It is often said that mathematics is a language. If so, group theory provides the proper vocabulary for discussing symmetry. In the same way, lattice theory provides the proper vocabulary for discussing order, and especially systems which are in any sense hierarchies. One might also say that just as group theory deals with permutations, so lattice theory deals with combinations. [Birkhoff]
Relational data analysis
The goal of the methods developed here is the fruitful application of methods from order and lattice theory in the context of statistical data analysis, especially in the context of discrete, potentially deficient data or data with non-standard scale of measurement (e.g., partially ordered scale of measurement). This ranges from more qualitative data analysis (e.g., the use of closure operators as a tool of qualitative data analysis) to quantitative, stochastic analysis, including stochastic generalizations of relational concepts (e.g., stochastic dominance for random variables with partially ordered scale of measurement). An essential requirement here is, in particular, to enable a stochastic analysis of the given data situation beyond a purely descriptive mathematical structural analysis of the data, also in the sense of statistical inference. Here, elements of Vapnik-Chervonenkis theory are used in connection with discrete regularization approaches.
The goal of the methods developed here is the fruitful application of methods from order and lattice theory in the context of statistical data analysis, especially in the context of discrete, potentially deficient data or data with non-standard scale of measurement (e.g., partially ordered scale of measurement). This ranges from more qualitative data analysis (e.g., the use of closure operators as a tool of qualitative data analysis) to quantitative, stochastic analysis, including stochastic generalizations of relational concepts (e.g., stochastic dominance for random variables with partially ordered scale of measurement). An essential requirement here is, in particular, to enable a stochastic analysis of the given data situation beyond a purely descriptive mathematical structural analysis of the data, also in the sense of statistical inference. Here, elements of Vapnik-Chervonenkis theory are used in connection with discrete regularization approaches.