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I'm interested in interaction effects in Random Forest. I try to implement Friedman H'statistic basically developped for Gradient boosting Friendman, 2001. But I have no idea how calculate (44) on the real data.

Part 8.1

If anybody has some examples or explanations?

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In your article $\hat F$ is defined by Eq. (39). Although it's a bit abstract, it says that the partial dependency function $\hat F_j$ is obtained by averaging over all variables but a subset $s$.

As an example, the one-variable dependency function for the variable $1$ is given by $$\hat F_1(x_1) = \frac{1}{n}\sum_{i=1}^n \hat f \left (x_{1}, x_{i2}, \dots, x_{im} \right ) ,$$ where $\hat f$ is your model prediction function (Random Forest in your case), $n$ is the number of samples, $m$ is the number of features (I've included $\hat f$ instead of $\hat F$ for the predictor to avoid confusion). Here $x_1$ is a variable, and the $x_{i2}, \dots, x_{im}$ values are taken from your samples.

For example, if you have two variables and two samples $(x_1=0,x_2=0)$ and $(x_1=1, x_2=1$) in your dataset (this is a very dummy, pedagogical example), the partial dependence function for the first variable reads: $$ \hat F_1 (x_1) = \frac{1}{2}\left [ \hat f(x_1, x_2=0) + \hat f(x_1, x_2=1) \right ]. $$

Similarly, you can compute the two-variables partial dependence function as $$\hat F_{12}(x_1, x_2) = \frac{1}{n}\sum_{i=1}^n \hat f \left (x_{1}, x_{2}, \dots, x_{im} \right )$$ (same as before, but note that the $i$ index is not under $x_2$ anymore).

The two equation I've just made explicit are the only two terms that appear in the equation (44), thus you should be able to compute the $H$-statistic with them.

Let me know if it is still unclear, and check out this for more exegesis on the $H$-statistic.

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