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...

Germany is a forerunner in developing renewable energies. It is therefore of considerable interest to investigate the impact of switch to renewables on the market during transition era. The aim of this study is in two parts: 1) Investigating the volatility; and 2) Conducting a descriptive study on the evolution of daily profiles and emergence of non-positive prices. In terms of volatility quantification, the following characteristics of EPEX prices should be recognized: 1) Covering the whole year (24/7); 2) Taking non-positive values; 3) Depending on calendar information; and 4) Changing according to demand and supply availability. We, therefore, propose a robust and generic approach to account for diurnal or seasonal patterns of human activities in volatility analysis. An important distinction of our work is in introducing an alternative representation (as matrices) for quasi-periodic price data. We, herein, propose a new notion of volatility using a matrix decomposition technique, namely the singular value decomposition (SVD). Our observations indicate price volatility reduction, in the recent years. The second part of this article provides evidences of effect of renewables on daily price profiles – emergence of non-positive prices and also shifts of peak price values to hours where solar is less available.

Scenario-based probabilistic forecasting models have been explored extensively in the literature in recent years. The performance of such models evidently depends to a large extent on how different input (temperature) scenarios are being generated. This paper proposes a generic framework for probabilistic load forecasting using an ensemble of regression trees. A major distinction of the current work is in using matrices as an alternative representation for quasi-periodic time series data. The singular value decomposition (SVD) technique is then used herein to generate temperature scenarios in a robust and timely manner. The strength of our proposed method lies in its simplicity and robustness, in terms of the training window size, with no need for subsetting or thresholding to generate temperature scenarios. Furthermore, to systematically account for the non-linear interactions between different variables, a new set of features is defined: the first and second derivatives of the predictors. The empirical case studies performed on the data from the load forecasting track of the Global Energy Forecasting Competition 2014 (GEFCom2014-L) show that the proposed method outperforms the top two scenario-based models with a similar set-up.

We address the problem of data-driven pattern identification and outlier detection in time series. To this end, we use singular value decomposition (SVD) which is a well-known technique to compute a low-rank approximation for an arbitrary matrix. By recasting the time series as a matrix it becomes possible to use SVD to highlight the underlying patterns and periodicities. This is done without the need for specifying user-defined parameters. From a data mining perspective, this opens up new ways of analyzing time series in a data-driven, bottom-up fashion. However, in order to get correct results, it is important to understand how the SVD-spectrum of a time series is influenced by various characteristics of the underlying signal and noise. In this paper, we have extended the work in earlier papers by initiating a more systematic analysis of these effects. We then illustrate our findings on some real-life data.

Many time series in smart energy systems exhibit two different timescales. On the one hand there are patterns linked to daily human activities. On the other hand, there are relatively slow trends linked to seasonal variations. In this paper we interpret these time series as matrices, to be visualized as images. This approach has two advantages: First of all, interpreting such time series as images enables one to visually integrate across the image and makes it therefore easier to spot subtle or faint features. Second, the matrix interpretation also grants elucidation of the underlying structure using well-established matrix decomposition methods. We will illustrate both these aspects for data obtained from the German day-ahead market.