Self-IntroductionApplied Fractional Calculus

Workshop Series

- Zhigang, Lian/Link
- MESA (Mechatronics, Embedded Systems and

Automation)Lab - School of Engineering,
- University of California, Merced
- E zlian2_at_ucmerced.edu Phone2092598023
- Lab CAS Eng 820 (T 228-4398)

Jun 30, 2014. Monday 800-1800 PM Applied

Fractional Calculus Workshop Series _at_ MESA Lab _at_

UCMercedu

Cuckoo Search with Levy and Mittag-Leffler

distribution

Outline

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1. Random distribution

1.1 Levy distribution

A Lévy flight is a random walk in which the

step-lengths have a probability distribution that

is heavy-tailed. The "Lévy" in "Lévy flight" is a

reference to the French mathematician Paul Lévy.

In probability theory and statistics,

the Lévy distribution, named after Paul Lévy, is

a continuous probability distribution for a

non-negative random variable.

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Broadly speaking, flights is a random walk

by step size follows distribution, and

walking direction is uniform distribution. CS

algorithm used Mantegna rule with distribution

to choose optional step vector. In the Mantegna

rule, step size s design as The ,

follows normal distribution, i.e

, here,

,

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Levy stable distributions are a rich class

of probability distributions and have many

intriguing mathematical properties. The class is

generally defined by a characteristic function

and its complete specification requires four

parameters Stability index

Skewness parameter Scale parameter

Location parameter with varying ranges

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The Curve of Levy distribution

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1.2 The Mittag-Leffler distribution

Pillai (1990) introduced the Mittag-Leffler

distribution in terms of Mittag-Leffler

functions. A random variable with support over is

said to follow the generalized Mittag-Leffler

distri-bution with parameters and if its Laplace

transform is given by The cumulative

distribution function (c.d.f.) corresponding to

above is given by

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1.3 Other distribution

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2. HCSPSO search

1)A Hybrid CS/PSO Algorithm for Global

Optimization

Iterative equation

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2) The pseudo-code of the CS/PSO is presented as

bellow

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3) Hybrid CSPSO flow

The algorithm flow

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3.New Cuckoo search

3.1 New Cuckoo Search method

based on the obligate brood parasitic

behavior of some cuckoo species in combination

with the Levy flight behavior of some birds and

fruit flies, at the same time, combine particle

swarm optimization (PSO), evolutionary

computation technique.

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3.2 New Cuckoo Search(Lian and Chen)

1) Iterative equation

2)The pseudo-code of the CS/PSO is presented as

bellow

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3) New CS with the Levy and Mittag-Leffler

distritution

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4. Experiment

4.1 Experiment function

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4.2 Experiment with large size

1) Simulation data

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2) The Graph of Convergence

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4.3 Experiment with different distributions

1) Improve test functions

The above test function , have

same characteristic of optimization solution

, which is their imperfection. In the

experimental process, we found algorithm with

high probability random coefficient

generation mode close to 0, it is easy to make

close to 0, so it is easy to converge to 0.

This caused problem is algorithm search

performance surface phenomena is powerful, in

fact this false appearance is mad by the defects

test function cause algorithm make strong

fake image.

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2) Test

To fund the best performance of algorithm

with different random coefficient generate by

Levy and Mittag-Leffler distribution. We will

take the main random coefficients with different

distribution generate, in which and from 0 to 2

with 0.1 step changes, research and analysis the

performance of different distribution random

parameters how to influence algorithm.

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we find the algorithm with random

coefficient generated by Mittag-Leffler

distributionand approximately equal 1 and 1 is

efficient, and by Levy distribution and

approximately equal 0.8 and 1.2 is efficient.

Again verify, the PSO algorithm is based on

Uniform distribution, c1 and c2 approximately

equal 1.8 and 1.6 is efficient.

The PSO, CS HCSPSO and NCS algorithm with

random generate of different Uniform, Levy and

Mittag-Leffler distributions and solve the test

function, in which and from 0 to 2 with

0.1 step changes, and for the X axis, for Y

axis, the optimal value as Z axis, the

three-dimensional graphics are as following.

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4.4 Solution

- Descine one efficient optization tool
- Find test function have big imperfection
- Find Uniform, Levy and Mittag-Leffler

distribution effective used in different

algortihm.

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Future work

- Base on the NCS, look for more efficient

optimization?

- The NCS and FC like the combination

of optimization tools, looking for more efficient?

- The application of NCS in the new object, solving

other optimization problems?

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Sincerely hope that you give me some advices!

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