Log-linear model

A log-linear model is a mathematical model that takes the form of a function whose logarithm equals a linear combination of the parameters of the model, which makes it possible to apply (possibly multivariate) linear regression. That is, it has the general form
 * $$\exp \left(c + \sum_{i} w_i f_i(X) \right)$$,

in which the $f_{i}(X)$ are quantities that are functions of the variable $X$, in general a vector of values, while $c$ and the $w_{i}$ stand for the model parameters.

The term may specifically be used for:
 * A log-linear plot or graph, which is a type of semi-log plot.
 * Poisson regression for contingency tables, a type of generalized linear model.

The specific applications of log-linear models are where the output quantity lies in the range 0 to ∞, for values of the independent variables $X$, or more immediately, the transformed quantities $f_{i}(X)$  in the range −∞ to +∞. This may be contrasted to logistic models, similar to the logistic function, for which the output quantity lies in the range 0 to 1. Thus the contexts where these models are useful or realistic often depends on the range of the values being modelled.