linkfüggvények

Linkfüggvények, often translated as link functions or linkage functions, are a fundamental concept in generalized linear models (GLMs). They are a crucial component that connects the linear predictor of a model to the expected value of the response variable. In simpler terms, a link function transforms the expected value of the response variable, which is typically constrained to a certain range (e.g., probabilities between 0 and 1), into a value that can be modeled linearly.

The general form of a GLM is E(Y) = g⁻¹(Xβ), where E(Y) is the expected value of the response variable Y, X is the matrix of predictor variables, β is the vector of coefficients, and g is the link function. The term Xβ is the linear predictor. The link function g(μ) = Xβ, where μ = E(Y), establishes a relationship between the expected value of the response and the linear combination of predictors. The inverse of the link function, g⁻¹, is used to recover the expected value of the response from the linear predictor.

Different link functions are appropriate for different types of response variables and distributions. For instance, the logit link is commonly used for binary response variables (logistic regression), where the expected value is a probability. The log link is often used for count data (Poisson regression), as it ensures that the predicted values are positive. Other common link functions include the probit link and the identity link (used in standard linear regression where g(μ) = μ). The choice of link function is guided by the assumed distribution of the response variable and the desired properties of the model.