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igmoidal
functions are used as the activation functions for both the
hidden and output layers. Let opk’ and opm be the output of the
hidden node k and the output node m from the input pattern.
􀈦nk’ is the network weight for the input node n and the hidden
node k, and 􀈦km is the network weight for the hidden node k
and the output node m. xpn is the input value in the input node
n for the input pattern p, and tpm is the target output value in
the output node m for the input pattern p. The symbol
Δ represents the difference between the current and new
value in the next iteration. The standard BP algorithm is
shown as follows [8]:
A. Initialization
Initialize all weights and refer to them as the current
weights 􀈦km (0) and 􀈦nk’ (0) for all k, m and n. Set the learning
rate and the momentum factor a to small positive values (e.g.,
0.1). Set the error threshold to a very small positive value and
the iteration number i=0.
B. Forward Pass
Select an input pattern xp={xp1,xp2,…,xpn} from the
training set and compute opm (i) and opk’(i) by using the
following equations:
􀂸􀂹
􀂷
􀂨􀂩
􀂧
= 􀂦=
K
k
pm km pk o i f i o i
1
( ) ω ( ) '( ) (1)
Solar Radiation Prediction Based on the Artificial Neutral Networks
Using the BP Algorithm
Gang Yang, Student Member, IEEE & Yongxian Du
S
217
978-1-4244-1888-6/08/$25.00c 2008 IEEE
􀂸􀂹
􀂷
􀂨􀂩
􀂧
= 􀂦=
N
n
pk nk pn o i f i x
1
'( ) ω '( ) (2)
where the Sigmoidal function is used as the activation
function. Use the desired target tp={tp1, tp2,…tpm} associated
with xp to compute the squared error for all input patterns,
shown as follows:
( )2
1 1
( )
2
( ) 1􀂦􀂦
= =
= −
P
p
M
m
pm pm E i t o i (3)
If E (i) is not greater than the error threshold, then the
algorithm is completed and the convergence is met;
otherwise, go to Backward Pass step.
C. Backward Pass
Compute the changes of the weights for the next
iteration Δ 􀈦km(i+1 ) and Δ 􀈦nk’(i+1 ) using the following:
( ) '( ) ( )
( )
( )
( 1) ( )
1
i o i a k i
i
i
i E i
pk km
P
p
pm
km
km
km
μ δ ω
α ω
ω
ω μ
= + Δ
+ Δ
∂
Δ + = − ∂
􀂦=
(4)
'( ) '( )
'( )
'( )
'( 1) ( )
1
i x a k i
i
i
i E i
pn nk
P
p
pk
nk
nk
nk
μ δ ω
α ω
ω
ω μ
= + Δ
+ Δ
∂
Δ + = − ∂
􀂦=
(5)
Where
(i) (t o (i))o (i)(1 o (i)) pm pm pm pm pm δ = − − (6)
'( ) '( )(1 '( )) ( ) ( )
1
i o i o i i i km
M
m
pk pk pk pm ω δ δ 􀂦=
= − (7)
Update the weights for the next iteration by using
􀈦km(i+1 ) =􀈦km(i )+ Δ 􀈦km(i+1 ) and 􀈦nk’(i+1 )= 􀈦nk’(i
)+ Δ 􀈦nk’(i+1 ). Set i=i+1 and go to the Forward Pass step.
D. Prediction Bbased on BP Neutral Networks
The purpose of the prediction based on neutral network is
to use the neutral network to a approach to a time series or the
deformation of a time series. It uses the first m values of a
time series {xp1,xp2,…,xpm} to predict the next s values of a
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