Topics for BA or MA Theses

Working Group FDA

Please contact Fabian Scheipl if you’re interested in one of these BA or MA thesis topics or if you want to discuss related ideas of your own.

Last update: 2024-03-26

Topic Area: Improving tidyfun and related packages

tidyfun is an R package for functional data analysis currently under development. Some of the issues tracked on Github for this and its underlying infrastructure package tf could also be good topics for theses.
For BA theses, we would keep the focus on refactoring/evaluating/describing existing implementations or applying those to real data, for MA more novel developments and detailed theory along with clean and performant implementations would be expected as well.

Topic: Implementing and comparing quantile methods for functional data (BA/MA)

The functional data literature contains many possible definitions of “function-valued quantiles”. We would pick out some of the most relevant/interesting of these, summarize the relevant theory behind them, implement them for use within tidyfun, and perform a comparison based on real and/or synthetic data sets.

A minimal BA thesis in this topic area would be re-implementing, documenting and validating (most of) the methods in the rainbow package integrated into / as an add-on package for tf & tidyfun.

Topic: Implementing and comparing registration/alignment methods (BA/MA)

Functional data contains both vertical (amplitude - how large is the peak/valley) and horizontal (phase - where is the peak/valley) variability. The latter requires more sophisticated mathematical theory and complex algorithms to deal with. Potential tasks here include:

• defining & implementing additional data structures, classes & methods to represent & visualize aligned functions along with their corresponding warping functions
• writing glue code for using registration packages like fdasrvf, registr, DTW methods with tf vectors
• … or (re-)implementing (simpler) alignment methods (like fda’s landmark alignment or alignment based on FPC 1 (“continuous registration”))
• implementation of summary statistics, visualizations, diagnostics etc for the results of registration/alignment procedures

Stretch goals here include implementing methods for noisy and/or sparse and/or non-Gaussian/discrete functional data and accommodating functional fragments/unequal domains with functions of different observed lengths. Excellent review of (mostly) SRVF framework: Wu et al (2023, ch. 3 f)

Topic: Implementing multivariate functions & scalar fields (MA, maybe BA)

Extend tf-classes and methods for

• multivarate functions with vector outputs ($f:\mathbb R \to \mathbb R^d$ for $d>1$)
• scalar fields ($f:\mathbb R^q \to \mathbb R$ for $q>1$)

This is a large SWE task - scope would probably be one of the above, limited to either extending tfd or tfb, and may require major refactoring of tf to make such an extension work smoothly and consistently (e.g. it probably requires definining new classes and logic for arg-“vectors” and function domains).

Topic: Representation and computation for probability densities in Bayes space (BA, maybe MA)

The Bayes Space paradigm developed by v.d. Boogart, Hron, Egozcue and others (e.g. v.d. Boogart et al. (2014), Hron et al. (2016)) provides a way to represent probability measures so that their addition and multiplication are well defined, enabling simple summary statistics (means etc) as well as methods such as PCA or linear regression for probability-density-valued data – i.e. the unit of observation is represented by an entire probability distribution, not a single value, and the inferential goal is typically to understand how other covariates are associated with changes in these distributions. This has many interesting applications, for example see Meier et al, (2021) for differential effects of family formation on gender-specific income distributions in East and West Germany or (Menafoglio et al, 2021) for an application to groundwater monitoring.
A thesis on this topic would

• summarize the necessary theoretic background and literature
• implement functionality for tf and tidyfun that represents density data and performs arithmetic operations as well as basic statistics in Bayes space,
• apply this to an interesting real-world data set (or: replicate a published analysis in this context with the new implementation).

Topics: Write tidyfun scripts for Craniceanu et al’s “Functional Data Analysis with R” / Ramsay et als’s “Functional Data Analysis with R and MATLAB”

Both of these books contain many chapters, data sets and case studies that could also be done (mostly) using tidyfun and/or refund.
We’ll select some of them, you’ll identify and implement missing functionality in tidyfun with my help, and write them up with all the necessary theoretical background and some extensions, in an online document / as vignettes for tidyfun.

Books: Craniceanu et al. (2024), Ramsay et al. (2009)

Topic: Registration based on Peak-Persistence Diagrams (MA, BA very maybe)

Summarize, implement & evaluate SRVF-based function registration using the peak-persistence diagrams of Kim, Dasgupta, Srivastava (2023). This topic would involve some more advanced and interesting maths and algorithms like differential geometry, topology, dynamic programming optimization. The paper to implement is bleeding edge state of the art, so this makes an excellent topic for people considering a PhD and looking for a thesis topic that might turn into something publishable. Potential tasks would include:

• summarizing the maths behind these methods
• implementation of the algorithms and visualizations from the paper in R, preferably using infrastructure of / integrated into tf/tidyfun
• benchmarking against other registration approaches available in R
• application to real world datasets (e.g. mouse brain stem audiograms, bodyweight fitness movement patterns, …)

Stretch goals would include extending this to either non-Gaussian/discrete functional data or accommodating functional fragments/unequal domains with functions of different observed lengths, based on ideas we’d develop together.

Topic Area: Improving manifun (BA/MA)

manifun is a small, unpublished R package for dimension reduction and embedding visualization (primarily) for functional data. Possible tasks include implementing suitable interfaces to mlr3 and/or tidyfun. Implementing AUMVC framework could be included in this topic area as well.

Topic: Improved interactive visualization of functional data embeddings (MA)

The central goal of the project is to improve existing and implement new embedding (i.e. dimension reduction) visualization approaches. Fairly flexible, interactive versions of the kind of visualization shown below: that includes e.g. tooltips/interactive highlighting when hovering over specific data points, brushing for selecting and highlighting specific embedding regions or curves, etc have already been implemented in a previous MA thesis (EmbedIt, Jennert 2023).

Thesis goals could include:

• Re-implementing EmbedIt based on more performant software like D3.js or refactoring it for better responsiveness etc
• Adding interactive 3D visualizations
• Implementing the “grand tour” and other classic multivariate exploration tools (c.f. tourr)
• Adding pre-processing and embedding steps to the existing app

Topic Area: (FDA) Outlier Detection

Beyond the methodological/theoretical topics below, we could develop more applied thesis topics in this context together with external partners that deal with large functional data sets such as the German Mouse Clinic (e.g. auditory brain stem response curves) or with (partners of) Prof. Christian Müller’s group at the Institute of Statistics.

Topic: Realistic functional outlier benchmark datasets (BA)

Realistic evaluation of outlier detection should use real datasets with real outliers. Usually, this is done by selecting all majority class observations from a labeled dataset and contaminating them with a few randomly sampled instances from other minority classes. This approach yields “false” negatives/positives unless the minority class is really sufficiently and consistently different from the majority class observations. The goal of this project is to investigate under which circumstances this “unless” applies by comparing two approaches:

• use only datasets from the mlr-fda classification benchmark (pdf) that were predicted very accurately to generate outlier detection benchmark data
• for the generated benchmark datasets, use detailed observation-level mlr-fda benchmark results to pick only those minority (and maybe also majority?) class observations that were consistently labeled correctly

Additionally, we are interested in how these results are affected by measures of dataset structure like separability ( pdf) and intrinsic dimensionality (pdf, CRAN).

Topic: Implementing the AUMVC framework (MA)

The area-under-the-mass-volume-curve (AUMVC, pdf) can be used to tune outlier detection algorithms. Yet, a major caveat is that it relies on MC simulations for approximating integrals and is thus not applicable to high-dimensional settings. Combining it with dimension reduction and manifold learning may allow to solve this issue.
The goal of this project is to implement the AUMVC framework in manifun and to conduct initial experiments. The central questions to be answered are:

• How robust is the AUMVC approach to the ambient dimensionality of the data? This could be assessed by comparing results on image and functional data.
• A further question, which may be investigated: are there any distance measures for complex data such as images, which can be used to induce suitable bias for AUMVC to work on (embeddings of) such data?
• stretch goal: can AUMVC be adapted to sample only from the relevant space so it scales better to (nominally) high-dimensional data? e.g. by simulating data uniformly from the (convex hull) of the observed data or from lower dimensional (but: almost losslessly compressed) representations/embeddings of the data?

Topic: Criterion for structural outlyingness (BA)

Multi-dimensional scaling (MDS) can be used to represent the outlier structure of functional datasets (pdf). However, MDS embeddings represent the entire data structure (… hopefully, at least), not just structural outlyingness. Since MDS embedding dimensions are sorted by decreasing amount of “explained structure”, this might lead to components of structural outlyingness being represented only in “late” embedding dimensions in datasets with few outliers and complex structured variation of high rank.
You would develop and evaluate a procedure for identifying embedding dimensions (or 2D-subspaces of such embeddings) in which structural outlyingness is reflected, i.e. the goal is to find relevant (combinations of) embedding dimensions for outlier visualization/detection (and, possibly, for tuning AUMVC if embeddings are too high-dimensional, see below). Possible approaches:

• use something like HiCS (Link; this is computationally heavy)
• run Local Outlier Factor (LOF) or similar methods on each 2D-subspace and pick the ones where upper-tail (!) dependencies with global LOF scores are maximal – “upper-tail” only because the strength of association between global LOF scores and corresponding subspace LOF scores for low/intermediate values is irrelevant for this issue.

Topic: Multivariate Functional Outlier Visualization and Detection (BA)

Replicate simulation study and application examples of Aléman-Gomez et al. (2021) with our geometric-topological approach and compare results.

Topic Area: Cluster Analysis

Topic: Embedding-based cluster analysis of overlapping / fuzzy clusters (MA)

Both UMAP and t-SNE, state-of-the-art manifold learning methods, can be used to detect and represent the cluster structure of complex, high-dimensional data. However, which method is more suitable for the task (or specific aspects of the task) under which conditions remains an open question. Answering this question is complicated since several hyperparameters need to be tuned for both methods and the underlying task is unsupervised (i.e., tuning is hard…).
While there are indications that UMAP leads to “better” clusterings (smaller intra-cluster distances, larger inter-cluster distances) when high-dimensional data consists of clearly separate clusters, the situation is less clear in situations with very close or even overlapping clusters. The focus of this project is to assess the latter setting and tasks include the following:

• set up extensive synthetic experiments to obtain an initial understanding of the problem. Possible factors to assess:
  • Overlap/Separability as a function of mean and variance of the underlying data generating processes
  • Parameter sensitivity w.r.t. to dimensionality and number of observations
  • Relevant parameters for improving separability
• investigate whether measures of separability (pdf) can be used to reliably infer the structure of data set

Topic Area: Regression Models with/for functional data

Refactoring refund::pffr & refund::pfr (MA)

These fairly old and rather badly written functions implement very general classes of penalized regression models (GAMs and GAMMs) for functional responses and/or predictors. Your thesis would be to re-write them from scratch with my help, using best practices for R programming like proper unit tests, input validation, and extensive documentation. This could also include developing a more stream-lined, consistent formula interface and developing better methods to deal with factor covariates and interaction effects as well as writing up some interesting case studies to be published as a vignette accompanying the package. See Scheipl & Greven (2017) for a review of the underlying methodology.