3
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I would like to fit a function to a set of data:

data = {
  {8, 0.138177}, {16, 0.0789974},
  {32, 0.0479225}, {64, 0.0351548},
  {128, 0.0299134}, {1024, 0.0277813}}

I tried something like:

nlm = 
  NonlinearModelFit[
    data,
    {a + b Log[c x], c > 0},
    {a, b, c}, x]

But I still can't get an ideal result. The plot looks like this:

Plot[nlm["BestFit"], {x, 0, 1024}, PlotRange -> {{0, 1030}, {0, 0.2}}, BaseStyle -> {FontFamily -> "Times", 16}, Frame -> True, FrameStyle -> Black, FrameLabel -> {{"Relative error", ""}, {"Time increments", ""}}, Epilog -> {{Red, PointSize[Large], Point@data}}]

enter image description here

Maybe I'm not using the correct fitting model here, so I got stuck for a while. Any help or suggestion will be appreciated!

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6
  • 3
    $\begingroup$ Please include a Plot of your result (curve and data superimposed) and describe what would constitute an ideal result. $\endgroup$
    – bbgodfrey
    Dec 28, 2021 at 3:05
  • $\begingroup$ Fitting 4 parameters (a, b, c, and error variance) with just 6 data points is likely (even with a "bad looking" fit) to be overly optimistic. Please consider including single prediction confidence bands to get an idea of the associated error in fit: Plot[ nlm["SinglePredictionBands"], {x,8,1024}, PlotRange->All]. $\endgroup$
    – JimB
    Dec 28, 2021 at 3:38
  • 2
    $\begingroup$ An alternative function to try is a + b/x^c. $\endgroup$
    – JimB
    Dec 28, 2021 at 3:39
  • $\begingroup$ Thanks! @JimB. It worked as following your suggestion now. I think I focused on the wrong function too long previously. $\endgroup$
    – J. W
    Dec 28, 2021 at 3:47
  • 3
    $\begingroup$ Note that your model was also not great from a fitting perspective because your parameters were correlated: $a+b\log{c x}=a+b(\log{c}+\log{x})=a+b\log{c}+b\log{x}$ which would essentially be the same as using $a+b\log{x}$ directly. $\endgroup$
    – MarcoB
    Dec 28, 2021 at 3:49

1 Answer 1

8
$\begingroup$
Clear["Global`*"]

data = {{8, 0.138177}, {16, 0.0789974}, {32, 0.0479225},
   {64, 0.0351548}, {128, 0.0299134}, {1024, 0.0277813}};

Using a different model

(nlm = NonlinearModelFit[data, {a + b*x^c, c < 0},
    {a, b, c}, x]) // Normal

(* 0.0257578 + 1.20405/x^1.13815 *)

Plotting,

Plot[nlm[x], {x, 8, 1024},
 Epilog -> {Red, AbsolutePointSize[4],
   Tooltip[Point[#], #] & /@ data},
 PlotRange -> {0.02, 0.14}]

enter image description here

On a log scale,

LogLinearPlot[nlm[x], {x, 8, 1024},
 Epilog -> {Red, AbsolutePointSize[4],
   Tooltip[Point[{Log[#[[1]]], #[[2]]}], #] & /@ data},
 PlotRange -> {0.02, 0.14}]

enter image description here

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1
  • $\begingroup$ Thanks a lot! I was just addicted to the wrong function too long to get the correct result $\endgroup$
    – J. W
    Dec 28, 2021 at 3:49

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