naspaclust: A hybridisation of spatial fuzzy clustering with nature-inspired optimisation

Bahrul Ilmi Nasution; Rezzy Eko Caraka; Robert Kurniawan

cs.NEOct 12, 2026

naspaclust: A hybridisation of spatial fuzzy clustering with nature-inspired optimisation

Authors: Bahrul Ilmi Nasution, Rezzy Eko Caraka, Robert Kurniawan

Organizations: Department of Social Statistics The University of Manchester, Manchester, M13 9PL, Manchester, UK · Research Center for Data and Information Sciences, Research Organisation for Electronics and Informatics National Research and Innovation Agency Bandung, 40135, West Java, Indonesia · School of Economics and Business Telkom University Bandung, 40257, West Java, Indonesia · Department of Statistical Computation Polytechnics Statistics STIS Jakarta, 13330, West Java, Indonesia

Abstract

We present an overview of R package called naspaclust, a tool for spatial fuzzy clustering improvement by hybridising the Fuzzy Geographically Weighted Clustering (FGWC) with nature-inspired optimisation algorithms. There are seven optimisation algorithms in general, namely Artificial Bee Colony, Flower Pollination Algorithm, Gravitational Search Algorithm, Harris-Hawk Optimisation, Intelligent Firefly Algorithm, Particle Swarm Optimisation, and Teaching-Learning Based optimisation. The optimisation algorithms available do not only use the basic algorithms, but also the modified version of several parts such as randomisation terms and updating rules. Furthermore, we also provide additional options by providing more distance metrics options based on the rdist package. This study showed the implementation of the naspaclust package in various cases, as well as comparing the performance of the algorithms in the scope of basic algorithms. The naspaclust package allows to improve the performance of fuzzy clustering, mainly in spatial context.

Figures & tables

Update type	Formula
1	$10^{-5}+(\alpha-10^{-5})\times e^{t}$
2	$\alpha\times rand^{t}$
3	$1-(\frac{10^{-4}}{9})^{\frac{1}{max_{iter}}}$
4	$\left(1,11\times{10}^{-4}\right)^{\frac{5}{max_{iter}}}{\times\alpha}$
5	$\alpha\times\frac{t}{max_{iter}}$

Table 1: Randomisation constant update formula

Inertia Weight	Formula
Constant Inertia Weight	$w=w_{max}$
Chaotic Inertia Weight	$w=(w_{max}-w_{min})\times(1-\frac{iter}{max_{iter}})+w_{min}*z$
Simulated Annealing Inertia Weight	$w_{min}+(w_{max}-w_{min})\times 0.95^{iter-1}$
Natural Exponent 1 Inertia Weight	$w_{min}+(w_{max}-w_{min})\times exp(-\frac{iter}{\frac{max_{iter}}{10}})$
Natural Exponent 2 Inertia Weight	$w_{min}+(w_{max}-w_{min})\times exp(-\frac{iter}{\frac{max_{iter}}{4}}^{2})$

Table 2: Inertia Weight Strategies

Algorithm	Parameters
Artificial Bee Colony	$n_{onlooker}=5,limit=10$
Flower Pollination Algorithm	$p=0.8,\gamma=1.2,\lambda=0.1$
Gravitational Search Algorithm	$G_{0}=2,v_{max}=0.7$
Harris-Hawk Optimisation	$q=0.5$
Intelligent Firefly Algorithm	$n_{intel}=2,\gamma_{f}=\beta_{0}=1$ , update type = 4
Particle Swarm Optimisation	$v_{max}=0.7,c_{1}=0.7,c_{2}=0.6,w_{max}=0.9,w_{min}=0.2,z=0.3$ , weight update = chaotic inertia weight

Table 3: Optimisation algorithms parameter used in the study

Panel of Figure 1 — Figure 1: Number of simulations with appropriate final number of clusters based on algorithm and number of clusters

Criteria	c	ABC	CLASSIC	FPA	GSA	HHO	IFA	PSO	TLBO
Obj. function	2	$1.359\times 10^{5}$	$1.368\times 10^{5}$	$1.362\times 10^{5}$	$1.362\times 10^{5}$	$1.355\times 10^{5}$	$1.363\times 10^{5}$	$1.361\times 10^{5}$	$\boldsymbol{1.333\times 10^{5}}$
	3	$1.103\times 10^{5}$	NA	$1.112\times 10^{5}$	$1.112\times 10^{5}$	$1.106\times 10^{5}$	$1.113\times 10^{5}$	$1.111\times 10^{5}$	$\boldsymbol{1.068\times 10^{5}}$
	4	$9.263\times 10^{4}$	NA	$9.627\times 10^{4}$	$9.629\times 10^{4}$	NA	$9.637\times 10^{4}$	$9.628\times 10^{4}$	$\boldsymbol{8.847\times 10^{4}}$
	5	$7.894\times 10^{4}$	NA	$8.615\times 10^{4}$	$8.623\times 10^{4}$	$8.591\times 10^{4}$	$8.627\times 10^{4}$	$8.537\times 10^{4}$	$\boldsymbol{7.638\times 10^{4}}$
	6	$7.062\times 10^{4}$	NA	$7.859\times 10^{4}$	$7.863\times 10^{4}$	NA	$7.865\times 10^{4}$	$7.875\times 10^{4}$	$\boldsymbol{6.836\times 10^{4}}$
PC Index	2	$5.211\times 10^{-1}$	$5.000\times 10^{-1}$	$5.194\times 10^{-1}$	$5.193\times 10^{-1}$	$5.096\times 10^{-1}$	$5.108\times 10^{-1}$	$5.135\times 10^{-1}$	$\boldsymbol{5.878\times 10^{-1}}$

Table 4: Performance evaluation summary for Case 1

Criteria	c	ABC	CLASSIC	FPA	GSA	HHO	IFA	PSO	TLBO
Itera- tion	2	$1.399\times 10^{1}$	$1.924\times 10^{1}$	$1.065\times 10^{1}$	$\boldsymbol{1.058\times 10^{1}}$	$1.894\times 10^{1}$	$1.280\times 10^{1}$	$1.737\times 10^{1}$	$2.370\times 10^{1}$
	3	$2.000\times 10^{1}$	NA	$1.038\times 10^{1}$	$\boldsymbol{1.014\times 10^{1}}$	$1.707\times 10^{1}$	$1.261\times 10^{1}$	$1.251\times 10^{1}$	$2.714\times 10^{1}$
	4	$3.703\times 10^{1}$	NA	$1.039\times 10^{1}$	$\boldsymbol{1.031\times 10^{1}}$	NA	$1.267\times 10^{1}$	$1.504\times 10^{1}$	$3.048\times 10^{1}$
	5	$4.331\times 10^{1}$	NA	$1.047\times 10^{1}$	$1.037\times 10^{1}$	$\boldsymbol{1.000\times 10^{1}}$	$1.238\times 10^{1}$	$2.900\times 10^{1}$	$3.205\times 10^{1}$
	6	$4.422\times 10^{1}$	NA	$1.007\times 10^{1}$	$\boldsymbol{1.000\times 10^{1}}$	NA	$1.100\times 10^{1}$	$1.000\times 10^{1}$	$3.211\times 10^{1}$
Comp. time	2	$1.081\times 10^{0}$	$\boldsymbol{3.260\times 10^{-2}}$	$1.510\times 10^{-1}$	$2.922\times 10^{-1}$	$3.050\times 10^{-1}$	$7.712\times 10^{-1}$	$6.623\times 10^{-1}$	$1.277\times 10^{0}$

Table 5: Computational cost summary for Case 1

Criteria	c	ABC	CLASSIC	FPA	GSA	HHO	IFA	PSO	TLBO
Obj. function	2	$\boldsymbol{4.727\times 10^{4}}$	$5.809\times 10^{4}$	$5.691\times 10^{4}$	$5.778\times 10^{4}$	$4.942\times 10^{4}$	$5.789\times 10^{4}$	$5.707\times 10^{4}$	$4.891\times 10^{4}$
	3	$\boldsymbol{3.702\times 10^{4}}$	$4.635\times 10^{4}$	$4.493\times 10^{4}$	$3.929\times 10^{4}$	$3.810\times 10^{4}$	$4.120\times 10^{4}$	$3.946\times 10^{4}$	$3.810\times 10^{4}$
	4	$\boldsymbol{2.497\times 10^{4}}$	$3.793\times 10^{4}$	$3.465\times 10^{4}$	$3.357\times 10^{4}$	$3.011\times 10^{4}$	$3.619\times 10^{4}$	$3.645\times 10^{4}$	$3.179\times 10^{4}$
	5	$2.029\times 10^{4}$	$3.247\times 10^{4}$	$2.910\times 10^{4}$	$3.030\times 10^{4}$	$\boldsymbol{2.007\times 10^{4}}$	$3.141\times 10^{4}$	$3.247\times 10^{4}$	$2.646\times 10^{4}$
	6	$\boldsymbol{1.771\times 10^{4}}$	$3.036\times 10^{4}$	$2.806\times 10^{4}$	$2.828\times 10^{4}$	$2.091\times 10^{4}$	NA	$3.249\times 10^{4}$	$2.370\times 10^{4}$
PC Index	2	$\boldsymbol{7.869\times 10^{-1}}$	$5.000\times 10^{-1}$	$5.512\times 10^{-1}$	$5.198\times 10^{-1}$	$7.347\times 10^{-1}$	$5.136\times 10^{-1}$	$5.523\times 10^{-1}$	$7.482\times 10^{-1}$

Table 6: Performance evaluation summary for Case 2

Criteria	c	ABC	CLASSIC	FPA	GSA	HHO	IFA	PSO	TLBO
Itera- tion	2	$2.300\times 10^{1}$	$1.875\times 10^{2}$	$\boldsymbol{1.201\times 10^{1}}$	$1.742\times 10^{1}$	$2.034\times 10^{1}$	$1.268\times 10^{1}$	$3.573\times 10^{1}$	$2.373\times 10^{1}$
	3	$2.181\times 10^{1}$	$9.023\times 10^{1}$	$\boldsymbol{1.767\times 10^{1}}$	$5.814\times 10^{1}$	$2.271\times 10^{1}$	$5.098\times 10^{1}$	$3.233\times 10^{1}$	$2.347\times 10^{1}$
	4	$3.626\times 10^{1}$	$2.395\times 10^{1}$	$2.289\times 10^{1}$	$4.423\times 10^{1}$	$2.709\times 10^{1}$	$3.287\times 10^{1}$	$\boldsymbol{2.194\times 10^{1}}$	$2.312\times 10^{1}$
	5	$3.466\times 10^{1}$	$2.227\times 10^{1}$	$\boldsymbol{2.226\times 10^{1}}$	$4.062\times 10^{1}$	$2.899\times 10^{1}$	$4.060\times 10^{1}$	$2.300\times 10^{1}$	$2.376\times 10^{1}$
	6	$3.736\times 10^{1}$	$3.067\times 10^{1}$	$2.044\times 10^{1}$	$4.061\times 10^{1}$	$2.760\times 10^{1}$	NA	$\boldsymbol{1.308\times 10^{1}}$	$2.430\times 10^{1}$
Comp. time	2	$6.250\times 10^{-1}$	$1.227\times 10^{-1}$	$\boldsymbol{1.070\times 10^{-1}}$	$3.192\times 10^{-1}$	$1.734\times 10^{-1}$	$2.912\times 10^{-1}$	$3.173\times 10^{-1}$	$5.271\times 10^{-1}$

Table 7: Computational cost summary for Case 2

Criteria	c	ABC	CLASSIC	FPA	GSA	HHO	IFA	PSO	TLBO
Obj. function	2	$4.651\times 10^{5}$	$4.669\times 10^{5}$	$4.668\times 10^{5}$	$4.669\times 10^{5}$	$4.647\times 10^{5}$	$4.669\times 10^{5}$	$4.664\times 10^{5}$	$\boldsymbol{4.631\times 10^{5}}$
	3	$\boldsymbol{3.704\times 10^{5}}$	$3.813\times 10^{5}$	$3.812\times 10^{5}$	$3.812\times 10^{5}$	$3.769\times 10^{5}$	$3.812\times 10^{5}$	$3.808\times 10^{5}$	$3.725\times 10^{5}$
	4	$\boldsymbol{3.160\times 10^{5}}$	$3.302\times 10^{5}$	$3.301\times 10^{5}$	$3.301\times 10^{5}$	NA	$3.301\times 10^{5}$	$3.299\times 10^{5}$	$3.170\times 10^{5}$
	5	$\boldsymbol{2.673\times 10^{5}}$	$2.953\times 10^{5}$	$2.949\times 10^{5}$	$2.953\times 10^{5}$	NA	$2.953\times 10^{5}$	$2.950\times 10^{5}$	$2.709\times 10^{5}$
	6	$\boldsymbol{2.399\times 10^{5}}$	NA	$2.691\times 10^{5}$	$2.695\times 10^{5}$	NA	$2.695\times 10^{5}$	$2.693\times 10^{5}$	$2.470\times 10^{5}$
PC Index	2	$5.135\times 10^{-1}$	$5.000\times 10^{-1}$	$5.136\times 10^{-1}$	$5.151\times 10^{-1}$	$5.256\times 10^{-1}$	$5.030\times 10^{-1}$	$5.020\times 10^{-1}$	$\boldsymbol{5.384\times 10^{-1}}$

Table 8: Performance evaluation summary for Case 3

Criteria	c	ABC	CLASSIC	FPA	GSA	HHO	IFA	PSO	TLBO
Itera- tion	2	$2.374\times 10^{1}$	$1.824\times 10^{1}$	$\boldsymbol{1.058\times 10^{1}}$	$1.078\times 10^{1}$	$1.382\times 10^{1}$	$1.272\times 10^{1}$	$1.832\times 10^{1}$	$2.267\times 10^{1}$
	3	$3.295\times 10^{1}$	$2.121\times 10^{1}$	$\boldsymbol{1.061\times 10^{1}}$	$1.130\times 10^{1}$	$1.850\times 10^{1}$	$1.276\times 10^{1}$	$1.816\times 10^{1}$	$2.536\times 10^{1}$
	4	$3.403\times 10^{1}$	$2.500\times 10^{1}$	$1.072\times 10^{1}$	$\boldsymbol{1.060\times 10^{1}}$	NA	$1.276\times 10^{1}$	$1.791\times 10^{1}$	$2.938\times 10^{1}$
	5	$4.175\times 10^{1}$	$2.867\times 10^{1}$	$\boldsymbol{1.073\times 10^{1}}$	$1.132\times 10^{1}$	NA	$1.267\times 10^{1}$	$1.811\times 10^{1}$	$2.973\times 10^{1}$
	6	$3.769\times 10^{1}$	NA	$1.101\times 10^{1}$	$\boldsymbol{1.054\times 10^{1}}$	NA	$1.263\times 10^{1}$	$1.799\times 10^{1}$	$3.314\times 10^{1}$
Comp. time	2	$5.790\times 10^{1}$	$\boldsymbol{9.467\times 10^{-1}}$	$7.336\times 10^{0}$	$7.189\times 10^{0}$	$8.277\times 10^{0}$	$2.134\times 10^{1}$	$1.096\times 10^{1}$	$3.975\times 10^{1}$

Table 9: Computational cost summary for Case 3

Figure 2: Kruskal-Wallis results of the evaluation measurements difference between FGWC algorithms based on the number of cases and number of clusters. The values in the grid represents Kruskal-Wallis statistics, while in bracket is the degrees of freedom. The asterisk (*) sign denotes significant difference of values between algorithms.

Panel of Figure 3 — Figure 3: Spatial map on social vulnerability (case 3) clustering in Indonesia at district level using 3 clusters based on optimisation algorithms

Panel of Figure 4 — Figure 4: Spatial map on social vulnerability (case 3) clustering in Indonesia at district level using 3 clusters based on optimisation algorithms (Continued)

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