Modified Particle Swarm Optimization With Student Mutation Biology Essay

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Abstract- Particle Swarm Optimization (PSO) is a well known technique for optimization problems, but suffers from premature convergence. To prevent PSO from stagnation in local minima we present here a new mutation operator. The proposed mutation operator is based on the student T distribution. The experimental results on benchmark functions shows that proposed variant improve the performance of PSO.

Keywords- PSO, Student T distribution, Modified PSO, PSO with student T distribution, STPSO.

Introduction

Particle Swarm Optimization (PSO) is a stochastic based evolutionary technique inspired by bird flocking and fish schooling, developed by kannedy [1] in 1995.For optimization problems PSO gives better performance but it suffers from premature convergence. Like other evolutionary technique PSO is population based technique, initial population is randomly generated and called particle. A particle in PSO can be defined as Pi ϵ [a, b] where i=1, 2, 3…. D and a, b∈R, D is for dimensions and R is for real numbers [30]. Each particle contains its position and velocity. After initialization particles moved towards the new positions based on their own and neighbourhood experience. PSO has two model cognition model and social model. Cognition model used to find global best particle while social model is used for local best. In PSO each particle maintains two positions gbest and pbest. gbest is global best particle among all the particles while pbest is the particle's own best position. Following equations are used to update positions and velocity.

………………………………………..…... (i)

………...………………….. (ii)

Where Xi is the position, Vi is the velocity, Pbest is the personal best position and gbest is the global best position for PSO. Similarly r1 and r2 are two random numbers their range is chosen from [0, 1], w is the inertia weight, C1 and C2 are learning factors specifically the cognition and cognition component influential respectively.

Related Work

Mutation operators perform a vital role to improve the performance of PSO, our work is also about mutation operator for PSO, therefore below will discuss PSO with respect to mutation operators.

Li [2] claims that one mutation operator is not enough to improve the performance of PSO; they said that different mutation operators should be used at different stages to improve the performance of PSO. Therefore they present three mutation operators to improve the PSO performance.

………………………………….. (iii)

……………………………….…. (iv)

Where Xg and Vg used for the position and velocity of the global best particle δ and δg are the Cauchy random number with scale parameter 1.

b. Gaussian mutation

…………………………………... (v)

…………………………………. (vi)

Where Xg and Vg used for the position and velocity of the global best particle N and Ng denotes Gaussian distributions with mean 0 and variance 1.

c. Levy mutation

……………………………… (vii)

……………………..……… (viii)

Where L (α) and Lg (α) are random numbers generated from the Levy distribution with a parameter α. Author set α=1.3

According to author each mutation operator will be selected according to its selection ratio, initially they set selection ratio as 1/3. The mutation ratio of an operator increased in case of higher fitness values of offspring and vice versa.

They implement their techniques on 7 benchmark functions and compare the results with traditional PSO and FEP. They also compare the results of all three new mutation operators separately as well and find the adaptive mutation result better than others.

Tang [3] proposed an adaptive mutation operator for PSO to improve its performance as following

… (ix)

……………….. (x)

Where is the ith vector of the global best particle, , and is the minimum and maximum values of the ith dimension in the current search space respectively, rand is used to generate random number within [0,1] and t=1,2,3…. represents generations. They implement their techniques on 8 benchmark function with 4 uni-models and 4 multimodals. They compare their technique with 4 other techniques including traditional PSO. The performance of proposed technique remains well in 6 functions while in two functions

Liu [4] proposed new variant of PSO using dynamic inertia weight with mutation. Author used linearly decreased inertia weight update das following

……………………………...….. (xi)

Where maxt is the maximum number of iteration and t is the current iteration number.

Pant [5] proposed a new mutation operator in PSO; they used the sobol sequence to perform mutation. The author claim that qausi random sequence are more better then pseudo random sequences, they are more able to cover the search space than to pseudo random sequences, Qausi random sequence include Vander Corput, Sobol, Faure and Halton. Author used the sobol sequence for mutation and is called as sobol mutation operator. They present two versions of PSO SOM-QPSO1 and SOM-QPSO2. In SOM-QPSO1 the best particle of the swarm is mutated while in SOM-QPSO2 worst particle of the swarm is mutated. The SOM operator is defined as

SOM=R1+ (R2/LnR1)

Where R1 and R2 are random number in sobol sequence.

The results of QPSO is compared with some other variants of PSO using three benchmark function using different population size, dimensions and generations. The performance of QPSO remains well in most of the cases.

Wu [6] introduce power distribution is PSO to improve its performance and to prevent it from stagnation of local minima. The defined the power mutation as

…………………………. (xi)

and

Where and [] is the boundary of the decision variables in the current search space, r is random number between 0, 1 and s is calculated as

……………….…………………… (xii)

P is the index of distribution function as set as 0.25. After mutating the gbest compare it with original and best one is replaced. Author used 10 benchmark functions to compare the PMPSO with some other variants of PSO and find the PMPSO better than other techniques.

To escape from local optima [19] presented Cauchy mutation in PSO. Author mutates the global best and compares it with original best one is replaced as global best particle. The global best particle is mutated as

………………… ......……………………… (xiii)

Where N is a Cauchy distributed function with scale parameter t=1, N(xmin, xmax) is a random number with in (xmin, xmax) of defined domain of test function and

………………… (xiv)

Where V[j][i] is the ith velocity vector of jth particle in the population popsize is the size of population.

Author compares their techniques with some other variant using 10 benchmark functions and finds CPSO better than others.

Pant [7] introduce beta distribution in PSO and proposed two versions of PSO AMPSO1 and AMPSO2. In AMPSO1 they mutate the global best particle while in AMPSO2 they mutate local best particle. The particle is mutated as

………………….…. (xv)

Where, N(0,1) is normally distributed function with mean 0 and standard deviation 1, that a different random number is generated for each value of j, and are set as and respectively and value of is originally set as 3.Betarand() is a random number generated by beta distribution with parameter less than 1. Author compares the presented technique with traditional PSO and some other variants of PSO.

Wang [8] coupled opposition based PSO with Cauchy mutation. The proposed method will accelerate the convergence of PSO. It employs the opposition based learning for every particle and then uses the Cauchy Mutation on the best particle. This mutation operator is used to increase to the escaping probability from local optimum. The comparative study of PSO and OPSO on 4 unimodel and 4 multi model functions represents that on unimodel functions OPSO could have faster convergence where as on multi model functions it provides the better global search ability.

Inertia weight is very important parameter of PSO to balance the exploration and exploitation trade off, therefore many researchers work on inertia weight to improve its performance. Pant [9] proposed new inertia weight scheme as

2……………………………………... (xvi)

Where rand is the random number having gaussion distribution.

Imran et al [10] proposed a variant of PSO in which OPSO is coupled with power mutation operator. They initialize swarm opposition based then employ power mutation on the global best particle if mutated give better result than replaced with original.

MODIFIED PSO

From above study it has been observed that mutating the global best particle using different distribution cause to increase the performance of PSO. But yet it is need to investigate to prevent PSO to stagnate in local minima. Therefore author present two new versions of PSO, STPSO1, and STPSO2. The two versions differ from each other that in STPSO1 local best particle is mutated while in STPSO2 global best particle mutated. Further in this paper STPSO2 will be discussed in detail.

The global best particle is mutated as.

…………………… (xvii)

Where

Where are the boundaries of the current search space and is the random number generated by student t distribution.

BENCH MARK FUNCTIONS

EXPERIMENTAL SETTING

.

For all techniques following experimental setting ware used during experiments.

CPSO represent Cauchy mutation PSO, AMPSO is used for adaptive mutation PSO and STPSO is PSO with student t mutation.

CONCLUSION

From above given results it can be observed that performance of STPSO is significantly well from PSO, CPSO and AMPSO in function f1, f2 and f3. The performance of all techniques remains same for function f4. The fitness of f5 is vary when less number of dimensions and iterations then STPSO perform well, when dimensions kept 20 with 1500 iterations AMPSO perform well but when increase dimensions and iteration to 30, 2000 respectively than performance of CPO is better. For function f6 performance of CPSO is slightly better than STPSO when less number of iterations but when increased number of iterations and dimension STPSO perform well then other techniques. For function f7 the average fitness value reaches to zero in case of STPSO which is global minima.

Over all we have 21 cases. In 14 cases STPSO perform well while in 3 cases performance of all techniques remains same. CPSO perform slightly better in 3 cases and AMPSO perform well in just one case.

FUTURE WORK

Presented variant of PSO is implemented just for seven functions, so it is required to implement and compare with other techniques for all available benchmark functions in the literature.

In presented technique linear decreasing inertia weight is used so a different inertia weight scheme can also be used. The size of population kept 30 for all experiments, 40 and 60 population size can also be used with presented techniques to make it clear.

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