Hybrid ARIMA-Spatial Autocorrelation (Moran Index and LISA) for Covid-19 Vaccination in All Indonesian Provinces

Nur'ainul Miftahul Huda, Nurfitri Imro'ah


Numerous issues arise from stochastic processes with temporal and spatial index parameters. From 2020, Covid-19 has occurred worldwide. Combining time series with geographical analysis is crucial. ARIMA and spatial autocorrelation analysis using Moran’s Index and LISA are prominent models for the two analyses. ARIMA predicts future values. The ARIMA model is applied to all recorded locations since it involves a stochastic process with a time and location parameter index. Then the prediction results at each location were examined using spatial autocorrelation, starting with the Moran index to see global relationships, then LISA (to look at the relationship between locations locally, to see which locations have a significant effect). The Queen Contiguity weight matrix calculates spatial autocorrelation (assuming that locations that are directly adjacent to each other have a spatial effect). Spatial autocorrelation will divide each place into four quadrants: High-High (HH), High-Low (HL), Low-High (LH), and Low-Low (LL). This approach was applied to 2021 Indonesian vaccination rates in all 34 provinces (354 days). Hence, the ARIMA model was applied to the 34 provinces, and each location received three forecasting. Moran’s Index revealed spatial autocorrelation in the 354th and 355th time forecasts. LISA shows that Aceh (LL), West Sumatra (LH), South Sumatra (HH), Lampung (LH), and North Maluku (LL) influence other provinces (LH).


Forecasting; Spatial; Autocorrelation; Hybrid

Full Text:



Y. Chen, “An analytical process of spatial autocorrelation functions based on moran’s index,” PLOS ONE, vol. 16, no. 4, p. e0249589, 2021. DOI:10.1371/journal.pone.0249589

Y. Chen, “Fractal analytical approach of urban form based on spatial correlation function,” Chaos, Solitons & Fractals, vol. 49, pp. 47–60, 2013. DOI:10.1016/j.chaos.2013.02.006

D. Z. Sui, “Tobler’s first law of geography: A big idea for a small world?” Annals of the Association of American Geographers, vol. 94, no. 2, pp. 269–277, 2004. DOI:10.1111/j.1467-8306.2004.09402003.x

B. Zheng, X. Lin, D. Yin, and X. Qi, “Does tobler’s first law of geography apply to internet attention? a case study of the asian elephant northern migration event,” PLOS ONE, vol. 18, no. 3, p. e0282474, 2023. DOI:10.1371/journal.pone.0282474

D. A. Griffith, “Spatial autocorrelation,” Encyclopedia of Social Measurement, pp. 581–590, 2005. DOI:10.1016/B0-12-369398-5/00334-0

A. Getis, “Spatial pattern analysis,”Encyclopedia of Social Measurement, pp. 627–632, 2005. DOI:10.1016/B0-12-369398-5/00336-4

R. Haining, “Spatial autocorrelation,”International Encyclopedia of the Social & Behavioral Sciences, pp. 14763–14768, 2001. DOI:10.1016/B0-08-043076-7/02511-0

S.-I. Lee, “Correlation and spatial autocorrelation,”Springer, 2017, pp. 360–368. ISBN 978-3-319-17884-4. DOI:10.1007/978-3-319-17885-1_1524

R. Westerholt, “Exploratory statistical analysis of spatial structures in urban datasets,” Metropolitan Research: Methods and Approaches, pp. 37–62, 2022. DOI:10.14361/9783839463109-003

D. A. Griffith and G. Arbia, “Detecting negative spatial autocorrelation in georeferenced random variables,” International Journal of Geographical Information Science, vol. 24, no. 3 pp. 417–437, 2010. DOI:10.1080/13658810902832591

Y. H. Chou, “Map resolution and spatial autocorrelation,” Geographical Analysis, vol. 23, no. 3, pp. 228–246, 1991. DOI:10.1111/j.1538-4632.1991.tb00236.x

M. Almeida-Neto and T. M. Lewinsohn, “Small-scale spatial autocorrelation and the interpretation of relationships between phenological parameters,” Journal of Vegetation Science, vol. 15, no. 4, pp. 561–568, 2004. DOI:10.1111/j.1654-1103.2004.tb02295.x

G. W. Mueller-Warrant, G. W. Whittaker, and W. C. Young, “Gis analysis of spatial clustering and temporal change in weeds of grass seed crops,” Weed Science, vol. 56, no. 5, pp. 647–669, 2008. DOI:10.1614/WS-07-032.1

A. Getis, “A history of the concept of spatial autocorrelation: A geographer’s perspective,” Geographical Analysis, vol. 40, no. 3, pp. 297–309, 2008. DOI:10.1111/j.1538-4632.2008.00727.x

L. M. Scott, “Spatial pattern, analysis of,” International Encyclopedia of the Social & Behavioral Sciences (Second Edition), pp. 178–184, 2015. DOI:10.1016/B978-0-08-097086-8.72064-2

J. K. Ord and A. Getis, “Testing for local spatial autocorrelation in the presence of global autocorrelation,” Journal of Regional Science, vol. 41, no. 3, pp. 411–432, 2002. DOI:10.1111/0022-4146.00224

A. Fotheringham and P. Rogerson, The SAGE Handbook of Spatial Analysis. SAGE Publications, Ltd, 2009. ISBN 9781412910828

A. R. Holt, M. Mears, L. Maltby, and P. Warren, “Understanding spatial patterns in the production of multiple urban ecosystem services,” Ecosystem Services, vol. 16, pp. 33–46, 2015. DOI:10.1016/j.ecoser.2015.08.007

J. Dubé and D. Legros, “Spatial autocorrelation,” John Wiley & Sons, Ltd., Spatial Econometrics Using Microdata, 2014, pp. 59–91. ISBN 9781119008651. DOI:10.1002/9781119008651.ch3

A. Abdulhafedh, “A novel hybrid method for measuring the spatial autocorrelation of vehicular crashes: Combining moran’s index and getis-ord statistic,” Open Journal of Civil Engineering, vol. 07, no. 2, pp. 208–221, 2017. DOI:10.4236/ojce.2017.72013

A. Getis, “Reflections on spatial autocorrelation,” Regional Science and Urban Economics, vol. 37, no. 4, pp. 491–496, 2007. DOI:10.1016/j.regsciurbeco.2007.04.005

L. Anselin, “Local indicators of spatial association-lisa,” Geographical Analysis, vol. 27, no. 2, pp. 93–115, 1995. DOI:10.1111/j.1538-4632.1995.tb00338.x

M. Osadebey, M. Pedersen, D. Arnold, and K. Wendel-Mitoraj, “Local indicators of spatial autocorrelation (lisa): Application to blind noise-based perceptual quality metric index for magnetic resonance images.” Journal of imaging, vol. 5, no. 1, pp. 1–23, 2019. DOI:10.3390/jimaging5010020

M. A. Oliver, “The variogram and kriging,” Springer, Berlin, Heidelberg, 2010, pp. 319–352. ISBN 978-3-642-03646-0. DOI:10.1007/978-3-642-03647-7_17

C. V. Deutsch, “Geostatistics,” Academic Press, 2003, pp. 697–707. ISBN 978-0-12-227410-7. DOI:10.1016/B0-12-227410-5/00869-3

P. Singh and P. Verma, “A comparative study of spatial interpolation technique (idw and kriging) for determining groundwater quality,” Elsevier, 2019, pp. 43–56. ISBN 978-0-12-815413-7. DOI:10.1016/B978-0-12-815413-7.00005-5

A. Soltani and S. Askari, “Exploring spatial autocorrelation of traffic crashes based on severity,” Injury, vol. 48, no. 3, pp. 637–647, 2017. DOI:10.1016/j.injury.2017.01.032

A. Getis, “Reflections on spatial autocorrelation,” Regional Science and Urban Economics, vol. 37, no. 4, pp. 491–496, 2007. DOI:10.1016/j.regsciurbeco.2007.04.005

N. M. Huda, N. Imro’ah, and R. Mailanda, “Spatial autocorrelation using moran’s index to map the confirmed positive of covid-19 cases in java,” AIP Conf. Proc., vol. 2588, no. 1, p. 050006, 2023. DOI:10.1063/5.0112014

X. Zhou and H. Lin, “Geary’s c,” Springer, 2008, pp. 329–330. ISBN 978-0-387-30858-6. DOI:10.1007/978-0-387-35973-1_446

F. Rossi and G. Becker, “Creating forest management units with hot spot analysis (getis-ord gi*) over a forest affected by mixed-severity fires,” Australian Forestry, vol. 82, no. 4, pp. 166–175, 2019. DOI:10.1080/00049158.2019.1678714

D. R. S. Saputro, Y. Widyaningsih, P. Widyaningsih, Sutanto, and Widiastuti, “Spatio-temporal patterns of dengue hemorrhagic fever (dhf) cases with local indicator of spatial association (lisa) and cluster map at areas risk in java-bali indonesia,” AIP Conf. Proc., vol. 2326, no. 1, p. 020027, 2021. DOI:10.1063/5.0040334

N. M. Huda, U. Mukhaiyar, and U. S. Pasaribu, “Forecasting dengue fever cases using autoregressive distributed lag model with outlier factor,” AIP Conf. Proc., vol. 2268, no. 1, p. 020005, 2020. DOI:10.1063/5.0018450

A. James and V. Tripathi, “Time series data analysis and arima modeling to forecast the short-term trajectory of the acceleration of fatalities in brazil caused by the corona virus (covid-19),” PeerJ, vol. 9, p. e11748, 2021. DOI:10.7717/peerj.11748

N. M. Huda and N. Imro’ah, “Determination of the best weight matrix for the generalized space time autoregressive (gstar) model in the covid-19 case on java island, indonesia,” Spatial Statistics, vol. 54, p. 100734, 2023. DOI:10.1016/j.spasta.2023.100734

O. J. Watson, G. Barnsley, J. Toor, A. B. Hogan, P. Winskill, and A. C. Ghani, “Global impact of the first year of covid-19 vaccination: a mathematical modelling study,” The Lancet Infectious Diseases, vol. 22, no 9, pp. 1293–1302, 2022. DOI:10.1016/S1473-3099(22)00320-6

M. Manaqib, M. Mahmudi, and G. Prayoga, “Mathematical Model and Simulation of the Spread of COVID-19 with Vaccination, Implementation of Health Protocols, and Treatment,” Jambura Journal of Biomathematics (JJBM), vol. 4, no. 1, pp. 69–79, 2023. DOI:10.34312/jjbm.v4i1.19162

F. Firmansyah and Y. M. Rangkuti, “Sensitivity Analysis and Optimal Control of Covid 19 Model,” Jambura Journal of Biomathematics (JJBM), vol. 4, no. 1, pp. 95–102, 2023. DOI:10.34312/jjbm.v4i1.19025

S. O. S. P. Ahaya, E. Rahmi, and N. Nurwan, “Analisis dinamik model SVEIR pada penyebaran penyakit campak,” Jambura Journal of Biomathematics (JJBM), vol. 1, no. 2, pp. 57–64, 2020. DOI:10.34312/jjbm.v1i2.8482

S. M. Moghadas, T. N. Vilches, K. Zhang, C. R. Wells, A. Shoukat, B. H. Singer, L. A. Meyers, K. M. Neuzil, J. M. Langley, M. C. Fitzpatrick, and A. P. Galvani, “The impact of vaccination on covid-19 outbreaks in the united states.” Clinical Infectious Diseases, vol. 73, no. 12, pp. 2257–2264, 2021. DOI:10.1093/cid/ciab079

G. E. P. Box, G. M. Jenkins, G. C. Reinsel, and G. M. Ljung, "Time Series Analysis: Forecasting and Control, 5th Edition," Wiley, 2015, p. 712. ISBN 978-1-118-67491-8

DOI: https://doi.org/10.37905/jjbm.v4i2.20915

Copyright (c) 2023 Nur'ainul Miftahul Huda, Nurfitri Imro'ah

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

Jambura Journal of Biomathematics (JJBM) has been indexed by:


 Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri Gorontalo
Jl. Prof. Dr. Ing. B. J. Habibie, Moutong, Tilongkabila, Kabupaten Bone Bolango 96554, Gorontalo, Indonesia
 Email: editorial.jjbm@ung.ac.id
 +6281356190818 (Call/SMS/WA)
 Jambura Journal of Biomathematics (JJBM) by Department of Mathematics Universitas Negeri Gorontalo is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.  Powered by Public Knowledge Project OJS.