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1482 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
t
0
time very far from the t time, t 0
→∞, x( t− t0
) has
no effect on yt (); means that the predicted value of yt ()
is irrelevant to x( t− t0
).
In the prediction of chaos traffic flow chaotic time
series, t′ = t+ ( m− 1) τ , T ( T > 0 ) is forward prediction
step, x( t′ + T)
represents the output associated with the
input signal x()
t and the delay time τ , then
N l −1
1
l1 l
2 N li 0 ∑ 1 1 1
l1
= 0
x( t′ + T) = f( x , x , x ) = h + h ( l ) x( t−lτ
)
Nl
−1N
1
2 l −
2
∑∑
+ h ( l , l ) x( t−lτ
) x( t−l
τ )
l1= 0 l2=
0
2 1 2 1 2
Nl −1N 3 l −1N
1
3 l −
3
∑∑∑ h3( l1, l2, l3) xt ( l1τ) xt ( l2τ) xt ( l3τ)
(5)
l1= 0 l2= 0 l3=
0
+ − − − +
note
Nmax = max( N , N , N , N ) , ( i = 1, 2, 3, ),
when n≥
N
li
max
l1 l2 l3
l i
, the same to meet the input traffic flow
signal x = xt ( − lτ ) is irrelevant to yt () , then the
formula (4) can be written as:
i
Nmax
−1
l1 l
2 N li 0 ∑ 1 1 1
l1
= 0
x( t′ + T) = f( x , x , x ) = h + h ( l ) x( t−lτ
)
Nmax
−1Nmax
−1
∑ ∑
+ h ( l , l ) x( t−lτ
) x( t−lτ
)
l1= 0 l2=
0
Nmax −1Nmax −1Nmax
−1
∑ ∑ ∑
l1= 0 l2= 0 l3=
0
2 1 2 1 2
+ h ( l , l , l ) x( t−lτ) x( t−lτ) x( t− lτ)
+
3 1 2 3 1 2 3
Know from the above analysis of the traffic flow
functional systems, the power series expansion item of
prediction results are in fact only related to Know from
the above analysis of the traffic flow functional systems,
the power series expansion item of prediction results are
in fact only related to summation form all the products of
the Input signal and the first power delay time signal.
This means that the value of
Nmax = max( Nl , N , , )
1 l
N
2 l
N
3 l i
, ( i = 1, 2, 3, ) is only
related with the number of input signal and the delay time
signal, which is the minimum embedding dimension m
of phase space, so Nmax = max( Nl , N , , )
1 l
N
2 l
N
3 l i
= m.
Such traffic flow chaotic time series Volterra series
model is finalized by the formula (5) as follows:
m−1
l1 l
2 N li 0 ∑ 1 1 1
l1
= 0
x( t′ + T) = f( x , x , x ) = h + h ( l ) x( t−lτ
)
m−1 m−1 m−1
∑∑∑
m−1 m−1
∑∑
+ h ( l , l ) x( t−lτ
) x( t−lτ
)
2 1 2 1 2
l1= 0l2=
0
+ h( l, l, l) xt ( −lτ) xt ( −lτ) xt ( − lτ)
+
3 1 2 3 1 2 3
l1= 0l2= 0l3=
0
m−1m−1m−1 m−1
∑∑∑ ∑
+ h (, l l, l, , l xt−lτ)( xt−lτ)( xt−lτ) xt ( −lτ)
(7)
m 1 2 3 m 1 2 3
m
l1= 0l2= 0l3= 0 lm=
0
(6)
III. TRAFFIC FLOW TIME SERIES VOLTERRA NEURAL
NETWORK MODEL (VNNTF)
A. Representation of Nonlinear Systems Using Artificial
Neural Network
Has proven that the BP neural network with one
hidden layer can approximate any continuous bounded
non-linear system, therefore, generally selected to contain
a three-layer back propagation BP network with one
hidden layer to approximate nonlinear systems. A single
output three-layer back propagation neural network is
shown in Figure 1. In the figure, the input vector
T
x = [ x , x , x ] at moment n can obtain by the
k k,0 k,1 k,
M
delay of x( k ), where x
,
= xk ( − m)
, the input of the l
km
hidden unit ( l = 1, 2, , L) is
Z
= S ( u ); ulk ,
= ∑ wlm ,
xkm
,
(8)
lk , l lk ,
M
m=
0
A single output three-layer back propagation neural
network is shown in Figure 1.
x k,0
x km ,
x kM ,
w L , m
w 1,0
w
,0
w l
L,0
w 1,m
w lm ,
w 1,M
w lM ,
w L , M
U 1,k
U lk ,
U L , k
S()
⋅
S()
⋅
S()
⋅
Z 1,k
Z lk ,
Z L , k
input hidden layer output
Figure 1. Three layer neural networks in response to M+1 input and
single output system
If the implicit function selected the sigmoid function,
then
1
Sl( u
,
) = l k
1 + exp[ − λ( u − θ )]
(9)
Where, θ
l
is the threshold of the unit n, If the output unit
is linear summation unit, the output at moment n is
y
L
r l
lk ,
r 1
r L
k l l,
k
l = 1
l
Z
= ∑ rZ
(10)
The output of each hidden unit to expand into a Taylor
series at the threshold θ
l
:
Z = ϕ ( u ) =∑ d ( θ ) u
i
(11)
l, k l l, k i l l,
k
i=
0
where, d ( θ ) is the commencement of the coefficient,
i
l
the value associated with
M
lk , lm , km ,
m=
0
∞
θ
l
y k
, and because of
u = ∑ w x , then the output of the neural network
is
L ∞
M M
∑∑ ∑ ∑
y = r d ( θ ) ⋅ w w x x (12)
k l i l l, m1 l, mi
k, m1
k,
mi
l= 1 i= 0 m1
= 0 mi
= 0
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