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1452 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
M of the adaptive RBF neural network filtering is
determined.
The dimension m of chaotic time series is
calculated by the way of G- P algorithms, and
the delay time τ is calculated by the selfcorrelation
method. For the overall description
of the dynamics characteristic of the original
system by the Takens' delay-coordinate phase
reconstruct theory, a chaotic series demand
m≥ 2d
+ 1 variances at least, so the number of
the input nerve cells of the adaptive RBF neural
network filtering is M = m ; The reconstruction
phase space vector number is 200, Then, the
200 phase space vectors to make a simple
normalized, the normalized as
[ x() t −mean( x())]/[max( t x()) t − min( x())]
t
,
t = 1, 2, 200 , and making the value is owned by a range
of -1 / 2 to 1/2.
Step2) The adaptive filtering is initialized and the
weights are vested the initial values. RBF neural network
vector weighting parameters w is initialized, where the
weight vector w in each component take random function
between 0 and 1; and the learning rate η is initialized at
the same time, where η = 0.0002 . β and γ are the
learning rate adjustment factors, 0< β < 1, γ > 1 , for
example, β = 0.75, γ = 1.05 .
Step3) Using the above the initialization network and
the pretreatment traffic flow time series, the first training
network is carried out.
Step4) The error is calculated. If the error is in the
scope of the permission, the error is calculated and it
turns into Step4), otherwise it continues; the error
function formula:
200
1 2
E( θ ) = ( y( t) − y( t))
(9)
∑
2 t = 1
Set the maximum error is E max
= 0.035 , if E < Emax
,
the storage RBF neural network parameter use w ;
otherwise, then a second training network will be
required.
Step5) Adjust the adaptive learning rate If A previous
training error is recorded as En
− 1
, the current error is
recorded as E n
, then Calculate the ratio of E
n
to En
− 1
,
En
Setting constants k = 1.04 , if > k = 1.04 , then
En
− 1
substitute βη for η to reduce learning rate; otherwise,
replace η with γη to increase learning rate.
Step6) In the adaptive RBF neural network filtering for
the chaotic time series prediction in Figure 1,
x( k) = x( t)
t = 1, 2, , N is the input, yk ˆ( ) = xt ˆ( ) is the
output.
Introduce nonlinear feedback into the weighting
formal to adopt Chaos Mechanisms, due to the nonlinear
feedback is vector form of weighting variables. In order
to facilitate understanding, respectively, gives the vector
w and its weighting formal, as follows.
Note
Δ w l ( t+ 1) = w l ( t+ 1) −w l ( t)
,
ji ji ji
which represents the current value of weighting variables,
then
1
Δ w l ( t+ 1) = w l ( t+ 1) − w l () t = −ηδ l+
() t x
l () t
ji ji ji j i
In order to speed up the learning process, in the right to
l
join a momentum term αΔw () t , then
l 1
( 1) l +
( ) l ( ) l
ji
t ηδ
j
t
i
t α
ji
( t)
ji
Δ w + = − x + Δw (10)
where α is inertia factor. As a constant, the weight of
amendments is linear, not introduce chaos mechanism.
then we Introduce a nonlinear feedback (chaos
mechanism on the right):
1
Δ w l ( t+ 1) = − ηδ l+
( t) x l ( t) + g( Δ w l ( t+
1)) (11)
ji j i ji
Expand this equation into scalar form as follow:
l l+
1 l l
⎧Δ wji ( t+ 1) = − ηδ
j
( t) xi ( t) + g( Δwji
( t))
⎪
l l+
1
l l
⎪Δ wji ( t+ 1 + τ) =− ηδ
j
( t+ τ) xi ( t+ τ) + g( Δ wji
( t+
τ))
⎪
l l+
1
l l
⎪Δ wji ( t+ 1+ 2 τ ) = − ηδ
j
( t+ 2) xi ( t+ 2 τ ) + g( Δ wji
( t+
2 τ ))
⎨
⎪
⎪ l l+
1
l
⎪
Δ wji ( t+ 1 + ( m− 1) τ ) =− ηδ
j
( t+ ( m− 1) τ ) xi
( t+ ( m−1) τ )
⎪ l
⎩
+ g(
Δwji
( t+ ( m−1) τ ))
(12)
where, feedback can take a variety of vector functions,
for example:
2
g( x) = tanh( px)exp( − qx )
or
g( x) = pxexp( − q x)
,
in the study, p = 0.7 , q = 0.1.
Step7) Using the new learning rate in Step5) and RBF
network parameters with nonlinear feedback in Step6) to
calculate the new value, and train network again, then get
the error and enter into Step4), repeated training until the
relative error in traffic meet E < Emax
.
Step8) Output of each stored network parameters and
training error curve.
V. EXAMPLE ANALYSIS AND CONCLUSIONS
A. Model and Data
In this paper, the chaotic time series is the object of
study of the numerical simulation in Lorenz dynamic
system. In 1963, the meteorologist Lorenz describe the
evolution of the weather by three-dimensional
autonomous equations; when the parameter σ = 10 ,
8
r = 28 , b = , the long-term changes in the weather
3
unpredictable, that is, the system presents a chaotic state,
and for the first time given a strange attractor. The
attractors are shown in Figure 2 (a), Figure 2 (b), Figure 2
(c) and Figure 2 (d):
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