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Find Fit for Nonlinear Data

Posted 5 years ago
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I am attempting to fit a function to data (pasted below) which has an s-shaped appearance. What parameter's should I use?
 {{0, 2.52}, {1., 2.83}, {2., 3}, {3., 3.2}, {4.1, 3.35}, {5.,
   3.47}, {6, 3.57}, {7, 3.66}, {8, 3.76}, {8.5, 3.81}, {9,
   3.85}, {9.5, 3.89}, {10.1, 3.94}, {10.5, 3.98}, {11, 4.01}, {11.5,
   4.06}, {12, 4.09}, {12.5, 4.15}, {13, 4.19}, {13.5, 4.25}, {14,
   4.3}, {14.5, 4.35}, {15, 4.41}, {15.6, 4.47}, {16, 4.53}, {16.5,
   4.6}, {17., 4.68}, {17.5, 4.77}, {18, 4.85}, {18.5, 4.96}, {19,
   5.11}, {19.55, 5.34}, {19.7, 5.44}, {19.9, 5.58}, {20.1,
   5.91}, {20.3, 6.27}, {20.5, 7.14}, {20.6, 7.14}, {20.8,
   7.81}, {20.9, 8.32}, {21, 7.75}, {21.2, 9.07}, {21.4, 9.49}, {21.5,
  9.71}, {21.6, 9.83}, {21.8, 10}, {22, 10.18}, {22.1, 10.21}, {22.2,
  10.25}, {22.3, 10.27}, {22.5, 10.3}, {22.7, 10.42}, {22.9,
  10.47}, {23.1, 10.52}, {23.3, 10.59}, {23.5, 10.63}, {23.7,
  10.67}, {24, 10.74}, {24.2, 10.78}, {24.4, 10.8}, {24.6,
  10.82}, {24.8, 10.84}, {25, 10.87}}
18 Replies
Posted 5 years ago
If I subtract about 0.1*x from each y the result looks much more like the usual logistic curve and the parameters for logistic might be easy to estimate.
However I cannot alter the data, it's for a lab. Would it be logical to try to fit a sigmoid function (f(x)=a/(b+e^-x)) to the data? If so, how would I find and input the parameters? Thank you for your help on this matter
Following Bill,
 In[1]:= data = {{0, 2.52}, {1., 2.83}, {2., 3}, {3., 3.2}, {4.1,
     3.35}, {5., 3.47}, {6, 3.57}, {7, 3.66}, {8, 3.76}, {8.5,
     3.81}, {9, 3.85}, {9.5, 3.89}, {10.1, 3.94}, {10.5, 3.98}, {11,
     4.01}, {11.5, 4.06}, {12, 4.09}, {12.5, 4.15}, {13, 4.19}, {13.5,
     4.25}, {14, 4.3}, {14.5, 4.35}, {15, 4.41}, {15.6, 4.47}, {16,
     4.53}, {16.5, 4.6}, {17., 4.68}, {17.5, 4.77}, {18, 4.85}, {18.5,
     4.96}, {19, 5.11}, {19.55, 5.34}, {19.7, 5.44}, {19.9,
     5.58}, {20.1, 5.91}, {20.3, 6.27}, {20.5, 7.14}, {20.6,
     7.14}, {20.8, 7.81}, {20.9, 8.32}, {21, 7.75}, {21.2,
    9.07}, {21.4, 9.49}, {21.5, 9.71}, {21.6, 9.83}, {21.8, 10}, {22,
    10.18}, {22.1, 10.21}, {22.2, 10.25}, {22.3, 10.27}, {22.5,
    10.3}, {22.7, 10.42}, {22.9, 10.47}, {23.1, 10.52}, {23.3,
    10.59}, {23.5, 10.63}, {23.7, 10.67}, {24, 10.74}, {24.2,
    10.78}, {24.4, 10.8}, {24.6, 10.82}, {24.8, 10.84}, {25, 10.87}};

In[2]:= dplot = ListPlot[data, PlotStyle -> Red];

In[3]:= eq = .1 x + b/(1 + Exp[-c (x - x0)]) + d;

In[4]:= pars = FindFit[data, eq, {b, c, d, x0}, x]

Out[4]= {b -> 5.38732, c -> 2.16503, d -> 2.92208, x0 -> 20.7642}

In[5]:= Show[Plot[eq /. pars, {x, 0, 25}], dplot, PlotRange -> All]


I put the 0.1 x in manually -- Mathematica does not seem very good at finding the value for itself.

Best,
David
Careful initial setting for parameters will make model work and will find linear trend automatically. Paying attention to residue behaviour can help you to improve the model.

model = k x + b Tanh[c (x - x0)] + d;

fit = NonlinearModelFit[data, model, {{k, .1}, b, c, d, {x0, 20}}, x];

Plot[fit[x], {x, 0, 25}, Epilog -> {{Red, PointSize[.01], Point[data]},
  Inset[fit["ParameterTable"], {9, 9}]}, Frame -> True, PlotLabel -> fit[x]]

ListPlot[Thread[{data[[All, 1]], fit["FitResiduals"]}], Frame -> True, AspectRatio -> 1/7, Filling -> 0]

Thank you for your help on this matter!
Sincerely,
Alejandro
Vitaliy,
That is very cool!
David
David,

How did you determine that k should be approx. 0.1 and that x0 should be close to 20? Was this determined through trial and error?

Sincerely,
Alejandro
Posted 5 years ago
I initially proposed the 0.1*x. That was estimated by calculating the slope between x==3 and x==18, subtracting that from the original data, plotting the result and concluding that was close enough. I tried small variations in that and it didn't seem to make a lot of difference.
Yes -- I followed Bill on 0.1 x, and with that FindFit actually found the breakpoint value of x0. But inspection of the data plot and the function pretty well shows the break in the exponential to be at about 20. (Note that Vitaliy used (x0,20}, but I didn't assign a start to x0, but I fixed the 0.1. I suspect making estimates of this sort of thing is important where parameter values control fluxuations that extreme.
I would like to propose a different way of doing the non-linear fitting using Quantile regression with B-splines. The advantages of the approach are that it does not need guessing of the fit functions and it requires less experimentation.

Load the package QuantileRegression (explained here http://community.wolfram.com/groups/-/m/t/170894 ).

After several experiments with the number of knots we can find a regression quantile using 3d order B-splines of 18 knots that approximates the data well.
qfunc = Simplify[QuantileRegression[data, 18, {0.5}, InterpolationOrder -> 2][[1]]];
qfGr = Plot[qfunc[x], {x, Min[data[[All, 1]]], Max[data[[All, 1]]]}, PlotPoints -> 700];
Show[{dGr, qfGr}, Frame -> True, ImageSize -> 700]


Here are the relative errors:
ListPlot[Map[{#[[1]], (qfunc[#[[1]]] - #[[2]])/#[[2]]} &, data],
Filling -> Axis, Frame -> True, ImageSize -> 500]


If we select a non-uniform grid of knots according to gradients of the data we get a good approximation with one attempt.
knots =
Join[
  Rescale[Range[0, 1, 0.2], {0, 1}, {Min[data[[All, 1]]], Max[data[[All, 1]]] 3/4}],
  Rescale[Range[0, 1, 0.1], {0, 1}, {Max[data[[All, 1]]] 3/4,  Max[data[[All, 1]]]}]
]
qfunc = QuantileRegression[data, knots, {0.5}][[1]];

The grid lines in the plot below are made with knots.
qfGr = Plot[qfunc[x], {x, Min[data[[All, 1]]], Max[data[[All, 1]]]}];
Show[{dGr, qfGr}, GridLines -> {knots, None}, GridLinesStyle -> Directive[GrayLevel[0.8], Dashed],
Frame -> True, ImageSize -> 700]


Here are the relative errors:
ListPlot[Map[{#[[1]], (qfunc[#[[1]]] - #[[2]])/#[[2]]} &, data],
Filling -> Axis, Frame -> True, ImageSize -> 500]


Here is the approximation function after using PiecewiseExpand and Simplify:
qfunc = Evaluate[Simplify[PiecewiseExpand[qfunc[[1]]]]] &;
qfunc
Hi Everyone,
I have a new set of data I am trying to fit a function to.
{{0, 0}, {5, 0.3}, {10, 0.6}, {15, 0.25}, {20, 0.14}, {25, 0.1}, {30,
  0.05}, {35, 0.02}, {40, 0.01}}
I believe it to be Lognormal but cannot seem to fit a function to it.
Thanks in advance.

Sincerely,
Alejandro
LogNormalDistribution[] without further preparation is not a good hit 
In[9]:= braun = NonlinearModelFit[dd,
   PDF[LogNormalDistribution[\[Mu], \[Sigma]], x], {{\[Sigma], 0.5}, {\[Mu], 1}}, x];

In[12]:=Plot[braun[x], {x, 0, 40}, Epilog -> {PointSize[Medium], Point[dd]},  PlotRange -> All]
as you see here


something around Planck's radiation law will do better
In[19]:= braun = NonlinearModelFit[
dd, {\[Alpha] x^3/(Exp[\[Beta] x] - \[Gamma]), {\[Beta] >
      0.1, \[Alpha] > 0, \[Gamma] < 1}},
   {{\[Alpha], 0.5}, {\[Beta], 1}, {\[Gamma], 0.8}}, x];

In[20]:= Plot[braun[x], {x, 0, 40}, Epilog -> {PointSize[Medium], Point[dd]},  PlotRange -> All]

as seen here


experiment and suck a bit fun out of it ...
Thank you Udo!
I altered the equation a bit and came up with what is attached. However it is giving me multiple values for each parameter, is there any way to limit it down to just one?
Attachments:
With EDA you mean http://www.wolfram.com/products/applications/eda/  (Experimental Data Analyst 1.3)? I don't have that package, seemingly one has to buy it first.
I did use EDA, it works just like NonlinearRegression but adds in the residual plot and provides a little more information. I figured out what the equation is, the first number provided after the parameter is the value used in the equation, I do not know what the second number represents but I have a feeling it is related to the error of the function.
Thanks again for your help.

Sincerely,
Alejandro Braun
One needs data between x = 0 and x = 5 and x = 10 to clarify the model; tried also PearsonDistribution but it is biased by the first data point {0., 0.} so look at this

Thanks Ugo! How would I interpret the fitted function? If it is a piecewise function as I suspect, how would I integrate this? I took the top line after the bracket and graphed it from x>0 to x=40 but the function reached its apex around f(x)=.09. Thanks in advance!

Sincerely,
Alejandro Braun
One interpretes it with Normal[], e.g.
In[56]:= Clear[b]
         b[x_] := Normal[braun]
and plots b to get  the same result as seen before
Plot[b[x], {x, 0, 40}, Epilog -> {PointSize[Large], Point[dd]},
PlotRange -> All]
in defining b[x_] one must remember the last argument given to NonlinearModelFit[], the x.  If you want to see, what it is, enter
In[63]:= InputForm[Normal[braun]]
Out[63]//InputForm=
7.236018556056977*Piecewise[
  {{(1.1202753191820644*^6*(x^(-1))^5.271377059086114)/
     E^(43.06464219042515/x), x > 0}}, 0]
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