Singular value decomposition for time series analysis with applications to smart energy systems

In a world replete with observations (physical as well as virtual), many data sets are represented by time series. In its simplest form, a time series is a set of data collected sequentially, usually at fixed intervals of time. In a number of applications, the mean and the variance of the time series is time-invariant and there is no seasonality in the data (such time series is called stationary). However, in many more applications, e.g., time series that are related to smart energy systems, the data have non-stationary characteristics. This thesis focuses primarily on matrices as an alternative representation of the latter type of time series, in order to take advantage of matrix decomposition methods. The rationale is straightforward: numerically stable matrix decomposition techniques enable us to extract underlying patterns in the data and use them to construct approximations of the corresponding time series. In particular, we will focus on singular value decomposition (SVD) as a powerful and numerically stable matrix factorization technique. Therefore, as the first step in this thesis, the SVD and its geometrical interpretation are extensively studied, in order to acquire a firm understanding of how it performs. That in turn enables us to look at different problems in time series analysis from a fresh perspective. For most of the applications of SVD in various fields, it is important to understand the properties of the SVD of a matrix whose entries show some degree of random fluctuations. Therefore, to determine how the noise level affects the singular value spectrum, it is essential to study the singular value decomposition of random matrices. As we will explain in the introductory chapter, one of the early applications of the SVD in time series analysis is in periodicity detection of the time series data. Therefore, we explore how the geometry of a matrix (the position of the data points with respect to the origin) and the aspect ratio of the matrix (the ratio between the number of columns and the number of rows) can affect its SVD results. Matrix factorisation techniques such as principal component analysis (PCA) and singular value decomposition (SVD) are both conceptually simple and effective. However, it iswell-known that they are sensitive to the presence of noise and outliers in input data. One way to mitigate this sensitivity is to introduce regularisation. To this aim, we hark back to the interpretation of SVD and PCA in terms of low-rank approximations, which involve the minimisation of specific functionals. We then derive algorithms for the minimisation of the regularised version of such functionals...