Funktionsformer

Funktionsformer refers to the assumed mathematical form that describes how a dependent variable depends on one or more explanatory variables in a statistical or mathematical model. It specifies the functional relationship and how parameters translate inputs into outputs. Choosing a functional form is a modeling decision that affects interpretability, extrapolation, and the risk of misfit. Parametric models fix a particular family of functions with a finite set of parameters, enabling estimation from data, while nonparametric approaches relax this assumption.

Common functional forms include linear (y = α + βx), polynomial (y = α + β1x + β2x^2 + ...), logarithmic (y = α + β log x), exponential (y = α e^{βx}), and power forms (y = α x^β). In multivariate settings one writes y = f(x1, x2, ..., xk; θ) with various interactions. Other widely used shapes are logistic and Gompertz functions for limited dependent variables or growth processes, and transformations such as Box-Cox to stabilize variance or linearize relationships.

Model selection involves theory, interpretability, and empirical fit. Criteria such as AIC, BIC, cross-validation, and residual diagnostics guide the choice. Estimation often uses linear or nonlinear regression, generalized linear models, or maximum likelihood depending on the form. Transformations like logarithms, reciprocal transformations, or Box-Cox can convert nonlinear relationships into easier-to-handle forms and improve stability.

Limitations include misspecification, restricted extrapolation, and identifiability issues when parameters are not separately identifiable. A good functional form balances theoretical justification, data support, and simplicity.

See also: parametric versus nonparametric modeling; regression; generalized linear models; nonlinear optimization.