# Exponential Regression Model

I need to model my data ($(x,y)$ pairs) using the following exponential function:

$$f(x) = \exp((x + a)/b) - c$$

So, I need to find $a, b, c$ coefficients that are the best fit for my data. What is the algorithm that can solve this problem?

• It depends on the probability model for the errors. Typically the observed value of $f(x)$ differs from the true $f(x)$; otherwise all the data points would fit the curve perfectly. You'd have a probability density $g(y)$ depending on $a$, $b$, $c$ and possible an error variance $\sigma^2$ or the like. Plug the data points into that function, and then view it as a function of $a,b,c,\sigma$. Find the values of $a,b,c,\sigma$ that maximize that function. Sep 12, 2012 at 16:23
• Are the $x$ values spaced equidistantly in the data? This is important for generalized Prony/Pisarenko methods. Aug 19, 2016 at 8:09

Assume that somehow you are able to find the value of $c$, and only $a$ and $b$ have to be found. Let $y=\log(f(x)+c)$. Then $y=a\,x+b/a$, so that simple linear regression will give the value of $a$ and $b$.
But how do we find $c$? One possibility is to try different values of $c$ until you get the best fit. Another one, is to observe that $$-c=\begin{cases}\lim_{x\to-\infty}f(x) & \text{if }a>0,\\ \lim_{x\to+\infty}f(x) & \text{if }a<0\end{cases}.$$ If the data decrease as $x$ increases, you can guess $c$ looking at your data for large values of $x$. If the data increase with $x$, look at large negative values of $x$.
• I think if $|a|$ is relatively larger than $c$ linear term will be dominant and the solution that you suggested will be a good one. The other way around is another thing. Sep 13, 2012 at 13:49