In inlabru-org/inlabru: Bayesian Latent Gaussian Modelling using INLA and Extensions

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Hierarchical basis

Let $d(a, b)$ denote the distance between two nodes, $a$, and $b$. Let $V_K={v_k;k=1,\dots,K}$ be the set of points included at step $k$ of the algorithm

1. Let $v_1$ be the initial node and let $V_1={v_1}$
2. Let $D=\max_v d(v_1, v)$
3. Let $\wt{b}^D_1\equiv b^D_1\equiv 1$
4. Let $B_1 = \wt{b}^D_1$
5. For $K=1,\dots,N_V-1$:
   1. Let $D=\max_v \min_{1\leq k \leq K} d(v_k, v)$
   2. Find $v_{K+1} = \argmax_v \min_{1\leq k \leq K} d(v_k, v)$
   3. Let $b^{D}_k(v) = \max(0, D - d(v_k, v))$ for all $1\leq k \leq K$.
   4. Normalise $\wt{b}^D_k(v) = b^D_k(v) / \sum_{j=1}^{K+1} b^D_j(v)$
   5. Let $V_{K+1}=V_K\cup{v_{K+1}}$ and $B_{K+1} = \wt{b}^D_{K+1}$

Equivalent variational adjustment

$$ \begin{aligned} u &\sim N(\mu_u,Q_u^{-1}) \ u &= Bv \ v &\sim N(B^{-1}\mu_u,(B^\top Q_u B)^{-1}) \ \eta &= A u = ABv \ Q_{u|y} &= Q_u + A^\top Q_{y|u} A \ Q_{u|y} \mu_{u|y} &= Q_u\mu_u + A^\top Q_{y|u} y = b_u \ Q_{v|y} &= B^\top Q_u B + B^\top A^\top Q_{y|u} A B \ Q_{v|y} \mu_{v|y} &= Q_v\mu_v + B^\top A^\top Q_{y|u} y = b_v = B^\top b_u \end{aligned} $$ Variational adjustment equations: $$ \begin{aligned} Q_{v|y} \mu_{v|y} &= b_v + \delta \ B^\top Q_{u|y} B B^{-1} \mu_{u|y} &= B^\top b_u + \delta \ Q_{u|y} \mu_{u|y} &= b_u + B^{-\top} \delta \end{aligned} $$

inlabru-org/inlabru documentation built on May 5, 2024, 4:31 p.m.