Can Mathematica interpolate non-uniform scatter data?
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From: Lawrence Teo on 26 Jan 2010 06:26 Hi all, Can Mathematica interpolate non-uniform scatter data? Like ListInterpolation or Interpolation functions... z = {{0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`}, {0.`, , 0.12467473338522769`, 0.18640329676226988`, 0.24740395925452294`, , 0.36627252908604757`, 0.42367625720393803`, 0.479425538604203`}, {0.`, 0.12467473338522769`, 0.24740395925452294`, 0.36627252908604757`, 0.479425538604203`, 0.5850972729404622`, 0.6816387600233341`, , 0.8414709848078965`}, {0.`, 0.18640329676226988`, 0.36627252908604757`, 0.5333026735360201`, 0.6816387600233341`, 0.806081108260693`, 0.9022675940990952`, 0.9668265566961802`, 0.9974949866040544`}, {0.`, 0.24740395925452294`, 0.479425538604203`, 0.6816387600233341`, 0.8414709848078965`, 0.9489846193555862`, 0.9974949866040544`, 0.9839859468739369`, 0.9092974268256817`}}; g = ListInterpolation[z, {{0, 1}, {0, 2}}]; g[0.5, 0.5] However, the previous interpolation will give us a lot of NULL in the expression...
From: dh on 27 Jan 2010 01:42 to do it. Anyway, here is it: now we can do the interpolation: Lawrence Teo wrote: > Hi all, > > Can Mathematica interpolate non-uniform scatter data? > Like ListInterpolation or Interpolation functions... > > z = {{0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`}, {0.`, , > 0.12467473338522769`, 0.18640329676226988`, > 0.24740395925452294`, , 0.36627252908604757`, > 0.42367625720393803`, 0.479425538604203`}, {0.`, > 0.12467473338522769`, 0.24740395925452294`, 0.36627252908604757`, > 0.479425538604203`, 0.5850972729404622`, 0.6816387600233341`, , > 0.8414709848078965`}, {0.`, 0.18640329676226988`, > 0.36627252908604757`, 0.5333026735360201`, 0.6816387600233341`, > 0.806081108260693`, 0.9022675940990952`, 0.9668265566961802`, > 0.9974949866040544`}, {0.`, 0.24740395925452294`, > 0.479425538604203`, 0.6816387600233341`, 0.8414709848078965`, > 0.9489846193555862`, 0.9974949866040544`, 0.9839859468739369`, > 0.9092974268256817`}}; > g = ListInterpolation[z, {{0, 1}, {0, 2}}]; > g[0.5, 0.5] > > However, the previous interpolation will give us a lot of NULL in the > expression... >
From: Lawrence Teo on 27 Jan 2010 06:23 Hi DH, Did you miss out anything in your reply? How to do the interpolation with non-uniform scattered data (with holes)? On Jan 27, 2:42 pm, dh <d...(a)metrohm.com> wrote: > to do it. Anyway, here is it: > > now we can do the interpolation: > > Lawrence Teo wrote: > > Hi all, > > > Can Mathematica interpolate non-uniform scatter data? > > Like ListInterpolation or Interpolation functions... > > > z = {{0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`}, {0.`, , > > 0.12467473338522769`, 0.18640329676226988`, > > 0.24740395925452294`, , 0.36627252908604757`, > > 0.42367625720393803`, 0.479425538604203`}, {0.`, > > 0.12467473338522769`, 0.24740395925452294`, 0.36627252908604757= `, > > 0.479425538604203`, 0.5850972729404622`, 0.6816387600233341`, , > > 0.8414709848078965`}, {0.`, 0.18640329676226988`, > > 0.36627252908604757`, 0.5333026735360201`, 0.6816387600233341`, > > 0.806081108260693`, 0.9022675940990952`, 0.9668265566961802`, > > 0.9974949866040544`}, {0.`, 0.24740395925452294`, > > 0.479425538604203`, 0.6816387600233341`, 0.8414709848078965`, > > 0.9489846193555862`, 0.9974949866040544`, 0.9839859468739369`, > > 0.9092974268256817`}}; > > g = ListInterpolation[z, {{0, 1}, {0, 2}}]; > > g[0.5, 0.5] > > > However, the previous interpolation will give us a lot of NULL in the > > expression...
From: dh on 29 Jan 2010 07:43 lately my post get mutilated, we are still seraching the reason. try again: to do it. Anyway, here is it: now we can do the interpolation: Lawrence Teo wrote: > On Jan 27, 2:42 pm, dh <d...(a)metrohm.com> wrote: >> to do it. Anyway, here is it: >> now we can do the interpolation: >> Lawrence Teo wrote: >>> Hi all, >>> Can Mathematica interpolate non-uniform scatter data? >>> Like ListInterpolation or Interpolation functions... >>> z = {{0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`}, {0.`, , >>> 0.12467473338522769`, 0.18640329676226988`, >>> 0.24740395925452294`, , 0.36627252908604757`, >>> 0.42367625720393803`, 0.479425538604203`}, {0.`, >>> 0.12467473338522769`, 0.24740395925452294`, 0.36627252908604757= > `, >>> 0.479425538604203`, 0.5850972729404622`, 0.6816387600233341`, , >>> 0.8414709848078965`}, {0.`, 0.18640329676226988`, >>> 0.36627252908604757`, 0.5333026735360201`, 0.6816387600233341`, >>> 0.806081108260693`, 0.9022675940990952`, 0.9668265566961802`, >>> 0.9974949866040544`}, {0.`, 0.24740395925452294`, >>> 0.479425538604203`, 0.6816387600233341`, 0.8414709848078965`, >>> 0.9489846193555862`, 0.9974949866040544`, 0.9839859468739369`, >>> 0.9092974268256817`}}; >>> g = ListInterpolation[z, {{0, 1}, {0, 2}}]; >>> g[0.5, 0.5] >>> However, the previous interpolation will give us a lot of NULL in the >>> expression... > >
From: JH on 29 Jan 2010 07:50
On 27 ene, 12:23, Lawrence Teo <lawrence...(a)yahoo.com> wrote: > Hi DH, > > Did you miss out anything in your reply? > How to do the interpolation with non-uniform scattered data (with > holes)? > > On Jan 27, 2:42 pm, dh <d...(a)metrohm.com> wrote: > > > to do it. Anyway, here is it: > > > now we can do the interpolation: > > > Lawrence Teo wrote: > > > Hi all, > > > > Can Mathematica interpolate non-uniform scatter data? > > > Like ListInterpolation or Interpolation functions... > > > > z = {{0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`, 0.`}, {0.`, , > > > 0.12467473338522769`, 0.18640329676226988`, > > > 0.24740395925452294`, , 0.36627252908604757`, > > > 0.42367625720393803`, 0.479425538604203`}, {0.`, > > > 0.12467473338522769`, 0.24740395925452294`, 0.366272529086047= 57= > `, > > > 0.479425538604203`, 0.5850972729404622`, 0.6816387600233341`,= , > > > 0.8414709848078965`}, {0.`, 0.18640329676226988`, > > > 0.36627252908604757`, 0.5333026735360201`, 0.6816387600233341= `, > > > 0.806081108260693`, 0.9022675940990952`, 0.9668265566961802`, > > > 0.9974949866040544`}, {0.`, 0.24740395925452294`, > > > 0.479425538604203`, 0.6816387600233341`, 0.8414709848078965`, > > > 0.9489846193555862`, 0.9974949866040544`, 0.9839859468739369`= , > > > 0.9092974268256817`}}; > > > g = ListInterpolation[z, {{0, 1}, {0, 2}}]; > > > g[0.5, 0.5] > > > > However, the previous interpolation will give us a lot of NULL in the > > > expression... _______________________________________________________________________ Hello, Lawrence: As far as I know, the answer is NOT YET (for more than one independent variable). The subsequent question is obvious: WHY? (I don't know, it's strange to me). When dealing with data you know we have several options: 1. "Total" interpolation (look for a unique function --generally a polynomial-- that passes thru all the points, as the Lagrange or Hermite polynomials (1D) or generalizations for more dimensions, as Shepard's. In Mathematica, for 1 indep. vble. we have InterpolatingPolynomial[]). 2. "Local" or piecewise interpolation (as the Mathematica: Interpolation[]) 3. Fit (approximation of data). In Mathematica: Fit, FindFit, ... For the first type, I have implemented some algorithms due to Shepard (in 1968, not so modern!). Here I copy my implementation of one: points= Table[xi = \[Pi] (2 Random[] - 1);yi = \[Pi] (2 Random[] - 1); {xi, yi, Sin[xi] Cos[yi]},{9}]; note: few points, please graphPoints = ListPointPlot3D[points, PlotStyle -> PointSize[0.04]]; Clear[n, vi, v]; n := Dimensions[points][[1]]; Array[vi, n]; Do[ vi[i] = (\!\( \*UnderoverscriptBox[\(\[Product]\), \(j = 1\), \(i - 1\)] \*SuperscriptBox[\(( \*SuperscriptBox[\((x - \(points[[j]]\)[[1]])\), \(2\)] + \*SuperscriptBox[\((y - \(points[[j]]\)[[ 2]])\), \(2\)])\), \(2\)]\)) (\!\( \*UnderoverscriptBox[\(\[Product]\), \(j = i + 1\), \(n\)] \*SuperscriptBox[\(( \*SuperscriptBox[\((x - \(points[[j]]\)[[1]])\), \(2\)] + \*SuperscriptBox[\((y - \(points[[j]]\)[[ 2]])\), \(2\)])\), \(2\)]\)), {i, 1, n}]; v = \!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(vi[i]\)\); funcionShepard2[xx_, yy_] := \!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\((\(points[[i]]\)[[3]]\ \*FractionBox[\(vi[i]\), \(v\)])\)\) /. {x -> xx, y -> yy} graphSup = Plot3D[funcionShepard2[xx, yy], {xx, -\[Pi], \[Pi]}, {yy, -\[Pi], \[Pi]}, PlotRange -> {{-\[Pi], \[Pi]}, {-\[Pi], \[Pi]}, {-1, 1}}, ClippingStyle -> None] Show[graphPoints, graphSup] _____________________ For nD (n>1) Piecewise Interpolation of scattered (irregular) data, one way is triangulation (in Mathematica we have DelaunayTriangulation [], see Computational Geometry Package). But the interpolation algorithm is not included in Mathem (I think). _____________________ In a way, we can say Mathem. DOES HAVE (local) interpolation of scattered data (2 indep. vbles), in List3DPlot[...], we can choose the interpolation order, and Mathem. interpolates for a pretty drawing: BUT THEY DON'T LET US USING THAT INTERPOLATION (Why???) _____________________ I hope this has been useful for you. JH |