 different types either manually or automatically for every time series analysed. However several recent studies show that an underlying statistical model may have a form completely different than the one assumed by specific exponential smoothing models. These modelling questions motivate our research. We propose a model without strictly defined "level", "trend" and "seasonality". The model greatly simplifies the selection procedure, distinguishing only between seasonal and non-seasonal time series. Although we call it "Complex Exponential Smoothing" (CES), due to the use of complex-valued functions, its usage simplifies the forecasting procedure. In this thesis we first discuss the main properties of CES and propose an underlying statistical model. We then extend it in order to take seasonality into account and conduct experiments on real data to compare its performance with several well-known univariate forecasting models. We proceed to discuss the parameters estimation for exponential smoothing and propose a "Trace Forecast Likelihood" function that allows estimating CES components more efficiently. Finally we show that Trace Forecast Likelihood has desirable statistical properties, is connected to shrinkage and is generally advisable to use with any univariate model