Based on the article and the Search Agent Sky review of it, here's the answer to your question: **Yes, the `gp_model` (GPModel) uses the residuals of the tree model's prediction as its observations — but only indirectly, through the iterative training algorithm.** Here's the precise mechanism: ### The Model Structure The GPBoost model is: ``` y = f(X) + Zu + ε ``` where: - `f(X)` = tree-boosting ensemble (fixed effects) - `Zu` = random effects (spatial GP + grouped) - `ε` = i.i.d. noise ### The Two-Step Iterative Training Algorithm The article describes the training as alternating between two steps: 1. **Boosting step**: Given current estimates of the random effects **u**, the tree ensemble `f(X)` is grown via gradient boosting to fit the **response adjusted for random effects** — i.e., `y - Zu`. So the trees are trained on the partial residuals after removing the GP's contribution. 2. **GP step (Random effects step)**: Given the current tree ensemble `f(X)`, the GP covariance parameters (σ², ρ, τ²) and grouped random effect variance are re-estimated via maximum likelihood, and the random effects are predicted via **Best Linear Unbiased Prediction (BLUP)**. ### What the GP Model "Sees" as Observations In the GP step, the BLUP formula for predicting the random effects is: ``` û = Zᵀ Σ⁻¹ (y - f(X)) ``` So the **observations that the GP model uses** to estimate the random effects are indeed the **residuals of the tree model's prediction**: `y - f(X)`. These are the "partial residuals" — what's left after accounting for the tree ensemble's predictions. However, there's an important nuance: the **covariance function** itself (the exponential kernel on the spatial coordinates) does **not** take these residuals as input. The covariance function operates on the **spatial coordinates** (the 242 region centroids: Longitude, Latitude) to compute the covariance matrix Σ. The residuals `y - f(X)` are then used as the response vector in the BLUP equation to predict the random effects. ### In Summary | What | What it uses as input | |---|---| | **Covariance function** (kernel) | Spatial coordinates (Long, Lat) — computes distances between region centroids | | **GP parameter estimation** (σ², ρ, τ²) | The full likelihood using all 484 data points, given current `f(X)` | | **BLUP prediction of random effects** | The residuals `y - f(X)` (tree model's prediction residuals) | So to directly answer your question: **Yes, the gp_model uses the residuals of the tree model's prediction as the observations for predicting the random effects via BLUP.** The tree ensemble `f(X)` is held fixed during the GP step, and the GP model estimates the random effects from what's left over.