Smoothing, decomposition and forecasting of multidimensional and functional time series using regularisation

This thesis by publication is built around three articles which are at different stages of publication. All three articles have in common the concept of regularisation and they provide various applications of this concept in the field of functional data analysis. The thesis consists of five chapters. The first chapter is an introduction and it sets the historical context for concepts such as complexity and regularisation. It also looks at different forecasting problems from the point of view of complexity and considers regularisation as a practical means of reducing complexity in statistical models. The second part is an article “Bivariate data with ridges: two-dimensional smoothing of mortality rates”, which applies the concept of regularisation to a problem of smoothing mortality rates in particular and to any bivariate data in general. The article proposes an innovative approach for smoothing which allows for data to have abrupt “two-dimensional” changes as well as “ridges” – one dimensional statistically significant effects. The third part is an article “Low-dimensional decomposition, smoothing and forecasting of sparse functional data”. This article proposes an innovative approach of dealing with bivariate data which allows the data to be sparse. The article demonstrates this approach by applying it to two different problems. The first is related to sparse medical data. The second is related to forecasting where the values, which need to be forecasted, are considered as missing values. The fourth part is an article “STR: A Seasonal-Trend Decomposition Procedure Based on Regression”. Proposing a new approach of decomposing seasonal time series, it sets a new level of simplicity and generality in this field. The last part concludes the thesis. It outlines the main ideas which drove my research as well as my understanding of my main contributions. It also discusses new possible research directions.