From the comments in Curve Fitting 4:
If you want to try a different flavor of gum, consider simultaneously fitting exponential + sine models:
y1(t) = y1(0) * exp(kt)
y2(t) = A * sin(((t-b)/T) * 2 * pi)
y(t) = y1(t) + y2(t)You’ve probably already thought of this, of course.
Ned’s been peeking ahead, so maybe I shouldn’t “reward” him. 
But here it is anyway. The procedures are similar. I fitted an exponential with a sine of the residuals, a sine with an exponential of the residuals, and fitting both a sine and exponential simultaneously. I know that there are pre-packaged R methods for fitting exponentials, but I use the method of looking for best fit in by looping through the parameter space.
The form of the exponential that I used was: y1= -a/100 + (bb/10) * exp((k/10000)*(YEARS-1880))
Exp a = 149
Exp bb = 11
Exp k = 40
Sine Amplitude A = 8
Sine Phase Shift b = 24
Sine Period T = 56
Sine Amplitude A = 16
Sine Phase Shift b = 24
Sine Period T = 50
Exp a = 130
Exp bb = 10
Exp k = 34
Exp a = 145
Exp bb = 11
Exp k = 37
Sine Amplitude A = 9
Sine Phase Shift b = 34
Sine Period T = 61
Ned
2011 January 12 at 11:36 am
That was quick. It looks similar to my version. Thank you!
I am curious about where you are leading us with all of this curve-fitting. Are you just doing “exploratory data analysis” here, or are you building up to something? No need to actually answer that if you don’t want to; I will wait and see.
Ron Broberg
2011 January 12 at 11:47 am
I figured that before I went snipe-hunting, I should do some target practise first.
Part of it is just to get better at R. Another is to get more familiar with equations which are statistically justifiable. A third is to walk slowly towards better tools: Fourier, Arima, ACF, PCA. Confidence intervals in complex eqns. Auto-correlation. There is lots of things right for me to learn some basics and not-so-basics.
But, no, I don’t have a pre-planned conclusion in mind. This is stream-of-consciousness, baby-steps in the dark stuff. But I would like to be able to better analyze the claims of 1500,1100,800,180,60,22,11 or whatever cycles. So there will be some stuff along those lines eventually.