Linear model of coregionalization and co-kriging across 38 Indonesian provinces: moderate spatial dependence (SDP = 68.78%) with the highest predicted stunting in the east

The study places stunting prevalence and its nine determinants inside a single positive semidefinite multivariate spatial covariance structure, fitting all direct and cross semivariograms jointly rather than pairwise. The authors state the contribution is integrated application rather than methodology, since the LMC, the Goulard–Voltz algorithm, and the co-kriging equations are standard; what is added is a single valid LMC fitted across all ten variables for stunting in Indonesia, with coregionalization matrices checked and reported. Data are provincial aggregates for 38 provinces from the Indonesian Ministry of Health's Indonesia Health Profile 2024, all variables from one source and one reference year; model choice rested on the smallest weighted residual sum of squares and best cross-validation performance, and the C0 and C1 matrices were eigen-decomposed, with C1 having ten strictly positive eigenvalues (smallest 1.44) and C0 a smallest eigenvalue of 0.002, satisfying positive semidefiniteness.

Stunting prevalence shows moderate spatial dependence, with SDP = 68.78%, so neighbouring provinces tend to record similar prevalence. Earlier Indonesian stunting studies largely used geographically weighted or segmentation-based models describing how regression relationships vary across space without explicitly modelling spatial cross-covariance among indicators; this study quantifies the spatially structured share of variation for each variable through the dimensionless SDP index, enabling cross-variable comparison. SDP is computed from the nugget and partial sill (for Y: nugget 18.36, partial sill 40.45, total sill 58.81); X9, X6, X5, and X1 had SDP values of 85.43%, 84.84%, 84.03%, and 78.83%, classed as strong, while X2, X3, X4, X7, and X8 were moderate.

Cross-semivariograms show positive cross-spatial dependence between low birth weight and stunting and negative dependence with exclusive breastfeeding coverage and pediatric health service coverage, while poverty was weak and mixed in sign in the full sample and clearly positive once the two 0% provinces were excluded. These are co-spatial patterns among provincial aggregates, and the authors repeatedly stress they are not causal effects and cannot be read as individual-level risk or protective factors; the sign change for poverty is reported separately and assessed through a sensitivity analysis. Cross-semivariance ranges include Y–X1 from −0.47 to 8.45, Y–X6 from −44.62 to 3.15, and Y–X9 from −69.64 to 34.16; Y–X5 had a negative nugget and positive spherical component, making the full-sample pattern scale-dependent; excluding the two 0% provinces left SDP moderate (68.78% to 63.80%), the effective range near 600 km, and the cross-covariance signs unchanged.

LMC-based co-kriging produces a continuous, uncertainty-aware provincial prediction surface with higher predicted values in eastern Indonesia and lowers prediction error relative to univariate ordinary kriging. Unlike ordinary kriging, which uses only the autocovariance of stunting prevalence, co-kriging borrows strength from correlated secondary variables to improve point prediction; the authors also map prediction standard errors so that uncertainty is visible. Leave-one-out cross-validation gave a mean error of 0.264 and an RMSE of 6.423 percentage points, about 30% of the national mean (21.09%) and below the between-province standard deviation (7.68); prediction standard errors ranged from 5.6 to 7.7 percentage points (mean 6.8), smallest in Java and largest in Aceh, East Nusa Tenggara, Maluku, and southern Papua; univariate ordinary kriging gave an RMSE of 7.699, so co-kriging lowered it by about 16.6%.

The study is aimed at provincial-level decision settings: it uses provincial aggregate data to describe how stunting prevalence is arranged across the country, helping policymakers decide where to concentrate programmes already known to work (such as strengthened first-1,000-days nutrition, higher exclusive breastfeeding coverage, low birth weight prevention through improved maternal nutrition, expanded maternal and child healthcare, and reinforced social protection), not what works. Its intended audience is Indonesian provincial health and nutrition planning, and similar ecological studies needing multivariate geostatistical prediction with quantified uncertainty; because the determinants cluster geographically rather than varying independently, the authors note that multisectoral policies integrating health, education, sanitation, and poverty alleviation are appropriate in these provinces.

The authors list several points requiring caution: the analysis rests on aggregate data for only 38 provinces, a small sample for geostatistical estimation, so national-scale results should be regarded as exploratory; all associations are ecological, and the cross-semivariograms and coregionalization matrices describe co-spatial dependence between mapped aggregates and cannot be read as individual-level relationships or causal effects; each province is represented by its centroid, so the prediction surface is a centroid-based approximation rather than district-level or local prevalence; the 0% values reported for Highland Papua and Central Papua most likely reflect incomplete reporting, and retaining them may bias predicted prevalence for eastern Indonesia downward; the model was assessed only by internal leave-one-out cross-validation, without external or temporal validation; no formal significance-based cluster detection was performed, so high-prevalence areas are not statistically identified hotspots; and the moderate spatial dependence implies that about 31% of the variation is not captured by the spatial structure, pointing to unmodeled local or non-spatial determinants that future work could address through finer district-level data or hybrid regression-kriging approaches. In addition, antenatal care coverage exceeds 100% in several provinces because its denominator is a projected number of pregnancies rather than an actual count, so the authors advise interpreting its relationship with stunting with caution.