Coupling function

In the statistics  and there in particular in generalized linear models, one is coupling function , even Link function , logic function or connection function called a function that by the linear predictor described systematic component and by the expectation value of the response variables described stochastic component of the distribution of the Art couples that: . There are many commonly used coupling functions and their selection depends on several considerations. Every exponential family has a unique canonical (natural) coupling function, which is given by . The coupling function is often non-linear. ${\displaystyle g(\cdot )}$ ${\displaystyle \eta _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}}$ ${\displaystyle \mu =\operatorname {E} (Y_{i})}$ ${\displaystyle Y_{i}}$ ${\displaystyle g(\mu )=\eta _{i}}$ ${\displaystyle \eta _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}}$

definition

This function couples the stochastic with the systematic component by transforming the expected value . The function is called the coupling function. It is assumed to be monotonous and differentiable. It applies ${\displaystyle \mu =\operatorname {E} (Y_{i})}$ ${\displaystyle g(\cdot ):\mathbb {R} \rightarrow \mathbb {R} }$

${\displaystyle g(\mu )=\eta _{i}=\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}\quad i=1,\ldots ,n}$.

From this representation it can be seen that the expected value of the -th observation depends on fixed but unknown regression parameters. A coupling function is called canonical if the linear predictor for all of them coincides with the distribution parameter . In other words, in the canonical coupling function, the coupling function is defined via by expressing the natural parameter in terms of . ${\displaystyle \mu _{i}}$ ${\displaystyle i}$ ${\displaystyle \beta _{0},\beta _{1},\beta _{2},\ldots ,\beta _{k}}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle \eta _{i}=\theta _{i}}$ ${\displaystyle g(\mu )=\theta }$ ${\displaystyle \theta }$ ${\displaystyle \mu }$

example

If you select for the coupling function of the natural logarithm , then always result positive expectation values: . ${\displaystyle g=\ln }$ ${\displaystyle \mu =\exp \left(\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}\right)}$

Examples of different coupling functions

If the logit transformation for the expected value of the response variable is selected as the coupling function , the logistic regression model is obtained ${\displaystyle \log \left(\operatorname {odds} (\cdot )\right)}$ ${\displaystyle \mu =\operatorname {E} (Y_{i})}$

${\displaystyle \log \left({\frac {\mu }{1-\mu }}\right)=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}}$.

If the inverse function of the distribution function of the normal distribution is chosen as the coupling function, the probit model is obtained ${\displaystyle \Phi ^{-1}(\cdot )}$

${\displaystyle \Phi ^{-1}(\mu )=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}}$.

Canonical (natural) coupling function

The canonical coupling function plays a special role among the coupling functions . It transforms the expected value of to the real-valued (unknown) distribution parameter of density, the so-called canonical (natural) parameter. Every exponential family has a unique canonical (natural) coupling function. The canonical coupling function can in principle be freely selected, except for the requirement that it should be invertible. It is defined by:, where represents a (known) twofold differentiable function (see exponential family ). From the fact that and holds, it follows that . Thus, when using the canonical coupling function, the linear predictor and the distribution parameter coincide. In general, the estimators are greatly simplified when using the canonical coupling function. An important property of the canonical coupling function defined by is that it can be scaled with a factor without losing the property of coinciding with the linear predictor: ${\displaystyle Y_{i}}$ ${\displaystyle \theta _{i}}$ ${\displaystyle g(\mu )\colon ={b^{\prime }}^{-1}(\mu )}$ ${\displaystyle {b^{\prime }}^{-1}(\cdot )}$ ${\displaystyle \operatorname {E} (Y_{i})=g^{-1}(\eta _{i})}$ ${\displaystyle \operatorname {E} (Y_{i})=b^{\prime }(\theta _{i})}$ ${\displaystyle g(\mu )\colon ={b^{\prime }}^{-1}(b^{\prime }(\theta _{i}))=\theta _{i}=\eta _{i}}$ ${\displaystyle \eta _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}}$ ${\displaystyle \theta _{i}}$ ${\displaystyle g={b^{\prime }}^{-1}}$ ${\displaystyle c\in \mathbb {R} }$

${\displaystyle c\cdot g(\mu )={\tilde {\eta }}_{i}=\mathbf {x} _{i}^{\top }{\tilde {\boldsymbol {\beta }}}\Rightarrow \theta _{i}=\mathbf {x} _{i}^{\top }{\tilde {\boldsymbol {\beta }}}}$,

where represents an unknown scaled parameter vector and the row of the experiment plan matrix belonging to the i-th observation . ${\displaystyle {\tilde {\boldsymbol {\beta }}}}$ ${\displaystyle {\tilde {\boldsymbol {\beta }}}=({\tilde {\beta }}_{0},{\tilde {\beta }}_{1},\ldots ,{\tilde {\beta }}_{k})^{\top }}$ ${\displaystyle \mathbf {x} _{i}^{\top }}$

Connection to the classic linear model

If the identity function is chosen as the coupling function , the equation of the classical linear model is obtained . ${\displaystyle g(\mu )=\mu }$ ${\displaystyle \mu =\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}}$

Response function

Especially in generalized linear models, the inverse of the coupling function becomes

${\displaystyle \mu =h(\eta )}$ With ${\displaystyle h=g^{-1}}$

Response function , or response function ( English response function called). The response function converts the linear combination of the explanatory variables into the (conditional) expected value . ${\displaystyle \mu =\operatorname {E} (Y_{i})}$

Suitable response functions are all distribution functions of continuous random variables, e.g. B. that of the standard normal distribution or that of the logistic distribution.

application

A coupling function transforms the probabilities of the levels of a categorical response function into an unlimited continuous scale. Once the transformation is complete, the relationship between the predictors and the response function can be modeled using nonlinear regression. For example, a binary response function can have two unique values. If these values ​​are converted into probabilities, the response variable ranges from 0 to 1. The log coupling function turns a linear relationship into an exponential one and the logit coupling function turns it into a sigmoid one.

Individual evidence

1. ↑ link function. Glossary of statistical terms. In: International Statistical Institute . June 1, 2011, accessed July 4, 2020 . 
2. ↑ Ludwig Fahrmeir , Thomas Kneib , Stefan Lang, Brian Marx: Regression: Models, Methods and Applications , Springer Verlag 2007., p. 109.
3. ^ Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx: Regression: models, methods and applications. Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2 , p. 304.
4. ^ Rencher, Alvin C., and G. Bruce Schaalje: Linear models in statistics. , John Wiley & Sons, 2008., p. 514.
5. ^ Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx: Regression: models, methods and applications. Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2 , p. 304.
6. ↑ Torsten Becker, et al .: Stochastic risk modeling and statistical methods. Springer Spectrum, 2016. p. 308.
7. ↑ Torsten Becker, et al .: Stochastic risk modeling and statistical methods. Springer Spectrum, 2016. p. 308.
8. ^ Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx: Regression: models, methods and applications. Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2 , p. 301.