ISSN: 1693-6930
TELKOMNIKA Vol. 14, No. 3A, September 2016 : 208 – 216
211 where
is the variance of the inertial weight.
max
and
min
are the maximal and minimal average value of the inertial weight, respectively.
v
is the random number in the range of 0 to 1. is the coefficient to be set.
In PSO algorithm, acceleration coefficients
1
c and
2
c are used for adjusting particles own experience and social experience. In the early evolution stage, it is hoped that particle
increases its own experience information and performs a global search, avoiding trapped into the local optima. During the later stage, particle is hoped to strengthen social experience
information for the local precise search. In TVACPSO algorithm,
1
c and
2
c are time-varying coefficients, which can improve the global search in the early part of the evolution and
encourage the particles to converge towards the global optima at the end of the evolution. The acceleration coefficients are defined as [17, 18]:
1,fin 1,ini
1 1,ini
max
c c
k c
c k
9
2, 2,
2 2,
max fin
ini ini
c c
k c
c k
10
where
1,ini
c ,
1, fin
c ,
2,ini
c and
2, fin
c are initial and final values of acceleration coefficients,
respectively. TOOPSO algorithm introduces two oscillating factors into the evolutionary equation to
adjust the influence of the acceleration coefficients on the velocity, which effectively overcomes the premature problem and then increase the evolutionary speed. In TOOPSO algorithm, each
particle updates its position as follows [19, 20]:
1 1
1 1
1 1
1 2
2 2
2
1 1
k k
k k
k k
k j
j j
j j
g j
j
V w V
c r P
S S
c r P
S S
11 where
1
and
2
are oscillating factors. They adjust the global and local search ability by given different values in different stages. If
max
0.5 k
k
,
1
and
2
are taken as:
1 1 1
1 1
2 1
c r c r
,
2 2 2
2 2
2 1
c r c r
12 If
max
0.5 k
k
,
1
and
2
are taken as:
1 1 1
1 1
2 1
c r c r
,
2 2 2
2 2
2 1
c r c r
13
2.3. SVM Optimized by IPSO
When using SVM to solve the prediction problem, we need to select the appropriate kernel function according to the characteristics of the problem. Presently, some kernel functions
have been used in SVM, including linear kernel function, polynomial kernel function, radial basis function RBF kernel function, sigmoid kernel function, and wavelet kernel function. RBF kernel
function only contains one parameter and can used for sample data with arbitrary distribution by choosing the appropriate parameter. In this paper, we select the RBF kernel function with the
following expression.
2 2
|| ||
, exp
2
t t
x x
x x
K
14 where
2
is the kernel parameter. Thus, two parameters, penalty factor C and kernel parameter
2
are needed to be chose in SVM.
TELKOMNIKA ISSN: 1693-6930
Forecasting Range Volatility using SVM with Improved PSO Algorithms Yigang Liang 212
To obtain good forecasting performance of SVM, this paper uses the three IPSO algorithms to optimize the parameters of SVM. That is, the construction and forecasting process
of SVM are embedded into the optimization steps of IPSO algorithms. Each particle represents a group of parameters
2
, C
and each particle looks for the global optimal solution in the two- dimensional search space composed of
2
, C
based on the fitness value. The steps of the IPSO algorithms to optimize the parameters of SVM for forecasting range volatility are
described as follows: Step 1: Preprocessing the sample data. Normalize the whole sample into [0,1]. And
then the whole data are divided into training samples and checking samples. Step 2: Initializing particle swarm. Randomly generate the initial m sets of particles
encompassing the parameters
2
, C
. Set the parameters of IPSO algorithms, such as the maximal and minimal inertia weight, the acceleration coefficients, the maximal generation
number and so on. Step 3: Defining fitness function. The fitness function of particles is defined by the
training error of the constructed model.
2 1
1 ˆ
Fitness
l t
t t
r R
l
15 where
ˆ
t
r
is the forecasted range volatility in training samples,
t
R
is the corresponding daily actual range.
l
is the number of training samples. Step 4: Particles evolutionary. Calculate the fitness value of each particle based on
equation 15. Search for the individual optimal position of each particle and the global positional of the particle swarm. Update the inertia weight or acceleration coefficients.
Step 5: Stopping criterion judgment. Judge whether the maximal generation number kmax is reach. If kmax is satisfied, terminate the optimization process and give the optimal
parameters
2
, C
, otherwise, k=k+1, back to step 3.
Step 6: Constructing forecasting model. SVM-IPSO models are constructed through the obtained optimal parameters
2
, C
and are used to forecast range volatility. After that, the
forecasted range volatility values are transformed into the original range volatility forecasts.
3. Empirical Research 3.1. Data Description
