How to determine best fit from two?

Question

I'm studying some variances ($\sigma^2$), that in my case, they must depend by velocity squared $v^2$.

So, in my experimental proofs, I have plotted a graph of $\sigma^2/v^2$, and I was hoping to find a horizontal line.

But, my data seem to be constants in $\sigma^2/v$, and now I find myself in times of trouble: I have to choose which one is the best fit.

In each cases ($\sigma^2/v^2$ or $\sigma^2/v$), I obtain a nearly straight line.

So, I look the equation of best fit lines as $y = mx +q$, where $m$ is his angular coefficient.

The angular coefficient $m$ is less, in the case of $\sigma^2/v^2$ (by about an order of magnitude), but in the case of $\sigma^2/v$, the range of the best fit line (from its maximum and minimum in the graph) is relatively more lower.

At this point, to choose the best fit, the best solution is:

• Choose the case where $m$ is lower.
• Or I have to choose the dynamic case (maximum value minus minimum value), where the best fit line is more flat in the graph (not by absolute value)?

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Here are an example: in the first graph $m$ is greater (see the equation in the plot), but the line varies of 0.01.

In the second case $m$ is lower, but the line varies of 0.15.

What is the best fit? The first or the second?

EDIT

For Matt L. - Your approximation error $e$ is very similar to the $R^2$ parameter, that I define so (now I use your notation):

$$R^2 = 1 - \frac{\sum_i \left[d_i-f(v_i)\right]^2}{\sum_i \left[d_i-\overline{d_i}\right]^2}$$,

where $\overline{d_i}$ is the average value of $d_i$.

Maybe you would like say that I must simply see where I have the minimum error $e$ or the best fit parameter, e.g. $R^2$.

Answer 1

It looks like a straight line is not a great approximation in either case. One thing which is unclear to me is that the y-axis in the top plot is $\sigma^2/v$ whereas in the second plot it is just $\sigma^2$ (as far as I can see), and not $\sigma^2/v^2$ as I expected.

What you need to figure out is what you should actually expect. When you say that $\sigma^2$ should depend on $v^2$ then this doesn't say much. From what you do it looks like that you expect those two quantities to be proportional:

$$\sigma^2=\alpha v^2$$

because then you would ideally have a constant ratio $\sigma^2/v^2$. Is this the case?

Note that if you model $\sigma^2/v$ as a linear function of $v$, then you have

$$\sigma^2/v=mv+q$$

which is equivalent to

$$\sigma^2=mv^2+qv$$

So you actually model $\sigma^2$ by a quadratic function of the variable $v$.

To determine the best fit, it seems natural to compare the normalized approximation errors of the different fits. Given that $d_i$ are your measured data points, and $v_i$ are the discrete velocities corresponding to the data $d_i$, the approximation error can be defined as

$$\epsilon=\frac{\sum_i \left[d_i-f(v_i)\right]^2}{\sum_id_i^2}$$

where $f(v)$ is the approximation function, which could be linear as you've suggested in your question, i.e. $f(v)=mv+q$.