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Assume that we have two normalized $n\times 1$ vectors $\bf x$ and $\bf y$. In terms of Pearson correlation, these two signals are uncorrelated if ${\bf x}^T {\bf y} = 0$. Now, assume that ${\bf x}^T {\bf y} \neq 0$. If we perform the following projection: $$ \widehat{\bf x} = {\bf x} - ({\bf x}^T {\bf y}) {\bf y}, $$ we get $\widehat{\bf x}^T {\bf y} = 0$.

In all the above computations, we use Pearson correlation.

My questions is if we can perform such projection for Kendall Tau correlation. In other words, if we have two normalized $n\times 1$ vectors $\bf x$ and $\bf y$ such that their Kendall Tau correlation is not zero, can we perform a projection of $\bf x$ to get $\widehat{\bf x}$ such that the Kendall Tau correlation of $\widehat{\bf x}$ and $\bf y$ is zero?

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Covariance is linear, so a linear projection can be used to zero it out.

Concordance is not linear, so a linear projection won't (in general) work to zero it out.

However, one can still come up with vectors which have zero Kendall correlation.

Specifically, if $\hat{\beta}^K$ is the slope estimate for the Theil-Sen regression of $y$ on $x$, then the Kendall correlation of $x$ and $r=y-\hat{\beta}^K x$ will be 0.

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  • $\begingroup$ This is what I was looking for, thanks. Is there an extension of this on higher-dimensional data. Say, I want to make y uncorrelated with two vectors x1 and x2. $\endgroup$ – MMM Mar 17 '15 at 13:02
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    $\begingroup$ That one might be best asked as a new question. $\endgroup$ – Glen_b Mar 17 '15 at 13:10

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