{"id":1428,"date":"2021-04-29T15:40:46","date_gmt":"2021-04-29T13:40:46","guid":{"rendered":"https:\/\/www.hsu-hh.de\/mathstat\/?page_id=1428"},"modified":"2026-02-17T11:20:48","modified_gmt":"2026-02-17T10:20:48","slug":"ordinale-zeitreihen","status":"publish","type":"page","link":"https:\/\/www.hsu-hh.de\/mathstat\/forschung\/projekte\/ordinale-zeitreihen","title":{"rendered":"Ordinale Zeitreihen: Modellierung, Vorhersage und Kontrolle"},"content":{"rendered":"\n<h3 class=\"wp-block-heading\">Projektpartner:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/uwm.edu\/business\/people\/alwan-layth-c\/\" rel='nofollow'><abbr title=\"Professor\">Prof.<\/abbr> <abbr title=\"Doktor\">Dr.<\/abbr> Layth C. Alwan<\/a>, University of Wisconsin, USA,<\/li>\n\n\n\n<li><a href=\"http:\/\/yunus.hacettepe.edu.tr\/~mtestik\/\" rel='nofollow'><abbr title=\"Professor\">Prof.<\/abbr> <abbr title=\"Doktor\">Dr.<\/abbr> Murat Caner Testik<\/a>, Hacettepe Universitesi Ankara.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">DFG-Projekt<\/h2>\n\n\n\n<p>&nbsp;<br>Dreij\u00e4hriges Projekt, gef\u00f6rdert durch die <a href=\"http:\/\/www.dfg.de\/\" rel='nofollow'>Deutsche Forschungsgemeinschaft (DFG)<\/a> \u2013 Projektnummer 516522977, und vorbereitet im Rahmen eines <a href=\"#iff\" rel='nofollow'>IFF-Projekts<\/a>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Projektziele:<\/h3>\n\n\n\n<p>Eine ordinale Zeitreihe ist eine zeitliche Abfolge diskretwertiger Beobachtungen, deren Wertebereich qualitativ ist und aus einer endlichen Zahl an geordneten Kategorien besteht. Ordinale Zeitreihen sind in den verschiedensten Situationen mit wirtschaftswissenschaftlichem Kontext anzutreffen und k\u00f6nnen vielf\u00e4ltige Formen hinsichtlich ihrer Abh\u00e4ngigkeitsstruktur oder Randverteilung aufweisen. Letztgenannte Merkmale k\u00f6nnen mit Hilfe analytischer Werkzeuge, wie sie j\u00fcngst f\u00fcr ordinale Zeitreihen entwickelt wurden, herausgearbeitet werden. Im Anschluss daran w\u00e4re nun eine ad\u00e4quate Modellierung der ordinalen Zeitreihe n\u00f6tig, welche wiederum als Basis f\u00fcr die Vorhersage der Zeitreihe oder f\u00fcr die statistische Kontrolle des weiteren Verlaufs genutzt werden k\u00f6nnte. An dieser Stelle setzt das geplante Forschungsvorhaben ein, denn weder f\u00fcr die Modellierung noch f\u00fcr die Vorhersage oder Kontrolle ordinaler Zeitreihen gibt es bis dato ein hinreichendes Repertoire an ma\u00dfgeschneiderten Methoden. Stattdessen werden zumeist Ans\u00e4tze f\u00fcr nominale Zeitreihen verwendet (die dann aber die nat\u00fcrliche Anordnung der Kategorien au\u00dfer Acht lassen), oder solche f\u00fcr quantitative Zeitreihen (die dann aber implizit eine metrische Struktur unterstellen).<\/p>\n\n\n\n<p>Ziel des geplanten Forschungsvorhabens ist es, ein umfassendes Paket an Methoden zur stochastischen Modellierung, Vorhersage und Kontrolle ordinaler Zeitreihen zu entwickeln. Dabei sollen bereits existierende Verfahren aufgegriffen und die noch offenen Bereiche durch neue eigene Beitr\u00e4ge geschlossen werden. Der erste Schritt w\u00e4re dabei die Entwicklung eines Baukastens aus m\u00f6glichst verschiedenen Modellen, durch die eine breite Palette an stochastischen Eigenschaften abgedeckt wird. Bei allen sich ergebenden Modelltypen ist neben der eigentlichen Modelldefinition und den stochastischen Modelleigenschaften stets auch die Frage der Modellanpassung (Identifikation, Sch\u00e4tzung, Validierung) zu ber\u00fccksichtigen. Daran schlie\u00dfen die Teilprojekte zur Vorhersage und Kontrolle an, welche die zuvor neu entwickelten Modelle bereits einbeziehen sollen. In puncto Vorhersage sollen ad\u00e4quate Kriterien zur Bewertung der Vorhersagequalit\u00e4t hergeleitet werden (welche also der ordinalen Natur der Daten gerecht werden), um dann mit diesen die Performanz der verschiedenen Vorhersageans\u00e4tze (Punkt-, Bereichs- und PMF-Vorhersagen) eingehend empirisch zu untersuchen. Hinsichtlich der Kontrolle sollen Kontrollkarten f\u00fcr seriell abh\u00e4ngige ordinale Prozesse entwickelt und untersucht werden, wobei neben stichprobenbasierten Karten vor allem Einzelwert-Ged\u00e4chtniskarten im Fokus stehen, welche bis dato g\u00e4nzlich in der Fachliteratur fehlen. F\u00fcr alle vorgeschlagenen Verfahren werden Performanz und Anwendbarkeit eingehend untersucht, sowohl durch umfassende vergleichende Simulationsstudien wie auch durch Anwendung auf in den Wirtschaftswissenschaften relevante reale Datenbeispiele.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Projektlaufzeit:<\/h3>\n\n\n\n<p>April 2023 &#8211; M\u00e4rz 2026.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Projektresultate:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Wei\u00df, C.H., Schnurr, A. (2024):<br>Generalized ordinal patterns in discrete-valued time series: nonparametric testing for serial dependence.<br><a href=\"https:\/\/www.tandfonline.com\/toc\/gnst20\/36\/3\" rel='nofollow'>Journal of Nonparametric Statistics<\/a> 36(3), 573-599 (<a href=\"https:\/\/doi.org\/10.1080\/10485252.2023.2231565\" rel='nofollow'>open access<\/a>).<br>\u00a0<br><em>Abstract:<\/em> We provide a new testing procedure to detect serial dependence in time series. Our method is based solely on the ordinal structure of the data. We explicitly allow for ties in the data windows we consider. Consequently, we use generalized ordinal patterns, that is, Cayley permutations. Unlike in the classical case, the pattern distribution is not uniform under the null hypothesis of serial independence. In our new framework, the underlying distribution has to be taken into account and we overcome this problem by a bootstrap procedure. The applicability of our method is supported by a simulation study and two real-world data examples.<br>\u00a0<\/li>\n\n\n\n<li>Wei\u00df, C.H. (2024):<br>Ordinal compositional data and time series.<br><a href=\"https:\/\/journals.sagepub.com\/toc\/smja\/24\/6\" rel='nofollow'>Statistical Modelling<\/a> 24(6), 561-580 (<a href=\"https:\/\/doi.org\/10.1177\/1471082X231190971\" rel='nofollow'>open access<\/a>).<br>\u00a0<br><em>Abstract:<\/em> There are several real applications where the categories behind compositional data (CoDa) exhibit a natural order, which, however, is not accounted for by existing CoDa methods. For various application areas, it is demonstrated that appropriately developed methods for ordinal CoDa provide valuable additional insights and are, thus, recommended to complement existing CoDa methods. The potential benefits are demonstrated for the (visual) descriptive analysis of ordinal CoDa, for statistical inference based on CoDa samples, for the monitoring of CoDa processes by means of control charts, and for the analysis and modelling of compositional time series. The novel methods are illustrated by a couple of real-world data examples.<br>\u00a0<\/li>\n\n\n\n<li>Jahn, M., Wei\u00df, C.H. (2024):<br>Nonlinear GARCH-type models for ordinal time series.<br><a href=\"https:\/\/link.springer.com\/journal\/477\/volumes-and-issues\/38-2\" rel='nofollow'>Stochastic Environmental Research and Risk Assessment<\/a> 38(2), 637-649 (<a href=\"https:\/\/doi.org\/10.1007\/s00477-023-02591-1\" rel='nofollow'>open access<\/a>).<br>\u00a0<br><em>Abstract:<\/em> Despite their relevance in various areas of application, only few stochastic models for ordinal time series are discussed in the literature. To allow for a flexible serial dependence structure, different ordinal GARCH-type models are proposed, which can handle nonlinear dependence as well as kinds of an intensified memory. The (logistic) ordinal GARCH model accounts for the natural order among the categories by relying on the conditional cumulative distributions. As an alternative, a conditionally multinomial model is developed which uses the softmax response function. The resulting softmax GARCH model incorporates the ordinal information by considering the past (expected) categories. It is shown that this latter model is easily combined with an artificial neural network response function. This introduces great flexibility into the resulting neural softmax GARCH model, which turns out to be beneficial in three real-world time series applications (air quality levels, fear states, cloud coverage).<br>\u00a0<\/li>\n\n\n\n<li>Wei\u00df, C.H., Swidan, O. (2025a):<br>Weighted Discrete ARMA Models for Categorical Time Series.<br><a href=\"https:\/\/onlinelibrary.wiley.com\/toc\/14679892\/2025\/46\/3\" rel='nofollow'>Journal of Time Series Analysis<\/a> 46(3), 505-529 (<a href=\"https:\/\/doi.org\/10.1111\/jtsa.12773\" rel='nofollow'>open access<\/a>).<br>\u00a0<br><em>Abstract:<\/em> A new and flexible class of ARMA-like (autoregressive moving average) models for nominal or ordinal time series is proposed, which are characterized by using so-called &#8222;weighting operators&#8220; and are, thus, referred to as weighted discrete ARMA (WDARMA) models. By choosing an appropriate type of weighting operator, one can model, for example, nominal time series with negative serial dependencies, or ordinal time series where transitions to neighbouring states are more likely than sudden large jumps. Essential stochastic properties of WDARMA models are derived, such as the existence of a stationary, ergodic, and phi-mixing solution as well as closed-form formulae for marginal and bivariate probabilities. Numerical illustrations as well as simulation experiments regarding the finite-sample performance of maximum likelihood estimation are presented. The possible benefits of using an appropriate weighting scheme within the WDARMA class are demonstrated by a real-world data application.<br>\u00a0<\/li>\n\n\n\n<li>Wei\u00df, C.H., Swidan, O. (2025b):<br>Hidden-Markov Models for Ordinal Time Series.<br><a href=\"https:\/\/link.springer.com\/journal\/10182\/volumes-and-issues\/109-2\" rel='nofollow'>AStA Advances in Statistical Analysis<\/a> 109(2), 217-239 (<a href=\"https:\/\/doi.org\/10.1007\/s10182-024-00514-1\" rel='nofollow'>open access<\/a>).<br>\u00a0<br><em>Abstract:<\/em> A common approach for modeling categorical time series are Hidden-Markov models (HMMs), where the actual observations are assumed to depend on hidden states in their behavior and transitions. Such categorical HMMs are even applicable to nominal data, but suffer from a large number of model parameters.<br>In the ordinal case, however, the natural order among the categorical outcomes offers the potential to reduce the number of parameters while improving their interpretability at the same time. The class of ordinal HMMs proposed in this article link a latent-variable approach with categorical HMMs. They are characterized by parametric parsimony and allow the easy calculation of relevant stochastic properties, such as marginal and bivariate probabilities. These points are illustrated by numerical examples and simulation experiments, where the performance of maximum likelihood estimation is analyzed in finite samples. The developed methodology is applied to real-world data from a health application.<br>\u00a0<\/li>\n\n\n\n<li>Wei\u00df, C.H., Swidan, O. (2025c):<br>Soft-clipping Autoregressive Models for Ordinal Time Series.<br><a href=\"https:\/\/onlinelibrary.wiley.com\/toc\/15264025\/2025\/41\/3\" rel='nofollow'>Applied Stochastic Models in Business and Industry<\/a> 41(3), e70015 (<a href=\"https:\/\/doi.org\/10.1002\/asmb.70015\" rel='nofollow'>open access<\/a>).<br>\u00a0<br><em>Abstract:<\/em> The linear autoregressive models are among the most popular models in the practice of time series analysis, which constitutes an incentive to adapt them to ordinal time series as well. Our starting point for modeling ordinal time series data is the latent variable approach to define a generalized linear model. This method, however, typically leads to a non-linear relationship between the past observations and the current conditional cumulative distribution function (cdf). To overcome this problem, we use the soft-clipping link to obtain an approximately linear model structure and propose the wide and flexible class of soft-clipping autoregressive (scAR) models. The constraints imposed on the model parameters allow us to identify relevant special cases of the scAR model family. We study the calculation of transition probabilities as well as approximate formulae for the cdf. Our proposals are illustrated by numerical examples and simulation experiments, where the performance of maximum likelihood estimation as well as model selection is analyzed. The novel model family is successfully applied to a real-world ordinal time series from finance.<br>\u00a0<\/li>\n\n\n\n<li>Jahn, M., Wei\u00df, C.H. (2025):<br>Modeling multivariate ordinal time series.<br>Accepted for publication in <a href=\"https:\/\/www.tandfonline.com\/action\/showAxaArticles?journalCode=cjas20\" rel='nofollow'>Journal of Applied Statistics<\/a> (<a href=\"https:\/\/doi.org\/10.1080\/02664763.2025.2575034\" rel='nofollow'>open access<\/a>).<br>\u00a0<br><em>Abstract:<\/em> In this paper, several regression-type models for multivariate ordinal time series are developed. The regression equations are inspired by existing GARCH-type models for univariate discrete-valued time series and include feedback terms in addition to the usual lagged observations to model the memory behavior. The corresponding terms from other individuals (components) are represented by weighted averages which are calculated based on a proximity matrix. The marginal conditional distributions are either binomial (employing the simplifying rank-count formulation) or multinomial. The approach can be generalized to obtain VARMA-type models to allow for more specific dependence between individuals. Additionally, different copulas are considered to model possible cross-dependence explicitly. The main data example concerns the daily air quality (ordinal) in three cities in North China. Here, a spatial dimension is present, which can be exploited in the definition of the proximity matrix and the copulas.<br>\u00a0<\/li>\n\n\n\n<li>wird fortgesetzt!<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"iff\">IFF-Projekt<\/h2>\n\n\n\n<p>&nbsp;<br>Einj\u00e4hriges Projekt, gef\u00f6rdert durch die <a href=\"https:\/\/www.hsu-hh.de\/forschung\/\">Interne Forschungsf\u00f6rderung (IFF2021)<\/a> der <abbr title=\"Helmut Schmidt Universit\u00e4t\">HSU<\/abbr> Hamburg.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Projektziele:<\/h3>\n\n\n\n<p>Das IFF-Projekt befasst sich mit der Modellierung, Vorhersage und Kontrolle von ordinalen Zeitreihen. Eine ordinale Zeitreihe ist hierbei eine qualitative Zeitreihe, bei der die m\u00f6glichen kategorialen Beobachtungswerte eine nat\u00fcrliche Anordnung aufweisen. Obwohl ordinale Zeitreihen in diversen Anwendungen vorkommen, ist die wissenschaftliche Literatur noch weit davon entfernt, ein vollst\u00e4ndiges Box-Jenkins-Programm f\u00fcr diese anzubieten, <abbr title=\"das hei\u00dft\">d.h.<\/abbr> Verfahren zu allen der Bereiche Analyse, Modellierung, Vorhersage und Kontrolle von ordinalen Zeitreihen. Nachdem in Vorarbeiten des Antragsstellers bereits der Bereich der Analyse ordinaler Zeitreihen etabliert wurde, soll nun mit einem IFF-Projekt ein DFG-Antrag vorbereitet werden, der die drei \u00fcbrigen Bereiche der Modellierung, Vorhersage und Kontrolle ausarbeitet. In der IFF-Phase stehen dabei umfassende Literaturrecherchen und Konzeptentwicklungen im Zentrum. Ferner sollen (in einer weiteren Vorarbeit) exemplarisch erste neue Kontrollkarten f\u00fcr ordinale Prozesse entwickelt und mit existierenden Kontrollkarten verglichen werden.<\/p>\n\n\n\n<p>Einen \u00dcberblick \u00fcber das IFF-Projekt bietet folgendes <a href=\"https:\/\/www.hsu-hh.de\/mathstat\/wp-content\/uploads\/sites\/781\/2023\/03\/Poster_IFF21_Weiss_A1Design.pdf\">Poster<\/a>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Projektlaufzeit:<\/h3>\n\n\n\n<p>Juli 2021 &#8211; Juni 2022.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Publikationen:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Ottenstreuer, S., Wei\u00df, C.H., Testik, M.C. (2023).<br>A review and comparison of control charts for ordinal samples.<br><a href=\"https:\/\/www.tandfonline.com\/toc\/ujqt20\/55\/4\" rel='nofollow'>Journal of Quality Technology<\/a> 55(4), pp. 422-441, 2023. (<a href=\"https:\/\/doi.org\/10.1080\/00224065.2023.2170839\" rel='nofollow'>Open Access<\/a>)<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Projektpartner: DFG-Projekt &nbsp;Dreij\u00e4hriges Projekt, gef\u00f6rdert durch die Deutsche Forschungsgemeinschaft (DFG) \u2013 Projektnummer 516522977, und vorbereitet im Rahmen eines IFF-Projekts. Projektziele: Eine ordinale Zeitreihe ist eine zeitliche Abfolge diskretwertiger Beobachtungen, deren [&hellip;]<\/p>\n","protected":false},"author":98,"featured_media":0,"parent":647,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-1428","page","type-page","status-publish","hentry","category-forschung"],"_links":{"self":[{"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/pages\/1428","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/users\/98"}],"replies":[{"embeddable":true,"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/comments?post=1428"}],"version-history":[{"count":33,"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/pages\/1428\/revisions"}],"predecessor-version":[{"id":2759,"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/pages\/1428\/revisions\/2759"}],"up":[{"embeddable":true,"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/pages\/647"}],"wp:attachment":[{"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/media?parent=1428"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/categories?post=1428"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.hsu-hh.de\/mathstat\/wp-json\/wp\/v2\/tags?post=1428"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}