Frontiers in Pathology and Genetics (FPG) Volume 2, 2014
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Modifed MEM Method Using Hybrid Genetic Algorithm Applied to Wave Directional Spectrum Estimation Zheng Wei*1 institute of oceanographic instrumentation, Shandong academy of sciences , Qingdao, China honestzheng123@163.com
*1
Abstract Wave directional spectrum estimation needs to measure the same quantity of the different locations, then the crossspectrum of different locations is calculated, finally wave direction distribution is estimated based on nonlinear equation between cross-spectrum and wave direction distribution. In the paper, a hybrid genetic algorithm is applied to modified MEM algorithm, the algorithm avoids the complicated Bessel function calculation and also eigenvalue and eigenvector calculation. Simulation result shows that the hybrid genetic algorithm has better precision and also high calculation speed. Keywords Wave Directional Spectrum; Nonlinear Equation; Modified MEM Algorithm; Hybrid Genetic Algorithm
Wave Directional Spectrum Estimation Now the wave directional spectrum estimation method is mostly based on the relationship between cross-spectrum and wave direction distribution, it is assumed that the wave amplitude is superposed by a large number of small composition waves of different frequencies, different phases, and different amplitudes. The cross-spectrum between any two wave quantities has relationship with transfer function of the corresponding wave quantity and wave directional spectrum. In the paper, the wave height is measured, so the transfer function is one, so (1) is get:
ϕmn = (ω ) S ( w) ∫−ππ exp[−ik ( xn − xm )]D (ω , θ )dθ
(1)
ϕmn (ω ) is the cross-spectrum of m and n wave quantity,
Introduction
k is wave number, xm and xn are location vector of m
Wave directional spectrum has been the central issue of wave forecasting, marine engineering and ocean remote sensing. Now the method of estimating wave directional spectrum is mostly based on the nonlinear equations of cross-spectrum and wave direction distribution, such as parameterization method (PDM), maximum likelihood method (MLM), maximum entropy method (MEM), Bayesian method (BDM). PDM has a poor precision; MLM has better precision as well as fast computation, so it is used mostly in engineering applications; MEM and BDM have best precision, but both methods need solving nonlinear equations, and both methods calculate slowly, so they are seldom used in engineering application. The paper succeeds in the application of a hybrid genetic algorithm to wave directional spectrum estimation, the hybrid genetic algorithm makes both use of the global search ability of genetic algorithm and the local search ability of the nonlinear least-squares algorithm, and the hybrid genetic algorithm have good precision as well as good calculation speed.
and n points, D(k , ω ) is wave direction distribution of the circular frequency ω . Wave directional spectrum can be expressed by the product of wave frequency spectrum and wave direction distribution. S (ω , θ ) = S (ω ) D(ω , θ )
(2)
S (ω , θ ) is two dimensional wave directional spectrum, S (ω ) is one dimensional wave frequency spectrum, D(ω , θ ) is the wave direction distribution, and D(ω , θ )
satisfies (3): π ∫−π D(ω , θ )dθ = 1
(3)
Modified MEM Method The equation (1) can be rewritten as: = φ j ∫−ππ q j (θ ) D(ω ,θ )d= θ j 1, M + 1
(4)
M N ( N − 1) , N is the number of wave quantity And=
measurement points.
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Frontiers in Pathology and Genetics (FPG) Volume 2, 2014
1, , M / 2 cos(krj cos( β j − θ )) j = q j =sin(krj ) cos( β j − θ )) j =M / 2 + 1, M = j M +1 1
ϕmn (ω ) Re [ϕ (ω )ϕ (ω )]1/ 2 j = 1, , M / 2 m n ϕmn (ω ) = = j M / 2 + 1, M φ j Im 1/ 2 [ϕm (ω )ϕn (ω )] 1 = j M +1 rj =
β j = tan −1[( ym − yn ) / ( xm − xn )]
(6)
(8) direction
E = − ∫−ππ D(ω , θ ) ln D(ω , θ )dθ
(9)
Minimizing E subject to the constraints imposed by the cross-spectrum matrix (4), so wave direction distribution should be represented as: M +1
) exp{−1 + ∑ ν j q j (θ )} D(ω , θ= j =1
(10)
ν j is Lagrange multipliers chosen to ensure (4) is correct, and (10) and (5) are combined to get nonlinear equa-tions [Okey Nwogu,1989]: M +1
π ∫−π exp{−1 + ∑ ν j q j (θ )}qi (θ )dθ =φi (ω ) j =1
(11)
i 1, M + 1 =
The following equations can be written based on (11): φ − π q (θ ) exp{−1 + L µ q (θ )}dθ = 0 = f ( x) ∑ j j 1 1 ∫−π 1 j =1 L φ − π q (θ ) exp{−1 + ∑ µ j q j (θ )}dθ = 0 = f 2 ( x) ∫ 2 2 − π j =1 (12) L φ − ∫ π q (θ ) exp{−1 + ∑ µ j q j (θ )}dθ = 0 = f M +1 ( x) M +1 −π M +1 j =1
A new algorithm should be presented to solve the above nonlinear equation, the benefit of the algorithm is that it do not needs to calculate the complicated Bessel function and also eigenvalue and eigenvector of MEM, but the initial value of iterations can not be decided as MEM, so the paper makes use of a hybrid genetic algorithm to solve (12). First, the hybrid genetic algorithm uses genetic algorithm to find the optimal solution close to the final value. Second, it uses a algorithm with a strong search ability to do second optimization. 2
X = {x1 , x2 xm } min{ f1 ( X ) 2 + f 2 ( X ) 2 + f m ( X ) 2 }
(7)
( xm − xn ) 2 + ( ym − yn ) 2
The entropy associated with wave distribution function can be expressed by
(5)
The equation (12) can transform to nonlinear least squares problem [Tian Qiaoyu et al,2007]: (13)
So the nonlinear least squares algorithm is selected to do the second to second Optimization. Nonlinear least squares algorithm needs a good assumptive initial value and then solve the problem through iterations. Now Levenberg-Marquardt(LM) algorithm is used mostly. The local search ability of LM algorithm is excellent, but it is sometimes likely to search the local optimal value rather than the global optimum value [Wen Zaiwen, 2004]. Hybrid Genetic Algotithm applied to Modified MEM Method Genetic algorithm is an adaptive search algorithm for global probability optimization based on simulation of biological genetic and evolutionary processes in the natural environment. Firstly, simulation of biological chromosomes is done based on an encoding method; secondly, the size of a random population is decided and the definition of the fitness evaluation function; thirdly, according to different individual fitness evaluation function, the probability of preservation to the next generation is determined; fourthly, a group of solutions is determined to meet their own requirements by choosing crossover and mutation operations. Genetic algorithm uses a multi-point search to avoid a local optimum, and it can quickly converge to the solution near the optimal solution value, but it will converge to the optimal solution for a long time, it has a poor local search ability [Zhou Ming et al, 2002]. Considering the respective advantages of the traditional algorithm and modern intelligent genetic algorithm, the realization of the combination of genetic algorithm and traditional algorithm should be done. There are two hybrid genetic algorithms now. On the one hand, the traditional algorithm optimization process is considered as a genetic factor of genetic algorithm to speed up the process of convergence speed. On the other hand, the first optimization is done based on genetic algorithm; there is a second optimization using tradition algorithm [Luo YaZhong et al, 2005]. In this paper, LUO YAZHONG the hybrid genetic algorithm is the second case above. First, genetic algorithm is used. With the quick converge ability, a solution closer to the optimal solution value
Frontiers in Pathology and Genetics (FPG) Volume 2, 2014
can be gotten with a short calculation time. Second, LM algorithm is used to do the second optimization, a good solution can be gotton through the strong local search ability of LM. The hybrid genetic algorithm can be seen in Figure. 1: Two initial value(mean wave direction and distribution parameter )
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Gs ( f , θ ) = ∑ G ( si ) cos 2 s ( i
i
θ − θi 2
)
[ ( s + 1)] / [2 π Γ( s + 0.5)] G ( S ) =Γ
(16) (17)
And Γ is gamma function, s is distribution parameter, θ is mean wave direction.In fact, signal is mixed with noise, the true wave distribution is assumed: G = ( f , θ ) Gs ( f , θ ) + Gn ( f , θ )
(18)
Gn ( f , θ ) stands for the isotropic background noise.
Coding
NSR(Noise ratio signal) [Guan Changlong et al, 1995] is defined π
Choose population P(N)
NSR =
∫−π GN (θ )dθ π ∫−π Gs (θ )dθ
(19)
In order to compare the performance of estimating ability, direction distribution error coefficient [Joan Oltman Shay et al, 1984] is used:
Choosing,crossover,muation
Population P(N+1)
∑ Desti (θ ) − Dtrue (θ ) (20) ∑ Dtrue (θ ) is the estimated distribution, Dtrue is assuming D' =
Fitness funciton is less than a scaler C ?
no
Desti
distribution.
yes
Peak error coefficient is used: θ esti is the estimated
Decoding values
angle, θtrue is assuming angle.
θ' =
Second Optimization using LevenbergMarquardt method
The two calulation values meet the requirements ?
θ esti − θture θtrue
(21)
no
yes Calculation end
FIGURE 1 HYBRID GENETIC ALGORITHM
In the Figure 1, the stopping criteria of the genetic algorithm is set according to fitness function [Wu GuoHui et al, 2007 and G..R.Liu et al, 2002]. F fitness = − min{ f1 ( X ) 2 + f 2 ( x) 2 f m ( X ) 2 } > C1 C1 is scaler
(14)
(a) single peak error coefficient
And the stopping criteria of second optimization is set as: max | xk +1 − xk |< C2
C2 is scaler
(15)
xk +1 and xk are calculation result of the k and k + 1
iteration result. In the paper, C1 = −10 ， C2 = 10−6 , in the paper, the [0,1] random uniform value is chosen as the initial value. Simulation Wave direction distribution choose the COS-2S form
(b) direction distribution error coefficient FIGURE 2 SINGLE PEAK ESTIMATION
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From Figure 2 (a), the peak error coefficient of MEM fluctuates slowly, and its value is very small, the maximal estimation peak error coefficient is less than 0.03, while the peak error coefficient of modified MEM is not as well as MEM. From Figure. 2 (b), we can see that the trend of the two curves is same, but the direction distribution error of coefficient modified MEM is slightly better than MEM. There is similar simulation result under the bimodal peak condition from the figure 3.
Frontiers in Pathology and Genetics (FPG) Volume 2, 2014
angle domain, while peak error coefficient is only restricted within certain angles. Through a large number of simulations, it also shows modified MEM using hybrid genetic algorithm calculates very fast, and seldom affected by the assumed initial value. Conclusions The modified MEM algorithm using hybrid genetic algorithm is improved on traditional MEM algorithm, it avoids calculations of complicated Bessel function and calculations of eigenvalue and eigenvector, the hybrid genetic algorithm has good converge speed and also strong local search ability. Simulation shows that application of the hybrid algorithm to wave directional spectrum estimation is feasible; the application has a reference value in engineering. REFERENCES
(a) left peak error coefficient comparison
Okey Nwogu. “‘Maximum entropy estimation of directional wave spectra from an array of wave probes.” Applied Ocean Research. Vol. 11 , 1989, pp.176-182. Tian Qiaoyu, Gu Zhongbi, Zhou Xinzhi. “‘Solving systems of nonlinear equation with hybrid genetic algorithm.” Computer technology and develepment. Vol. 17, 2007, pp.10-12. Wen Zaiwen. “‘Least squares and their applications.” Institute of computational mathematics. Academy of Mathematics and Systems Sciences, Chinese Academy Science. Master thesis. 2004, pp.51-66.
(b) right peak error coefficient
Zhou Ming, Sun Shu Dong. “‘Genetic algorithms theory and application.” National defense industry press. 2002, pp.32-130. Luo Yazhong, Yuan Duancai, Tang Guojin. “‘Hybrid genetic algorithm for solving systems of nonlinear equations.” Chinese Journal of Computational Mechanics. Vol. 22, 2005, pp.109-114. Wu Guohui, Dai Jiyang, Wu Yinhua, Zhu Guomin. “‘A new hybrid
genetic
algorithm
for
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nonlinear
equations.”Journal of Nanchang Institute of Aeronautical (c) direction distribution error coefficient FIGURE 3 BIMODAL PEAK ESTIMATION
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Technology(Natural Science). Vol. 22, 2007, pp.5-9. G..R.Liu, X.Han, K.Y.Lam. “‘A combined genetic algorithm and nonlinear least squares method for material characterization using elastic waves.” Comput. Methods Appl. Mech. Engrg. Vol. 191, 2002, pp.9162-1916.. Guan Changlong, Wen Shengchang, Zhang Dacuo. An
Frontiers in Pathology and Genetics (FPG) Volume 2, 2014
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