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Forecasting for Discrete Time Processes based on Causal Band-limited
Approximation
Nikolai Dokuchaev
Department of Mathematics & Statistics, Curtin University, GPO Box U1987, Perth, 6845 Western Australia, Australia
Keywords:
Band-limited Processes, Discrete Time Processes, Causal Filters, Low-pass Filters, Forecasting.
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 Eu-
clidean 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.
1 INTRODUCTION
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 Eu-
clidean norm by the trace of a band-limited discrete
time process. Since an unique extrapolation to fu-
ture times of the optimal approximating band-limited
process can be interpreted as an optimal forecast, this
task has many practical applications. It is well known
that it is not possible to find an ideal low-pass causal
linear time-invariant filter. In continuous time setting,
it is known that the distance of the set of ideal low-
pass filters from the set of all causal filters is positive
(Almira and Romero, 2008) and that the optimal ap-
proximation of the ideal low-pass filter is not possible
(Dokuchaev, 2012c). Our goal is to substitute the so-
lution of these unsolvable problems by solution of an
easier problem in discrete time setting such that the
filter is not necessary time invariant. Our motivation
is that, for some problems, the absence of time in-
variancy for a filter can be tolerated. For example,
a typical approach to forecasting in finance is to ap-
proximate the known path of the stock price process
by a process allowing an unique extrapolation that can
be used as a forecast. This has to be done at current
time; at future times, forecasting rule can be amended
according to new data collected.
We suggest to approximate discrete time pro-
cesses by the discrete time band-limited processes.
More precisely, we suggest to approximate the known
historical path of the process by the trace of a band-
limited process. The approximating sequence does
not necessary match the underlying process at sam-
pling points. This is different from classical sampling
approach; see, e.g., (Jerry, 1977). Our approach is
close to the approach from (Ferreira, 1995b) and (Fer-
reira, 1995a), where the estimate of the error norm is
given. The difference is that, in our setting, it is guar-
anteed that the approximation generates the error of
the minimal Euclidean norm.
We obtain analyze existence and uniqueness of an
optimal approximation. The optimal process is de-
rived in time domain in a form of sinc series.The ap-
proximating band-limited process can be interpreted
as a causal and linear filter that is not time invari-
ant. The filter obtained is not time invariant; as a
consequence, the coefficients of these series and have
to be changed dynamically, to accommodate the cur-
rent flow of observations. An unique extrapolation
to future times of the optimal approximating band-
limited process can be interpreted as an optimal fore-
cast at any given time. This paper develops further the
approach suggested in (Dokuchaev, 2011) where the
continuous time setting was considered. We extend
now this approach on discrete time processes. Some
related results can be found in (Dokuchaev, 2012b)
and (Dokuchaev, 2012d) for discrete time processes
that are band-limited or close to band-limited.
2 DEFINITIONS
For a Hilbert space H, we denote by (·,·)
H
the cor-
280
Dokuchaev N..
Forecasting for Discrete Time Processes based on Causal Band-limited Approximation.
DOI: 10.5220/0004243700820085
In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 82-85
ISBN: 978-989-8565-40-2
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)