I am struggling with the understanding of a saturated model.
As far as I know, the saturated model is the model that have as many parameter as the data points. But I don't know how to build it or what is the exact form of the saturated model.
For illustration, I have the example as follow:
$(Y, X_1, X_2, X_3, X_4)$ = $(1, 2, 3, 4,5); (2, 3,4, 5,6); (3,4,5, 6,7), (3, 5, 6,7,8)$
where $Y$ is a Poisson distribution, $X_1, X_2, X_3, X_4$ are the independent variable. The link function is log.
I think the saturated model may have the form:
$log(Y) = coef_1 * X_1 + coef_2 * X_2 + coef_3 * X_3 + coef_4 * ???$
(4 coefficients/ parameters as we have 4 data points)
And to fill in $???$, I think we have many way to choose : $X_1 * X_2, X_1/X_3$ or even $X_1* X_2 * X_3 / X_4$...
So I would like to have 2 question please:
How many saturated model are their given a dataset for a GLM model ? (i.e. we fix the hypothesis that Y follow some distributions and also fix the dataset) and what is its form ? Is there any general principle to construct the saturated model ?
If there are more than 1 saturated models, what is the "real saturated" model ? (because as I know, the saturated model is defined to be the model that fit perfectly the dataset)
Thank you very much for your help!