15 Nearest Neighbours

The nearest neighbours interpolation is performed considering multiple closest samples locations (neighbours) in the neighbourhood of the unknown location. The operation is repeated for all unknown locations in the 2D spatial domain, i.e. \(\forall\, ({\mathbf u}) \in\, {\mathbf D}^2(x,y)\).

NOTE: Please, check that the value interpolated at \(({\mathbf u})\) (red point) is the nearest neighbours interpolation of the 6 closest observations (green points).

Given a generic unknown location \(({\mathbf u})\) as in the figure above, the formula is the following:

\(\hat{z}({\mathbf u}) = \displaystyle\sum_{\alpha=1}^{n({\mathbf u})} \lambda_{\alpha}({\mathbf u}) z({\mathbf u}_{\alpha})\)

Weights must sum up to one:

\(\sum_{\alpha=1}^{n({\mathbf u})} \lambda_{\alpha}({\mathbf u}) = 1\)

The nearest neighbors interpolation is conducted by assigning a weight, \(\lambda_{\alpha}({\mathbf u})=\frac{1}{n({\mathbf u})}\,,\;\; \forall\, \alpha=1,...,n({\mathbf u})\), to the \(n({\mathbf u})\) observation locations that are closest to the unknown location \(({\mathbf u})\).