I have a power-law-ish data series and need to back my way into a formula for it, such that it can be generated for any number of points. This is the n=20 version (value rounded to nearest 0.25). Could someone provide me with a formula that fits or a pointer to how I could derive it myself?
20 10 7.5 6 5.25 4.75 4.25 3.75 3.5 3.25 3 2.75 2.5 2.5 2.25 2.25 1.75 1.75 1.5 1.5
Assuming that $x_i=i$, a scatter plot of the data reveals an hyperbolic trend. As Raymond Manzoni noticed, if we do the same using the reciprocals of the $y$'s, "almost" a linear trend seems to appear.
So, the simplest model to regress could be $$y=\frac 1{a_0+a_1x}$$ Using least squares, we get $$y=\frac{1}{0.0164378+0.0351191\, x}$$ to which corresponds a sum of squares equal to $5.7851$ and $R^2=0.9923$.
As Raymond Manzoni did, we could add more terms and fit a model $$y=\frac 1{\sum_{k=0}^n a_i x^i}$$ and, varying $n$ try to find a model. For example, using $n=2$, we should get $$y=\frac{1}{0.00898542+0.0425887\, x-0.000865475 \,x^2}$$ to which corresponds a sum of squares equal to $2.9419$ and $R^2=0.9961$.
But, in order to limit the number of parameters, we can also try something like $$y=\frac 1{a_0+a_1x^{a_2}}$$ This will lead to $$y=\frac{1}{-0.0300556+0.0802496\,x^{0.647925}}$$ to which corresponds a sum of squares equal to $1.1558$ and $R^2=0.9985$. This is quite significantly better that the previous model.
You should notice that I did not work with thre reciprocals of the $y_i$'s because what is measured is $y$ and not any of its possible transforms.