Spatial Dependence and Co-Kriging Prediction of Stunting Prevalence Using the Linear Model of Coregionalization

Zakiyah Mar'ah, Isma Muthahharah, Mahendri Deayu Putri

Abstract


Stunting remains a major public health challenge in Indonesia, marked by substantial regional disparities. This study analyzed the spatial dependence structure of stunting prevalence and its determinants using the Linear Model of Coregionalization (LMC) and co-kriging, which exploit spatial cross-correlation among variables rather than autocorrelation alone. Secondary data for 38 provinces in 2024 from the Indonesian Health Profile comprised stunting prevalence and nine explanatory variables, modeled jointly through empirical semivariograms and cross-semivariograms. The selected model combined a nugget effect with a spherical structure of 600 km effective range (LMC = Nug + Sph(600 km)); the nugget was non-zero for every variable and reported explicitly. Stunting prevalence showed moderate spatial dependence (SDP = 68.78%). Low birth weight showed positive cross-spatial dependence with stunting, whereas exclusive breastfeeding and pediatric health service coverage showed negative dependence. The poverty pattern was weak and mixed in the full sample but turned clearly positive once the two provinces reporting 0% prevalence were excluded. These are co-spatial patterns among provincial aggregates, not causal effects. Co-kriging revealed distinct clustering, with East Nusa Tenggara, parts of Papua, and Sulawesi showing relatively high predicted prevalence. Leave-one-out cross-validation gave a mean error of 0.264 and an RMSE of 6.423 percentage points, close to the between-province standard deviation (7.68), with prediction standard errors of 5.6–7.7, indicating moderate accuracy. Excluding the 0% values left the dependence class, effective range, and cross-covariance signs unchanged. Multivariate geostatistics thus offers spatial evidence for geographically targeted, ecological-level stunting interventions.


Keywords


co-kriging; linear model of coregionalization; multivariate geostatistics; spatial dependence; stunting prevalence

Full Text:

PDF

References


[1] T. Susanto, T. Saifudin, and N. Chamidah. “Spatial modeling of child malnutrition in Indonesia using geographically weighted multivariate regression”. BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 20, no. 3, pp. 1837–1854, 2026. DOI: https://doi.org/10.30598/barekengvol20iss3pp1837-1854.

[2] A. Meily and W. Nurasih. “Determinan stunting pada anak di Indonesia: Sebuah tinjauan literatur sistematis”. MEJORA Medical Journal Awatara, vol. 3, no. 4, pp. 199–205, 2025. DOI: https://doi.org/10.61434/mejora.v3i4.287.

[3] Kementerian Kesehatan Republik Indonesia. Survei Status Gizi Indonesia (SSGI) 2024. Technical Report. Jakarta: Badan Kebijakan Pembangunan Kesehatan, Kementerian Kesehatan Republik Indonesia, 2025.

[4] A. J. Tanesib and A. Kusumayati. “Determinan kejadian stunting pada balita di Provinsi Nusa Tenggara Timur: Analisis data Survei Kesehatan Indonesia 2023”. Jurnal Promotif Preventif, vol. 8, no. 6, pp. 1542–1554, 2025. DOI: https://doi.org/10.47650/jpp.v8i6.2121.

[5] A. Widodo and B. A. Herlambang. “Sistem informasi geografis sebaran stunting dan tingkat kemiskinan di Provinsi Jawa Tengah tahun 2023”. Jurnal Penelitian Sistem Informasi, vol. 2, no. 1, pp. 55–63, 2024. DOI: https://doi.org/10.54066/jpsi.v2i1.1457.

[6] P. A. Dowd and E. Pardo-Igúzquiza. “The many forms of co-kriging: A diversity of multivariate spatial estimators”. Mathematical Geosciences, vol. 56, no. 2, pp. 387–413, 2024. DOI: https://doi.org/10.1007/s11004-023-10104-7.

[7] S. De Iaco, D. E. Myers, and D. Posa. “The linear coregionalization model and the product-sum space-time variogram”. Mathematical Geology, vol. 35, no. 1, pp. 25–38, 2003. DOI: https://doi.org/10.1023/A:1022425111459.

[8] H. Pramoedyo, M. Mudjiono, A. A. Fernandes, D. Ardianti, and K. Septiani. “Determination of stunting risk factors using spatial interpolation geographically weighted regression kriging in Malang”. Mutiara Medika: Jurnal Kedokteran dan Kesehatan, vol. 20, no. 2, 2020. DOI: https://doi.org/10.18196/mm.200250.

[9] N. A. C. Cressie. Statistics for Spatial Data. Revised ed. New York: John Wiley & Sons, 1993.

[10] P. Goovaerts. Geostatistics for Natural Resources Evaluation. New York: Oxford University Press, 1997.

[11] T. Hengl, G. B. M. Heuvelink, and D. G. Rossiter. “About regression-kriging: From equations to case studies”. Computers & Geosciences, vol. 33, no. 10, pp. 1301–1315, 2007. DOI: https://doi.org/10.1016/j.cageo.2007.05.001.

[12] L. Anselin. Spatial Econometrics: Methods and Models. Dordrecht: Kluwer Academic Publishers, 1988.

[13] J. Besag, J. York, and A. Mollié. “Bayesian image restoration, with two applications in spatial statistics”. Annals of the Institute of Statistical Mathematics, vol. 43, no. 1, pp. 1–20, 1991. DOI: https://doi.org/10.1007/BF00116466.

[14] A. S. Fotheringham, C. Brunsdon, and M. Charlton. Geographically Weighted Regression: The Analysis of Spatially Varying Relationships. Chichester: John Wiley & Sons, 2002.

[15] A. S. Fotheringham, W. Yang, and W. Kang. “Multiscale geographically weighted regression (MGWR)”. Annals of the American Association of Geographers, vol. 107, no. 6, pp. 1247–1265, 2017. DOI: https://doi.org/10.1080/24694452.2017.1352480.

[16] H. Wackernagel. Multivariate Geostatistics: An Introduction with Applications. 3rd ed. Springer, 2003. DOI: https://doi.org/10.1007/978-3-662-05294-5.

[17] A. G. Journel and C. J. Huijbregts. Mining Geostatistics. London: Academic Press, 1978.

[18] B. W. Otok, Purhadi, R. Sriningsih, and D. S. Dila. “Segmentation of toddler nutritional status using REBUS and FIMIX partial least square in Southeast Sulawesi”. MethodsX, vol. 12, p. 102515, 2024. DOI: https://doi.org/10.1016/j.mex.2023.102515.

[19] F. R. Fadmi, B. W. Otok, Kuntoro, S. Melaniani, and R. Sriningsih. “Segmentation of stunting, wasting, and underweight in Southeast Sulawesi using geographically weighted multivariate Poisson regression”. MethodsX, vol. 12, p. 102736, 2024. DOI: https://doi.org/10.1016/j.mex.2024.102736.

[20] D. Payares-Garcia, F. Osei, J. Mateu, and A. Stein. “Multivariate Poisson cokriging: A geostatistical model for health count data”. Statistical Methods in Medical Research, vol. 33, no. 9, pp. 1637–1659, 2024. DOI: https://doi.org/10.1177/09622802241268488.

[21] Kementerian Kesehatan Republik Indonesia. Profil Kesehatan Indonesia 2024. Jakarta: Kementerian Kesehatan Republik Indonesia, 2025.

[22] M. C. Ribeiro, A. J. Sousa, and M. J. Pereira. “A coregionalization model can assist specification of geographically weighted Poisson regression: Application to an ecological study”. Spatial and Spatio-temporal Epidemiology, vol. 17, pp. 1–13, 2016. DOI: https://doi.org/10.1016/j.sste.2016.02.001.

[23] F. H. Zhao, A. X. Zhu, and C. Z. Qin. “Spatial distribution pattern analysis using variograms over geographic and feature space”. Geo-spatial Information Science, vol. 28, no. 4, pp. 1774–1788, 2025. DOI: https://doi.org/10.1080/10095020.2024.2439371.

[24] P. Fabijańczyk, J. Zawadzki, and T. Magiera. “Magnetic susceptibility as a proxy for soil contamination: Cross-variogram analysis across contrasting land use areas”. Environmental Science and Pollution Research, vol. 32, no. 45, pp. 25917–25931, 2025. DOI: https://doi.org/10.1007/s11356-025-37158-w.

[25] S. Yu, L. Zhao, and S. Li. “Leveraging deep learning for automated experimental semivariogram fitting”. Atmosphere, vol. 16, no. 2, p. 191, 2025. DOI: https://doi.org/10.3390/atmos16020191.

[26] K. N. Kyzyurova. “On linear model of coregionalization”. arXiv preprint, 2019.

[27] Z. Li, X. Zhang, K. C. Clarke, G. Liu, and R. Zhu. “An automatic variogram modeling method with high reliability fitness and estimates”. Computers & Geosciences, vol. 120, pp. 48–59, 2018. DOI: https://doi.org/10.1016/j.cageo.2018.07.011.

[28] E. J. Seidel and M. S. de Oliveira. “A classification for a geostatistical index of spatial dependence”. Revista Brasileira de Ciência do Solo, vol. 40, 2016. DOI: https://doi.org/10.1590/18069657rbcs20160007.

[29] G. Caringi and P. Secchi. “Fixed rank co-kriging: A model for multivariate spatial prediction”. arXiv preprint, 2026. URL: http://arxiv.org/abs/2603.21748.

[30] D. Khodoli Zangeneh, H. Amanipoor, and S. Battaleb-Looie. “Evaluation of heavy metal contamination using cokriging geostatistical method (case study of Abteymour Oilfield in Southern Iran)”. Applied Water Science, vol. 13, no. 10, 2023. DOI: https://doi.org/10.1007/s13201-023-01980-9.

[31] M. Goulard and M. Voltz. “Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix”. Mathematical Geology, vol. 24, no. 3, pp. 269–286, 1992. DOI: https://doi.org/10.1007/BF00893750.

[32] N. Akseer, T. Vaivada, O. Rothschild, K. Ho, and Z. A. Bhutta. “Understanding multifactorial drivers of child stunting reduction in exemplar countries: A mixed-methods approach”. The American Journal of Clinical Nutrition, vol. 112, pp. 792S–805S, 2020. DOI: https://doi.org/10.1093/ajcn/nqaa152.

[33] UNICEF. The State of the World’s Children 2023. Technical Report. United Nations Children’s Fund, 2023.

[34] M. de Onis and F. Branca. “Childhood stunting: A global perspective”. Maternal & Child Nutrition, vol. 12, pp. 12–26, 2016. DOI: https://doi.org/10.1111/mcn.12231.

[35] T. Beal, A. Tumilowicz, A. Sutrisna, D. Izwardy, and L. M. Neufeld. “A review of child stunting determinants in Indonesia”. Maternal & Child Nutrition, vol. 14, no. 4, e12617, 2018. DOI: https://doi.org/10.1111/mcn.12617.




DOI: https://doi.org/10.18860/cauchy.v11i2.44312

Refbacks

  • There are currently no refbacks.


Copyright (c) 2026 Zakiyah Mar'ah, Isma Muthahharah, Mahendri Deayu Putri

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Editorial Office
Mathematics Department,
Maulana Malik Ibrahim State Islamic University of Malang
Gajayana Street 50 Malang, East Java, Indonesia 65144
e-mail: cauchy@uin-malang.ac.id

Creative Commons License
CAUCHY: Jurnal Matematika Murni dan Aplikasi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.