Figuring out the best parameterization for this kind of model is a bit tricky until you get the hang of it. Let the function be
y_t = y_0 + alpha * E^t where uppercase Y_t denotes an observed value and lower case y_t is a predicted value. Index the times by t_1 .... t_n WLOG assume that t_1 is the smallest time and t_n is the largest time. Now we already have a good idea what the predicted values y_1 and y_n should be as we have observations for them. We have the two equations y_1 = y_0 + alpha * E^t_1 y_n = y_0 + alpha * E^t_n we can solve these for alpha in terms of y_1,y_n,and E giving alpha = (y_n-y_1)/(E^t_n -E^t_1) (1) and solve for y_0 as y_0 = y_1 - alpha * E^t_1 using (1) for alpha Now use the good estimates Y_1 and Y_n as the starting values for y_1 and y_n and try some "reasonable value for E (say 0.1 < E < 100) Do the minimization in two stages first holding y_1 and y_n fixed and then estimate y_1,y_n,and E together. This converges in less than a second using AD Model Builder and the starting values (large value for E. 2018.34 2778.47 exp(10.0) where I deliberately used exp(10) as a large initial value for E. The parameter estimates together with the est std devs are 1 y_1 1.9994e+03 3.9177e-01 2 y_n 2.7881e+03 6.7557e-01 3 log_E 5.6391e-01 1.2876e-03 4 alpha 6.0130e-04 1.9398e-05 5 y_0 1.9906e+03 4.5907e-01 6 E 1.7575e+00 4.3935e-02 There are problems for E near 1 which need to be dealt with if you have to go there, but that is just a technicality. This idea also works well for a logistic say 3 4 or 5 parameter. -- David A. Fournier P.O. Box 2040, Sidney, B.C. V8l 3S3 Canada Phone/FAX 250-655-3364 http://otter-rsch.com ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.