Abstract
The use of constraint optimization has recently proven to be a successful approach to providing solutions to various NP-hard search and optimization problems in data analysis. In this work we extend the use of constraint optimization systems further within data analysis to a central problem arising from the analysis of multivariate data, namely, determining minimum-width multivariate confidence intervals, i.e., the minimum-width confidence band problem (MWCB). Pointing out drawbacks in recently proposed formalizations of variants of MWCB, we propose a new problem formalization which generalizes the earlier formulations and allows for circumvention of their drawbacks. We present two constraint models for the new problem in terms of mixed integer programming and maximum satisfiability, as well as a greedy approach. Furthermore, we empirically evaluate the scalability of the constraint optimization approaches and solution quality compared to the greedy approach on real-world datasets.
This work was financially supported by Academy of Finland (grants 251170 COIN, 276412, 284591, 288814); Tekes (Revolution of Knowledge Work); and DoCS Doctoral School in Computer Science and Research Funds of the University of Helsinki.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In case of ties, both elements get the same rank r and the next greatest element gets rank \(r+1\).
- 2.
- 3.
References
Asín, R., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E.: Cardinality networks: a theoretical and empirical study. Constraints 16(2), 195–221 (2011)
Biere, A., Heule, M., van Maaren, H., Walsh, T.: Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press, Amsterdam (2009)
Davies, J., Bacchus, F.: Exploiting the power of mip solvers in maxsat. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 166–181. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39071-5_13
Gardner, M.J., Altman, D.G.: Confidence intervals rather than P values: estimation rather than hypothesis testing. Br. Med. J. (Clin. Res. Ed.) 292(6522), 746–750 (1986)
Goldberger, A.L., Amaral, L.A.N., Glass, L., Hausdorff, J.M., Ivanov, P.C., Mark, R.G., Mietus, J.E., Moody, G.B., Peng, C.K., Stanley, H.E.: PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 101(23), e215–e220 (2000)
Guilbaud, O.: Simultaneous confidence regions corresponding to Holm’s step-down procedure and other closed-testing procedures. Biom. J. 50(5), 678 (2008)
Hyndman, R.J., Fan, Y.: Sample quantiles in statistical packages. Am. Stat. 50(4), 361–365 (1996)
IBM ILOG: CPLEX optimizer (2017). http://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/
Kolsrud, D.: Time-simultaneous prediction band for a time series. J. Forecast. 26(3), 171–188 (2007)
Korpela, J., Oikarinen, E., Puolamäki, K., Ukkonen, A.: Multivariate confidence intervals. In: Proceedings of SDM, pp. 696–704. SIAM (2017)
Korpela, J., Puolamäki, K., Gionis, A.: Confidence bands for time series data. Data Min. Knowl. Discov. 28(5–6), 1530–1553 (2014)
Koshimura, M., Zhang, T., Fujita, H., Hasegawa, R.: QMaxSAT: a partial Max-SAT solver. J. Satisf. Boolean Model. Comput. 8(1/2), 95–100 (2012)
Lichman, M.: UCI machine learning repository (2013). http://archive.ics.uci.edu/ml
Liu, W., Jamshidian, M., Zhang, Y., Bretz, F., Han, X.: Some new methods for the comparison of two linear regression models. J. Stat. Plan. Inference 137(1), 57–67 (2007)
Mandel, M., Betensky, R.A.: Simultaneous confidence intervals based on the percentile bootstrap approach. Comput. Stat. Data Anal. 52(4), 2158–2165 (2008)
Menne, M., Durre, I., Korzeniewski, B., McNeal, S., Thomas, K., Yin, X., Anthony, S., Ray, R., Vose, R., Gleason, B., Houston, T.: Global Historical Climatology Network – Daily (GHCN-Daily), version 3.11 (2012)
Menne, M., Durre, I., Vose, R., Gleason, B., Houston, T.: An overview of the global historical climatology network-daily database. J. Atmos. Ocean. Technol. 29, 897–910 (2012)
Moody, G.B., Mark, R.G.: The impact of the MIT-BIH arrhythmia database. IEEE Eng. Med. Biol. Mag. 20(3), 45–50 (2001)
Morgado, A., Dodaro, C., Marques-Silva, J.: Core-guided MaxSAT with soft cardinality constraints. In: O’Sullivan, B. (ed.) CP 2014. LNCS, vol. 8656, pp. 564–573. Springer, Cham (2014). doi:10.1007/978-3-319-10428-7_41
Nuzzo, R.: Scientific method: statistical errors. Nature 506, 150–152 (2014)
Schüssler, R., Trede, M.: Constructing minimum-width confidence bands. Econ. Lett. 145, 182–185 (2016)
Staszewska-Bystrova, A., Winker, P.: Constructing narrowest pathwise bootstrap prediction bands using threshold accepting. Int. J. Forecast. 29(2), 221–233 (2013)
Trafimow, D., Marks, M.: Editorial. Basic Appl. Soc. Psychol. 37(1), 1–2 (2015)
Wolf, M., Wunderli, D.: Bootstrap joint prediction regions. J. Time Ser. Anal. 36(3), 352–376 (2015)
Woolston, C.: Psychology journal bans P values. Nature 519, 9 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Berg, J., Oikarinen, E., Järvisalo, M., Puolamäki, K. (2017). Minimum-Width Confidence Bands via Constraint Optimization. In: Beck, J. (eds) Principles and Practice of Constraint Programming. CP 2017. Lecture Notes in Computer Science(), vol 10416. Springer, Cham. https://doi.org/10.1007/978-3-319-66158-2_29
Download citation
DOI: https://doi.org/10.1007/978-3-319-66158-2_29
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66157-5
Online ISBN: 978-3-319-66158-2
eBook Packages: Computer ScienceComputer Science (R0)