Re: Automate datafitting to a series of parameterized function
- To: mathgroup at smc.vnet.net
- Subject: [mg70303] Re: Automate datafitting to a series of parameterized function
- From: Peter Pein <petsie at dordos.net>
- Date: Thu, 12 Oct 2006 05:37:34 -0400 (EDT)
- References: <egi1h9$j6r$1@smc.vnet.net>
Peng Yu schrieb: > Suppose I have some data x_i, y_i. Let us generated them by > > coef = Table[Random[Real, {-1, 1}], {i, 0, 4}] > poly = Table[x^n, {n, 0, 4}] > data = Table[{x, coef.poly}, {x, 0, 1, .01}] > > Let us forget how we generated the data. Now, we want to find an > function to fit the data. The simplest way to do is using Tayler > series. But the problem is I don't know how many terms I should keep. > One way I can do is to try for different number of terms. I start by > only 1 term (0th order). > > f[x_, a0_] := a0 > f[x_] := Evaluate@NonlinearFit[data, f[x, a0], {x}, {a0}] > error = Apply[(f[#1] - #2)^2 &, data, {1}]; > avererror = Apply[Plus, error]/Length[data] > maxerror = Apply[Max, error] > > If the above one doesn't work, I'll try 2 terms (0th and 1st order). > f[x_, a0_, a1_] := a0 + a1 x > f[x_] := Evaluate@NonlinearFit[data, f[x, a0, a1], {x}, {a0, a1}]; > error = Apply[(f[#1] - #2)^2 &, data, {1}]; > avererror = Apply[Plus, error]/Length[data] > maxerror = Apply[Max, error] > > I can try even higher orders. But the problem is that ever time I have > to define the function f[x_,a0_...] and supply the right function and > parameter list to NonlinearFit. I feel it is especially difficult to > parameterize the number of parameters to a function. > > Can some give some idea on how to do it if it is possible in > Mathematica? Of cause I could write a python program to generate the > mathematica code, and call math.exe in the command line mode. But that > is just to much, I just want to stick with mathematica. > > Thanks, > Peng > Hi Peng, there is no need to - state all the a_k explicitly in the function call. The approximations can be built dynamically. - use NonlinearFit; Fit is sufficient. In[1]:= n = 4; SeedRandom[12] coef = Table[2*Random[] - 1, {n + 1}] data = Transpose[{#1, (coef . #1^Range[0, n] & ) /@ #1}]&[0.01*Range[100]]; lpl = ListPlot[data, PlotStyle -> {AbsolutePointSize[4]}] Out[3]= (* the coefficients *) {0.465631143384559, 0.2213501334248431, -0.9824888769638274, 0.27138778246826445, 0.7190072574890038} Out[5]= Graphics[] In[6]:= f[k_] := Fit[data, x^Range[0, k], x] The errors for k=0..6: In[7]:= Chop[Norm/@Table[f[k]/.x->#1&/@First/@data - Last/@data, {k, 0, 6}]] Out[7]= {0.6339511113440388, 0.6050342778548943, 0.3273264300746273, 0.034187091387071745, 0, 0, 0} In[8]:= DisplayTogether[lpl, Plot[Evaluate[f /@ Range[0, n]], {x, 0, 1}, PlotStyle -> Table[Hue[(1 - j/n)/2], {j, 0, n}]], ImageSize -> {640, 480}, Background -> Black] Out[8]= Graphics[] Peter P.S.: Have a look at Interpolation too!