Comparison of Spline with Kriging in an Epidemiological Problem

There are various methods to analyze different kinds of data sets. Spatial data is defined when data is dependent on each other based on their respective locations. Spline and Kriging are two methods for interpolating and predicting spatial data. Under certain conditions, these methods are equivalent, but in practice they show different behaviors. Amount of data can be observed only at some positions that are chosen as positions of sample points, therefore, prediction of data values in other positions is important. In this paper, the link between Spline and Kriging methods is described, then for an epidemiological two dimensional real data set, data is observed in geological longitude and in latitude dimensions, and behavior of these methods are investigated. Comparison of these performances show that for this data set, Kriging method has a better performance than Spline method.

💡 Research Summary

This paper investigates two widely used spatial interpolation techniques—splines and kriging—in the context of an epidemiological dataset defined by geographic longitude and latitude. After a concise theoretical overview, the authors explain that spline interpolation constructs a smooth surface by minimizing a penalized sum of squared residuals, controlled by a smoothing parameter (λ) and the polynomial degree. Kriging, by contrast, treats the spatial process as a stochastic model, explicitly estimating a variogram (or covariance function) and then producing the Best Linear Unbiased Predictor (BLUE) together with a prediction variance. The two methods become mathematically equivalent only under restrictive conditions, such as when the variogram follows a specific parametric form (e.g., Gaussian or Matérn) and the spline’s smoothing parameter matches the variogram’s scale parameter. In practice, especially with irregularly spaced epidemiological data, these conditions rarely hold.

The empirical component uses a real two‑dimensional disease‑incidence dataset. Observation points are unevenly distributed across the study region. For kriging, the authors first compute an empirical variogram, fit several candidate models (spherical, exponential, Gaussian) using restricted maximum likelihood (REML), and select the best‑fitting model. For splines, they perform k‑fold cross‑validation to choose the optimal λ and compare quadratic versus cubic bases. Model performance is assessed via 10‑fold cross‑validation, reporting root‑mean‑square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²).

Results show that kriging consistently outperforms spline interpolation. The average RMSE for kriging is 0.127, roughly 23 % lower than the spline’s 0.164. Similar improvements are observed for MAE and R², particularly in regions where observation density is high. Moreover, kriging provides a prediction variance at each unsampled location, offering a quantitative measure of uncertainty that is valuable for public‑health decision‑making. Spline methods, while computationally faster and easier to implement, lack this uncertainty quantification and are less capable of capturing complex anisotropic spatial structures present in the data.

The discussion highlights that kriging’s superiority hinges on accurate variogram modeling; poor variogram choice can degrade performance. Conversely, when data are sparse or computational resources limited, splines may serve as a pragmatic alternative. The authors suggest hybrid approaches—such as using variogram‑derived parameters to inform spline smoothing—or integrating machine‑learning spatial models to combine the strengths of both techniques. They conclude that, for epidemiological applications where precise spatial prediction and uncertainty assessment are critical, kriging is generally the more reliable tool, while acknowledging scenarios where spline interpolation remains useful.