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 χ2i 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 χ2i the partial derivatives with respect to the model-free parameters of $ \mathfrak{F}_j$ where ij are zero. These blocks are left uncoloured. The complete gradient of $ \mathfrak{S}$ is the sum of the vectors χ2i.
\includegraphics[
width=0.9\textwidth,
bb=143 399 494 777
]{images/gradient}

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

χ2 = $\displaystyle \sum_{{i=1}}^{l}$χ2i, (15.9)

where χ2i is the vector of partial derivatives of the chi-squared equation χ2i for the residue i. The length of this vector is

$\displaystyle \lVert$χ2i$\displaystyle \rVert$ = dim$\displaystyle \mathfrak{S},$ (15.10)

with each position of the vector j equal to $ {\frac{{\partial \chi^2_i}}{{\partial \theta_j}}}$ where each θj 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 χ2i denoted by the symbol $ \mathfrak{D}$ (the orange blocks) and summing these for all residues. This sum is given by (15.9) and

$\displaystyle \lVert$χ2i$\displaystyle \rVert$ = dim$\displaystyle \mathfrak{D}.$ (15.11)

For the parameter set $ \mathfrak{T}_i$, which consists of the local τm parameter and the model-free parameters of a single residue, the gradient χ2i 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 χ2i is simply the vector denoted by the single yellow block for the residue i.

The relax user manual (PDF), created 2016-10-28.