Define a polynomial function.
Polynomial functions are one of many functions used to model global trends across one or n-dimensional spaces. In spatial analysis, polynomial functions are often used in modeling trends in a variable across a two-dimensional space. This may serve many purposes including (1) describing overall changes in a variable as a function of location, (2) removing global trend from the data for the purpose of exposing the residuals (a crucial step when exploring spatial autocorrelation in the data), (3) and interpolation. Polynomial functions are intended to capture the first order effect of an underlying process whereby the variation in a variable is defined by absolute location and not by its value in neighboring locations. In spatial analysis, these mathematical functions typically consist of a combination of non-negative integer powers of both the x and y coordinate values. These functions are parametric in that the equation’s form is first predefined and its coefficients are then derived by fitting the model to the data. In practice one should seek the lowest polynomial order when modeling a spatial trend and avoid capturing localized variation of the data with the model.