Page 303 - Contributed Paper Session (CPS) - Volume 7

CPS2105 Hermansah et al.
the actual target outcome and the neuron outputs. Through this
process the network “learns”.
Forecasting is done based on the best model chosen by fulfilling the above
stage. The general form of output from the neural network is as follows [10]:
y = f (w 01 + ∑ q w f (v + ∑ p x v )) (5)
i ij
2
j1 1
0j
i=1
j=1
with the sigmoid activation function the following formula is obtained,
−(v +∑ p x v )
0j
i ij
q
1−e
i=1
y = f (w 01 + ∑ j=1 w ( p )) (6)
j1
2
t
i ij
0j
1+e −(v +∑ i=1 x v )
c. Wavelet model
By using the MODWT, a discrete time series {X , t = 1, 2, … , N} can be
t
written with the following form:
̃
̃
X = S J 0 ,t + ∑ J 0 D , t = 1, 2, … , N. (7)
t
j,t
j=1
̃
̃
The first part S = {S , t = 1, 2, … , N} which presents the tendency of series
J 0 J 0 ,t
and which is characterized by slow dynamics, and the second part components
̃ j ̃ j,t 0 t
D = {D , t = 1, 2, … , N}, j = 1, 2, … , J , present the local details of the series X
and is characterized by fast dynamics especially for low levels.
̂
To compute the predicted value X N+h of X, it suffices then to do this for
̂
̂
̃
̃
̃
̃
S and the D , i.e, to evaluate S N+h and D j,N+h , j = 1, 2, … , J . To do this, let us
0
j
J 0
write:
̂ J 0 ,N+h = f (S J 0 ,N , S J 0 ,N−1 , … , S ̃ J 0 ,N−p 0 ) (8)
̃
̃
̃
S
0
and similarly
̂ j,N+h = f (D , D j,N−1 , … , D j,N−p j ), j = 1, 2, … , J (9)
̃
̃
̃
̃
D
j
0
j,N
where f (j = 1, 2, … , J ) is the estimator, each estimator f may have its proper
j
0
j
order p . The choice of f , f , … , f is related to the dynamic behavior of the
j
j
0 1
series to be predicted.
Based on linear ARMA model theory, the tendency and details can be
approximated as following form:
S ̃ ̂ J 0 ,N = φ 1 S ̃ J 0 ,N−1 + ⋯ + φ p 0 J 0 ,N−p 0 + e N + θ 1 e N−1 + ⋯ + θ q 0 N−q 0 (10)
S ̃
e
and
D ̃ ̂ j,N = φ j1 D ̃ j,N−1 + ⋯ + φ jp j D ̃ j,N−p j + e N + θ j1 e N−1 + ⋯ + θ jq j N−q j (11)
e
By using the notation above, Eqs. (3) and (4) can be written as follows:
̂ J 0 ,N = [φ(B) − 1]S J 0 ,N + θ(B)e (12)
̃
̃
S
N
and
̂ j,N = [φ (B) − 1]D j,N + θ (B)e (13)
̃
̃
D
N
j
j
Hence, the MODWT-ARMA prediction model is
̂
̂
̂ N+h = S J 0 ,N+h + ∑ J 0 D j,N+h (14)
̃
̃
X
j=0
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