Pattern recognition in spaces of probability distributions for the analysis of edge-localized modes in tokamak plasmas

Magnetically confined fusion plasmas provide several data analysis challenges due to the occurrence of massive data sets, substantial measurement uncertainty, stochasticity and data dimensionality, and often nonlinear interactions between measured quantities. Recently, methods from the fields of machine learning and probability theory - some standard, some more advanced - have come to play an increasingly important role in analyzing data from fusion experiments. The capabilities offered by such methods to efficiently extract, possibly in real time, additional information from the data that is not immediately apparent to human experts, has attracted attention from an increasing number of researchers. In addition, innovative methods for real-time data processing can play an important role in plasma control, in order to ensure safe and reliable operation of the machine. Pattern recognition is a discipline within the information sciences that concerns the exploration of structure in (multidimensional) data sets using computer-based methods and algorithms. In this doctoral work, pattern recognition techniques are developed and applied to data from tokamak plasmas, in order to contribute to a systematic analysis of edge-localized modes (ELMs). ELMs are magnetohydrodynamic (MHD) instabilities occurring in the edge region of high-confinement (H-mode) fusion plasmas. The type I ELMy H-mode is the reference scenario for operation of the next-step fusion device ITER. On the one hand, ELMs have a beneficial effect on plasma operation through their role in impurity control. On the other hand, ELMs eject energy and particles from the plasma and, in ITER, large unmitigated ELMs are expected to cause intolerable heat loads on the plasma-facing components (PFCs). In interpreting experiments focused on ELM understanding and control, a significant challenge lies in handling the measurement uncertainties and the inherent stochasticity of ELM properties. In this work, we employ probabilistic models (distributions) for a quantitative data description geared towards an enhanced systematization of ELM phenomenology. Hence, we start from the point of view that the fundamental object resulting from the observation of a system is a probability distribution, with every single measurement providing a sample from this distribution. We argue that, particularly for richly stochastic phenomena like ELMs, the probability distribution of physical quantities contain significantly more information compared to mere averages. Consequently, in exploring the patterns emerging from the various ELM regimes and relations, we need methods that can handle the intrinsic probabilistic nature of the data. The original contributions of this work are twofold. First, several novel pattern recognition methods in non-Euclidean spaces of probability distribution functions (PDFs) are developed and validated. The second main contribution lies in the application of these and other techniques to a systematic analysis of ELMs in tokamak plasmas. In regard to the methodological aims of the work, we employ the framework of information geometry to develop pattern visualization and classification methods in spaces of probability distributions. In information geometry, a family of probability distributions is considered as a Riemannian manifold. Every point on the manifold represents a single PDF and the distribution parameters provide local coordinates on the manifold. The Fisher information plays the role of a Riemannian metric tensor, enabling calculation of geodesic curves on the surface. The length of such curves yields the geodesic distance (GD) on probabilistic manifolds, which is a natural similarity (distance) measure between PDFs. Equipped with a suitable distance measure, we extrapolate several distance-based pattern recognition methods to the manifold setting. This includes k-nearest neighbor (kNN) and conformal predictor (CP) methods for classification, as well as multidimensional scaling (MDS) and landmark multidimensional scaling (LMDS) for data visualization (dimensionality reduction). Furthermore, two new classification schemes are developed: a distance-to-centroid classifier (D2C) and a principal geodesic classifier (PGC). D2C classifies on the basis of the minimum GD to the class centroids and PGC considers the shape of the class on the manifold by determining the minimum distance to the principal geodesic of each class. The methods are validated by their application to the classification and retrieval of colored texture images represented in the wavelet domain. Both methods prove to be computationally efficient, yield high accuracy and also clearly exhibit the adequacy of the GD and its superiority over the Euclidean distance, for comparing PDFs. This also aids in demonstrating the utility and adaptability of the developed methods to a wide range of applications other than ELMs, which are the prime focus of analysis in this work. The second main goal of the work targets ELM analysis at three fronts, using pattern recognition and probabilistic modeling : i). We first concentrate on visualization of ELM characteristics by creating maps containing projections of multidimensional ELM data, as well as the corresponding probabilistic models. Such maps can provide physicists and machine operators with a convenient means and a useful tool for plasma monitoring and for studying data patterns reflecting key regimes and their underlying physics. In particular, GD-based MDS is used for representing the complete distributions of the multidimensional data characterizing the operational space of ELMs onto two-dimensional maps. Clusters corresponding to type I and type III ELMs are identified and the maps enable tracking of trends in plasma parameters across the operational space. It is shown that the maps can also be used with reasonable accuracy for predicting the values of the plasma parameters at a certain point in the operational space. ii). Our second application concerns fast, standardized and automated classification of ELM types. ELM types have so far been identified and characterized on an empirical and phenomenological basis. The presented classification schemes are aimed at complementing the phenomenological characterization using standardized methods that are less susceptible to subjective interpretation, while considerably reducing the effort of ELM experts in identifying ELM types. To this end, different classification paradigms (parametric and non-parametric) are explored and put to use. Discriminant analysis (DA) is used for determining a linear separation boundary between type I and III ELMs in terms of global plasma parameters, which can then be used for the prediction of ELM types as well as the study of ELM occurrence boundaries and ELM physics. However, DA makes an assumption about the underlying class distribution and presently cannot be applied in spaces of probability distributions, leading to a sub-optimal treatment of stochasticity. This is circumvented by the use of GD-based CP and kNN classifiers. CP provides estimates of its own accuracy and reliability and kNN is a simple, yet powerful classifier of ELM types. It is shown that a classification based on the distribution of ELM properties, namely inter-ELM time intervals and the distribution of global plasma parameters, is more informative and accurate than the classification based on average parameter values. iii). Finally, the correlation} between ELM energy loss (ELM size) and ELM waiting times (inverse ELM frequency) is studied for individual ELMs in a set of plasmas from the JET tokamak upgraded with the ITER-like wall (ILW). Typically, ELM control methods rely on the empirically observed inverse dependence of average ELM energy loss on average ELM frequency, even though ELM control is targeted at reducing the size of individual ELMs and not the average ELM loss. The analysis finds that for individual ELMs the correlation between ELM energy loss and waiting times varies from zero to a moderately positive value. A comparison is made with the results from a set of carbon-wall (CW) JET plasmas and nitrogen-seeded ILW JET plasmas. It is found that a high correlation between ELM energy loss and waiting time comparable to CW plasmas is only found in nitrogen-seeded ILW plasmas. Furthermore, most of the unseeded JET ILW plasmas have ELMs that are followed by a second phase referred to as the slow transport event (STE). The effect of the STEs on the distribution of ELM durations is studied, as well as their influence on the correlation between ELM energy loss and waiting times. This analysis has a clear outcome for the optimization of ELM control methods, while presenting insights for an improved physics understanding of ELMs.