Exponential regression plus a constant

Question

I've been looking for a convenient way to fit an expression of the form $y=a\cdot b^x + c$ to a set of points in the (x; y) plane. The technique I'm looking for should be sufficiently "light" from a theoretical point of view (no calculus) and be feasible with "by hand calculation" at least in some easy case to show as an example (then I would use a PC). The purpose is didactic, and my aim is to use the technique with students in Italian secondary school (which would be equivalent to first-year college in terms of mathematical competence). The students would already know linear regression, so my first attempt was to try to linearize the model with the log function, but of course the trick doesn't work when there's a translation due to c. Does any of you have good ideas about how to deal with that?

Answer 1

So, you want to fit a model $$y=a \times b^x+c$$ based on $n$ data points $(x_i,y_i)$. As said, in principle, you cannot avoid nonlinear regression.

However, suppose that you assign a value to parameter $b$; so, defining $z_i=b^{x_i}$, the model reduces to $$y=a z+c$$ which is linear and the standard linear regression will provide $a(b)$, $c(b)$ and $SSQ(b)$.

So try different values of $b$ until you see a minimum of $SSQ(b)$. You can polish the solution to the level you want.