I have several variances ($\sigma^2$) which value depends on the velocity ($v$). As you can see in the graph, if increase the velocity, the variance does the same.
I am studying this dependency, but I do not know the exponent of the velocity. I have tried to write:
$$\sigma^2=q \cdot v^m$$
where, $q$ is a constant, and $m$ the exponent that I have to find.
I have rewritten this equation in logarithmic format:
$$\log(\sigma^2)=m \log(v)+\log(q)$$
now is a straight line equation.
I am looking for $m$ and $\log(q)$, via best fit. I am not a mathematician, I am an engineer, so I can use the tool for best fit, but this tool does work not as I would like.
Question: there is a way to compute the best fit in a logaritm chart, with strait line (first order) and know its slope ($m$) and y-intercept ($\log(q)$)?
Thank you and by,
Giacomo
I was looking for an equation of the type $\sigma^2 = q \cdot v^m$, which, in the logarithmic chart must be:
$$\log(\sigma^2)=m \log(v)+\log(q)$$
So, I have computed the $\log(\sigma^2)$ for each variance, and $\log(v)$ for each velocity (mine $y$ and $x$).
At this point, I have computed their best fit, looking $\log(\sigma^2)=m \log(v)+\log(q)$ like $y=mx+q$, so looking for a linear interpolation.
We can call $\hat{m}$ the best fit of $m$, and $\log(\hat{q})$ the best fit of $\log(q)$.
Eventually, my best fit equation is:
$$\hat{\sigma^2} = e^{\log(\hat{q})} \cdot v^\hat{m}=\hat{q}\cdot v^\hat{m}$$