Report

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: [email protected] 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 MESA Lab Cuckoo Search with L´evy and Mittag-Leffler distribution MESA Lab Outline 1 Random distribution 2 HCSPSO search 3 New Cuckoo search 4 Experiment Slide-4/1024 MESA Lab 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. AFC Workshop Series @ MESALAB @ UCMerced Slide-5/1024 MESA Lab 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 AFC Workshop Series @ MESALAB @ UCMerced Slide-6/1024 MESA Lab 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, AFC Workshop Series @ MESALAB @ UCMerced MESA Lab The Curve of L’evy distribution AFC Workshop Series @ MESALAB @ UCMerced Slide-8/1024 MESA Lab 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[etX ] (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 )) AFC Workshop Series @ MESALAB @ UCMerced Slide-9/1024 1.3 Other distribution AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-10/1024 2. HCSPSO search 1)A Hybrid CS/PSO Algorithm for Global Optimization Iterative equation: AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-11/1024 MESA Lab 2) The pseudo-code of the CS/PSO is presented as bellow: AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-13/1024 3) Hybrid CSPSO flow The algorithm flow: AFC Workshop Series @ MESALAB @ UCMerced MESA Lab MESA Lab MESA Lab Slide-15/1024 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 technique. (PSO), evolutionary AFC Workshop Series @ MESALAB @ UCMerced computation Slide-16/1024 MESA Lab 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 AFC Workshop Series @ MESALAB @ UCMerced Slide-17/1024 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-18/1024 MESA Lab 3) New CS with the L´evy and Mittag-Leffler distritution AFC Workshop Series @ MESALAB @ UCMerced Slide-19/1024 4. Experiment 4.1 Experiment function AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-20/1024 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-21/1024 4.2 Experiment with large size 1) Simulation data AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-22/1024 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-23/1024 2) The Graph of Convergence AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-24/1024 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-25/1024 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-26/1024 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-27/1024 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-28/1024 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-29/1024 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-30/1024 MESA Lab 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. AFC Workshop Series @ MESALAB @ UCMerced Slide-31/1024 n f1 ( x) x 2i i 1 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Slide-32/1024 MESA Lab 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. AFC Workshop Series @ MESALAB @ UCMerced Slide-33/1024 AFC Workshop Series @ MESALAB @ UCMerced MESA Lab MESA Lab 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 Slide-44/1024 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. AFC Workshop Series @ MESALAB @ UCMerced Slide-45/1024 MESA Lab 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? AFC Workshop Series @ MESALAB @ UCMerced MESA Lab Sincerely hope that you give me some advices! MESA Lab AFC Workshop Series @ MESALAB @ UCMerced