The approximation algorithm based on seeding method for functional $ k $-means problem†

Min Li; Yishui Wang; Dachuan Xu; Dongmei Zhang

doi:10.3934/jimo.2020160

Article Contents

2022, Volume 18, Issue 1: 411-426. Doi: 10.3934/jimo.2020160

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The approximation algorithm based on seeding method for functional $ k $-means problem^†

1.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
2.
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, 1068 Xueyuan Avenue, Shenzhen University Town, Shenzhen 518055, China
3.
Department of Operations Research and Information Engineering, Beijing University of Technology, Beijing 100124, China
4.
School of Computer Science and Technology, Shandong Jianzhu University, Jinan 250101, China

^* Corresponding author: Dongmei Zhang
^* Corresponding author: Dongmei Zhang

^† A preliminary version of this article appeared in Proceedings of COCOON 2019, pp. 387-396.

Received: February 29, 2020

Revised: June 30, 2020

Early access: November 2020

Published: January 2022

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Abstract

Different from the classical $ k $-means problem, the functional $ k $-means problem involves a kind of dynamic data, which is generated by continuous processes. In this paper, we mainly design an $ O(\ln\; k) $-approximation algorithm based on the seeding method for functional $ k $-means problem. Moreover, the numerical experiment presented shows that this algorithm is more efficient than the functional $ k $-means clustering algorithm.

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Mathematics Subject Classification: Primary: 90C27; Secondary: 68W25.

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Table 1. Notation Index

Notations	Meaning of symbols
$x_i(t)$ or $x_i^j(t)$	functional curve
$X(t)$ or $X^i(t)$ or $C^i(t)$	$d$-dimensional functional sample with functional curves as its components
$\mathfrak{F}^d(t)$	set of $d$-dimensional functional samples
$\Gamma(t)$ or $\Delta(t)$ or $C(t)$	subset of $\mathfrak{F}^d(t)$
$\mu(\Gamma(t))$	center of mass of functional samples of $\Gamma(t)$
$d(X^i(t), X^j(t))$	distance between the functional samples $X^i(t)$ and $X^j(t)$
$d(X^i(t), \Gamma(t))$	distance from the functional sample $X^i(t)$ to the subset of functional samples $\Gamma(t)$
$X(t)_{\Gamma(t)}$	the closest functional sample in $\Gamma(t)$ to the functional sample $X(t)$
$\Phi(\Gamma(t), C(t))$	potential function of $\Gamma(t)$ over $C(t)$
$\Gamma_{C(t)}^i(t)$	the $i$-th cluster of $\Gamma(t)$ with respect to $C(t)$

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Table 2. Comparison of Algorithm 1 and the functional $ k $-means algorithm in [15]

Data Set	Method	ARI	DBI	Initial Cost	Returned Cost	Time (s)
Simudata	SeedAlg	0.7919	0.8652	3338	1689	58
	FuncAlg	0.7975	0.8656	5465	1689	61
Sdata	SeedAlg	0.6651	1.1385	1667	712	192
	FuncAlg	0.6561	1.1384	2485	720	217

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