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Let, we have two power equations: $$y=k_1x^{a_1}$$ $$y=k_2x^{a_2}$$ Is there any way (analytical or numerical) to combine these two equations into one i.e. into the form: $$y=kx^a$$ Infact,what I am trying to do here is:

I have got around 20 equations like the first two equations above using regression analysis from experiments. Now, I am trying to combine all of those into just one equation to express relation between $y$ and $x$.

Any help would be highly appreciated. Thanks in advance for your help. :)

fnafis
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  • Not sure this is clear. The first two equations can't be met simultaneously (unless they coincide). If they did, you'd get $\frac {k_1}{k_2}=x^{a_1-a_2}$ where one side is constant and the other is not. I assume you mean you get all of these by regressing on different data sets, yes? So...try regressing on the combined data. – lulu Jul 18 '16 at 21:07
  • @lulu , yes, you are right. Those were from different data sets. By the way, thanks for your advice. My problem is solved. :) – fnafis Jul 18 '16 at 23:02

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What I suppose is that for each function $$y=k_i \,x^{a_i}$$ parameters $k_i$ and $a_i$ have been obtained by regression for a fixed value of another parameter (say $t$).

To combine all the equations together, I should analyze the variation of each parameter as a function of $t$ and, hoping that simple relations appear, regress the parameters.

Suppose that you notice that $k=\alpha +\frac \beta t$ and $a=\gamma+\delta t$; then the general model would be $$y=\left(\alpha +\frac \beta t \right)x^{\gamma+\delta t}$$

Now, reuse all the data sets and perform a nonlinear regression to get the best values of parameters $(\alpha,\beta,\gamma,\delta)$.

Claude Leibovici
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