Can someone tell me what would be the expression for calculating the partial dependence on a GLM model with family specified as Poisson?
From applying Friedman partial dependence estimation expression (page 27):
$$\bar{F}_l(\mathbf{z}_l) = \frac{1}{N}\sum_{i=1}^N\hat{F}(\mathbf{z}_l, \mathbf{z}_{i,l})$$
(where $\mathbf{z}_l$ represents the subset of predictors we are interested in calculating the partials) the following expression can be derived for a single predictor $k$ on a linear model $\hat{Y}_i = \beta_0 + \sum_{j=1}^p\beta_jx_{ji}$:
$$F_k(x) = \beta_0 + \beta_kx+\sum_{j=1,\,j\ne k}^p\beta_j<x_j>$$
where $<x_j>$ represents the $j$ - predictor mean $\frac{1}{N}\sum_i^N x_{ij}$. Is it possible to obtain some analogous expression for the "loglin" case, i.e.: $\hat{Y}_i = e^{\beta_0}e^{\sum_{j=1}^p\beta_jx_{ji}}$?
Additional related questions:
I've only seen examples of the partial dependence concept applied within the context of either random forests or gradient boosting techniques. On the other hand, for GLM models, people usually talk about marginal effects, which are defined using partial derivatives of the response with respect to the predictors:
- Does it make sense trying to also calculate partial dependence for GLM models?
- How are partial dependence and marginal effects related in general?
Thanks!!
