Local Polynomial Math
For each grid node, the neighboring data are identified by the user-specified sector search. Using only these identified data, a local polynomial is fit using weighted least squares, and the grid node value is set equal to this value. Local polynomials can be order 1, 2, or 3.
The form of these polynomials are:
|
Order 1 |
|
|
Order 2 |
|
|
Order 3 |
|
The weighted least squares function weights data closer to the grid node higher and data further away lower. The weighting function depends on the search ellipse, the power, and the specific data geometry. The actual calculations for the weights are somewhat involved. Define TXX,
|
|
where
|
|
is the angle of the search ellipse |
|
|
is search radius 1 |
|
|
is search radius 2 |
Define AXX,
|
|
Note that these values (AXX,
Next, consider a datum at location (Xi,
|
|
then
|
|
and finally,
|
|
where Wiis the weight for datum i andpis the specified power.
Let the collection of neighboring data be enumerated as
|
|
The local least squares parameters are computed by minimizing the weighted sum of the squared residuals:
|
|
