Kernelvektning

Kernelvektning, or kernel weighting, is a method in statistics and data analysis for assigning weights to observations based on their distance from a target point, using a kernel function. The weighted data are then used to estimate local properties such as a mean, regression function, or probability density.

In practice, weights are computed as w_i = K((x - x_i)/h), possibly normalized so that the weights sum to one. The parameter h, called the bandwidth, controls the width of the neighborhood: larger h yields smoother estimates with more bias, while smaller h gives less bias but higher variance. The kernel function K is non-negative and typically scaled so that it integrates to one.

Common kernel functions include the Gaussian, Epanechnikov, uniform (rectangular), triangular, and biweight kernels. While the exact shape of the kernel can affect results, the bandwidth often has a larger influence on the estimate than the particular kernel choice. Many kernels give broadly similar performance in typical settings.

Applications of kernel weighting include kernel density estimation, where weights estimate a density function at a point; kernel regression (such as the Nadaraya-Watson estimator) for estimating a conditional mean; and local polynomial regression. Kernel weighting is also used to weight observations in time series or spatial analyses and to handle boundary effects near the edge of the data domain.

Bandwidth selection is a central practical task, addressed by rules of thumb, cross-validation, plug-in methods, or adaptive bandwidth approaches. For large data sets, computational considerations arise, but several efficient algorithms and data structures exist to speed up kernel-weighted calculations.