HSIC documentation typos

[l-brevault]

Hello,

Here is a list of possible typos in the Sensitivity analysis using Hilbert-Schmidt Indepencence Criterion (HSIC) theory documentation page: HSIC theory documentation

• In the paragraph HSIC definition: "we can then define a dependence measure between $Y^k$ and $X^i$ " $\rightarrow$ it should be between $Y$ and $X^i$

• (in the following the tilde on $\tilde{L}$ is omitted for markdown reasons) where $L{i{j,k}}$ and $L{j,k}$ are computed as : $$L{i{j,k}} = (1-\delta{j,k})\kappa(Y^{(i)},Y^{(k)})$$ $$L{j,k} = (1-\delta{j,k})\kappa_i(X_i^{(j)},X_i^{(k)}) $$ $\rightarrow$ there is probably an inversion in $\kappa()$ and $\kappai()$, it should be: "computed as: $$L{i{j,k}} = (1-\delta{j,k})\kappa_i(X_i^{(j)},Xi^{(k)})$$ $$L{j,k} = (1-\delta_{j,k})\kappa(Y^{(i)},Y^{(k)}) $$

• There are issues in the indices to determine the coordinate of the vector $X$ in the entire documentation, sometimes it is $X^i$, sometimes it is $X_i$. See for instance first line $X=(X^1,\cdots,X^d)$ and the first equation after "Monte Carlo sampling or real life observations:"

• This can be defined as the the process of $\rightarrow$ repetition of the

• we wish to test the the following hypothesis $\rightarrow$ repetition of the

• From this samples $\rightarrow$ these samples

• We can generate a set of B independent permutations ${\tau_1,\cdots, \tauB }$ of $X^{(j)}{i_{(1\leq j \leq n)}}$ and compute the associated HSIC values $\rightarrow$ should it be B independent permutations ${\tau_1,\cdots, \tau_B }$ of $Y^{(j)}$ as in the following we have: $\hat{H}^{*b}:=\widehat{\text{HSIC}}(X^{(j)},Y^{(\taub(j))}){(1\leq j \leq n)}$

• We can then estimate the target HSIC value associated to the input variable $X_i$ as: $T\widehat{-HSI}C,i$ $\rightarrow$ issue on the HSIC hat and on the i index: $T-\widehat{HSIC}_i$

• Similar problem on hat and HSIC in the definition of $C\widehat{-HSI}C$

• Matrices $U$ et $I_n$ are not defined in $H_1$ and $H_2$ definition

Best, Mathieu and Loïc

[m-balesdent]

@vchabri or @JPelamatti, could you validate these changes to add them in the doc for v1.20?

[JPelamatti]

@l-brevault , @m-balesdent thanks for these comments. I agree with all of the points listed in your issue report. I'll start correcting the documentation right away