# Statistical descriptors and binning

## Global statistical descriptors

The final solution of the Monte-Carlo inversion algorithm, $$\mathcal{P}(\mathbf{D})$$, can be understood as the median of all bootstrap solutions, $$\mathcal{P}_{\mathit{n}_\mathrm{b}}(\mathbf{D})$$ with $1\leq \mathit\mathrm \leq \mathit\mathrm$. To quantify the main features of this final solution, one computes for instance the medians over bootstrap solutions of the per-bootstrap means $$\mathrm{Med}_{(\mathit{n}_\mathrm{b})}\,(\mathrm{E}[\chi]_{\mathit{n}_\mathrm{b}})$$, variances $$\mathrm{Med}_{(\mathit{n}_\mathrm{b})}\,(\mathrm{V}[\chi]_{\mathit{n}_\mathrm{b}})$$ and covariances $$\mathrm{Med}_{(\mathit{n}_\mathrm{b})}\,(\mathrm{C}[\chi,\chi^\prime]_{\mathit{n}_\mathrm{b}})$$ of $$\chi=\mathit{D}_\mathrm{iso}, D_\Delta^2$$, respectively referred to as the “size” and "shape" of the diffusion tensors building up $$\mathcal{P}(\mathbf{D})$$.

Here, the median operator “$$\mathrm{Med}$$” acts across bootstrap solutions, and $$\mathrm{E}[\,\cdot\,]_{\mathit{n}_\mathrm{b}}$$, $$\mathrm{V}[\,\cdot\,]_{\mathit{n}_\mathrm{b}}$$ and $$\mathrm{C}[\,\cdot,\cdot\,]_{\mathit{n}_\mathrm{b}}$$ denote the per-bootstrap average, variance and covariance over bootstrap solution $$\mathit{n}_\mathrm{b}$$, respectively. For simplicity, the median operator is implicitly omitted when addressing a statistical descriptor, thereby writing averages, variances and covariances as "$$\mathrm{E}[\chi]$$", “$$\mathrm{V}[\chi]$$” and "$$\mathrm{C}[\chi,\chi^\prime]$$", respectively.

While previous works have relied on means to compute averages across bootstrap solutions [@deAlmeidaMartins_Topgaard:2018 ; @Topgaard:2019], more recent works have employed medians instead, because of their enhanced robustness to statistical outliers [@Reymbaut_accuracy_precision:2020 ; @deAlmeidaMartins:2020].

## Bin-specific statistical descriptors

Given that the DTD method builds up $$\mathcal{P}(\mathbf{D})$$ as a discrete sum of components, all aforementioned statistical descriptors can be extracted within tissue-specific bins, i.e. subdivisions of $$\mathcal{P}(\mathbf{D})$$'s configuration space.

For instance, the “thin”, "thick" and “big” bins introduced in Refs.[@deAlmeidaMartins:2020 ; @Reymbaut_book_chapter:2020] aim to isolate the signal contributions from white matter, grey matter and cerebrospinal fluid, respectively. The boundaries of these bins, implemented as default bins in dVIEWR, are defined as follows:

• “thin” bin within $$\mathit{D}_\mathrm{iso} \in [0.1, 2.5] \;\mu\mathrm{m}^2/\mathrm{ms}$$ and $$\mathit{D}_\parallel/\mathit{D}_\perp \in [4, 1000]$$.
• “thick” bin within $$\mathit{D}_\mathrm{iso} \in [0.1, 2.5] \;\mu\mathrm{m}^2/\mathrm{ms}$$ and $$\mathit{D}_\parallel/\mathit{D}_\perp \in [0.01, 4]$$.
• “big” bin within $$\mathit{D}_\mathrm{iso} \in [2.5, 10] \;\mu\mathrm{m}^2/\mathrm{ms}$$ and $$\mathit{D}_\parallel/\mathit{D}_\perp \in [0.01, 1000]$$.