Least Squares Nonlinear Curve Fitting without the logs

What if we have what looks from the historical data to be a good nonlinear relationship between the variable we are trying to estimate and one or more drivers, but the relationship doesn’t fit well into one or more of our transformable functions? Do we: Throw in the proverbial towel, and make a ‘No bid’ recommendation because it’s too difficult and we don’t want to make a mistake? Think ‘What the heck; it’s close enough!’ and proceed with an assumption of a Generalised form of the nearest transformable function, estimate the missing constant, or just put up with the potential error? Shrug our shoulders, and say ‘Oh well! We can’t win the all!’, and go off for a coffee (other beverages are available), and compile a list of experts whose judgement we trust? Use a Polynomial Regression and only worry if we get bizarre predictions? Consider another nonlinear function, and find the ‘Best Fit Curve’ using the Least Squares technique from first principles? Apart from the first option which goes against the wise advice of Roosevelt, all of these might be valid options, although we should really avoid option iv (just because we can it doesn’t mean we should!) However, in this chapter, we will be looking at option (v): how we can take the fundamental principles that underpin Regression, and apply them to those stubborn curves that don’t transform easily. A word (or two) from the wise? "It is common sense to take a method and try it. If it fails, admit it frankly and try another; but above, all, try something." Franklin D Roosevelt 1882-1945 American president