MESA Lab
Self-Introduction
Applied 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
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Cuckoo Search with
L´evy and Mittag-Leffler
distribution
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Outline
1
Random distribution
2
HCSPSO search
3
New Cuckoo search
4
Experiment
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1. Random distribution
1.1 L’evy distribution
A Lévy flight is a random walk in which the steplengths have a probability distribution that is heavytailed. 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:
s

| |
1

The   ,
follows normal distribution, i.e
 ~ N (0,  2 ), ~ N (0, 2 ) ,
here,   { (1   ) sin(  / 2) } ,
1

[(1   ) / 2] 2(  1) / 2
 1
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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: 
0    2,1    1,   0,   
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The Curve of L’evy 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:
 (t )  E[etX ]  (1  t  )  ;0    1,   0.
The cumulative distribution function (c.d.f.) corresponding to
above is given by
(1) k (   k ) x ( k )
F , ( x)  P[ X  x]  
k 0 k!(  )(1   (   k ))

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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 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:
X i(t 1)  X i(t )     * Levy( )  (1   ) ( R1 ( Pi (t )  X i(t ) )  R2 ( Pg(t )  X i(t ) ))
2)The pseudo-code of the CS/PSO is presented as bellow
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3) New CS with the L´evy 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 f1  f5 , f11 have same characteristic of
optimization solution x  0 , 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
x 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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n
f1 ( x)   x 2 i
i 1
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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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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.
MESA Lab
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.
MESA Lab
Algorithm with Different Distribution Generate for Fa
PSO with Uniform Distribution
PSO with L´evy Distribution
4
5
x 10
Z
5
x 10
2
2
1.5
1.5
1
1
Z
10
Z
PSO with ML Distribution
5
x 10
0.5
0
2
0.5
0
2
1.5
1
0.5
0
Y
0
1
0.5
1.5
2
0
2
1.5
1
0.5
0
Y
X
CS with L´evy Distribution
0
1
0.5
1.5
1.5
2
1
0.5
0
Y
X
CS with ML Distribution
0
1
0.5
1.5
2
X
HCSPSO with L´evy Distribution
4
x 10
600
10
600
9.5
Z
Z
400
Z
400
200
200
0
2
9
2
1.5
1
0.5
0
Y
0
1
0.5
1.5
2
0
2
1.5
1
0.5
0
Y
X
HCSPSO with ML Distribution
0
1
0.5
1.5
2
1.5
1
0.5
0
Y
X
NCS with L´evy Distribution
0
1
0.5
1.5
2
X
NCS with ML Distribution
4
x 10
6
150
100
100
Z
Z
8
150
Z
10
50
4
2
2
50
0
2
1.5
1
0.5
Y
0
0
1
0.5
X
1.5
2
0
2
1.5
1
0.5
Y
0
0
1
0.5
X
1.5
2
1.5
1
0.5
Y
0
0
1
0.5
X
1.5
2
MESA Lab
Algorithm with Different Distribution Generate for Fb
PSO with Uniform Distribution
6
PSO with L´evy Distribution
4
x 10
6
PSO with ML Distribution
5
x 10
x 10
10
2
5
1
Z
Z
Z
4
2
0
2
0
2
0
2
2
1
1
1
Y
0 0
2
Y
X
CS with L´evy Distribution
0 0
Y
X
CS with ML Distribution
6
2
1
1
1
0
0
X
HCSPSO with L´evy Distribution
2000
1000
5
1000
0
2
0
2
2
1
0 0
Y
X
HCSPSO with ML Distribution
5
0
2
2
1
1
Y
Z
10
Z
2000
0 0
2
1
1
Y
X
NCS with L´evy Distribution
1
0
0
X
NCS with ML Distribution
x 10
1500
2
1000
1000
1
Z
1500
Z
3
Z
Z
x 10
500
0
2
2
1
Y
1
0 0
X
500
0
2
2
1
Y
1
0 0
X
0
2
2
1
Y
1
0
0
X
MESA Lab
Algorithm with Different Distribution Generate for Fc
PSO with Uniform Distribution
PSO with L´evy Distribution
150
100
100
Z
50
150
Z
Z
100
PSO with ML Distribution
50
0
2
50
0
2
1
Y
0 0
1
0.5
1.5
0
2
2
1
Y
X
1.5
1
Y
X
CS with ML Distribution
10
5
50
5
Z
51
0
2
49
2
1
Y
0 0
1
0.5
1.5
1
Y
HCSPSO with ML Distribution
0 0
1
0.5
1.5
1
Y
X
2
Y
0 0
1
0.5
X
1.5
X
Z
Z
Z
1
2
2
1
1
0
2
2
0.5
1.5
3
0.5
49
2
0 0
1
NCS with ML Distribution
1.5
50
2
X
2
NCS with L´evy Distribution
51
1.5
0
2
2
X
0 0
1
0.5
HCSPSO with L´evy Distribution
10
Z
Z
CS with L´evy Distribution
0 0
1
0.5
2
0
2
1
Y
0 0
1
0.5
X
1.5
2
1
Y
0 0
1
0.5
X
1.5
2
MESA Lab
Algorithm with Different Distribution Generate for Fd
PSO with L´evy Distribution
PSO with ML Distribution
-5000
-2000
-4000
-6000
-4000
-6000
-7000
-6000
Z
-2000
Z
Z
PSO with Uniform Distribution
-8000
-8000
-8000
-10000
2
-9000
2
-10000
2
1.5
1
0.5
0
Y
0
1
0.5
1.5
2
1.5
1
0.5
0
Y
X
0
1
0.5
1.5
2
1.5
1
0.5
CS with ML Distribution
CS with L´evy Distribution
Z
Z
-8000
-8500
-1000
-7500
-2000
-8000
-3000
-8500
-4000
-9000
2
1
0.5
0
Y
0
1
0.5
1.5
2
1
0.5
0
Y
X
0
1
0.5
1.5
2
1.5
1
0.5
Z
Z
Z
-1.24
-1.24
-1.25
-1.26
2
0.5
Y
0
0
1
0.5
X
1.5
2
X
-1.23
-1.25
1.5
2
4
-1.23
-8000
2
0.5
1.5
x 10
-1.22
-6000
0
1
NCS with ML Distribution
4
x 10
1
0
Y
X
NCS with L´evy Distribution
HCSPSO with ML Distribution
-4000
X
-9500
2
1.5
-2000
0.5
2
-9000
-5000
2
1.5
0
1.5
HCSPSO with L´evy Distribution
Z
-7500
0
Y
X
1
-1.26
2
1.5
1
0.5
Y
0
0
1
0.5
X
1.5
2
1.5
1
0.5
Y
0
0
1
0.5
X
1.5
2
MESA Lab
Algorithm with Different Distribution Generate for Fe
140
200
120
250
200
Z
Z
Z
300
100
100
0
2
80
2
1.5
1
0.5
0
Y
0
1
0.5
1.5
50
2
1.5
2
1
0.5
0
Y
X
0
1
0.5
1.5
1.5
2
1
0.5
Z
Z
Z
20
2
179
2
0.5
0
Y
0
1
0.5
1.5
1.5
2
1
0.5
0
Y
X
0
1
0.5
1.5
1
0.5
15
15
10
10
1.5
Y
0
0
1
0.5
X
1.5
2
X
0
2
0
2
179
2
0.5
2
5
5
179.5
0
1.5
NCS with ML Distribution
Z
180
0.5
0
Y
X
1
Z
181
1
1.5
2
NCS with L´evy Distribution
HCSPSO with ML Distribution
180.5
X
40
179.5
1.5
2
60
180
20
2
0.5
1.5
80
181
40
0
1
HCSPSO with L´evy Distribution
180.5
60
0
Y
X
CS with ML Distribution
80
1
150
100
CS with L´evy Distribution
Z
PSO with ML Distribution
PSO with L´evy Distribution
PSO with Uniform Distribution
1.5
1
0.5
Y
0
0
1
0.5
X
1.5
2
1.5
1
0.5
Y
0
0
1
0.5
X
1.5
2
MESA Lab
Algorithm with Different Distribution Generate for Ff
PSO with L´evy Distribution
20
20
19.9
18
19.8
16
19.7
14
2
19.6
2
1
Y
0
0
1
0.5
1.5
22
20
18
16
2
2
1
Y
X
PSO with ML Distribution
Z
22
Z
Z
PSO with Uniform Distribution
CS with L´evy Distribution
0
0
1
0.5
1.5
2
1
Y
X
CS with ML Distribution
18
18
Y
0
0
1
0.5
1.5
16
2
2
1
Y
X
HCSPSO with ML Distribution
0
0
1
0.5
1.5
2
1
Y
X
NCS with L´evy Distribution
Z
Z
18
17
4
4
3
3
2
2
1
16
2
Y
0
0
0.5
X
1.5
0
0.5
1.5
2
X
1
0
2
1
0
1
NCS with ML Distribution
Z
19
1
18
17
16
2
1
X
19
17
17
2
2
20
Z
19
0
1.5
HCSPSO with L´evy Distribution
Z
19
Z
20
0
1
0.5
0
2
2
1
Y
0
0
1
0.5
X
1.5
2
1
Y
0
0
1
0.5
X
1.5
2
MESA Lab
Algorithm with Different Distribution Generate for Fg
PSO with Uniform Distribution
PSO with ML Distribution
PSO with L´evy Distribution
1000
2000
2000
1500
Z
Z
Z
1500
500
1000
1000
0
2
500
500
2
0
2
2
1
2
1
1
Y
0
0
Y
X
CS with L´evy Distribution
0
0
Y
X
CS with ML Distribution
20
152
10
150
2
5
2
2
1
1
Y
0
0
Y
X
HCSPSO with ML Distribution
0
0
Y
X
151.5
3
151
2
0
0
X
X
2
1
0
2
Y
0
3
0
2
2
1
0
NCS with ML Distribution
1
150
2
1
Z
4
Z
152
1
2
1
1
NCS with L´evy Distribution
150.5
X
15
151
2
1
0
20
150.5
5
2
0
Z
Z
Z
10
1
HCSPSO with L´evy Distribution
151.5
15
Z
2
1
1
2
1
Y
1
0
0
X
2
1
Y
1
0
0
X
MESA Lab
Algorithm with Different Distribution Generate for Fh
PSO with Uniform Distribution
PSO with L´evy Distribution
8
9
x 10
x 10
2
2
5
1
1
0
2
1
Y
0
0
1
0.5
1.5
Z
10
Z
Z
PSO with ML Distribution
9
x 10
0
2
2
1
Y
X
CS with L´evy Distribution
0
0
1
0.5
1.5
0
2
2
1
Y
X
CS with ML Distribution
2
15
1
1
Y
0
0
1
0.5
1.5
1
Y
X
HCSPSO with ML Distribution
0
0
1
0.5
1.5
5
2
2
1
Y
X
NCS with L´evy Distribution
3
X
10
0
2
2
0.5
2
Z
15
Z
20
Z
3
5
2
0
1.5
HCSPSO with L´evy Distribution
20
10
0
1
0
0
1
0.5
1.5
2
X
NCS with ML Distribution
1
1
0.5
0.5
Z
Z
Z
2
1
0
2
1
Y
0
0
1
0.5
X
1.5
2
0
2
1
Y
0
0
1
0.5
X
1.5
2
0
2
1
Y
0
0
1
0.5
X
1.5
2
MESA Lab
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4.4 Solution
I.
Descine one efficient optization tool;
II. Find test function have big imperfection;
III. Find
Uniform,
L´evy
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?
2. The NCS and FC like the combination of optimization
tools, looking for more efficient?
3. The application of NCS in the new object, solving other
optimization problems?
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Sincerely hope that you
give me some advices!
MESA Lab
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