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Self-Introduction Applied Fractional Calculus Workshop Series

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## Self-Introduction Applied Fractional Calculus Workshop Series

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### 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@ucmerced.edu Phone:2092598023

Lab: CAS Eng 820 (T: 228-4398)

Jun 30, 2014. Monday 8:00-18:00 PM

Applied Fractional Calculus Workshop Series @ MESA Lab @ UCMercedu

1. Random distribution

1.1 L’evy 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.

AFC Workshop Series @ MESALAB @ UCMerced

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, ,

AFC Workshop Series @ MESALAB @ UCMerced

AFC Workshop Series @ MESALAB @ UCMerced

Le´vy 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 L’evy distribution

AFC Workshop Series @ MESALAB @ UCMerced

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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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.New Cuckoo search

3.1 New Cuckoo Search method

based on the obligate brood parasitic behavior of some cuckoo species in combination with the L´evy 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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AFC Workshop Series @ MESALAB @ UCMerced

3) New CS with the L´evy and Mittag-Leffler distritution

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AFC Workshop Series @ MESALAB @ UCMerced

4.2 Experiment with large size

1) Simulation data

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AFC Workshop Series @ MESALAB @ UCMerced

AFC Workshop Series @ MESALAB @ UCMerced

AFC Workshop Series @ MESALAB @ UCMerced

AFC Workshop Series @ MESALAB @ UCMerced

AFC Workshop Series @ MESALAB @ UCMerced

AFC Workshop Series @ MESALAB @ UCMerced

AFC Workshop Series @ MESALAB @ UCMerced

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.

AFC Workshop Series @ MESALAB @ UCMerced

AFC Workshop Series @ MESALAB @ UCMerced

2) Test

To fund the best performance of algorithm with different random coefficient generate by L´evy 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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AFC Workshop Series @ MESALAB @ UCMerced

we find the algorithm with random coefficient generated by Mittag-Leffler distributionand approximately equal 1 and 1 is efficient, and by L´evy 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, L´evy 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.

4.4 Solution

Descine one efficient optization tool;

Find test function have big imperfection;

Find Uniform,L´evy 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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