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I have the following sample data

data={-0.16952640034568625, -0.26279580767569677, -0.34698969646746414, -0.41925051155260096, -0.4827096382888264, -0.537535673345224, -0.5788064930492074, -0.6021747066904077, -0.6227212709656731, 
  -0.645218874215294, -0.6529681030876083, -0.680922499828511, -0.6990457831526136, -0.7171409355950619, -0.7551752118001522, -0.7567042771077576, -0.7855256227381958, -0.8192822457689661, 
  -0.9070805586340569, -0.9355711982148748, -0.9702794105459295, -0.9709478394058481, -1.00960304274483, -1.0449022765424374, -1.0568778288104164, -1.089960160477901, -1.1366859307252049, 
  -1.1536965306829752, -1.1739583080589295, -1.218790605784555, -1.2454443064015006, -1.2556730491342347, -1.3020486408969945, -1.3418965449194218, -1.3492791213811992, -1.3961112371912672, 
  -1.4415701870346849, -1.444004907160478, -1.4918390718475778, -1.525102287594224, -1.5405835817417768, -1.5643157209521064, -1.5897076791692157, -1.6397299682197652, -1.656067204344228, 
  -1.690364074785328, -1.7290189420463014, -1.7645899501306705, -1.7814318696151457, -1.821141175417192, -1.862144107248702, -1.8807315364476849, -1.916046576313208, -1.9582601701356717, 
  -2.014751276413228, -2.0188465647853437, -2.069687763952646, -2.121381464658644, -2.167215147969383, -2.1697380854449326, -2.215207289423022, -2.25750295652639, -2.2900413566371123, 
  -2.2903942320960273, -2.315418523785436, -2.326637389424847, -2.3393266664011994, -2.3621807136810262, -2.383913838102939, -2.4045837785699717, -2.4309069078167242, -2.441281250644293, 
  -2.462414537853321, -2.4871407077668786, -2.510408023054892, -2.5321943008486754, -2.5360753455730203, -2.553217387754695, -2.573080036902164, -2.5882285548479715, -2.59607485927888, 
  -2.602916762267663, -2.6172366828046867, -2.63102036476022, -2.6443027245568675, -2.648009131235619, -2.6609251531154725, -2.677376846682117, -2.6933543178580965, -2.693498920957068, 
  -2.7089179311530374, -2.7238797751148525, -2.733717960345198, -2.73810045626225, -2.7523913155308444, -2.7657009087431543, -2.769454907195716, -2.7790216819563263, -2.791532841390251, 
  -2.8002610285791865, -2.8037687440535124, -2.815639884564145, -2.826851315579978, -2.826898969699483, -2.8378849463648463, -2.8459393302115785, -2.8464883749525836, -2.8539163210552405, 
  -2.8628792969511387, -2.8629224952512264, -2.8712842930488, -2.8760217217933697, -2.877225966193879, -2.882953693493401, -2.8868502791450554}

which I wish to interpolate smoothly, but using

int = Interpolation[data, Method -> "Spline"]

I get the followingenter image description here

So these small kinks, cusps and wiggles are supposed to disappear with Spline but they persist. I would like to have a completely "polished" curve (slowly changing derivative).

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  • 2
    $\begingroup$ It's nicer to have a semicolon at the end of data = {...}. $\endgroup$
    – Michael E2
    Jun 27 at 14:16
  • 6
    $\begingroup$ Interpolation is producing a $C^2$ (continuous second derivative) interpolant. Plot @@ {int''[x], Flatten@{x, int@"Domain"}, PlotRange -> All}. You probably want to smooth the data or fit an approximant. There have been several questions about this before. Here is a recent one with links to others: mathematica.stackexchange.com/questions/269356/… $\endgroup$
    – Michael E2
    Jun 27 at 14:22

1 Answer 1

6
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There is a misunderstanding. Interpolation goes always through the given data points and is used to obtain values in between. "Smoothing" or "Filtering" is used to smooth noise data.

One of the simplest filter is e.g. the mean filter: MeanFilter[data, r]. "r" determines how strongly the data is smoothed (however, there are also more sophistic filters). E.g.:

ListLinePlot[{MeanFilter[data, 4]}]

enter image description here

To get the derivative you need a somewhat more sophisticated filter. E.g. "LowpassFilter". If you are interessted, you can read ""tutorial/DigitalFilterDesign" in the help. Here is an example.

fun = Interpolation[LowpassFilter[data, 0.1]];
Plot[fun[x], {x, 1, Length@data}]
Plot[fun'[x], {x, 1, Length@data}]

enter image description here

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  • $\begingroup$ Thanks. It seems that with this filter there are still some isolated points where the derivative is not defined, even if the degree is increased. $\endgroup$ Jun 27 at 14:46
  • $\begingroup$ Look at the addendum to my answer. $\endgroup$ Jun 27 at 15:11

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