M.Sc. Kevin Höhlein

Parallel Coordinates Plots (PCPs, eg., Stumpfegger et al., 2022) are a valuable tool for visualizing high-dimensional data. In contrast to well-known scatter plots and similar bivariate data visualizations, PCPs allow for displaying many parameters at the same time along parallel coordinate axes, which enables the user to identify multi-parameter relations in the data, which would remain hidden in other visualizations. PCPs are most powerful when used in an interactive setting, where the data analyst can interactively exchange the axis ordering or select subsets of the data for more detailed analysis. Especially the subselection process, also known as brushing, can improve the visibility of hidden structure in the data. Finding good brushes, however, can be a tedious task when dealing with very high-dimensional data, since a lot of brushing configurations have to be examined and compared manually. A partially automated brushing process, in which the user receives proposals of potentially interesting brushes could therefore increase the usability of PCPs in data-intensive analysis scenarios. The task in this thesis will be to explore one way of achieving such an automated brushing, based on neural-network based similarity learning. Building up on the idea of generative adversarial learning, a differentiable brushing approach is to be implemented and compared against competitor approaches for smart brushing from the relevant literature.

T-distributed stochastic neighbor embedding (tSNE) is a powerful tool for nonlinear dimension reduction with various applications in the analysis of high-dimensional data (van der Maaten and Hinton, 2008). The core idea of the algorithm lies in exploiting neighborhood relations between data points to identify local structure in high-dimensional datasets. Dimension reduction is achieved by seeking a lower-dimensional embedding space, in which point neighborhoods preserve the neighborhood structure of the high-dimensional data space as accurately as possible. Neighborhood information in both high- and low-dimensional spaces is expressed in terms of point-wise probability mass functions, which measure the likelihood of a reference data point picking a respective other point of the dataset as its nearest neighbor. The embedding search proceeds by updating the position of the embeding points iteratively such that the neighborhood probabilities in the embedding space for all points align with the neighborhood probabilities in the high-dimensional data space. In the original t-SNE algorithm, similarities between probability functions are measured according to the Kullback-Leibler (KL) divergence. While tSNE has proven very powerful in a variety of application scenarios, the KL divergence is known to have a number of problems, which are related to the structure of the similarity relations implied by this criterion. This raises the question whether a different similarity metric could further improve the achievable embedding quality. An alternative metric for comparing discrete probabilities functions is the Wasserstein distance, which was shown to imply smoother and often more expressive similarities in machine learning tasks (eg. Arjovsky et al., 2017). In this work, the task is to explore how the embedding quality of tSNE changes if the standard KL criterion in the tSNE algorithm is replaced with an analogon based on the Wasserstein distance. The new algorithm will need to be implemented and compared to standard tSNE with respect to embedding quality and runtime performance. In addition to the standard algorithm, the use of acceleration structures can be explored.

Deep learning methods have achieved impressive results in classifying and segmenting natural images, making deep learning models a powerful tools for scene understanding and and reasoning tasks in computer vision. Beyond classifying image contents according to a predefined classification scheme, recent research has focussed on unsupervised and self-supervised training of image segmentation models and learning scene understanding on a purely data-driven basis. Particularly interesting results have been achieved by Caron et al. (2021), who have shown that suitably designed neural network models can learn well-localised and informative image feature descriptors from image data without the need for expensive class labels. The impressive ability of such models to find structure in raw image data raises the question whether similar model architectures can be applied to gridded meteorological data to find reoccuring statistical patterns that have been overlooked by classical meteorological analyses. Within this maste thesis, students are expected implement and train neural network models similar to those of Caron et al. (2021), which shall be used to analyse gridded meteorological simulation data. The models are expected to yield feature representations of the physical field variables, which are to be analysed and visualised. Prior experience in implementing and training convolutional neural networks, as well as a thorough understanding of statistical data analysis, are strictly required.

Feature Attribution, i.e. attributing the prediction of a deep learning model to its input features (e.g. https://arxiv.org/abs/1703.01365), is among the most promising methods that allow peaking inside the black box of nonlinear statistical models, and enable the user to obtain more detailed knowledge on the reasoning processes behind deep learning predictions. In brief, the method relies on comparing the prediction for a particular input sample to the model prediction for a selected reference sample, and attributing differences in predictions to differences in the inputs. The feature attribution is obtained by integrating back-propagated gradients along a pre-defined path in input-space. More important feature differences then yield larger attribution values, whereas less important differences are assigned smaller attribution values. In many cases, the path between images is defined as just a linear connection between reference sample and the prediction sample in question. However, already in comparatively simple model settings, like image classification, for example, the direct connection may be far from optimal. The linear combination between a pair of RGB images, for example, usually does not correspond to an image, which matches the data distribution the statistical model has seen during training. Attribution results may thus be unreliable and not deliver accurate results.

The goal of this thesis will be to explore how generative deep learning models in combination with manifold learning approaches can be used to identify more informative, data-adaptive pathways for the integration and how this impacts the quality of the attribution results.

The visualization scalar fields, such as density fields, temperature maps, or pressure distributions, on spatially extended domains is an important problem in scientific data visualization. For scalar fields in 2 or 3 spatial dimensions, with or without explicit time dependence, simple visualization methods such as density or contour plots, quickly run into problems. Scalar field topology (see, e.g., Heine et al., 2016, for examples) offers a principled way to extract feature descriptors from the field data, which summarize local maxima and minima or gradients of the fields, and can be visualized concisely. Knowledge of such features can help scientists to gain a more intuitive und understanding of their data and improve the applicability of statistical analysis tasks.

The goal of this thesis is to apply methods from topology-based scalar field analysis to meteorological simulation data. Different analysis methods are to be compared, both with respect to computational feasibility and interpretability of the results.

Vector fields, i.e. vector-valued functions, which assign a direction to every point of a spatial domain, appear in many fields of science in engineering. Most naturally, they are used to describe the motion of liquids or gases in 2D or 3D spaces, with applications in aerodynamics, fluid mechanics or atmospheric research. More abstractly and in higher dimensional spaces, however, they can also be used to capture the temporal evolution of more general dynamical systems, opening up applications in electrical engineering or control theory. Oftentimes, vector fields are interpreted as representing a map of the temporal or spatial flow of particles, fluids or abstract system states.

To analyze vector fields, especially in 2D and 3D, a common approach consists in extracting prominent features from the flow to create a so-called topological skeleton, which provides qualitative information on how the flow behaves in different regions of space. While some of the features, like source points, sinks or separation lines between distinct laminar flow regimes, are easy to extract, periodic motion on closed curves in space is harder to identify. Specifically self-sustained periodic motions, however, are often of particular interest in dynamical system analysis, as they may give rise to damaging periodic forcings in mechanical applications or indicate undesired feedback loops in control systems.

The goal of this work is to examine how suitably parameterized neural networks in combination with simple, iterative optimization procedures can be used to identify curves of periodic motion in vector fields in 2D, 3D or higher dimensions, and how the curves can be tracked through the spatial domain under smooth changing of the vector fields.

With the increasing adoption of data-driven analysis methods in earth sciences, there is a growing need for a proper understanding of correlation patterns within meteorological and climatological datasets. Understanding potentially nonlinear correlation patterns is important for multiple different reasons. On the one hand, correlation patterns within or among datasets may hint towards causal relationships between datasets and may thus provide guidance on the way to more effective usage of the available data in new application tasks. One example for such data applications beyond the scope of the actual data product is connection quality data from cellular network operators, which can be used to locate and track rain showers in rural areas. On the other hand, correlation patterns in data may affect probability estimates, which are derived from data-driven statistical models. As an example for this, one can consider weather forecasts for high-impact weather, such as hail storms or heavy rainfalls. Though such events are fairly uncommon, they may cause severe damage when they occur. As a result, it is important to have precise estimates for the probability of such events to occur, in order to give out weather warnings if needed and avoid warnings when no danger is to be expected. Unresolved correlations in the forecast data can cause bias in these probability estimates and render the forecasts unnreliable. From a third perspective, knowing about correlations is also important, when datasets shall be split and used for training and validation of machine-learning models. Correlations between training and validation data should in general be avoided as far as possible in order to achive reliable model validation.

The purpose of this work will be to investigate spatio-temporal correlation patterns in meteorological datasets. Starting from bivariate linear correlation measures, which can be used, e.g., to describe spatial or temporal auto-correlations in low-dimensional data, the scope shall be broadened in the long-run to nonlinear correlation measures, such as mutual information, which have recently become applicable to high-dimensional datasets through the adoption of neural-network-based estimation methods (e.g., MINE: https://arxiv.org/abs/1801.04062). The tasks throughout the work include a literature review focussing on related and relevant work, application of the most promissing approaches to exemplary meteorological datasets, and visualization of the identified correlation patterns. A thorough knowledge of statistics is required as well as an independent, structured and well-focussed style of working.