MehrkomponentenFits

MehrkomponentenFits, or multi-component fitting, is a statistical approach to modeling data as a sum of multiple component functions. It is used when an observed signal or measurement arises from several overlapping sources, each with its own characteristic shape, scale, and location. The goal is to estimate the parameters of each component so that their sum best describes the data.

Common component choices include Gaussians, Lorentzians, Voigt profiles, exponentials, or physically motivated templates. The overall model is the sum of these components, each with parameters such as amplitude, position, and width, subject to optional constraints (non-negativity, orderings).

Estimation methods include nonlinear least squares to minimize residuals, maximum likelihood estimation under specified noise models, and Bayesian approaches for posterior distributions and priors. For mixture-like problems, expectation-maximization is a common algorithm because it handles latent component membership, while global optimization may be necessary due to non-convex objective functions.

Evaluation and challenges involve identifiability and potential label switching when components resemble each other. Overfitting is a risk with too many components; information criteria such as AIC or BIC, along with cross-validation, help determine an appropriate model size. Regularization or informative priors can stabilize estimates, and uncertainty quantification yields confidence or credible intervals for component parameters.

Applications and software: MehrkomponentenFits is used in spectroscopy, chromatography, imaging, and signal processing to separate overlapping signals and interpret underlying processes. Implementations are found within general scientific computing libraries (for nonlinear solvers and EM algorithms) and specialized packages such as lmfit, Stan, or PyMC, depending on the modeling approach.