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Opposition-based antlion optimizer using Cauchy distribution and its application to data clustering problem

  • Shail Kumar DinkarEmail author
  • Kusum Deep
Original Article
  • 50 Downloads

Abstract

This paper proposes an improved version of antlion optimizer (ALO) to solve data clustering problem. In this work, Cauchy distribution-based random walk is employed in place of uniform distribution to jump out of local optima as a first strategy. Then opposition-based learning model is utilized in conjunction with acceleration coefficient to overcome the slow convergence of classical ALO as second strategy to propose opposition-based ALO using Cauchy distribution (OB-C-ALO). The performance of the proposed OB-C-ALO is evaluated over a set of benchmark problems of different varieties of characteristics and analysed statistically by performing Wilcoxon rank-sum test. The proposed version then utilizes K-means clustering by refining the clusters formed using K-means as objective function. The algorithm is evaluated on six data sets of UCI machine learning repository and compared with classical ALO and recently developed version of ALO, namely OB-L-ALO, over benchmark test problems as well as data clustering problem and proved to be better in terms of performance achieved.

Keywords

Optimization Cauchy distribution Opposition-based learning Data clustering Intra-cluster variance 

List of symbols

\(S_{\text{ant}} = \left( {S_{A,1} ,S_{A,2} , \ldots S_{A,n} , \ldots ,S_{A,N} } \right)^{T}\)

Initial population of ant

\(S_{A,n} = \left( {S_{A,n}^{1} , \ldots S_{A,n}^{d} , \ldots ,S_{A,n}^{D} } \right)\)

nth ant

\(S_{A,n}^{d}\)

dth variable of the nth ant

\(T_{\text{ant}} = \left( {T_{A,1} ,T_{A,2} \ldots T_{A,n} , \ldots T_{A,N} } \right)\)T

Fitness matrix of ant

\(T_{A,n} = f\left( {S_{A,n}^{1} , \ldots ,S_{A,n}^{d} , \ldots ,S_{A,n}^{D} } \right)\)

Fitness value of nth ant

\(T_{\text{antlion}} = \left( {S_{AL,1} ,S_{AL,2} , \ldots ,S_{AL,n} , \ldots ,S_{AL,N} } \right)^{T}\)

Antlion population

\(T_{AL,n} = \left( {S_{AL,n}^{1} , \ldots S_{AL,n}^{d} , \ldots S_{AL,n}^{D} } \right)\)

nth antlion

\(S_{AL,n}^{d}\)

dth variable of the nth antlion

\(T_{\text{antlion}} = \left( {T_{AL,1} , \ldots ,T_{AL,n} , \ldots ,T_{AL,N} } \right)\)

Fitness matrix of antlion

\(T_{AL,n} = f\left( {S_{AL,n}^{1} , \ldots S_{AL,n}^{d} , \ldots S_{AL,n}^{D} } \right)\)

Fitness value of nth antlion

\(it_{\text{curr}} ,it_{ \hbox{max} }\)

Current and maximum iteration

L, U

Lower and upper bounds

\(S_{\text{sel}}\)

Selected antlion

\(S_{\text{elite}}\)

Elite (best) antlion

\(r_{\text{wA}}\)

Random walk around \(S_{\text{sel}}\)

\(r_{\text{wE}}\)

Random walk around \(S_{\text{elite}}\)

\(p_{\text{ac}}\)

Acceleration coefficient

\(p_{\hbox{max} } = 1,\,p_{\hbox{min} } - 0.00001\)

Max and min values of constant

\(X(x_{1} ,x_{2} , \ldots ,x_{D} )\)

Point in D-dimensional space

\(Y(y_{1} ,y_{2} , \ldots ,y_{D} )\)

Point in D-dimensional space

\(d\left( {X,Y} \right)\)

Distance between two points

\(T = \left( {t_{1} ,t_{2} , \ldots ,t_{n} } \right)\)

n data objects

\(C = \left\{ {c_{1} ,c_{1} , \ldots ,c_{k} } \right\}\)

Set of k-clusters

\(F_{i} = \left\{ {f_{1} , \ldots ,f_{p} ,f_{p + 1} , \ldots ,f_{k*p} } \right\}\)

Set of cluster centres

p

Number of features

k

Number of clusters

Notes

Acknowledgements

The first author is thankful to All India Council of Technical Education (AICTE), Government of India, for funding this research.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Bäck T, Fogel DB, Michalewicz Z (1997) Handbook of evolutionary computation. CRC Press, Boca RatonzbMATHGoogle Scholar
  2. 2.
    Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgezbMATHGoogle Scholar
  3. 3.
    Eiben AE, Smith JE (2003) Introduction to evolutionary computing, vol 53. Springer, HeidelbergzbMATHGoogle Scholar
  4. 4.
    Wolpert DH, Macready WG (1995) No free lunch theorems for search. Technical Report SFI-TR-95-02-010 (Santa Fe Institute)Google Scholar
  5. 5.
    Holland JH (1975) Adaptation in natural and artificial system. The University of Michigan Press, Ann ArborGoogle Scholar
  6. 6.
    Storn R, Price K (1997) Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11(4):341–359MathSciNetzbMATHGoogle Scholar
  7. 7.
    Das S, Suganthan PN (2011) Differential evolution: a survey of the state-of-the-art. IEEE Trans Evol Comput 15(1):4–31Google Scholar
  8. 8.
    Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings IEEE international conference neural networks, vol 4, pp 1942–1948Google Scholar
  9. 9.
    Dorigo M, Di Caro G (1999) Ant colony optimization: a new meta-heuristic. In: Proceedings of the 1999 congress on evolutionary computation, CEC 99, vol 2, pp 1470–1477Google Scholar
  10. 10.
    Karaboga D (2005) An idea based on honey bee swarm for numerical optimization (vol 200). Technical Report-tr06, Erciyes University, Engineering Faculty, Computer Engineering DepartmentGoogle Scholar
  11. 11.
    Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61Google Scholar
  12. 12.
    Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67Google Scholar
  13. 13.
    Krishnanand KN, Ghose D (2006) Glowworm swarm based optimization algorithm for multimodal functions with collective robotics applications. Multiagent Grid Syst 2(3):209–222zbMATHGoogle Scholar
  14. 14.
    Rashedi E, Nezamabadi-Pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179(13):2232–2248zbMATHGoogle Scholar
  15. 15.
    Yadav A, Deep K, Kim JH, Nagar AK (2016) Gravitational swarm optimizer for global optimization. Swarm Evol Comput 31:64–89Google Scholar
  16. 16.
    Formato RA (2007) Central force optimization: a new metaheuristic with applications in applied electromagnetics. Prog Electromagn Res 77:425–491Google Scholar
  17. 17.
    Geem ZW, Kim JH, Loganathan GV (2001) A new heuristic optimization algorithm: harmony search. Simulation 76:60Google Scholar
  18. 18.
    Yang XS, Deb (2009) Cuckoo search via Lévy flights. In: World congress on nature and biologically inspired computing. NaBIC 2009, pp 210–214Google Scholar
  19. 19.
    Yılmaz S, Küçüksille EU (2015) A new modification approach on bat algorithm for solving optimization problems. Appl Soft Comput 28:259–275Google Scholar
  20. 20.
    Mirjalili S (2015) The antlion optimizer. Adv Eng Softw 83:80–98Google Scholar
  21. 21.
    Evangelou IE, Hadjimitsis DG, Lazakidou AA, Clayton C (2001) Data mining and knowledge discovery in complex image data using artificial neural networks. In: Proceedings of workshop complex reason. Geogr. Data, PaphosGoogle Scholar
  22. 22.
    Kamel MS, Selim SZ (1994) New algorithms for solving the fuzzy clustering problem. Pattern Recogn 27(3):421–428Google Scholar
  23. 23.
    Omran MG, Engelbrecht AP, Salman A (2004) Image classification using particle swarm optimization. In: Recent advances in simulated evolution and learning, pp 347–365Google Scholar
  24. 24.
    Anderberg MR (1973) Cluster analysis for application. Academic Press, New YorkzbMATHGoogle Scholar
  25. 25.
    Hartigan JA (1975) Clustering algorithms. Wiley, New YorkzbMATHGoogle Scholar
  26. 26.
    Devijver PA, Kittler J (1982) Pattern recognition: a statistical approach. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar
  27. 27.
    Jain AK, Dubes RC (1988) Algorithms for clustering data. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  28. 28.
    Han J, Kamber M (2001) Data mining: concepts and techniques. Academic Press, New YorkzbMATHGoogle Scholar
  29. 29.
    Ding C, He X (2002) Cluster merging and splitting in hierarchical clustering algorithms. In: Proceedings of IEEE international conference on data mining, ICDM 2003. pp 139–146Google Scholar
  30. 30.
    Selim SZ, Ismail MA (1984) K-means-type algorithms: a generalized convergence theorem and characterization of local optimality. IEEE Trans Pattern Anal Mach Intell 1:81–87zbMATHGoogle Scholar
  31. 31.
    Van der Merwe DW, Engelbrecht AP (2003) Data clustering using particle swarm optimization. In: The 2003 congress on evolutionary computation, CEC’03. vol 1, pp 215–220Google Scholar
  32. 32.
    Zhao M, Tang H, Guo J, Sun Y (2014) Data clustering using particle swarm optimization. In: Park JJ, Pan Y, Kim CS, Yang Y (eds) Future information technology, Springer, Berlin, pp 607–612Google Scholar
  33. 33.
    Karaboga D, Ozturk C (2011) A novel clustering approach: artificial bee colony (ABC) algorithm. Appl Soft Comput 11(1):652–657Google Scholar
  34. 34.
    Zhang C, Ouyang D, Ning J (2010) An artificial bee colony approach for clustering. Expert Syst Appl 37(7):4761–4767Google Scholar
  35. 35.
    Yan X, Zhu Y, Zou W, Wang L (2012) A new approach for data clustering using hybrid artificial bee colony algorithm. Neurocomputing 97:241–250Google Scholar
  36. 36.
    Shelokar PS, Jayaraman VK, Kulkarni BD (2004) An ant colony approach for clustering. Anal Chim Acta 509(2):187–195Google Scholar
  37. 37.
    Niknam T, Amiri B, Olamaei J, Arefi A (2009) An efficient hybrid evolutionary optimization algorithm based on PSO and SA for clustering. J Zhejiang Univ-Sci A 10(4):512–519zbMATHGoogle Scholar
  38. 38.
    Niknam T, Fard ET, Pourjafarian N, Rousta A (2011) An efficient hybrid algorithm based on modified imperialist competitive algorithm and K-means for data clustering. Eng Appl Artif Intell 24(2):306–317Google Scholar
  39. 39.
    Niknam T, Amiri B (2010) An efficient hybrid approach based on PSO, ACO and k-means for cluster analysis. Appl Soft Comput 10(1):183–197Google Scholar
  40. 40.
    Senthilnath J, Omkar SN, Mani V (2011) Clustering using firefly algorithm: performance study. Swarm Evol Comput 1(3):164–171Google Scholar
  41. 41.
    Yogarajan G, Revathi T (2018) Improved cluster based data gathering using ant lion optimization in wireless sensor networks. Wireless Pers Commun 98(3):2711–2731Google Scholar
  42. 42.
    Dua D, Graff C (2019) UCI Machine Learning Repository. University of California, School of Information and Computer Science, Irvine, CAGoogle Scholar
  43. 43.
    Wang GG, Deb S, Gandomi AH, Alavi AH (2016) Opposition-based krill herd algorithm with Cauchy mutation and position clamping. Neurocomputing 177:147–157Google Scholar
  44. 44.
    Elaziz MA, Oliva D, Xiong S (2017) An improved opposition-based sine cosine algorithm for global optimization. Expert Syst Appl 90:484–500Google Scholar
  45. 45.
    Mirjalili S (2016) SCA: a sine cosine algorithm for solving optimization problems. Knowl-Based Syst 96:120–133Google Scholar
  46. 46.
    Sharma H, Bansal JC, Arya KV (2013) Opposition based Lévy flight artificial bee colony. Memetic Comput 5(3):213–227Google Scholar
  47. 47.
    Dinkar SK, Deep K (2018) An efficient opposition based Lévy flight antlion optimizer for optimization problems. J Comput Sci 29:119–141Google Scholar
  48. 48.
    Rahnamayan S, Tizhoosh HR, Salama MM (2008) Opposition-based differential evolution. IEEE Trans Evol Comput 12(1):64–79Google Scholar
  49. 49.
    Ahandani MA, Alavi-Rad H (2012) Opposition-based learning in the shuffled differential evolution algorithm. Soft Comput 16(8):1303–1337Google Scholar
  50. 50.
    Ahandani MA, Alavi-Rad H (2015) Opposition-based learning in shuffled frog leaping: an application for parameter identification. Inf Sci 291:19–42Google Scholar
  51. 51.
    Chen K, Zhou F, Yin L, Wang S, Wang Y, Wan F (2018) A hybrid particle swarm optimizer with sine cosine acceleration coefficients. Inf Sci 422:218–241MathSciNetGoogle Scholar
  52. 52.
    Saremi S, Mirjalili S, Lewis A (2017) Grasshopper optimisation algorithm: theory and application. Adv Eng Softw 105:30–47Google Scholar
  53. 53.
    Yang XS (2014) Nature-inspired optimization algorithms. Elsevier, AmsterdamzbMATHGoogle Scholar
  54. 54.
    Goodfellow I, Pouget-Abadie J, Mirza M, Xu B, Warde-Farley D, Ozair S, Bengio Y (2014) Generative adversarial nets. In: Advances in neural information processing systems, pp 2672–2680Google Scholar
  55. 55.
    Chandrasekaran VK, Muthu RB (2018) Application of Cauchy mutated memetic particle swarm optimization algorithm to economic dispatch problem with practical constraints. WSEAS Trans Syst Control 13:80–87Google Scholar
  56. 56.
    Wu Q (2011) Hybrid forecasting model based on support vector machine and particle swarm optimization with adaptive and Cauchy mutation. Expert Syst Appl 38(8):9070–9075Google Scholar
  57. 57.
    Wang B, Wang S, Zhou XZ, Watada J (2016) Two-stage multi-objective unit commitment optimization under hybrid uncertainties. IEEE Trans Power Syst 31:2266–2277Google Scholar
  58. 58.
    Forcael E, González V, Orozco F, Vargas S, Pantoja A, Moscoso P (2014) Ant colony optimization model for tsunamis evacuation routes. Comput-Aided Civ Infrastruct Eng 29(10):723–737Google Scholar
  59. 59.
    Chen Y, Feng J, Wu Y (2012) Novel form-finding of tensegrity structures using ant colony systems. J Mech Robot 4(3):031001Google Scholar
  60. 60.
    Chen Y, Feng J, Wu Y (2012) Prestress stability of pin-jointed assemblies using ant colony systems. Mech Res Commun 41:30–36Google Scholar
  61. 61.
    Chen Y, Feng J (2012) Efficient method for Moore-Penrose inverse problems involving symmetric structures based on group theory. J Comput Civ Eng 28(2):182–190Google Scholar
  62. 62.
    Dinkar SK, Deep K (2018) Accelerated opposition-based antlion optimizer with application to order reduction of linear time-invariant systems. Arab J Sci Eng 44:1–29Google Scholar
  63. 63.
    Dinkar SK, Deep K (2019) A novel CPU scheduling algorithm based on ant lion optimizer. In: Bansal JC, Das KN, Nagar A, Deep K, Ojha AK (eds) Soft computing for problem solving, Springer, Singapore, pp 339–353Google Scholar
  64. 64.
    Yao P, Wang H (2016) Dynamic adaptive antlion optimizer applied to route planning for unmanned aerial vehicle. Soft Comput 21:1–14Google Scholar
  65. 65.
    Wang H, Li H, Liu Y, Li C, Zeng S (2007) Opposition-based particle swarm algorithm with Cauchy mutation. In: IEEE congress on evolutionary computation, CEC 2007. pp 4750–4756Google Scholar
  66. 66.
    Qin H, Zhou J, Lu Y, Wang Y, Zhang Y (2010) Multi-objective differential evolution with adaptive Cauchy mutation for short-term multi-objective optimal hydro-thermal scheduling. Energy Convers Manag 51(4):788–794Google Scholar
  67. 67.
    Ali M, Pant M (2011) Improving the performance of differential evolution algorithm using Cauchy mutation. Soft Comput 15(5):991–1007Google Scholar
  68. 68.
    Norman L, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  69. 69.
    Feller W (1971) An introduction to probability theory and its applications, vol 2, 2nd edn. Wiley, New York. ISBN 978-0-471-25709-7zbMATHGoogle Scholar
  70. 70.
    Dinkar SK, Deep K (2017) Opposition based Laplacian antlion optimizer. J Comput Sci 23:71–90MathSciNetGoogle Scholar
  71. 71.
    Rahnamayan S, Tizhoosh HR, Salama MM (2006) Opposition versus randomness in soft computing techniques. Appl Soft Comput 8(2):906–918Google Scholar
  72. 72.
    Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evol Comput 3(2):82–102Google Scholar
  73. 73.
    Van Den Bergh F, Engelbrecht AP (2006) A study of particle swarm optimization particle trajectories. Inf Sci 176(8):937–971MathSciNetzbMATHGoogle Scholar
  74. 74.
    Kishor A, Singh PK, Prakash J (2016) NSABC: non-dominated sorting based multi-objective artificial bee colony algorithm and its application in data clustering. Neurocomputing 216:514–533Google Scholar
  75. 75.
    Güngör Z, Ünler A (2007) K-harmonic means data clustering with simulated annealing heuristic. Appl Math Comput 184(2):199–209MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia

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