Maximum likelihood and other parameter estimations

First, a basic definition: A parameter is an unknown, fixed value that describes a characteristic. For example, a mean is a characteristic that describes the average over a population. The true mean is usually not known, but rather estimated.

When we fit a model to our data, we get parameters such as the regression coefficients (β's). In spatial stats, we use a (semi)variogram function to estimate the parameters range, sill, and nugget effect. An introduction to the semivariogram and its parameters may be found here.

A few common methods of parameter estimation used in spatial stats are the least squares (OLS or more commonly WLS) and the likelihood based methods (maximum likelihood MLE or restricted maximum likelihood REML).

Least squares methods fit a model by minimizing the distance between the observed data and the best fit line. Likelihood based methods use the observed data to estimate the population parameters using established distributions. When you code a likelihood estimation, you will input parameters and an underlying distribution. For example, for a spatial stats dataset, you would first investigate the semivariogram to estimate the nugget effect parameter and the distribution (i.e. exponential or linear), and then model the data via MLE.

In R, the likfit command in the package geoR models likelihood based methods. In the same package, variofit models least squares.