How (un)reasonable is graphic linear regression in a monolog graphic?

[DaTario]

TL;DR Summary

Consider two different data sets whose contexts are not related. One is ploted in a linear graphic and the other in a monolog graphic. Both data sets end up looking the same in each graphic. How (un)reasonable is to proceed to a graphic method of linear regression in a monolog graphic?

Hi All,

Consider two different data sets whose contexts are not related. One is ploted in a linear graphic and the other in a monolog graphic. Both data sets end up looking the same in each graphic. How (un)reasonable is to proceed to a graphic method of linear regression in a monolog graphic?

With respect to the figure below, where the lines were drawn by hand as a visual estimate of the best line to represent the data, is the straight line drawn in the monolog graphic usefull to produce any reliable information? The measurement of distances in a monolog graphic is in general rather different from what we do in linear graphics.

Best wishes,

DaTario

 

[renormalize]

TL;DR Summary: Consider two different data sets whose contexts are not related. One is ploted in a linear graphic and the other in a monolog graphic. Both data sets end up looking the same in each graphic. How (un)reasonable is to proceed to a graphic method of linear regression in a monolog graphic?

A linear-fit to a semi-log (monolog) graph corresponds to an exponential-fit on a linear graph. A semi-log straight-line-fit is then quite reasonable if your data happens to exhibit exponential growth or decay when plotted on a linear scale.

 

[DaTario]

A linear-fit to a semi-log (monolog) graph corresponds to an exponential-fit on a linear graph. A semi-log straight-line-fit is then quite reasonable if your data happens to exhibit exponential growth or decay when plotted on a linear scale.

Thank you, renormalize, but an important point in my question is: is the best straight line fitting in linear graph the best straight line fitting in semi-log ? (consider the two data sets I presented in the figure, which look identical)

As we are dealing here with a visual estimate of the best straight line, I am insterested in possible technical differences between drawing the two (best) lines as they appear in the figure I presented in the OP.

 

[renormalize]

Thank you, renormalize, but an important point in my question is: is the best straight line fitting in linear graph the best straight line fitting in semi-log ? (consider the two data sets I presented in the figure, which look identical)

As we are dealing here with a visual estimate of the best straight line, I am insterested in possible technical differences between drawing the two (best) lines as they appear in the figure I presented in the OP.

To answer, you have to tell us what your technical criteria are (using equations) for measuring goodness-of-fit.

 

[DaTario]

If I took the second data set and stick to the pairs $(x, ln(y))$, I could obtain by calculations the parameters of the best straight line to draw in the semi-log graph. Is this calculated straight line similar to the ones one usually draws by hand in a linear graph with points with the same relative position?

 

[renormalize]

If I took the second data set and stick to the pairs $(x, ln(y))$, I could obtain by calculations the parameters of the best straight line to draw in the semi-log graph. Is this calculated straight line similar to the ones one usually draws by hand in a linear graph with points with the same relative position?

I've created 3 graphs that hopefully address your question.
First, I pick 5 data pairs $(x_i,y_i)$ that mimic those in your original post (OP) and plot them in red, along with the least-squares best linear-fit to that data, shown in black:

Then I make a second data set $(x_i,\text{exp}(y_i))$ by exponentiating the y-values of the first set, and also take the exponential of the linear fit. I graph this second set and exponentiated fit on a semi-log plot:

As you can see, both sets of data and best-fits are visually identical, which is the situation you ask about in your OP. So you can use a straight-line to best fit a semi-log data plot and then plot that data and fit on a linear scale to display a best exponential-fit, as shown here:

Does this answer your question?

 

[DaTario]

Thank you, renormalize. It seems to be a very satisfactory answer to my question.

So it seems that we can conclude that even though the Euclidean notion of distance within the space created in a semi-log graph is not valid, the lines that minimize the sum of the square of the distance to the points in the linear and semi-log cases will coincide.

Perhaps this is due to the monotonicity of the exponential function, that is, to the fact that minimizing the sum of the squares of the distances also corresponds to minimizing the sum of the squares of the exponentials of the distances.