Problem: Find the "best fit" to (xn, yn) in the form y = K/(1+A exp(-r x))
Minimize E(K,A,r) = Σ {yn - K/(1+A exp(-r xn))}2
Solve for K, A and r from:
dE/dK = Σ {y - K/(1+A exp(-r x))} / (1+A exp(-r x)) = 0
dE/dA = Σ {y - K/(1+A exp(-r x))} exp(-rx) / (1+A exp(-r x))2 = 0
dE/dr = Σ {y - K/(1+A exp(-r x))} x exp(-rx) / (1+A exp(-r x))2 = 0
where, for convenience, we've dropped the "subscripts" n.
Mamma mia! No can do.
Instead, I've generated a spreadsheet, like so:
bell-stuff spreadsheet
RIGHT-click the link and Save Target or Save Link.