Research
My current main research interest lies in the connections between applied relational data analysis (order and lattice theory) and nonparametric statistics. Specifically, I am trying to utilize elements of statistical learning theory (Vapnik-Chervonenkis theory) for a fruitful stochastic analysis of relational methods. In particular, I am working on the development of regularization techniques for relational methods that are transparent in the sense that substantive science matters as well as measurement theory aspects can be explicitly taken into account.
In terms of applications, this covers very different areas such as knowledge discovery in data bases (e.g., subgroup discovery and association rules), generalizations of nonparametric regularized statistical tests which find application e.g., in the context of multidimensional stochastic dominance or spatial statistics, as well as applications in the context of classical machine learning (such as supervised classification). I am convinced that with this kind of research I contribute to strengthening the connections between theoretical statistics and applied machine learning, which is nowadays undoubtedly a key stone that guides modern artificial intelligence applications into a still more fruitful and lively direction.
Not only in technical terms, the theoretical concepts used here are closely related to Formal Concept Analysis, which provides powerful tools for both analysis and concrete computation. In order to make relational methods well applicable in practice, for example in the context of subgroup discovery, other computational techniques, in particular the application of mixed-integer linear programming or the approximation of hard computational problems (such as the approximation of optimization over a closure system by optimization over an approximating local ring of sets), are also a research topic.
In addition, I am generally interested in the theory of imprecise probabilities, especially in the context of the analysis of deficient data and partial identification, and in decision theory under complex uncertainty, here especially in the context of stochastic orders.
In terms of applications, this covers very different areas such as knowledge discovery in data bases (e.g., subgroup discovery and association rules), generalizations of nonparametric regularized statistical tests which find application e.g., in the context of multidimensional stochastic dominance or spatial statistics, as well as applications in the context of classical machine learning (such as supervised classification). I am convinced that with this kind of research I contribute to strengthening the connections between theoretical statistics and applied machine learning, which is nowadays undoubtedly a key stone that guides modern artificial intelligence applications into a still more fruitful and lively direction.
Not only in technical terms, the theoretical concepts used here are closely related to Formal Concept Analysis, which provides powerful tools for both analysis and concrete computation. In order to make relational methods well applicable in practice, for example in the context of subgroup discovery, other computational techniques, in particular the application of mixed-integer linear programming or the approximation of hard computational problems (such as the approximation of optimization over a closure system by optimization over an approximating local ring of sets), are also a research topic.
In addition, I am generally interested in the theory of imprecise probabilities, especially in the context of the analysis of deficient data and partial identification, and in decision theory under complex uncertainty, here especially in the context of stochastic orders.
Keywords
- Relational data analysis
- Formal concept analysis
- Vapnik-Chervonenlis theory
- Stochastic Orders
- Descriptive analysis and inference for deficient or non-standard data
- Partial identification
- Theories of imprecise probabilities
- Non-cardinal and partially ordered scales of measurement in the context of item response theory and the measurement of utility, especially under uncertainty and imprecise probabilities
Links
- Working group method(olog)ical foundations of statistics and its applications, headed by Thomas Augustin
- Jean Baccelli, University of Oxford
- Hannah Blocher (LMU Munich)
- Frank Coolen, Department of Mathematical Sciences, Durham University
- Scott Ferson (University of Liverpool)
- Christoph Jansen (LMU Munich)
- Munich Center for Mathematical Philosphy (MCMP)
- Society for Imprecise Probabilities: Theories and Applications (SIPTA)
- Krasymyr Tretiak (University of Liverpool)
- Lev Utkin, Institute of Applied Mathematics and Mechanics, Peter the Great St. Petersburg Polytechnic University
Publications
Work in Progress
- Hannah Blocher and Georg Schollmeyer (2023): Data depth functions for non-standard data by use of formal concept analysis (submitted)
- Georg Schollmeyer, Hannah Blocher, Christoph Jansen and Thomas Augustin (2023): On the analysis of epiontic data: a case study. Poster abstract. ISIPTA '23 (abstract)
- Successful and Unsuccessful Learning With and Without Guarantees: A short Didactical Remark on Idempotent Learners, Empirical Risk Minimization and a Look Beyond Idempotency (together with Christoph Jansen)
- Duality in Subgroup Discovery: Solving the Exhaustive Search Problem Using Formal Concept Analysis and Mixed Integer Linear Programming (Abstract)
- Starshaped subgroup discovery with uniform Vapnik-Chervonenkis-guarantees (together with Hannah Blocher and Christoph Jansen, Abstract)
- Multi-target decision making under conditions of severe uncertainty (together with Christoph Jansen and Thomas Augustin, Preprint).
- Robust statistical comparison of random variables with locally varying scale of measurement (with Christoph Jansen, Hannah Blocher, Julian Rodemann and Thomas Augustin, Abstract).
- Systematic Bias in Subgroup Discovery: A VC-analysis and a remedy
- Neural network model for imprecise regression (together with Krasymyr Tretiak and Scott Ferson, preprint)
- A Statistical Depth Function for Non–Standard Data based on Formal Concept Analysis (together with Hannah Blocher, Abstract)
- On depth functions and robustness in formal concept analysis (together with Hannah Blocher, Abstract)
- On the uniform control of the Vapnik-Chervonenkis dimension in subgroup discovery using formal concept analysis (preliminary draft)
- Regularization in exploratory data analysis: Rethinking Vapnik’s ’Rethinking statistical learning theory' in the context of social sciences and subgroup discovery
Papers
- Jansen, C.; Schollmeyer, G.; Blocher, H.; Rodemann, J.; Augustin,T. (2023): Robust statistical comparison of random variables with locally varying scale of measurement. In: Uncertainty in Artificial Intelligence (UAI 2023). PMLR. [preprint]
- Jansen, C.; Schollmeyer, G.; Augustin,T. (2023): Multi-target decision making under conditions of severe uncertainty. Forthcoming in: Torra, V. et al. (eds): Modeling Decisions for Artificial Intelligence. Lecture Notes in Computer Science, vol 13890. Springer.
[preprint], - Blocher, H.; Schollmeyer, G.: Jansen, C.; Nalenz, M. (2023): Depth functions for partial orders with a descriptive analysis of machine learning algorithms. In: Proceedings of the Thirteenth International Symposium on Imprecise Probabilities: Theories and Applications (ISIPTA '23). PMLR.
- Schollmeyer, G.; Bocher,H.; Jansen, C.; Augustin,T. (2023): On the Analysis of Epiontc Dta: A Case Study. Forthcoming in: Proceedings of the Thirteenth International Symposium on Imprecise Probabilities: Theories and Applications (ISIPTA '23). PMLR (to appear). [Abstract]
- Rodemann, J.; Jansen, C.; Schollmeyer, G.; Augustin, A. (2023): In all Likelihoods: How to Reliably Select Pseudo-Labeled Data for Self-Training in Semi-Supervised Learning. Forthcoming in: Proceedings of the Thirteenth International Symposium on Imprecise Probabilities: Theories and Applications (ISIPTA '23). PMLR (to appear).[preprint]
- Blocher, H.; Schollmeyer, G.; Jansen, C. (2022): Statistical models for partial orders based on data depth and formal concept analysis. In: Ciucci, D.; Couso, I.; Medina, J.; Slezak, D.; Petturiti, D.; Bouchon-Meunier, B.; Yager, R.R. (eds): Information Processing and Management of Uncertainty in Knowledge-Based Systems. Communications in Computer and Information Science, vol 1602, Springer.
[link, preprint] - Jansen, C.; Blocher, H.; Augustin, T.; Schollmeyer, G. (2022): Information efficient learning of complexly structured preferences: Elicitation procedures and their application to decision making under uncertainty . International Journal of Approximate Reasoning, 144 C: 69 - 91.
[link, early poster version, preprint] - Baccelli, J.; Schollmeyer, G.; Jansen, C. (2021): Risk Aversion over Finite Domains. Theory and Decision.
- Augustin, T.; Schollmeyer, G. (2021): Comment: On focusing, soft and strong revision of Choquet capacities and their role in statistics. Statistical Science, 36(2):205 – 209.
- Schollmeyer, G. (2021): Computing simple bounds for regression estimates for linear regression with interval-valued covariates. In Jasper De Bock, Andr ́es Cano, Enrique Mirand, and Seraf ́ın Moral, editors, Proceedings of the Twelfth International Symposium on Imprecise Probabilities: Theories and Applications, Proceedings of Machine Learning Research, Granada, Spain, 06–09 Jul 2021. PMLR.
- Kreiss, D.; Schollmeyer, D.; Augustin, T. (2021): Towards improving electoral forecasting by including undecided voters and interval-valued prior knowledge. In Jasper De Bock, Andr ́es Cano, Enrique Mirand, and Seraf ́ın Moral, editors, Proceedings of the Twelfth International Symposium on Imprecise Probabilities: Theories and Applications, Proceedings of Machine Learning Research, Granada, Spain, 06–09 Jul 2021. PMLR.
- Schollmeyer, G. (2019): A Short Note on the Equivalence of the Ontic and the Epistemic View on Data Imprecision for the Case of Stochastic Dominance for Interval-Valued Data. In: De Bock, J.; de Campos, C.; de Cooman, G.; Quaeghebeur, E.; Wheeler, G. (eds): Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications, in PMLR, Volume 103 of Proceedings of Machine Learning Research, pages 330-337
- Fuetterer, C.; Schollmeyer, G.; Augustin, T. (2019): Constructing Simulation Data with Dependency Structure for Unreliable Single-Cell RNA-Sequencing Data Using Copulas. In: De Bock, J.; de Campos, C.; de Cooman, G.; Quaeghebeur, E.; Wheeler, G. (eds): Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications, in PMLR, Volume 103 of Proceedings of Machine Learning Research, pages 216-224
- Jansen, C.; Schollmeyer, G.; Augustin, T. (2018): Concepts for decision making under severe uncertainty with partial ordinal and partial cardinal preferences. International Journal of Approximate Reasoning, 98: 112–131.
- Jansen, C.; Schollmeyer, G.; Augustin, T. (2018): A probabilistic evaluation framework for preference aggregation reflecting group homogeneity. Mathematical Social Sciences, 96: 49-62.
- Jansen, C.; Schollmeyer, G.; Augustin, T. (2017): Quantifying degrees of E-admissibility in decision making with imprecise probabilities. To appear in: Theory and Decision Library, Springer.
- Jansen, C.; Schollmeyer, G.; Augustin, T. (2017): Concepts for decision making under severe uncertainty with partial ordinal and partial cardinal preferences. In: Antonucci, A.; Corani, G.; Couso, I.; Destercke, S. (eds): Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications, Volume 62 of Proceedings of Machine Learning Research, pages 253–264, PMLR.
- Jansen, C.; Augustin, T.; Schollmeyer, G. (2017): Decision theory meets linear optimization beyond computation. In: Antonucci, A.; Cholvy, L.; Papini, O. (eds): Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2017.Lecture Notes in Computer Science, vol 10369. Springer.
- Plass, J., Cattaneo, M., Schollmeyer, G., Augustin, T. (2017): On the testability of coarsening assumptions: A hypothesis test for subgroup independence. International Journal of Approximate Reasoning, 90:292-306.
- Plass, J., Cattaneo, M., Schollmeyer, G., Augustin, T. (2016): Testing of coarsening mechanisms: Coarsening at random versus subgroup independence. In Maria Brigida Ferraro, Paolo Giordani, Barbara Vantaggi, Marek Gagolewski, Maria Angeles Gil, Przemyslaw Grzegorzewski, Olgierd Hryniewicz, editors, Advances in Intelligent Systems and Computing, pages 415 to 422. SMPS, 2016.
- Schollmeyer, G. (2015): On the Number and Characterization of the Extreme Points of the Core of Necessity Measures on Finite Spaces. ISIPTA '15, Proceedings of the Ninth International Symposium on Imprecise Probability: Theories and Applications.
- Plass, J. and Augustin, T. and Cattaneo, M. and Schollmeyer, G. (2015): Statistical modelling under epistemic data imprecision: Some results on estimating multinomial distributions and logistic regression for coarse categorical data. In Thomas Augustin, Serena Doria, Enrique Miranda, and Erik Quaeghebeur, editors, ISIPTA '15, Proceedings of the Ninth International Symposium on Imprecise Probability: Theories and Applications, pages 247 to 256. SIPTA.
- Schollmeyer, G., Augustin, T. (2015): Statistical modeling under partial identification: Distinguishing three types of identification regions in regression analysis with interval data. International Journal of Approximate Reasoning, 56: 224-248.
- Schollmeyer, G., Augustin, T. (2013): On Sharp Identification Regions for Regression Under Interval Data. ISIPTA '13, Proceedings of the Eighth International Symposium on Imprecise Probability: Theories and Applications
Technical Reports & Preprints
- A note on the connectedness property of union-free generic sets of partial orders. arXiv
- Jansen, C., Schollmeyer, G., Augustin, T. (2022): Multi-Target Decision Making under Conditions of Severe Uncertainty. arxiv.
- Schollmeyer, G., Jansen, C., Augustin, T. (2017): A simple descriptive method for multidimensional item response theory based on stochastic dominance. Technical Report 210, Department of Statistics, LMU Munich.
- Schollmeyer, G., Jansen, C., Augustin, T. (2017): Detecting stochastic dominance for poset-valued random variables as an example of linear programming on closure systems. Technical Report 209, Department of Statistics, LMU Munich.
- Schollmeyer, G. (2017): Application of lower quantiles for complete lattices to ranking data: Analyzing outlyingness of preference orderings. Technical Report 208, Department of Statistics, LMU Munich.
- Schollmeyer, G. (2017): Lower Quantiles for Complete Lattices. Technical Report 207, Department of Statistics, LMU Munich.
- Jansen, C., Schollmeyer, G., Augustin, T. (2016): Probabilistic Evaluation of Preference Aggregation Functions: A Statistical Approach in Social Choice Theory. Technical Report 193, Department of Statistics, LMU Munich.
- Schollmeyer, G., Augustin, T. (2013): On Sharp Identification Regions for Regression Under Interval Data. Technical Report 143, Department of Statistics, LMU Munich.
- Plass, J., Cattaneo, M., Augustin, T., Schollmeyer, G.; Heumann, C. (2017): Towards a reliable categorical regression analysis for non-randomly coarsened observations: An analysis with German labour market data. Technical Report 206, Department of Statistics, LMU Munich.
- Plass, J., Cattaneo, M., Schollmeyer, G., Augustin, T. (2017): On the testability of coarsening assumptions: A hypothesis test for subgroup independence. Technical Report 201, Department of Statistics, LMU Munich.
Theses
- Dissertation: Reliable statistical modeling of weakly structured information: contributions to partial identification, stochastic partial ordering and imprecise probabilities
- Diploma Thesis: Modellierung unsicheren Wissens durch kohärente Prävisionen
Other
- Habilitationsschrift
- Matthias Mohr, Markus Heydenreich, Georg Schollmeyer, Stefan Ufer: Berufsfeldbezug in den fachwissenschaftlichen Studienanteilen des Lehramtsstudiums Mathematik – Problemaufriss und Initiativen
Presentations
- Presentation Habilitation
- Counting Concepts Part (I+) II: Counting Concepts
- Counting Concepts Part I: Motivating Examples & Comments (05.12.2023)
- Generalizing rings of sets in a star-spangled manner: Star-shaped subgroup discovery
- On the Uniform Control of the Vapnik-Chervonenkis Dimension in Subgroup Discovery using Formal Concept Analysis (DAGStat 2022)
- Computing Simple Bounds for Regression Estimates for Linear Regression with Interval-valued Covariates (ISIPTA 2021, Slides, Poster)
- On depth functions and robustness in formal concept analysis: The (double-)peeling depth (together with Hannah Blocher, research seminar, 03.08.2021).
- Fallibilistic regularization for (semi-inductive) abduction by adjoining auxiliary co-outcomes (research seminar, 18.03.2021).
- Relational data analysis for weakly structured information: Utilizing linear and binary programming for computing supremum statistics on closure systems. ECDA 2018.
- Classification with stylized betweenness-relations allowing for regularization with uniform Vapnik-Chervonenkis-guarantees. DAGStat 2019.
- Decision making under severe uncertainty with partial ordinal and partial cardinal preferences. ISIPTA 2017.
- Quantile constructions for complete lattices. DAGStat 2016.
- Simple multivariate Kolmogorov-Smirnov type tests based on lattice-valued quantiles. 12th German Probability and Statistics Days 2016.
- Partial identification in linear models: Regression with interval-valued data. CM Statistics (ERCIM 2015), London, 12-14 December 2015.