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I have the following model $y \sim b_0 + Z \theta_0 + X \beta + \sum((X_j \circ Z) \theta_j)$

Where...

  • $Z$ is (n, k)
  • $X$ is (n, p)
  • $\beta$ is (p,)
  • $\theta$ is (p, k)
  • $\theta_0$ is (k,)
  • $\circ$ is the element wise multiplication
  • n is the number of samples
  • p is the number of predictors
  • k is the number of other predictors
  • $j$ is the jth predictor of predictors p

y is centered by subtracting the mean of y and $X$ and $Z$ are scaled so they have 0 mean and 1 std. My question is how do I transform $\beta$ and $\theta$ so that their values are the values they would have had if X, Z and y were not modified.

PS: I can guess for simpler models like $y = Xm + b$ the variables $m$ and $b$ can be modified by $m_{scaled} = m / \sigma_X$ and $b = \mu_y - (b + m \cdot X_{offset})$ . But I'm not sure if that universally correct.

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