scattereddata

Scattered data refers to samples that are irregularly distributed in a domain, as opposed to data on a regular grid. Such data are common in environmental measurements, survey data, computer graphics, and many scientific experiments. The irregular spacing can be in two or three dimensions and may vary in density across the domain.

The central problem is to estimate a function f on a domain from pairs (x_i, y_i) where x_i are sample locations and y_i are observed values. Tasks include interpolation (predicting values at new locations), smoothing or fitting (reducing noise), and surface reconstruction. Challenges include nonuniform spacing, variable density, and noise.

Common approaches include radial basis function interpolation (sum w_j phi(||x - x_j||)), Kriging (a geostatistical method that uses a covariance model), thin-plate splines, and Shepard-type weighted averages. Delaunay triangulation and Voronoi-based natural neighbor interpolation are used for piecewise surface construction. Partition-of-unity methods and local kriging or local RBFs address scalability.

Computational considerations include efficient nearest-neighbor searches, partitioning data, and sparse representations. Model selection and validation often rely on cross-validation, leave-one-out error, or information criteria. Robust variants handle outliers.

Applications include geostatistics, meteorology, environmental monitoring, computer graphics, 3D surface reconstruction, and scan data processing.