# Forecasting for Discrete Time Processes based on Causal Band-limited Approximation

### Nikolai Dokuchaev

#### Abstract

We study causal dynamic smoothing of discrete time processes via approximation by band-limited discrete time processes. More precisely, a part of the historical path of the underlying process is approximated in Euclidean norm by the trace of a band-limited process. We analyze related optimization problem and obtain some conditions of solvability and uniqueness. An unique extrapolation to future times of the optimal approximating band-limited process can be interpreted as an optimal forecast.

#### References

- Almira, J. and Romero, A. (2008). How distant is the ideal filter of being a causal one? In Atlantic Electronic Journal of Mathematics. 3 (1) 46-55.
- Dokuchaev, N. (2011). On causal band-limited mean square approximation. In Working paper. http://arxiv.org/abs/1111.6701.
- Dokuchaev, N. (2012a). Causal band-limited approximation and forecasting for discrete time processes. In Working paper. http://lanl.arxiv.org/abs/1208.3278.
- Dokuchaev, N. (2012b). On predictors for band-limited and high-frequency time series. In Signal Processing. 92, iss. 10, 2571-2575.
- Dokuchaev, N. (2012c). On sub-ideal causal smoothing filters. In Signal Processing. 92, iss. 1, 219-223.
- Dokuchaev, N. (2012d). Predictors for discrete time processes with energy decay on higher frequencies. In IEEE Transactions on Signal Processing. 60, No. 11, 6027-6030.
- Ferreira, P. G. S. G. (1995a). Approximating non-bandlimited functions by nonuniform sampling series. In SampTA'95.
- Ferreira, P. G. S. G. (1995b). Nonuniform sampling of nonbandlimited signals. In IEEE Signal Processing Letters. 2, Iss. 5, 89-91.
- Jerry, A. (1977). The shannon sampling theorem - its various extensions and applications: A tutorial review. In Proc. IEEE. 65, 11, 1565-1596.

#### Paper Citation

#### in Harvard Style

Dokuchaev N. (2013). **Forecasting for Discrete Time Processes based on Causal Band-limited Approximation** . In *Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-8565-40-2, pages 82-85. DOI: 10.5220/0004243700820085

#### in Bibtex Style

@conference{icores13,

author={Nikolai Dokuchaev},

title={Forecasting for Discrete Time Processes based on Causal Band-limited Approximation},

booktitle={Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2013},

pages={82-85},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004243700820085},

isbn={978-989-8565-40-2},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - Forecasting for Discrete Time Processes based on Causal Band-limited Approximation

SN - 978-989-8565-40-2

AU - Dokuchaev N.

PY - 2013

SP - 82

EP - 85

DO - 10.5220/0004243700820085