Electronic Journal of Statistics

Bayesian nonparametric forecasting of monotonic functional time series

Antonio Canale and Matteo Ruggiero

Full-text: Open access


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

Permanent link to this document

Digital Object Identifier

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

Export citation


  • 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 Forecasting32, 1340–1351.
  • Caron, F., Davy. M. and Doucet, A. (2007). Generalized Polya urn for time-varying Dirichlet process mixtures., In23rd 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 Processing56, 71–84.
  • Dahl, D.B. (2006). Model-based clustering for expression data via a Dirichlet process mixture, model.Bayesian inference for gene expression and proteomics201–218.
  • Dunson, D.B. (2006). Bayesian dynamic modelling of latent trait, distributions.Biostatistics7, 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 Statistics3, 1556–1566.
  • Ethier, S.N. and Kurtz, T.G. (1993). Fleming–Viot processes in population, genetics.SIAM Journal on Control and Optimization31, 345–386.
  • European Union (2003). Directive, 2003/54/ec.Official Journal of the European Union176, 37–55.
  • Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric, problems.Annals of Statistics1, 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&CP31, 20–28.
  • Gestore Mercati Energetici (2010). Italian natural gas trading platform operative details. Available in English at the, pagehttp://www.mercatoelettrico.org/En/Mercati/Gas/PGas.aspx.
  • Griffin, J.E. and Steel, M.F.J. (2010). Stick-breaking autoregressive, processes.Journal of Econometrics162, 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., InASA 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 Sciences100, 15324–15328
  • Marin, J.-M., Pudlo, P., Robert, C.P. and Ryder, R.J. (2012). Approximate Bayesian computational, methods.Statistics and Computing22, 1167–1180.
  • Mena, R.H. and Ruggiero, M. (2016). Dynamic density estimation with diffusive Dirichlet, mixtures.Bernoulli22, 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 Inference141, 3217–3230.
  • Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling., InProceedings of the 3rd International Workshop on Distributed Statistical Computing
  • Prünster, I. and Ruggiero, M. (2013). A Bayesian nonparametric approach to modeling market share, dynamics.Bernoulli19, 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 Analysis3, 339–366.
  • Ruggiero, M. and Walker, S.G. (2009a). Bayesian nonparametric construction of the Fleming–Viot process with fertility, selection.Statistica Sinica19, 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 Probability14, 501–517.