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thus potentially unbounded, what carries some considerable consequences if the
exogenous process is a non-stationary one. Namely, if characteristics of a mod-
elled object changes at time
t
then the group of best strategies would change
either. But if, until
t
, some strategies collected many positive payoffs and after
time
t
they are no longer profitable, then many steps with negative payoffs are
required to lower their utilities. Hence, after time
t
the system still uses poten-
tially ineffective strategies. The following example should provide some deeper
understanding of this issue.
Let us assume that after 1500 steps the system described by Eq. 5 is replaced
by the following one:
y
(
t
)=
3) +
ξ
(
t
). The
new
AR
(3) process is defined in such a way that the best MG strategies for this
process constitute also the set of the worst strategies of the previous process.
After time
t
= 1500 the predictor still uses strategy which is no longer ecient,
but is characterized by the highest
U
.Asaresult,the
Ψ
as a function of time
starts to decrease, as seen in Fig.3 (green dotted line).
In order to speed up the new appraisal of outdated
U
we introduced additional
parameter
λ
to the rule (3)
−
0
.
3
y
(
t
−
1)
−
0
.
2
y
(
t
−
2) + 0
.
6
y
(
t
−
U
α
n
(
t
+1)=
λU
α
n
(
t
)+
Φ
α
n
(
t
)
,
(7)
where
λ
[0
,
1]. If
λ
= 1 the strategies have an infinite memory of all previous
rewards, what corresponds to standard MG. If
λ
= 0, then there is no memory
effect. The intermediate values 0
<λ<
1 preserve increasing of
U
to infinity,
when the process is a stationary one, what considerably speeds up the time
of adaptation. The cost of the introduction of the additional parameter is a
need of its optimization. The heuristic analysis of correctness as a function of
λ
is presented in Fig. 4. The value
λ
=0
.
97 assures the best results, although
other values, that are close to it, work effectively either. Summing up, the
λ
-
GCMG model effectively follows changes in the predicted signal and significantly
outperforms the GCMG for a non-stationary process.
∈
6 Sign Prediction of Assets' Returns
In this section we present the assumed model of movements of share prices.
Given this we apply the
λ
-GCMG predictor to retrieve dependencies between
past and future samples of stocks prices taken from various markets.
6.1 Price Movements Model
We assume that the dynamics of returns is driven by two factors where one is
endogenous and another one exogenous. The model of signs of return rates is
sgn(r(t)) =
f
[sgn
r
(
t
−
1)
,...,
sgn
r
(
t
−
m
)] +
ξ
(
t
)
.
(8)
where the first term reflects a dependence on previous returns and the second
term represents the influence of external events. We are interesting only in signs
