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Using yaw angle $\theta$ and velocity $v$, the curvature $c$ can be calculated according to:

$$ c = \frac{\dot{\theta}}{v} $$

Given are discrete values for the yaw angle $\theta_t$, velocity $v_t$ and their corresponding standard deviations $\sigma_{\theta_t}$ and $\sigma_{v_t}$.

Can you explain me how to calculate the standard deviation of the curvature $\sigma_{c_t}$ using the given terms?

I though of using the ratio of the standard deviations for the yaw angle and the velocity but this seems to be wrong because there is no closed formula for the variance of a ratio. I cannot use a Taylor approximation since I don't have an analytical formula, only a few discrete values (approx. 800).

Note that the yaw angle and the velocity are uncorrelated.

To compute the angle derivative I use

$$ \dot{\theta} = \frac{\theta_{t+1} + \theta_{t}}{\Delta t} $$

With this in mind, the variance should be:

$$ var(\dot{\theta}) = var(\frac{\theta_{t+1} - \theta_{t}}{\Delta t}) = \frac{1}{\Delta t ^2} (var(\theta_{t+1}) + var(\theta_{t})) $$

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    If you have (or can compute) a random sample of values for $c$, you can estimate the standard deviation of the distribution of $c$ (assuming it exists) using the standard deviation of the data. – 2017-01-10
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    First of all you need to find an equation for the variance of the angle derivative... harder than the one for the ratio. – 2017-01-10
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    The equation over there is the relation between 3 random variables. The first thing is do you have the raw, paired data of $(\dot{\theta}, v)$? If you have this you can just form the ratio and calculate the sample variance to estimate the variance of $c$. If you do not have it, you have to make assumption on the dependency structure between them first. – 2017-01-11
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    If you assume they are independent, and if the value you observed are at regular time interval, you may try to estimate the derivative by finite differencing, and use e.g. bootstrap technique to estimate. – 2017-01-11
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    @BGM I don't have the raw paired data ($\dot{\theta},v$). I just have the yaw angle $\theta_t$, velocity $v_t$ and the corresponding standard deviations. Therefore I compute the angle derivative using finite differences (I'll add further details to my question). – 2017-01-11
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    @Robert Israel I am not sure but I don't think that a distribution for c exists since I only have values for $\theta_t$ and $v_t$ with their standard deviations at regular discrete time instances $\Delta t$. – 2017-01-11
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    @N74 I will add an equation for the variance of the angle derivative. – 2017-01-11
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    If a distribution for $c$ doesn't exist, what do you want the standard deviation of? – 2017-01-11
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    You said that yaw angle and speed are uncorrelated, but in the expression of the variance of the yaw derivative you are also conjecturing that two yaw angles are uncorrelated, and this is a strong condition on a continuous measurement. Also your equation gives the yaw derivative at $t+{1 \over 2}$, so it is not comparable with $v_t$. – 2017-01-13
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    @N74 the assumption is that $v_t$ is discrete though continuous and does not change abruptly. Therefore, It think the error that comes with my provided equation should not that big. Could you provide a better calculation for the yaw derivative? – 2017-01-20
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    @RobertIsrael I am not sure if I understood your first comment. How can I estimate the standard deviation of $c$ using the data I have: yaw angle and velocity $(\theta_t,v_t)$ with the corresponding standard deviations $(\sigma_{\theta}, \sigma_v)$? I just calculated the discrete curvature values using this information and now I am stuck finding an appropriate estimate for the standard deviation of $c$. So far I am just adding the two standard deviations of yaw angle and velocity since they are themselves not really exact $\sigma_c = \sigma_{theta} + \sigma_{v}$. Is this a possible estimation? – 2017-01-20
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    For the numerical derivation you can find many examples here: http://www.holoborodko.com/pavel/numerical-methods/numerical-derivative/central-differences/ . Then, I think you are not looking for variances, but for measurement error propagation, so find an easy reference here: https://www.google.it/url?sa=t&source=web&rct=j&url=http://virgo-physics.sas.upenn.edu/uglabs/lab_manual/Error_Analysis.pdf&ved=0ahUKEwj5xdefxdHRAhWEXSwKHYuZB3EQFgglMAQ&usg=AFQjCNFAWEQz6X2j5VKXjUMO1zXj_KZsTw&sig2=zFac-RHsU5TydvwD7wpxzQ – 2017-01-20

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