Nonlinear Regression

$The\; relationship\; between\; Y\; and\; X\; or\; Y\; and\; several\; Xs\; is\; not\; always\; linear\; in\; form.\; An\; example\; was\; given\; with\; the\; discussion\; offigure\; 5.\; In\; that\; example,\; however,\; a\; transformation\; was\; applied\; that\; resulted\; in\; a\; linear\; relationship.\; In\; some\; instances\; such\; a\; transformation\; may\; not\; exist\; and\; in\; others\; theoretical\; concerns\; may\; require\; analysis\; to\; be\; carried\; out\; with\; the\; untransformed\; equation.$

Development of size of biological organisms is well described by the equation

W = A (1 - e^Bt)^C

where W is some measure of organism size, t is time or age in the life of the organism, and A, B, and C are unknown parameters to be estimated from data. No transformation can be applied to this equation that would allow use of linear least squares methodology to estimate A, B, and C.

Least squares methodology can be used to solve nonlinear regression problems. For the above equation the least squares estimates of the parameters would be the solution of the minimization of

Application of calculus leads to three equations whose solution requires an iterative technique. For all but the simplest of cases, solving nonlinear least squares problems involves use of computer-based algorithms. A multitude of such algorithms exist emphasizing the number of problems whose valid solution requires the nonlinear least squares technique.