(* Exploring non-linear list square fit *)

(* To run on command line:  math -run "<<least_square.nonlinear.math" *)
(* The above line assumes that math kernel is aliased to "math" *)
(* For example, on my machine alias
math='/Applications/Mathematica.app/Contents/MacOS/MathKernel' *)
(* or simply use << least_square.nonlinear.math once math kernel is already running
*)

<<JavaGraphics`



(* << Statistics`NonlinearFit`   *) (* seems deprecated, not needed *) 

F[x_] := 0.5*Exp[0.1*x]; (* define function *) 


a = .0; b= 20.0; (* define the interpolating interval *) 

plotfunction = Plot[F[x], {x, a, b+20}, PlotStyle->{Hue[1], Thickness[0.01]}, PlotLabel->"Original function", DisplayFunction -> Identity ]

n=20; h = (b-a)/n;  (* define data points *)

(*
data = Table[N[{x, F[x]}], {x, a, b, h}] 
*) 

(* now introduce experimental uncertainity *)


SeedRandom[]
data = Table[N[{x + 5.1*Random[Real, {-1, 1} ], F[x] + 1.1*Random[Real, {-1, 1} ]}], {x, a, b, h}]



plotdata = ListPlot[data, PlotLabel -> "Data points", Prolog-> AbsolutePointSize
[5] ]



fit = NonlinearModelFit[data, alpha*Exp[beta*x], {alpha, beta}, x] (* Do non-linear least square fit *)

Print[]
Print[Normal[fit]]

plotfit = Plot[fit[x], {x, a, b+20}, PlotStyle->{Hue[0.5], Thickness[0.005]}, PlotLabel-> "Least square", DisplayFunction -> Identity ]

Show[plotfunction, plotdata, plotfit, PlotLabel -> FontForm["red - original f(x), blue - Least S. Fit ", {"Courier", 15}], Prolog-> AbsolutePointSize[5], DisplayFunction -> $DisplayFunction ]


(* Show only data and the fit *)

Show[plotdata, plotfit, PlotLabel -> FontForm["circles - data, blue - Least S. Fit ", {"Courier", 15}], Prolog-> AbsolutePointSize[5], DisplayFunction -> $DisplayFunction ]


Clear[data]

Clear[fit]