13 NULL model

We are going to interpolate (estimate) for unsampled locations the value of a variable. The simplest way would be to take the mean of all observations. We can consider this as a “Null-model” and we can compare other approaches to it.

In the following formula all the \(N\) available observations are used:

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

and the weights must sum up to one:

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

The mean of all observations \(z({\mathbf u})\) (global mean) can be calculated using the following weights:

\(\lambda_{\alpha}({\mathbf u})=\frac{1}{N}\,,\;\; \forall\, \alpha=1,...,N\)