Construction of the gradient

Figure 15.1: The construction of the model-free gradient ∇χ² for the global model $\mathfrak{S}$. For each residue i a different vector ∇χ²_i is constructed. The first element of the vector represented by the symbol ∂$\mathfrak{D}$ (the orange block) is the sub-vector of chi-squared partial derivatives with respect to each of the diffusion tensor parameters $\mathfrak{D}_j$. The rest of the elements, grouped into blocks for each residue denoted by the symbol ∂$\mathfrak{F}_i$, are the sub-vectors of chi-squared partial derivatives with respect to each of the model-free parameters $\mathfrak{F}_i^j$. For the residue dependent vector ∇χ²_i the partial derivatives with respect to the model-free parameters of $\mathfrak{F}_j$ where i≠j are zero. These blocks are left uncoloured. The complete gradient of $\mathfrak{S}$ is the sum of the vectors ∇χ²_i.

The construction of the gradient is significantly different for the models $\mathfrak{F}_i$, $\mathfrak{T}_i$, $\mathfrak{D}$, and $\mathfrak{S}$. In Figure 15.1 the construction of the chi-squared gradient ∇χ² for the global model $\mathfrak{S}$ is demonstrated. In this case

$$\sum_{{i=1}}^{l}$$ (15.9)

where ∇χ²_i is the vector of partial derivatives of the chi-squared equation χ²_i for the residue i. The length of this vector is

$$\lVert \rVert \mathfrak{S},$$ (15.10)

with each position of the vector j equal to ${\frac{{\partial \chi^2_i}}{{\partial \theta_j}}}$ where each θ_j is a parameter of the model.

The construction of the gradient ∇χ² for the model $\mathfrak{D}$ is simply a subset of that of $\mathfrak{S}$. This is demonstrated in Figure 15.1 by simply taking the component of the gradient ∇χ²_i denoted by the symbol ∂$\mathfrak{D}$ (the orange blocks) and summing these for all residues. This sum is given by (15.9) and

$$\lVert \rVert \mathfrak{D}.$$ (15.11)

For the parameter set $\mathfrak{T}_i$, which consists of the local τ_m parameter and the model-free parameters of a single residue, the gradient ∇χ²_i for the residue i is simply the combination of the single orange block and single yellow block of the index i (Figure 15.1).

The model-free parameter set $\mathfrak{F}_i$ is even simpler. In Figure 15.1 the gradient ∇χ²_i is simply the vector denoted by the single yellow block for the residue i.