Abstract
We propose two kernel based methods for detecting the time direction in empirical time series. First we apply a Support Vector Machine on the finite-dimensional distributions of the time series (classification method) by embedding these distributions into a Reproducing Kernel Hilbert Space. For the ARMA method we fit the observed data with an autoregressive moving average process and test whether the regression residuals are statistically independent of the past values. Whenever the dependence in one direction is significantly weaker than in the other we infer the former to be the true one. Both approaches were able to detect the direction of the true generating model for simulated data sets. We also applied our tests to a large number of real world time series. The ARMA method made a decision for a significant fraction of them, in which it was mostly correct, while the classification method did not perform as well, but still exceeded chance level.
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
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Balian, R. (1992). From microphysics to macrophysics. Berlin: Springer.
Breidt, F. J., & Davis, R. A. (1991). Time-reversibility, identifiability and independence of innovations for stationary time series. Journal of Time Series Analysis, 13(5), 379–390.
Brockwell, P. J., & Davis, R. A. (1991). Time series: Theory and methods (2nd ed.). Berlin: Springer.
Eichler, M., & Didelez, V. (2007). Causal reasoning in graphical time series models. In Proceedings of the 23nd Annual Conference on Uncertainty in Artifical Intelligence (pp. 109–116).
Jarque, C. M., & Bera, A. K. (1987). A test for normality of observations and regression residuals. International Statistical Review, 55(2), 163–172.
Gretton, A., Borgwardt, K., Rasch, M., Schölkopf, B., & Smola, A. (2007). A kernel method for the two-sample-problem. In Advances in Neural Information Processing Systems (Vol. 19, pp. 513–520). Cambridge, MA: MIT Press.
Gretton, A., Bousquet, O., Smola, A., & Schölkopf, B. (2005). Measuring statistical dependence with Hilbert–Schmidt norms. In Algorithmic Learning Theory: 16th International Conference (pp. 63–78). Berlin: Springer.
Hein, M., & Bousquet, O. (2005). Hilbertian metrics and positive definite kernels on probability measures. In Proceedings of AISTATS.
Kankainen, A. (1995). Consistent testing of total independence based on the empirical characteristic function. Ph.D. Thesis, University of Jyväskylä.
Mandelbrot, B. (1967). On the distribution of stock price differences. Operations Research, 15(6), 1057–1062.
Pearl, J. (2002). Causality. Cambridge: Cambridge University Press.
Peters, J. (2008). Asymmetries of time series under inverting their direction. Diploma Thesis.
Reichenbach, H. (1956). The direction of time. Berkeley: University of California Press.
Shimizu, S., Hoyer, P. O., Hyvärinen, A., & Kerminen, A. (2006). A linear non-gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7, 2003–2030.
Smola, A. J., Gretton, A., Song, L., & Schölkopf, B. (2007). A Hilbert space embedding for distributions. In Algorithmic Learning Theory: 18th International Conference (pp. 13–31). Berlin: Springer.
Spirtes, P., Glymour, C., & Scheines, R. (1993). Causation, prediction, and search. Berlin: Springer.
Sriperumbudur, B., Gretton, A., Fukumizu, K., Lanckriet, G., & Schölkopf, B. (2008). Injective hilbert space embeddings of probability measures. In COLT, 2008.
Weiss, G. (1975). Time-reversibility of linear stochastic processes. Journal of Applied Probability, 12, 831–836.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Peters, J., Janzing, D., Gretton, A., Schölkopf, B. (2009). Kernel Methods for Detecting the Direction of Time Series. In: Fink, A., Lausen, B., Seidel, W., Ultsch, A. (eds) Advances in Data Analysis, Data Handling and Business Intelligence. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01044-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-01044-6_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01043-9
Online ISBN: 978-3-642-01044-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)