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[
(
t
)
y
(
t
)
sgn(
y
(
t
))
<
(
t
)
sgn(
y
(
t
))
Fig. 1.
Block scheme presents the MG in the configuration of prediction
assume that samples of
ξ
are Independent and Identically Distributed (IID).
The model is supposed to retrieve dependencies between the past and future
values of sgn(
y
(
t
)). The delay block introduces a one-step delay to its every
input sample
y
(
t
) and forms the vector
n
)of
n>
1 past samples. The MG
model predicts the next sign of sample exploiting the information included in the
signs of previous movements. The block
f
compares the predicted signal with
a real output of the object and calculates the
correctness Ψ
(
t
). The correctness
Ψ
(
t
) represents the average success rate of prediction and is calculated as a
percentage of properly predicted signs of
y
, provided that all samples up to time
t
are considered:
y
(
t
−
t
Ψ
(
t
)=
1
t
δ
(sgn(
y
(
n
))
,
sgn(
y
(
n
)))
(4)
n
=1
where
δ
stands for Kronecker symbol.
Two types of objects were analyzed: the autoregressive stochastic processes
and the time series of real prices of shares. The former is mainly used to demon-
strate interesting properties of the model and to learn how to tune its parameters.
The latter one is used as an example of practical application of the predictor.
In the case of the autoregressive stochastic process, assuming that the definition
of the object is known, the maximal theoretical level of
Ψ
can be calculated
by placing
E
[
y
(
n
)] instead of
sgn(
y
(
n
)) in Eq. 4. The expected value
E
[
y
(
n
)] is
calculated recursively for each
n
according to the process definition.
5 Optimization of the Parameters
Initially, we apply the predictor to the third order autoregressive time series,
AR(3) defined as follows:
y
(
t
)=0
.
7
y
(
t
−
1)
−
0
.
5
y
(
t
−
2)
−
0
.
2
y
(
t
−
3) +
ξ
(
t
)
,
(5)
where
ξ
(
t
) is an instance of gaussian white noise. It was found numerically that
the process is characterized by
ar
[
Ψ
MAX
]
<
0
.
01,
where
t
= 3000 steps, and the average was taken over ten realizations.
[
Ψ
MAX
]=0
.
77, and
E
V
