Electronic Journal of Statistics

Bayesian nonparametric forecasting of monotonic functional time series

Antonio Canale and Matteo Ruggiero

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We propose a Bayesian nonparametric approach to modelling and predicting a class of functional time series with application to energy markets, based on fully observed, noise-free functional data. Traders in such contexts conceive profitable strategies if they can anticipate the impact of their bidding actions on the aggregate demand and supply curves, which in turn need to be predicted reliably. Here we propose a simple Bayesian nonparametric method for predicting such curves, which take the form of monotonic bounded step functions. We borrow ideas from population genetics by defining a class of interacting particle systems to model the functional trajectory, and develop an implementation strategy which uses ideas from Markov chain Monte Carlo and approximate Bayesian computation techniques and allows to circumvent the intractability of the likelihood. Our approach shows great adaptation to the degree of smoothness of the curves and the volatility of the functional series, proves to be robust to an increase of the forecast horizon and yields an uncertainty quantification for the functional forecasts. We illustrate the model and discuss its performance with simulated datasets and on real data relative to the Italian natural gas market.

Article information

Electron. J. Statist. Volume 10, Number 2 (2016), 3265-3286.

Received: December 2015
First available in Project Euclid: 16 November 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G00
Secondary: 62F15: Bayesian inference 60J22: Computational methods in Markov chains [See also 65C40]

Approximate Bayesian computation dependent processes Dirichlet process interacting particle system Moran model Polya urn prediction


Canale, Antonio; Ruggiero, Matteo. Bayesian nonparametric forecasting of monotonic functional time series. Electron. J. Statist. 10 (2016), no. 2, 3265--3286. doi:10.1214/16-EJS1190. https://projecteuclid.org/euclid.ejs/1479287221.

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  • Arratia, R., Barbour, A.D. and Tavarè, S. (2003)., Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics, Zürich.
  • Canale, A. and Vantini, S. (2016). Constrained functional time series: applications to the Italian gas market., International Journal of Forecasting 32, 1340–1351.
  • Caron, F., Davy. M. and Doucet, A. (2007). Generalized Polya urn for time-varying Dirichlet process mixtures. In, 23rd Conference on Uncertainty in Artificial Intelligence, Vancouver.
  • Caron, F., Davy, M., Doucet, A., Duflos, E. and Vanheeghe, P. (2008). Bayesian inference for linear dynamic models with Dirichlet process mixtures., IEEE Transactions on Signal Processing 56, 71–84.
  • Dahl, D.B. (2006). Model-based clustering for expression data via a Dirichlet process mixture model., Bayesian inference for gene expression and proteomics 201–218.
  • Dunson, D.B. (2006). Bayesian dynamic modelling of latent trait distributions., Biostatistics 7, 551–568.
  • Etheridge, A.M. (2009)., Some mathematical models from population genetics. École d’été de Probabilités de Saint-Flour XXXIX. Lecture Notes in Math. 2012. Springer.
  • Favaro, S., Ruggiero, M. and Walker, S.G. (2009). On a Gibbs sampler based random process in Bayesian nonparametrics., Electronic Journal of Statistics 3, 1556–1566.
  • Ethier, S.N. and Kurtz, T.G. (1993). Fleming–Viot processes in population genetics., SIAM Journal on Control and Optimization 31, 345–386.
  • European Union (2003). Directive 2003/54/ec., Official Journal of the European Union 176, 37–55.
  • Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems., Annals of Statistics 1, 209–230.
  • Foti, N.J., Futoma, J.D., Rockmore, D.N. and Williamson, S. (2013). A unifying representation for a class of dependent random measures., Artificial Intelligence and Statistics, Journal of Machine Learning Research W&CP 31, 20–28.
  • Gestore Mercati Energetici (2010). Italian natural gas trading platform operative details. Available in English at the page, http://www.mercatoelettrico.org/En/Mercati/Gas/PGas.aspx.
  • Griffin, J.E. and Steel, M.F.J. (2010). Stick-breaking autoregressive processes., Journal of Econometrics 162, 383–396.
  • Hjort, N.L., Holmes, C.C., Müller, P. and Walker, S.G., eds. (2010)., Bayesian Nonparametrics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge Univ. Press.
  • MacEachern, S.N. (1999). Dependent Nonparametric Processes. In, ASA Proceedings of the Section on Bayesian Statistical Science. American Statist. Assoc., Alexandria, VA.
  • MacEachern, S.N. (2000). Dependent Dirichlet processes., Tech. Rep., Ohio State University.
  • Marjoram, P., Molitor, J., Plagnol and V., Tavaré, S. (2012). Markov chain Monte Carlo without likelihood., Proceedings of the National Academy of Sciences 100, 15324–15328
  • Marin, J.-M., Pudlo, P., Robert, C.P. and Ryder, R.J. (2012). Approximate Bayesian computational methods., Statistics and Computing 22, 1167–1180.
  • Mena, R.H. and Ruggiero, M. (2016). Dynamic density estimation with diffusive Dirichlet mixtures., Bernoulli 22, 901–926.
  • Mena, R.H., Ruggiero, M. and Walker, S.G. (2011). Geometric stick-breaking processes for continuous-time Bayesian nonparametric modelling., Journal of Statistical Planning and Inference 141, 3217–3230.
  • Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In, Proceedings of the 3rd International Workshop on Distributed Statistical Computing
  • Prünster, I. and Ruggiero, M. (2013). A Bayesian nonparametric approach to modeling market share dynamics., Bernoulli 19, 64–92.
  • Ramsay, J. and Silverman, B. (2005)., Functional Data Analysis. Springer Series in Statistics. Springer.
  • Rodriguez, A. and Ter Horst, E. (2008). Bayesian dynamic density estimation., Bayesian Analysis 3, 339–366.
  • Ruggiero, M. and Walker, S.G. (2009a). Bayesian nonparametric construction of the Fleming–Viot process with fertility selection., Statistica Sinica 19, 707–720.
  • Ruggiero, M. and Walker, S.G. (2009b). Countable representation for infinite-dimensional diffusions derived from the two-parameter Poisson–Dirichlet process., Electronic Communications in Probability 14, 501–517.