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

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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.

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Abbreviations

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

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

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Acknowledgements

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

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  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    Shail Kumar Dinkar & Kusum Deep

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  1. Shail Kumar Dinkar
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Correspondence to Shail Kumar Dinkar.

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Dinkar, S.K., Deep, K. Opposition-based antlion optimizer using Cauchy distribution and its application to data clustering problem. Neural Comput & Applic 32, 6967–6995 (2020). https://doi.org/10.1007/s00521-019-04174-0

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