Say $a_x$, $a_y$, $b_x$, $b_y$ are signals with values in $\mathbb{R}$. Say they are independednt, normally distributed variables with given SDs $\sigma_{a_x}$, $\sigma_{a_y}$, $\sigma_{b_x}$, $\sigma_{b_y}$.

Say $f: \mathbb{R}^4\longrightarrow\mathbb{R}$ is a continuous function of these signals (i.e. $f(a_x, a_y, b_x, b_y)\in \mathbb{R}$ for all $(a_x, a_y, b_x, b_y)\in\mathbb{R}^4$.

  • General question

    What is the SD of $f(a_x, a_y, b_x, b_y)$?

  • Example/Question

Say $a$ and $b$ are vectors (say in $\mathbb{R}^2$) and $\sphericalangle$ is the angle between $a$ and $b$, say defined as $$\sphericalangle(a,b) := \mathrm{arcos}\left(\frac{ab}{\vert a\rvert \lvert b\rvert}\right)\quad\text{where}\quad a=(a_x, a_y)\quad\text{and}\quad b=(b_x, b_y).$$

Say $a_x, a_y, b_x, b_y$ are independent and normally distributed, have SDs $\sigma_{a_x}$, $\sigma_{a_y}$, $\sigma_{b_x}$, $\sigma_{b_y}$ respectively.

Then Gauss error proparation yields $$\sigma_\sphericalangle:= \sqrt{ { \left(\frac{\delta \sphericalangle}{\delta a_x} (a,b) |\mu \ \cdot \sigma_{a_x}\right) }^2 + { \left(\frac{\delta \sphericalangle}{\delta a_y} (a,b) |\mu \ \cdot \sigma_{a_y}\right) }^2 + { \left(\frac{\delta \sphericalangle}{\delta b_x} (a,b) |\mu \ \cdot \sigma_{b_x}\right) }^2 + { \left(\frac{\delta \sphericalangle}{\delta b_y} (a,b) |\mu \ \cdot \sigma_{b_y}\right) }^2}$$

as the error of $\sphericalangle$ (where $..|\mu$ shall note evaluation at average $\mu$).

  • Question

Is that $\sigma_\sphericalangle$ the SD of $\sphericalangle$?

  • 1
    $\begingroup$ i have never seen the symbol $\sphericalangle$ used as a variable name. $\endgroup$ – robert bristow-johnson Jun 16 '17 at 13:53
  • 1
    $\begingroup$ Regardless of a known result, need to know the mean values of your a's and b's $\endgroup$ – user28715 Jun 16 '17 at 14:23
  • $\begingroup$ Correct - I added $..|\mu$ to indicate that. $\endgroup$ – FWE Jun 16 '17 at 16:37
  • $\begingroup$ @robert bristow-johnson $\sphericalangle$ is not used as a variable name but as symbol for angle function $\sphericalangle(a,b)$. In the given example the $f$ from the general question is set to $f:=\sphericalangle$. $\endgroup$ – FWE Jun 16 '17 at 16:48

It is an estimation of the angle SD.


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