Ed's Big PlansArchive for the ‘Partial Derivatives’ tag
Partial Derivatives for Residuals of the Gaussian Function
I needed to get the partial derivatives for the residuals of the Gaussian Function this week. This is needed for a curve fit I’ll use later. I completely forgot about Maxima, which can do this automatically — so I did it by hand (Maxima is like Maple, but it’s free). I’ve included my work in this post for future reference. If you want a quick refresh on calculus or a step-by-step for this particular function, enjoy :D. The math below is rendered with MathJax.
The Gaussian Function is given by …
$$ f(x) = ae^{-\frac{(x-b)^2}{2c^2}} $$
- a, b, c are the curve parameters with respect to which we differentiate the residual function
- e is Euler’s number
Given a set of coordinates I’d like to fit (xi, yi), i ∈ [1, m], the residuals are given by …
$$ r_i = y_i – ae^{-\frac{(x_i-b)^2}{2c^2}} $$
We want to get …
$$ \frac{\partial{r}}{\partial{a}}, \frac{\partial{r}}{\partial{b}}, \frac{\partial{r}}{\partial{c}} $$
