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## Self-Introduction Applied Fractional Calculus Workshop Series

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Title: Self-Introduction Applied Fractional Calculus Workshop Series

1
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
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Cuckoo Search with Levy and Mittag-Leffler
distribution
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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.
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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
1. Descine one efficient optization tool
2. Find test function have big imperfection
3. Find Uniform, Levy and Mittag-Leffler
distribution effective used in different
algortihm.

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Future work
1. Base on the NCS, look for more efficient
optimization?
1. The NCS and FC like the combination
of optimization tools, looking for more efficient?
1. 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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