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Online estimation of discrete, continuous, and conditional joint densities using classifier chains

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Abstract

We address the problem of estimating discrete, continuous, and conditional joint densities online, i.e., the algorithm is only provided the current example and its current estimate for its update. The family of proposed online density estimators, estimation of densities online (EDO), uses classifier chains to model dependencies among features, where each classifier in the chain estimates the probability of one particular feature. Because a single chain may not provide a reliable estimate, we also consider ensembles of classifier chains and ensembles of weighted classifier chains. For all density estimators, we provide consistency proofs and propose algorithms to perform certain inference tasks. The empirical evaluation of the estimators is conducted in several experiments and on datasets of up to several millions of instances. In the discrete case, we compare our estimators to density estimates computed by Bayesian structure learners. In the continuous case, we compare them to a state-of-the-art online density estimator. Our experiments demonstrate that, even though designed to work online, EDO delivers estimators of competitive accuracy compared to other density estimators (batch Bayesian structure learners on discrete datasets and the state-of-the-art online density estimator on continuous datasets). Besides achieving similar performance in these cases, EDO is also able to estimate densities with mixed types of variables, i.e., discrete and continuous random variables.

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Notes

  1. Below we define the problem in a more general way to consider also drift and recurrent distributions, but we focus only on the most fundamental problem of estimating a single distribution from a stream in this paper.

  2. https://github.com/geilke/mideo.

  3. Please notice that we also compared the online density estimator with a corresponding batch version. The results are available in Online Resource 1.

  4. Please note that the problem of having too few examples for accurately estimating the CPTs could be less prominent when the CPTs are replaced by decision trees (Friedman and Goldszmidt 1996; Su and Zhang 2006).

  5. Unfortunately, even after several emails, the authors of RS-Forest did not respond to our request to share their program.

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Acknowledgements

We would like to thank the editor and the anonymous reviewers for their comments. They improved the presentation, readability, and quality of this paper substantially. We are particularly grateful to the anonymous reviewer who proposed the exponentiated gradient investment strategy for weighting the classifier chains.

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Authors and Affiliations

  1. Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128, Mainz, Germany

    Michael Geilke, Andreas Karwath & Stefan Kramer

  2. Department of Computer Science, The University of Waikato, Hamilton, 3240, New Zealand

    Eibe Frank

Authors
  1. Michael Geilke
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  2. Andreas Karwath
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  3. Eibe Frank
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  4. Stefan Kramer
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Correspondence to Michael Geilke.

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Responsible editor: Hendrik Blockeel.

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Geilke, M., Karwath, A., Frank, E. et al. Online estimation of discrete, continuous, and conditional joint densities using classifier chains. Data Min Knowl Disc 32, 561–603 (2018). https://doi.org/10.1007/s10618-017-0546-6

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