2014 7th International Conference on Intelligent Computation Technology and Automation
An improved nonlinear fitting method and its application in function approximation based on particle swarm algorithm XIAO Fei1, LIU Qiang1, JIA Bei1, WU Zeping2 1 Xi’an Communications Institute, Xi’an 710000, China,
[email protected] 2 College of Aerospace and Material Engineering, National University of Defense Technology, Changsha 410073, China converted into standard normal distribution by linear transformation. But it is inconvenient that its distribution value can only be obtained through the look-up table, because the distribution function does not have analytical expression. In order to increase the convenience during application, we try to use a known simple function to obtain the approximate expression, according to the function approximation method. In this way, the function approximation is essentially an optimization problem, as is shown in eq. (1), where the objective is to minimize approximation error.
Abstract: Standard normal distribution is widely used in engineering. But it is not so convenient during application, because the analytic expression of the distribution function does not exist. In this paper, the Particle Swarm Algorithm is applied in the nonlinear fitting process of standard normal distribution, and different modified methods are used to obtain analytical expressions of standard normal distribution function. According to the results, it is approved that the improved fitting method can get a better precision. Key words: nonlinear fitting method, particle swarm algorithm, function approximation
I.
min '
INTRODUCTION
x
Particle Swarm Optimization (PSO) is an optimization search algorithm based on swarm intelligence. As a bionic evolution algorithm, it is proposed by Kennedy and Eberhart in 1995, by researching on the birds’ flying behavior [1~3] . Although simple, but the PSO algorithm has profound intelligence background. Because of the fast convergence, high resolution quality, and strong robustness in multidimensional space function and dynamic targets optimization, the PSO is widely applied in function optimization, neural networks, pattern classification, fuzzy system control and other related fields [4,5]. As a part of the function theory, function approximation plays an important role in numerical calculation. In recent decades, it has always been one of the fundamental problems which are concerned by mathematic researchers and engineers. Thanks to the development of computer technology, mathematical tools are enriched, where the introduction of intelligent optimization algorithms explores new ideas to solve the problem of function approximation. Because intelligent optimization algorithms do not rely on the initial value, the minimal objective function, and can perform global search, it has been widely used in optimization fields. In this paper, particle swarm algorithm is used to fit the standard normal distribution function, and to find an approximation of normal distribution function. II.
(1)
Where, f ( x) is the normal distribution function, M ( x ) is its approximation function. III.
APPROXIMATION MODEL OF STANDARD NORMAL DISTRIBUTION FUNCTION
A. Approximation model based on S-function The approximation model of normal distribution function is established on the basis of S-function, as is shown in eq. (2), because its distribution function has the same description to S-function.
< ( x)
1 1 e g ( x)
(2)
where, g ( x ) correlates with x positively, and,
g (0) 0 ® ¯ g (f ) o f
(3)
And g ( x) can be described in eq. (4) n
g ( x)
¦a x
bi
i
(4)
i 1
where, n is the specified order, usually valued 1 or 2.
PROBLEM DESCRIPTION
B. Nonlinear fitting model According to the standard normal distribution table, a series of discrete values > xi , ) i @ can be obtained,
As one of the most common distributions in probability theory and mathematical statistics, normal distribution is widely used in engineering, because most distributions in nature world can be described in normal or incremental normal distribution, and any normal distribution can be 978-1-4799-6636-3/14 $31.00 © 2014 IEEE DOI 10.1109/ICICTA.2014.26
f ( x) M ( x)
76
where, c1 and c2 are learning factors; Z is the inertia
denoted > X , Y @ . In this way, the nonlinear fitting problem can be described as, find : g ( x) (5) min : Yl Y
weight; r1 0,1 and r2 0,1 are two random numbers, which are uniformly distributed in (0,1); i 1, 2,..., N . The learning factors are updated according to eq. (9)
c1
where, Yˆ [ yˆ1 , yˆ 2 ," , yˆ N ]T , yˆ i < ( xi ) , and < ( x) is shown in eq. (2). According to eq. (4), g ( x) is parameterized, and eq. (5) can be converted to the following parameter optimization model, find : ai , bi (6) min : Yl Y IV.
c2
i max _ step
2.0
(9)
The inertia factors are updated according to eq. (10)
w V.
0.9
i u 0.5 max _ step
(10)
SIMULATION AND ANALYSIS
In order to find the approximate expression function, the PSO algorithm is applied to solve the first-order and secondorder approximation of eq.(4). In the PSO algorithm, the number of particles is chosen 100, and the maximum iterations is 2000.
THE IMPROVED NONLINEAR FITTING MODEL BASED ON PARTICLE SWARM ALGORITHM
In PSO algorithm, each individual is considered as a particle without mass and volume in a multi-dimensional search space. These particles have their own double properties of position and velocity in the search space. According to their flying experience and their companions’, particles dynamically adjust the velocities. It is that, each particle adjusts the direction and velocity by its own optimal value and the groups’. So that the particle will not stop flying to the more optimized area according to adjustment, until the optimal solution is found. The flowchart of PSO algorithm is shown in Fig. 1,
A. First-order approximation In eq. (4), considering n 1 , so that the design variables are a1 and b1 . By choosing two objective functions that minimize both the maximum error and mean square error, using multi-objective optimization method, and the results are shown in Table 1. Table 1 results of the first-order approximation
maximum error iterations best value worst value mean value
0.00458 0.00458 0.00458
mean square error 10 0.00323 0.00323 0.00323
The optimal solution is that a 1.702 , b 1.705 and approximation function is,
)1 ( x )
