Construction of the Hessian

Figure 15.2: The model-free Hessian kite - a demonstration of the construction of the model-free Hessian ∇²χ² for the global model $\mathfrak{S}$. For each residue i a different matrix ∇²χ²_i is constructed. The first element of the matrix represented by the two symbols ∂$\mathfrak{D}$ (the red block) is the sub-matrix of chi-squared second partial derivatives with respect to the diffusion tensor parameters $\mathfrak{D}_j$ and $\mathfrak{D}_k$. The orange blocks are the sub-matrices of chi-squared second partial derivatives with respect to the diffusion parameter $\mathfrak{D}_j$ and the model-free parameter $\mathfrak{F}_i^k$. The yellow blocks are the sub-matrices of chi-squared second partial derivatives with respect to the model-free parameters $\mathfrak{F}_i^j$ and $\mathfrak{F}_i^k$. For the residue dependent matrix ∇²χ²_i the second partial derivatives with respect to the model-free parameters $\mathfrak{F}_l^j$ and $\mathfrak{F}_l^k$ where i≠l are zero. In addition, the second partial derivatives with respect to the model-free parameters $\mathfrak{F}_i^j$ and $\mathfrak{F}_l^k$ where i≠l are also zero. These blocks of sub-matrices are left uncoloured. The complete Hessian of $\mathfrak{S}$ is the sum of the matrices ∇²χ²_i.

The construction of the Hessian for the models $\mathfrak{F}_i$, $\mathfrak{T}_i$, $\mathfrak{D}$, and $\mathfrak{S}$ is very similar to the procedure used for the gradient. The chi-squared Hessian for the global models $\mathfrak{D}$ and $\mathfrak{S}$ is

$$\sum_{{i=1}}^{l}$$ (15.12)

Figure 15.2 demonstrates the construction of the full Hessian for the model $\mathfrak{S}$. The Hessian for the model $\mathfrak{D}$ is the sum of all the red blocks. The Hessian for the model $\mathfrak{T}_i$ is the combination of the single red block for residue i, the two orange blocks representing the sub-matrices of chi-squared second partial derivatives with respect to the diffusion parameter $\mathfrak{D}_j$ and the model-free parameter $\mathfrak{F}_i^k$, and the single yellow block for that residue. The Hessian for the model-free model $\mathfrak{F}_i$ is simply the sub-matrix for the residue i coloured yellow.