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MixedTrails: Bayesian Hypotheses Comparison on Heterogeneous Sequential Data

DOI:10.1007/s10618-017-0518-x
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Martin Becker at University of Wuerzburg
Martin Becker
Florian Lemmerich
Florian Lemmerich
Florian Lemmerich
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Philipp Singer at GESIS - Leibniz Institute of the Social Sciences
Philipp Singer
  • GESIS - Leibniz Institute of the Social Sciences
Markus Strohmaier
Markus Strohmaier
Markus Strohmaier
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Abstract and Figures

Sequential traces of user data are frequently observed online and offline, e.g.,as sequences of visited websites or as sequences of locations captured by GPS. However,understanding factors explaining the production of sequence data is a challenging task,especially since the data generation is often not homogeneous. For example, navigation behavior might change in different phases of a website visit, or movement behavior may vary between groups of user. In this work, we tackle this task and propose MixedTrails, a Bayesian approach for comparing the plausibility of hypotheses regarding the generative processes of heterogeneous sequence data. Each hypothesis represents a belief about transition probabilities between a set of states that can vary between groups of observed transitions.For example, when trying to understand human movement in a city, a hypothesis assuming tourists to be more likely to move towards points of interests than locals, can be shown to be more plausible with observed data than a hypothesis assuming the opposite. Our approach incorporates these beliefs as Bayesian priors in a generative mixed transition Markov chain model, and compares their plausibility utilizing Bayes factors. We discuss analytical and approximate inference methods for calculating the marginal likelihoods for Bayes factors,give guidance on interpreting the results, and illustrate our approach with several experiments on synthetic and empirical data from Wikipedia and Flickr. Thus, this work enables a novel kind of analysis for studying sequential data in many application areas.
llustrating example. In this figure, we show an illustrating soccer example: We are interested in a team's strategy in a specific game. We start with data on passes and shots (a). Using a simple Markov chain model, we can model these as transitions between states (b). The previously proposed HypTrails approach allows researchers to compare homogeneous hypotheses about sequential data that express beliefs in transition probabilities (d-g, strength indicated by line width). Utilizing Bayesian inference, it then determines the evidence of the data (b) under these hypotheses (d-g) and ranks the hypotheses based on their plausibility; in this case, the uniform hypothesis (d) is the relatively most plausible one. However, this HypTrails is limited to homogeneous data, and does not allow for more fine-grained hypotheses. Figure (c) reveals that splitting the data into halftimes allows for a way better explanation of the data: A hypothesis that assumes offense (e) in the first halftime and defense (g) in the second appears to be way more plausible. MixedTrails enables the comparison of such hypotheses on heterogeneous data.
llustrating example. In this figure, we show an illustrating soccer example: We are interested in a team's strategy in a specific game. We start with data on passes and shots (a). Using a simple Markov chain model, we can model these as transitions between states (b). The previously proposed HypTrails approach allows researchers to compare homogeneous hypotheses about sequential data that express beliefs in transition probabilities (d-g, strength indicated by line width). Utilizing Bayesian inference, it then determines the evidence of the data (b) under these hypotheses (d-g) and ranks the hypotheses based on their plausibility; in this case, the uniform hypothesis (d) is the relatively most plausible one. However, this HypTrails is limited to homogeneous data, and does not allow for more fine-grained hypotheses. Figure (c) reveals that splitting the data into halftimes allows for a way better explanation of the data: A hypothesis that assumes offense (e) in the first halftime and defense (g) in the second appears to be way more plausible. MixedTrails enables the comparison of such hypotheses on heterogeneous data.
… 
Hypotheses for heterogeneous sequence data. In MixedTrails , we formulate hypotheses about heterogeneous sequence data. e.g., in the soccer example, we define two hypotheses: the homogeneous hypothesis Hhom\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {hom}$$\end{document}a assumes that players just randomly pass the ball around; the heterogeneous hypothesis Hhet\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {het}$$\end{document}, b assumes an offensive strategy in the first half of the game and a defensive strategy in the second half, cf. Fig. 1. This is formalized based on two components: group assignment probabilitiesγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }$$\end{document}, i.e., probability distributions over a set of groups for each transition, and a belief matrix of group transition probabilitiesϕg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\phi }_g$$\end{document} for each group g. The soccer example features a special case, where group assignments are deterministic, i.e., the probabilities are either 0 or 1
Hypotheses for heterogeneous sequence data. In MixedTrails , we formulate hypotheses about heterogeneous sequence data. e.g., in the soccer example, we define two hypotheses: the homogeneous hypothesis Hhom\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {hom}$$\end{document}a assumes that players just randomly pass the ball around; the heterogeneous hypothesis Hhet\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {het}$$\end{document}, b assumes an offensive strategy in the first half of the game and a defensive strategy in the second half, cf. Fig. 1. This is formalized based on two components: group assignment probabilitiesγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }$$\end{document}, i.e., probability distributions over a set of groups for each transition, and a belief matrix of group transition probabilitiesϕg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\phi }_g$$\end{document} for each group g. The soccer example features a special case, where group assignments are deterministic, i.e., the probabilities are either 0 or 1
… 
This plot shows the MixedTrails results for the illustrating soccer example, i.e., marginal likelihood values of different hypotheses for increasing strengths of belief κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}. We observe that among the hypotheses without grouping, the uniform hypothesis performs best (a). However, far more plausible explanations can be obtained by heterogeneous hypotheses that assume different behavior in both halftimes (b). Finally, randomly splitting the data into groups leads to less plausible explanations (c)
This plot shows the MixedTrails results for the illustrating soccer example, i.e., marginal likelihood values of different hypotheses for increasing strengths of belief κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}. We observe that among the hypotheses without grouping, the uniform hypothesis performs best (a). However, far more plausible explanations can be obtained by heterogeneous hypotheses that assume different behavior in both halftimes (b). Finally, randomly splitting the data into groups leads to less plausible explanations (c)
… 
Synthetic data results. We compare homogeneous (Hlink\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {link}$$\end{document}) and heterogeneous hypotheses (Hlink-colored\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {link-colored}$$\end{document}, Hcolor\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {color}$$\end{document} and Hmem\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {mem}$$\end{document}) on three synthetic datasets (Dlink\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_\text {link}$$\end{document}, Dcolor\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_\text {color}$$\end{document} and Dmem\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_\text {mem}$$\end{document}). We observe that the hypotheses that are fitting the respective datasets work best, illustrating that the MixedTrails approach can identify the correct ordering of the defined hypotheses. For details on interpreting the plots, see Sect. 4.1
Synthetic data results. We compare homogeneous (Hlink\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {link}$$\end{document}) and heterogeneous hypotheses (Hlink-colored\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {link-colored}$$\end{document}, Hcolor\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {color}$$\end{document} and Hmem\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\text {mem}$$\end{document}) on three synthetic datasets (Dlink\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_\text {link}$$\end{document}, Dcolor\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_\text {color}$$\end{document} and Dmem\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_\text {mem}$$\end{document}). We observe that the hypotheses that are fitting the respective datasets work best, illustrating that the MixedTrails approach can identify the correct ordering of the defined hypotheses. For details on interpreting the plots, see Sect. 4.1
… 
Probabilistic group assignments on synthetic data. The violet, mixed hypothesis, using probabilistic group assignment probabilities, is the most plausible one for increasing concentration factors as it directly models the processes underlying the data. The violet, naive hypothesis illustrates the integral role of the mixing step, as skipping it significantly reduces the performance of a hypothesis even though the underlying processes were correctly understood. Further details are discussed in Sect. 4.2
+2
Probabilistic group assignments on synthetic data. The violet, mixed hypothesis, using probabilistic group assignment probabilities, is the most plausible one for increasing concentration factors as it directly models the processes underlying the data. The violet, naive hypothesis illustrates the integral role of the mixing step, as skipping it significantly reduces the performance of a hypothesis even though the underlying processes were correctly understood. Further details are discussed in Sect. 4.2
… 
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Data Min Knowl Disc (2017) 31:1359–1390
DOI 10.1007/s10618-017-0518-x
MixedTrails: Bayesian hypothesis comparison on
heterogeneous sequential data
Martin Becker1·Florian Lemmerich2·
Philipp Singer2·Markus Strohmaier2·
Andreas Hotho1
Received: 18 December 2016 / Accepted: 5 June 2017 / Published online: 7 July 2017
© The Author(s) 2017
Abstract Sequential traces of user data are frequently observed online and offline,
e.g., as sequences of visited websites or as sequences of locations captured by GPS.
However, understanding factors explaining the production of sequence data is a chal-
lenging task, especially since the data generation is often not homogeneous. For
example, navigation behavior might change in different phases of browsing a website
or movement behavior may vary between groups of users. In this work, we tackle
this task and propose MixedTrails, a Bayesian approach for comparing the plausibil-
ity of hypotheses regarding the generative processes of heterogeneous sequence data.
Each hypothesis is derived from existing literature, theory, or intuition and represents
a belief about transition probabilities between a set of states that can vary between
groups of observed transitions. For example, when trying to understand human move-
ment in a city and given some data, a hypothesis assuming tourists to be more likely
Responsible editors: Kurt Driessens, Dragi Kocev, Marko Robnik-Šikonja, Myra Spiliopoulou
BMartin Becker
becker@informatik.uni-wuerzburg.de
Florian Lemmerich
florian.lemmerich@gesis.org
Philipp Singer
me@philippsinger.com
Markus Strohmaier
markus.strohmaier@gesis.org
Andreas Hotho
hotho@informatik.uni-wuerzburg.de
1Data Mining and Information Retrieval Group, University of Würzburg, Würzburg, Germany
2GESIS - Leibniz Institute for the Social Sciences, Cologne, Germany
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... To establish if one of the hypotheses is better than another we compare the marginal likelihood (the higher, the better) across a range of concentration factors (scaled by the number of states). For more details we refer to [2,9]. ...
... For future work, it may be interesting to extend the transition models as applied in this work, for example, by further investigating the influence of the road network in the context of the data provided by OpenStreetMap, or by formulating heterogeneous hypotheses to explain the overall behavior of users during the APIC campaign [2]. Finally, by employing data from other participatory sensing campaigns as well as subjective information from users may provide the necessary background information to formulate and compare hypotheses that enable further insights into human navigation behavior as well as their incentives and goals in the context of environmental studies. ...
Air Trails -Urban Air Quality Campaign Exploration Patterns
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Air pollution in urban areas has become a major issue and has attracted significant public attention. As a consequence, many citizens have started campaigns for measuring the air quality of their personal environment using mobile devices. In this study, we adapt HypTrails-a Bayesian method for comparing hypotheses about human trails-in order to investigate mobility patterns from such campaigns. In particular, we derive an approach to apply HypTrails to continuous, temporally dense navigation paths as is characteristic for GPS tracks. This allows us to directly study the behavioral processes of participants. We showcase our method on the citizen science campaign APIC (the AirProbe International Challenge) yielding promising results: We find differing mobility patterns of users in restricted and unrestricted environments, and extend previous work by showing that roads and road types play an important role explaining the observed paths. This gives first insights into movement patterns of urban air quality exploration. Ultimately, we believe that our approach can help to better interpret data collected in the context of participatory sensing campaigns, and to develop new theories about the motivational processes of volunteers.
... This is called sequential pattern mining, where the goal is to develop algorithms that quickly find the most frequent subsequences in large sequence data [1,24,18,38]. Some work addresses this problem based on statistical methods, e.g., using Markov modeling techniques [22,36,45,10,44,34,23], hypothesis testing [39,6,43], or information-theoretic methods to detect "surprising" subsequences [25,13,7]. Applications include the detection of common patterns in user trajectories [44,35], testing hypotheses about generative processes of trajectory data [39], or finding clusters in sequence data [10,34]. ...
... In the below experiments, we compare HYPA to a simple frequency-based anomaly detection (FBAD) of our own design. We note that despite similar problem settings, the methods for hypothesis testing on human trails presented in [39,6] are not directly comparable with our work because the output is Bayesian evidence for a hypothesis on an entire dataset (a single number), whereas we are interested in edge-level analysis. Further, we did not compare with a method like [36] because, while based on detecting significant deviations from a Markov chain model, this method assumes that the data is given as one long sequence and detects anomalous subsequences, which does not correspond to any of the datasets we analyze here. ...
HYPA: Efficient Detection of Path Anomalies in Time Series Data on Networks
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... This is closely related to sequential pattern mining [1], e.g., algorithms to quickly find the most frequent subsequences in large sequence data [20,43]. Other works address this problem based on statistical methods, e.g., using Markov modelling techniques [12,25,37,39,50,53], hypothesis testing [8,44], or information-theoretic methods to detect "surprising" subsequences [9,15,27]. Applications include the detection of common patterns in user trajectories [38,50], testing hypotheses about generative processes of trajectory data [44], or finding clusters in click streams and other sequence data [12,37]. ...
... In the below experiments, we compare HYPA to a simple frequencybased anomaly detection (FBAD) of our own design. We note that despite similar problem settings, the methods for hypothesis testing on human trails presented in [8,44] are not directly comparable with our work because the output is Bayesian evidence for a hypothesis on an entire dataset (a single number), whereas we are interested in edge-level analysis. However, in future work we could use HYPA to generate hypotheses to be tested using these methods. ...
Detecting Path Anomalies in Time Series Data on Networks
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The unsupervised detection of anomalies in time series data has important applications, e.g., in user behavioural modelling, fraud detection, and cybersecurity. Anomaly detection has been extensively studied in categorical sequences, however we often have access to time series data that contain paths through networks. Examples include transaction sequences in financial networks, click streams of users in networks of cross-referenced documents, or travel itineraries in transportation networks. To reliably detect anomalies we must account for the fact that such data contain a large number of independent observations of short paths constrained by a graph topology. Moreover, the heterogeneity of real systems rules out frequency-based anomaly detection techniques, which do not account for highly skewed edge and degree statistics. To address this problem we introduce a novel framework for the unsupervised detection of anomalies in large corpora of variable-length temporal paths in a graph, which provides an efficient analytical method to detect paths with anomalous frequencies that result from nodes being traversed in unexpected chronological order.
... This is closely related to sequential pattern mining [1], e.g., algorithms to quickly find the most frequent subsequences in large sequence data [20,43]. Other works address this problem based on statistical methods, e.g., using Markov modelling techniques [12,25,37,39,50,53], hypothesis testing [8,44], or information-theoretic methods to detect "surprising" subsequences [9,15,27]. Applications include the detection of common patterns in user trajectories [38,50], testing hypotheses about generative processes of trajectory data [44], or finding clusters in click streams and other sequence data [12,37]. ...
... In the below experiments, we compare HYPA to a simple frequencybased anomaly detection (FBAD) of our own design. We note that despite similar problem settings, the methods for hypothesis testing on human trails presented in [8,44] are not directly comparable with our work because the output is Bayesian evidence for a hypothesis on an entire dataset (a single number), whereas we are interested in edge-level analysis. However, in future work we could use HYPA to generate hypotheses to be tested using these methods. ...
Detecting Path Anomalies in Time Series Data on Networks
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  • May 2019
The unsupervised detection of anomalies in time series data has important applications, e.g., in user behavioural modelling, fraud detection, and cybersecurity. Anomaly detection has been extensively studied in categorical sequences, however we often have access to time series data that contain paths through networks. Examples include transaction sequences in financial networks, click streams of users in networks of cross-referenced documents, or travel itineraries in transportation networks. To reliably detect anomalies we must account for the fact that such data contain a large number of independent observations of short paths constrained by a graph topology. Moreover, the heterogeneity of real systems rules out frequency-based anomaly detection techniques, which do not account for highly skewed edge and degree statistics. To address this problem we introduce a novel framework for the unsupervised detection of anomalies in large corpora of variable-length temporal paths in a graph, which provides an efficient analytical method to detect paths with anomalous frequencies that result from nodes being traversed in unexpected chronological order.
... Like Lemmerich et al. (2016); Song (2017); Bueno et al. (2020), we take an approach where candidate subgroups are evaluated using a bottom-up, heuristic search through the descriptive space. In comparison, Becker et al. (2017) take a top-down approach where the subgroups are hypothesised beforehand based on theory and evaluated using Bayes Factors. Kiseleva et al. (2013) hypothesise two groups of sequences based on descriptive information and distributional characteristics. ...
Mining sequences with exceptional transition behaviour of varying order using quality measures based on information-theoretic scoring functions
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Discrete Markov chains are frequently used to analyse transition behaviour in sequential data. Here, the transition probabilities can be estimated using varying order Markov chains, where order k specifies the length of the sequence history that is used to model these probabilities. Generally, such a model is fitted to the entire dataset, but in practice it is likely that some heterogeneity in the data exists and that some sequences would be better modelled with alternative parameter values, or with a Markov chain of a different order. We use the framework of Exceptional Model Mining (EMM) to discover these exceptionally behaving sequences. In particular, we propose an EMM model class that allows for discovering subgroups with transition behaviour of varying order. To that end, we propose three new quality measures based on information-theoretic scoring functions. Our findings from controlled experiments show that all three quality measures find exceptional transition behaviour of varying order and are reasonably sensitive. The quality measure based on Akaike’s Information Criterion is most robust for the number of observations. We furthermore add to existing work by seeking for subgroups of sequences, as opposite to subgroups of transitions. Since we use sequence-level descriptive attributes, we form subgroups of entire sequences, which is practically relevant in situations where you want to identify the originators of exceptional sequences, such as patients. We show this relevance by analysing sequences of blood glucose values of adult persons with diabetes type 2. In the experiments, we find subgroups of patients based on age and glycated haemoglobin (HbA1c), a measure known to correlate with average blood glucose values. Clinicians and domain experts confirmed the transition behaviour as estimated by the fitted Markov chain models.
... Zhang et al. [121] propose a learning scheme that offers a recursive algorithm to explore the distribution of class density for the Bayesian estimation and an automated approach to select powerful discriminant functions for the classification of high-dimensional data, while Celotto [122] proposes a unified visual approach to compare and classify a large subset of Bayesian confirmation measures. In the work of Becker et al. [123], analytical and approximate inference methods are discussed to calculate the marginal probabilities of Bayes factors, providing guidance on the interpretation of results and offering new types of analysis to study sequential data in many application areas. ...
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... In order to extract the exceptional transition behaviors, the proposed algorithm mines for subgroups whose fitted markov transition' matrix significantly differs (using an adapted manhattan distance) from the one computed over the entire population. Similarly, HypTrails (Singer et al., 2015) extended to MixedTrails (Becker et al., 2017) operationalizes bayesian model comparsion on simple markov chains (HypTrails) and hetergeneous mixed comparison markov chains (MixedTrails). Although, these methods do not consider descriptive attributes to extract groups but rather evaluate the transition behavior of an input group. ...
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Conference Paper
  • Aug 2016
We present a new method for detecting interpretable subgroups with exceptional transition behavior in sequential data. Identifying such patterns has many potential applications, e.g., for studying human mobility or analyzing the behavior of internet users. To tackle this task, we employ exceptional model mining, which is a general approach for identifying interpretable data subsets that exhibit unusual interactions between a set of target attributes with respect to a certain model class. Although exceptional model mining provides a well-suited framework for our problem, previously investigated model classes cannot capture transition behavior. To that end, we introduce first-order Markov chains as a novel model class for exceptional model mining and present a new interestingness measure that quantifies the exceptionality of transition subgroups. The measure compares the distance between the Markov transition matrix of a subgroup and the respective matrix of the entire data with the distance of random dataset samples. In addition, our method can be adapted to find subgroups that match or contradict given transition hypotheses. We demonstrate that our method is consistently able to recover subgroups with exceptional transition models from synthetic data and illustrate its potential in two application examples. Our work is relevant for researchers and practitioners interested in detecting exceptional transition behavior in sequential data.
What Makes a Link Successful on Wikipedia?
Article
  • Nov 2016
While a plethora of hypertext links exist on the Web, only a small amount of them are regularly clicked. Starting from this observation, we set out to study large-scale click data from Wikipedia in order to understand what makes a link successful. We systematically analyze effects of link properties on the popularity of links. By utilizing mixed-effects hurdle models supplemented with descriptive insights, we find evidence of user preference towards links leading to the periphery of the network, towards links leading to semantically similar articles, and towards links in the top and left-side of the screen. We integrate these findings as Bayesian priors into a navigational Markov chain model and by doing so successfully improve the model fits. We further adapt and improve the well-known classic PageRank algorithm that assumes random navigation by accounting for observed navigational preferences of users in a weighted variation. This work facilitates understanding navigational click behavior and thus can contribute to improving link structures and algorithms utilizing these structures.
The pagerank citation ranking: Bringing order to the web
Article
  • Jan 1999
Bayes factors
Article
  • Jan 1995
Photowalking the City: Comparing Hypotheses About Urban Photo Trails on Flickr
Conference Paper
  • Dec 2015
Understanding human movement trajectories represents an important problem that has implications for a range of societal challenges such as city planning and evolution, public transport or crime. In this paper, we focus on geo-temporal photo trails from four different cities (Berlin, London, Los Angeles, New York) derived from Flickr that are produced by humans when taking sequences of photos in urban areas. We apply a Bayesian approach called HypTrails to assess different explanations of how the trails are produced. Our results suggest that there are common processes underlying the photo trails observed across the studied cities. Furthermore, information extracted from social media, in the form of concepts and usage statistics from Wikipedia, allows for constructing explanations for human movement trajectories.
A Bayesian test for the hot hand phenomenon
Article
  • Feb 2016