Localityweighted

Localityweighted refers to a class of weighting schemes in statistics, data analysis, and geospatial science where observations contribute to a statistic or model output in proportion to their locality or proximity to a target point. In localityweighted methods, observations that lie closer to the point of interest receive greater influence than distant observations, allowing models to capture local variation and heterogeneity.

Common forms of locality weighting are distance-based. Inverse distance weighting (IDW) assigns weights w = 1 / d^p, where d is the distance to the target and p controls the rate of decay. Kernel-based weights use functions such as Gaussian, Epanechnikov, or tri-cubic kernels to assign weights that decline with distance and are often governed by a bandwidth parameter that defines the locality. Local regression techniques, such as LOESS or LOWESS, apply locality weighting to fit simple models (often linear) within a neighborhood for each target point, producing a smooth, flexible fit.

Locality weighting also plays a key role in kernel density estimation and kernel regression, where a kernel function determines the influence of nearby observations within a chosen bandwidth. In machine learning, locality-aware or locally weighted models rely on nearby data to make predictions, enabling adaptation to spatial or feature-space heterogeneity.

Applications of localityweighted methods include spatial interpolation and prediction (estimating values at unsampled locations using nearby data), environmental and meteorological modelling, image smoothing and denoising, and certain instance-based learning approaches. Important considerations include selecting an appropriate distance metric and bandwidth, handling edge effects, avoiding bias from outliers, and balancing computational cost with accuracy. Cross-validation is commonly used to choose tuning parameters that control locality.