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I am trying to transform samples $x_1$ that have already been sampled from a von Mises distribution $\mathcal{V}_1(\mu_1,\kappa_1)$ to match a another von Mises distribution $\mathcal{V}_2(\mu_2, \kappa_2)$.

With normal distributions $\mathcal{N}_1(\mu_1, \sigma_1^2)$ and $\mathcal{N}_2(\mu_2, \sigma_2^2)$, we would do :

$$ x_{1\to2}(\mu_1, \mu_2, \sigma_1, \sigma_2) = \frac{x_1 - \mu_1}{\sigma_1} \cdot \sigma_2 + \mu_2 $$

Is there a solution to do this in the case of a von Mises distribution ? As a subsidiary question, does this operation have a name ?

EDIT : transforming $x_1$ to a new location $\mu_2$ is not a problem since it corresponds to translating the pdf, and therefore the sampled variables $x_1$. We just need to care about reformating them between $[-\pi,\pi]$. It is in fact a rotation. $$ x_{1\to2}(\mu_1, \mu_2, \kappa_1, \kappa_2=\kappa_1) = (x_1 - \mu_1 + \mu_2)[-\pi,\pi] $$

Updating the concentration/dispersion around the mean is however more difficult.

If we use a scaling factor, we end up with the tails of the pdfs being altered (sorry my terrible drawing).

Kiord
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  • Wouldn’t a simple transformation $X_2 := a X_1 + b$ do the trick? Pick $a$ and $b$ appropriately. – A rural reader Oct 03 '23 at 19:39
  • I think scaling the variables would result in the "tails" of the pdf to accumulate because the domain is periodic. The resulting pdf would not be a von Mises pdf. And the new variables would not be von Mises samples. – Kiord Oct 03 '23 at 19:53

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