I've read a paper about Huff model, and I have a question for linearization technique of Multiplicative model.
How does following linearization work?
$U_{ij} = X_{1j}^\alpha X_{2j}^\beta X_{1j}^{\gamma X_{2j}}\ $ (dependency of $X_1$ and $X_2$)
$P_{ij} = U_{ij} / \sum_j U_{ij} $
$logcenter Z = lc(Z) = \log(Z / \tilde Z)$ ($\tilde Z$ is geometric mean of Z)
$lc(P_{ij}) = \alpha\ lc(X_{1j}) + \beta\ lc(X_{2j}) + \gamma\ lc(X_{1j}) lc(X_{2j})$
I already understood the independent term, but I cannot understand the linearization of interaction term($X_{1j}^{\gamma X_{2j}}$)
- My reference
[1]Braeye, T., Quoilin, S., & Hens, N. (2019). Incidence estimation from sentinel surveillance data; a simulation study and application to data from the Belgian laboratory sentinel surveillance. BMC Public Health, 19(1), 1-18.