arXiv subjects

Christian Hennig

Publications and source records attributed to Christian Hennig.

At least 19 recordsLinked to original sources

arXiv2026

A unified approach to outlier identification for mixed-type data

We present an outlier identification method for mixed type data sets comprising continuous and ordinal variables. We define outliers based on using a multivariate Gaussian distribution as reference distribution for non-outliers, with a latent Gaussian assumed for ordinal variables. The proposed algorithm is based on the robust Minimum Covariance Determinant estimator for estimating the parameters of the multivariate Gaussian for the non-outliers. This is extended to account for the fact that the full Gaussian information underlying the ordinal variables is not observed. A breakdown theorem shows that replacing observations will noty stop extreme enough outliers from being identified. The effectiveness of our approach is demonstrated via simulations on synthetic data with various types of contamination, achieving high detection and low false positive rates. Practical relevance is illustrated through an application to Airbnb listing data containing both continuous and ordinal attributes.

stat.ME↗

arXiv2024

The quantile-based classifier with variable-wise parameters

Quantile-based classifiers can classify high-dimensional observations by minimising a discrepancy of an observation to a class based on suitable quantiles of the within-class distributions, corresponding to a unique percentage for all variables. The present work extends these classifiers by introducing a way to determine potentially different optimal percentages for different variables. Furthermore, a variable-wise scale parameter is introduced. A simple greedy algorithm to estimate the parameters is proposed. Their consistency in a nonparametric setting is proved. Experiments using artificially generated and real data confirm the potential of the quantile-based classifier with variable-wise parameters.

stat.ME↗

arXiv2024

Approaches to biological species delimitation based on genetic and spatial dissimilarity

The delimitation of biological species, i.e., deciding which individuals belong to the same species and whether and how many different species are represented in a data set, is key to the conservation of biodiversity. Much existing work uses only genetic data for species delimitation, often employing some kind of cluster analysis. This can be misleading, because geographically distant groups of individuals can be genetically quite different even if they belong to the same species. We investigate the problem of testing whether two potentially separated groups of individuals can belong to a single species or not based on genetic and spatial data. Existing methods such as the partial Mantel test and jackknife-based distance-distance regression are considered. New approaches, i.e., an adaptation of a mixed effects model, a bootstrap approach, and a jackknife version of partial Mantel, are proposed. All these methods address the issue that distance data violate the independence assumption for standard inference regarding correlation and regression; a standard linear regression is also considered. The approaches are compared on simulated meta-populations generated with SLiM and GSpace - two software packages that can simulate spatially-explicit genetic data at an individual level. Simulations show that the new jackknife version of the partial Mantel test provides a good compromise between power and respecting the nominal type I error rate. Mixed-effects models have larger power than jackknife-based methods, but tend to display type I error rates slightly above the significance level. An application on brassy ringlets concludes the paper.

q-bio.PE↗

arXiv2023

Probability Models in Statistical Data Analysis: Uses, Interpretations, Frequentism-As-Model

Note: Published now as a chapter in "Handbook of the History and Philosophy of Mathematical Practice" (Springer Nature, editor B. Sriraman, https://doi.org/10.1007/978-3-030-19071-2_105-1). The application of mathematical probability theory in statistics is quite controversial. Controversies regard both the interpretation of probability, and approaches to statistical inference. After having given an overview of the main approaches, I will propose a re-interpretation of frequentist probability. Most statisticians are aware that probability models interpreted in a frequentist manner are not really true in objective reality, but only idealisations. I argue that this is often ignored when actually applying frequentist methods and interpreting the results, and that keeping up the awareness for the essential difference between reality and models can lead to a more appropriate use and interpretation of frequentist models and methods, called "frequentism-as-model". This is elaborated showing connections to existing work, appreciating the special role of independently and identically distributed observations and subject matter knowledge, giving an account of how and under what conditions models that are not true can be useful, giving detailed interpretations of tests and confidence intervals, confronting their implicit compatibility logic with the inverse probability logic of Bayesian inference, re-interpreting the role of model assumptions, appreciating robustness, and the role of "interpretative equivalence" of models. Epistemic probability shares the issue that its models are only idealisations, and an analogous "epistemic-probability-as-model" can also be developed.

stat.OT↗

arXiv2023

Consistency for constrained maximum likelihood estimation and clustering based on mixtures of elliptically-symmetric distributions under general data generating processes

The consistency of the maximum likelihood estimator for mixtures of elliptically-symmetric distributions for estimating its population version is shown, where the underlying distribution $P$ is nonparametric and does not necessarily belong to the class of mixtures on which the estimator is based. In a situation where $P$ is a mixture of well enough separated but nonparametric distributions it is shown that the components of the population version of the estimator correspond to the well separated components of $P$. This provides some theoretical justification for the use of such estimators for cluster analysis in case that $P$ has well separated subpopulations even if these subpopulations differ from what the mixture model assumes.

math.ST↗

arXiv2023

Choice of trimming proportion and number of clusters in robust clustering based on trimming

So-called "classification trimmed likelihood curves" have been proposed as a useful heuristic tool to determine the number of clusters and trimming proportion in trimming-based robust clustering methods. However, these curves needs a careful visual inspection, and this way of choosing parameters requires subjective decisions. This work is intended to provide theoretical background for the understanding of these curves and the elements involved in their derivation. Moreover, a parametric bootstrap approach is presented in order to automatize the choice of parameter more by providing a reduced list of "sensible" choices for the parameters. The user can then pick a solution that fits their aims from that reduced list.

stat.ME↗

arXiv2023

Some issues in robust clustering

Some key issues in robust clustering are discussed with focus on Gaussian mixture model based clustering, namely the formal definition of outliers, ambiguity between groups of outliers and clusters, the interaction between robust clustering and the estimation of the number of clusters, the essential dependence of (not only) robust clustering on tuning decisions, and shortcomings of existing measurements of cluster stability when it comes to outliers.

stat.ML↗

arXiv2023

Benchmarking in cluster analysis: A white paper

Note: A revised version of this is now published. Please cite and read (it's open access): Van Mechelen, I., Boulesteix, A.-L., Dangl, R., Dean, N., Hennig, C., Leisch, F., Steinley, D., Warrens, M. J. (2023). A white paper on good research practices in benchmarking: The case of cluster analysis. WIREs Data Mining and Knowledge Discovery, e1511. https://doi.org/10.1002/widm.1511 To achieve scientific progress in terms of building a cumulative body of knowledge, careful attention to benchmarking is of the utmost importance. This means that proposals of new methods of data pre-processing, new data-analytic techniques, and new methods of output post-processing, should be extensively and carefully compared with existing alternatives, and that existing methods should be subjected to neutral comparison studies. To date, benchmarking and recommendations for benchmarking have been frequently seen in the context of supervised learning. Unfortunately, there has been a dearth of guidelines for benchmarking in an unsupervised setting, with the area of clustering as an important subdomain. To address this problem, discussion is given to the theoretical conceptual underpinnings of benchmarking in the field of cluster analysis by means of simulated as well as empirical data. Subsequently, the practicalities of how to address benchmarking questions in clustering are dealt with, and foundational recommendations are made.

stat.OT↗

arXiv2023

Onset of a conceptual outline map to get a hold on the jungle of cluster analysis

The domain of cluster analysis is a meeting point for a very rich multidisciplinary encounter, with cluster-analytic methods being studied and developed in discrete mathematics, numerical analysis, statistics, data analysis, data science, and computer science (including machine learning, data mining, and knowledge discovery), to name but a few. The other side of the coin, however, is that the domain suffers from a major accessibility problem as well as from the fact that it is rife with division across many pretty isolated islands. As a way out, the present paper offers a thorough and in-depth review of the clustering domain as a whole under the form of an outline map based on an overarching conceptual framework and a common language. With this framework we wish to contribute to structuring the clustering domain, to characterizing methods that have often been developed and studied in quite different contexts, to identifying links between methods, and to introducing a frame of reference for optimally setting up cluster analyses in data-analytic practice.

stat.OT↗

arXiv2023

Should we test the model assumptions before running a model-based test?

Statistical methods are based on model assumptions, and it is statistical folklore that a method's model assumptions should be checked before applying it. This can be formally done by running one or more misspecification tests of model assumptions before running a method that requires these assumptions; here we focus on model-based tests. A combined test procedure can be defined by specifying a protocol in which first model assumptions are tested and then, conditionally on the outcome, a test is run that requires or does not require the tested assumptions. Although such an approach is often taken in practice, much of the literature that investigated this is surprisingly critical of it. Our aim is to explore conditions under which model checking is advisable or not advisable. For this, we review results regarding such "combined procedures" in the literature, we review and discuss controversial views on the role of model checking in statistics, and we present a general setup in which we can show that preliminary model checking is advantageous, which implies conditions for making model checking worthwhile.

stat.ME↗

arXiv2023

Parameters not empirically identifiable or distinguishable, including correlation between Gaussian observations

Note: Accepted version, published in Statistical Papers, https://doi.org/10.1007/s00362-023-01414-3. It is shown that some theoretically identifiable parameters cannot be empirically identified, meaning that no consistent estimator of them can exist. An important example is a constant correlation between Gaussian observations (in presence of such correlation not even the mean can be empirically identified). Empirical identifiability and three versions of empirical distinguishability are defined. Two different constant correlations between Gaussian observations cannot even be empirically distinguished. A further example are cluster membership parameters in $k$-means clustering. Several existing results in the literature are connected to the new framework. General conditions are discussed under which independence can be distinguished from dependence.

math.ST↗

arXiv2022

Clustering of football players based on performance data and aggregated clustering validity indexes

We analyse football (soccer) player performance data with mixed type variables from the 2014-15 season of eight European major leagues. We cluster these data based on a tailor-made dissimilarity measure. In order to decide between the many available clustering methods and to choose an appropriate number of clusters, we use the approach by Akhanli and Hennig (2020). This is based on several validation criteria that refer to different desirable characteristics of a clustering. These characteristics are chosen based on the aim of clustering, and this allows to define a suitable validation index as weighted average of calibrated individual indexes measuring the desirable features. We derive two different clusterings. The first one is a partition of the data set into major groups of essentially different players, which can be used for the analysis of a team's composition. The second one divides the data set into many small clusters (with 10 players on average), which can be used for finding players with a very similar profile to a given player. It is discussed in depth what characteristics are desirable for these clusterings. Weighting the criteria for the second clustering is informed by a survey of football experts.

stat.AP↗

arXiv2022

Validation of cluster analysis results on validation data: A systematic framework

Cluster analysis refers to a wide range of data analytic techniques for class discovery and is popular in many application fields. To judge the quality of a clustering result, different cluster validation procedures have been proposed in the literature. While there is extensive work on classical validation techniques, such as internal and external validation, less attention has been given to validating and replicating a clustering result using a validation dataset. Such a dataset may be part of the original dataset, which is separated before analysis begins, or it could be an independently collected dataset. We present a systematic structured framework for validating clustering results on validation data that includes most existing validation approaches. In particular, we review classical validation techniques such as internal and external validation, stability analysis, hypothesis testing, and visual validation, and show how they can be interpreted in terms of our framework. We precisely define and formalise different types of validation of clustering results on a validation dataset and explain how each type can be implemented in practice. Furthermore, we give examples of how clustering studies from the applied literature that used a validation dataset can be classified into the framework.

stat.ME↗

arXiv2021

A comparison of different clustering approaches for high-dimensional presence-absence data

Presence-absence data is defined by vectors or matrices of zeroes and ones, where the ones usually indicate a "presence" in a certain place. Presence-absence data occur for example when investigating geographical species distributions, genetic information, or the occurrence of certain terms in texts. There are many applications for clustering such data; one example is to find so-called biotic elements, i.e., groups of species that tend to occur together geographically. Presence-absence data can be clustered in various ways, namely using a latent class mixture approach with local independence, distance-based hierarchical clustering with the Jaccard distance, or also using clustering methods for continuous data on a multidimensional scaling representation of the distances. These methods are conceptually very different and can therefore not easily be compared theoretically. We compare their performance with a comprehensive simulation study based on models for species distributions. This has been accepted for publication in Ferreira, J., Bekker, A., Arashi, M. and Chen, D. (eds.) Innovations in multivariate statistical modelling: navigating theoretical and multidisciplinary domains, Springer Emerging Topics in Statistics and Biostatistics.

stat.ME↗

arXiv2021

Inference for the proportional odds cumulative logit model with monotonicity constraints for ordinal predictors and ordinal response

The proportional odds cumulative logit model (POCLM) is a standard regression model for an ordinal response. Ordinality of predictors can be incorporated by monotonicity constraints for the corresponding parameters. It is shown that estimators defined by optimization, such as maximum likelihood estimators, for an unconstrained model and for parameters in the interior set of the parameter space of a constrained model are asymptotically equivalent. This is used in order to derive asymptotic confidence regions and tests for the constrained model, involving simple modifications for finite samples. The finite sample coverage probability of the confidence regions is investigated by simulation. Tests concern the effect of individual variables, monotonicity, and a specified monotonicity direction. The methodology is applied on real data related to the assessment of school performance.

econ.EM↗

arXiv2021

An empirical comparison and characterisation of nine popular clustering methods

Nine popular clustering methods are applied to 42 real data sets. The aim is to give a detailed characterisation of the methods by means of several cluster validation indexes that measure various individual aspects of the resulting clusters such as small within-cluster distances, separation of clusters, closeness to a Gaussian distribution etc. as introduced in Hennig (2019). 30 of the data sets come with a "true" clustering. On these data sets the similarity of the clusterings from the nine methods to the "true" clusterings is explored. Furthermore, a mixed effects regression relates the observable individual aspects of the clusters to the similarity with the "true" clusterings, which in real clustering problems is unobservable. The study gives new insight not only into the ability of the methods to discover "true" clusterings, but also into properties of clusterings that can be expected from the methods, which is crucial for the choice of a method in a real situation without a given "true" clustering.

stat.ME↗

arXiv2020

An adequacy approach for deciding the number of clusters for OTRIMLE robust Gaussian mixture based clustering

We introduce a new approach to deciding the number of clusters. The approach is applied to Optimally Tuned Robust Improper Maximum Likelihood Estimation (OTRIMLE; Coretto and Hennig 2016) of a Gaussian mixture model allowing for observations to be classified as "noise", but it can be applied to other clustering methods as well. The quality of a clustering is assessed by a statistic $Q$ that measures how close the within-cluster distributions are to elliptical unimodal distributions that have the only mode in the mean. This nonparametric measure allows for non-Gaussian clusters as long as they have a good quality according to $Q$. The simplicity of a model is assessed by a measure $S$ that prefers a smaller number of clusters unless additional clusters can reduce the estimated noise proportion substantially. The simplest model is then chosen that is adequate for the data in the sense that its observed value of $Q$ is not significantly larger than what is expected for data truly generated from the fitted model, as can be assessed by parametric bootstrap. The approach is compared with model-based clustering using the Bayesian Information Criterion (BIC) and the Integrated Complete Likelihood (ICL) in a simulation study and on two datasets of scientific interest. Keywords: parametric bootstrap; noise component; unimodality; model-based clustering

stat.ME↗

arXiv2020

Clustering with the Average Silhouette Width

The Average Silhouette Width (ASW; Rousseeuw (1987)) is a popular cluster validation index to estimate the number of clusters. Here we address the question whether it also is suitable as a general objective function to be optimized for finding a clustering. We will propose two algorithms (the standard version OSil and a fast version FOSil) and compare them with existing clustering methods in an extensive simulation study covering the cases of a known and unknown number of clusters. Real data sets are also analysed, partly exploring the use of the new methods with non-Euclidean distances. We will also show that the ASW satisfies some axioms that have been proposed for cluster quality functions (Ackerman and Ben-David (2009)). The new methods prove useful and sensible in many cases, but some weaknesses are also highlighted. These also concern the use of the ASW for estimating the number of clusters together with other methods, which is of general interest due to the popularity of the ASW for this task.

stat.ML↗