gnls

Generalized nonlinear least squares (GNLS) is a statistical modeling framework used to fit nonlinear relationships when the observed errors exhibit non-constant variance or correlation. It extends traditional nonlinear least squares by allowing a specified variance–covariance structure for the residuals, rather than assuming independent, homoscedastic errors. This makes GNLS well suited for data where variance changes with the level of the fitted values or with other covariates, and for data where observations are correlated, such as repeated measures or time series.

In a GNLS model, the mean response is modeled by a nonlinear function of parameters, often written as f(x, beta), where beta denotes the parameter vector. The residuals are assumed to have a variance-covariance matrix that is specified through a variance function and a correlation structure. The estimation proceeds by generalized least squares, typically via iterative reweighting or maximum likelihood approaches, updating both the mean parameters and, when desired, the variance and correlation components. Software implementations usually provide options to select different variance functions (to capture heteroscedasticity) and correlation structures (to capture autocorrelation), along with choices for estimation methods such as ML or REML.

GNLS is commonly used in fields with complex error patterns, including ecology, pharmacokinetics, and environmental science, as well as in longitudinal or repeated-measures studies where simple nonlinear regression would be inappropriate due to heteroscedasticity or correlation among observations. It is implemented in several statistical environments, with R’s nlme package providing the gnls function as a prominent example. See also nonlinear least squares, generalized least squares, and variance function modeling.