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mathematical optimization - Reordering numerically calculated eigenvalues assuming smooth dependence on a parameter


As was discussed in a different question here on SE (link) if you compute eigenvalues numerically of a matrix which depends on a parameter y, the resulting plot of the eigenvalues will yield non-smooth plots. Assuming that I know that my eigenvalues will transform smoothly under the change of y I would like to reorder them, so that when I plot them the color coding is correct.


Secondly I would also like to apply the reordering to the eigenvectors as I would like to have a look at how they transform under changes of y, too.


In this link george2079 does already partially answer my question by applying a minimization procedure which punishes large curvature via a cost function. However, I have trouble adapting it to my situation and I am not sure if this is the best approach (so if you have a better idea feel free to post it as an answer).



My Matrix:


m = {{-0.576 Cos[y], 0, 0, 0. + 0.06 I, 0, 0.369858, 0, 0, 0,    0, -0.0906385, 0, 0, -0.265868, 0.0366083, 0.0157771, -0.185737,    0.0349767, 0.0435434, -0.276945, (0.288 - 0.498831 I) Sin[y],    0. - 0.03 I, 0.03, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    0}, {0, -0.576 Cos[y], 0. - 0.03 I, 0, 0, 0, -0.170921, 0, 0, 0, 0,    0.234664, 0.234164, 0, 0, -0.185737, 0.205344, -0.239136, -0.249447,    0.143353, 0. + 0.03 I, (0.288 - 0.498831 I) Sin[y], 0, -0.03,    0.0519615, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0,    0. + 0.03 I, -0.576 Cos[y], 0, 0, 0, 0, 0.369858, 0, 0, 0,    0.234164, -0.0357261, 0, 0, 0.0349767, -0.239136, -0.0707859,    0.185737, 0.248295, -0.03, 0, (0.288 - 0.498831 I) Sin[y],    0. - 0.03 I, 0. - 0.0519615 I, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    0, 0, 0}, {0. - 0.06 I, 0, 0, -0.576 Cos[y], 0, 0, 0,    0, -0.244138, -0.0422717, -0.265868, 0, 0, 0.216359, 0.0211358,    0.0435434, -0.249447, 0.185737, 0.0660567, 0.159894, 0, 0.03,    0. + 0.03 I, (0.288 - 0.498831 I) Sin[y], 0, 0, 0, 0, 0, 0, 0, 0, 0,    0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, -0.576 Cos[y], 0, 0,    0, -0.0422717, -0.195326, 0.0366083, 0, 0,    0.0211358, -0.195326, -0.276945, 0.143353, 0.248295, 0.159894,    0.293945, 0, -0.0519615, 0. + 0.0519615 I,    0, (0.288 - 0.498831 I) Sin[y], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    0, 0, 0}, {0.369858, 0, 0, 0, 0, -0.576 Cos[y], 0, 0, 0. + 0.06 I,    0, -0.0906385, 0, 0, 0.265868, -0.0366083, 0.0157771, 0.185737,    0.0349767, -0.0435434, 0.276945, 0, 0, 0, 0,    0, (0.288 + 0.498831 I) Sin[y], 0. - 0.03 I, 0.03, 0, 0, 0, 0, 0, 0,    0, 0, 0, 0, 0, 0}, {0, -0.170921, 0, 0, 0, 0, -0.576 Cos[y],    0. - 0.03 I, 0, 0, 0, 0.234664, -0.234164, 0, 0, 0.185737, 0.205344,    0.239136, -0.249447, 0.143353, 0, 0, 0, 0, 0,    0. + 0.03 I, (0.288 + 0.498831 I) Sin[y], 0, -0.03, 0.0519615, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0.369858, 0, 0, 0,    0. + 0.03 I, -0.576 Cos[y], 0, 0, 0, -0.234164, -0.0357261, 0, 0,    0.0349767, 0.239136, -0.0707859, -0.185737, -0.248295, 0, 0, 0, 0,    0, -0.03, 0, (0.288 + 0.498831 I) Sin[y], 0. - 0.03 I,    0. - 0.0519615 I, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0,    0, -0.244138, -0.0422717, 0. - 0.06 I, 0, 0, -0.576 Cos[y], 0,    0.265868, 0, 0, 0.216359,    0.0211358, -0.0435434, -0.249447, -0.185737, 0.0660567, 0.159894, 0,    0, 0, 0, 0, 0, 0.03, 0. + 0.03 I, (0.288 + 0.498831 I) Sin[y], 0,    0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, -0.0422717, -0.195326, 0,    0, 0, 0, -0.576 Cos[y], -0.0366083, 0, 0, 0.0211358, -0.195326,    0.276945, 0.143353, -0.248295, 0.159894, 0.293945, 0, 0, 0, 0, 0,    0, -0.0519615, 0. + 0.0519615 I, 0, (0.288 + 0.498831 I) Sin[y], 0,    0, 0, 0, 0, 0, 0, 0, 0, 0}, {-0.0906385, 0, 0, -0.265868,    0.0366083, -0.0906385, 0, 0, 0.265868, -0.0366083, -0.576 Cos[y], 0,    0, 0. + 0.06 I, 0, 0.0911964, 0, 0.356683, 0, 0, 0, 0, 0, 0, 0, 0,    0, 0, 0, 0, -0.576 Sin[y], 0. - 0.03 I, 0.03, 0, 0, 0, 0, 0, 0,    0}, {0, 0.234664, 0.234164, 0, 0, 0, 0.234664, -0.234164, 0, 0,    0, -0.576 Cos[y], 0. - 0.03 I, 0, 0, 0, -0.208851, 0,    0.0722586, -0.286706, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    0. + 0.03 I, -0.576 Sin[y], 0, -0.03, 0.0519615, 0, 0, 0, 0, 0}, {0,    0.234164, -0.0357261, 0, 0, 0, -0.234164, -0.0357261, 0, 0, 0,    0. + 0.03 I, -0.576 Cos[y], 0, 0, 0.356683, 0, 0.343409, 0, 0, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0, -0.03, 0, -0.576 Sin[y], 0. - 0.03 I,    0. - 0.0519615 I, 0, 0, 0, 0, 0}, {-0.265868, 0, 0, 0.216359,    0.0211358, 0.265868, 0, 0, 0.216359, 0.0211358, 0. - 0.06 I, 0,    0, -0.576 Cos[y], 0, 0, 0.0722586, 0, -0.00936269, -0.319789, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0, 0, 0.03, 0. + 0.03 I, -0.576 Sin[y], 0, 0,    0, 0, 0, 0}, {0.0366083, 0, 0, 0.0211358, -0.195326, -0.0366083, 0,    0, 0.0211358, -0.195326, 0, 0, 0, 0, -0.576 Cos[y], 0, -0.286706,    0, -0.319789, 0.293945, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -0.0519615,    0. + 0.0519615 I, 0, -0.576 Sin[y], 0, 0, 0, 0,    0}, {0.0157771, -0.185737, 0.0349767, 0.0435434, -0.276945,    0.0157771, 0.185737, 0.0349767, -0.0435434, 0.276945, 0.0911964, 0,    0.356683, 0, 0, 1.02645, 0, 0, 0. + 0.53 I, 0, 0, 0, 0, 0, 0, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0, 0, 0. - 0.265 I, 0.265, 0, 0}, {-0.185737,    0.205344, -0.239136, -0.249447, 0.143353, 0.185737, 0.205344,    0.239136, -0.249447, 0.143353, 0, -0.208851, 0,    0.0722586, -0.286706, 0, 1.02645, 0. - 0.265 I, 0, 0, 0, 0, 0, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. + 0.265 I, 0, 0, -0.265,    0.458993}, {0.0349767, -0.239136, -0.0707859, 0.185737, 0.248295,    0.0349767, 0.239136, -0.0707859, -0.185737, -0.248295, 0.356683, 0,    0.343409, 0, 0, 0, 0. + 0.265 I, 1.02645, 0, 0, 0, 0, 0, 0, 0, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0, -0.265, 0, 0, 0. - 0.265 I,    0. - 0.458993 I}, {0.0435434, -0.249447, 0.185737, 0.0660567,    0.159894, -0.0435434, -0.249447, -0.185737, 0.0660567, 0.159894, 0,    0.0722586, 0, -0.00936269, -0.319789, 0. - 0.53 I, 0, 0, 1.02645, 0,    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.265,    0. + 0.265 I, 0, 0}, {-0.276945, 0.143353, 0.248295, 0.159894,    0.293945, 0.276945, 0.143353, -0.248295, 0.159894, 0.293945,    0, -0.286706, 0, -0.319789, 0.293945, 0, 0, 0, 0, 1.02645, 0, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -0.458993, 0. + 0.458993 I,    0, 0}, {(0.288 + 0.498831 I) Sin[y], 0. - 0.03 I, -0.03, 0, 0, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.576 Cos[y], 0, 0,    0. - 0.06 I, 0, 0.369858, 0, 0, 0, 0, -0.0906385, 0, 0, -0.265868,    0.0366083, 0.0157771, -0.185737, 0.0349767,    0.0435434, -0.276945}, {0. + 0.03 I, (0.288 + 0.498831 I) Sin[y], 0,    0.03, -0.0519615, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    0.576 Cos[y], 0. + 0.03 I, 0, 0, 0, -0.170921, 0, 0, 0, 0, 0.234664,    0.234164, 0, 0, -0.185737, 0.205344, -0.239136, -0.249447,    0.143353}, {0.03, 0, (0.288 + 0.498831 I) Sin[y], 0. - 0.03 I,    0. - 0.0519615 I, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    0. - 0.03 I, 0.576 Cos[y], 0, 0, 0, 0, 0.369858, 0, 0, 0,    0.234164, -0.0357261, 0, 0, 0.0349767, -0.239136, -0.0707859,    0.185737, 0.248295}, {0, -0.03,    0. + 0.03 I, (0.288 + 0.498831 I) Sin[y], 0, 0, 0, 0, 0, 0, 0, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0. + 0.06 I, 0, 0, 0.576 Cos[y], 0, 0, 0,    0, -0.244138, -0.0422717, -0.265868, 0, 0, 0.216359, 0.0211358,    0.0435434, -0.249447, 0.185737, 0.0660567, 0.159894}, {0, 0.0519615,    0. + 0.0519615 I, 0, (0.288 + 0.498831 I) Sin[y], 0, 0, 0, 0, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.576 Cos[y], 0, 0,    0, -0.0422717, -0.195326, 0.0366083, 0, 0,    0.0211358, -0.195326, -0.276945, 0.143353, 0.248295, 0.159894,    0.293945}, {0, 0, 0, 0, 0, (0.288 - 0.498831 I) Sin[y],    0. - 0.03 I, -0.03, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.369858, 0,    0, 0, 0, 0.576 Cos[y], 0, 0, 0. - 0.06 I, 0, -0.0906385, 0, 0,    0.265868, -0.0366083, 0.0157771, 0.185737, 0.0349767, -0.0435434,    0.276945}, {0, 0, 0, 0, 0, 0. + 0.03 I, (0.288 - 0.498831 I) Sin[y],    0, 0.03, -0.0519615, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -0.170921, 0,    0, 0, 0, 0.576 Cos[y], 0. + 0.03 I, 0, 0, 0, 0.234664, -0.234164,    0, 0, 0.185737, 0.205344, 0.239136, -0.249447, 0.143353}, {0, 0, 0,    0, 0, 0.03, 0, (0.288 - 0.498831 I) Sin[y], 0. - 0.03 I,    0. - 0.0519615 I, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.369858, 0,    0, 0, 0. - 0.03 I, 0.576 Cos[y], 0, 0, 0, -0.234164, -0.0357261, 0,    0, 0.0349767, 0.239136, -0.0707859, -0.185737, -0.248295}, {0, 0, 0,    0, 0, 0, -0.03, 0. + 0.03 I, (0.288 - 0.498831 I) Sin[y], 0, 0, 0,    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -0.244138, -0.0422717, 0. + 0.06 I,    0, 0, 0.576 Cos[y], 0, 0.265868, 0, 0, 0.216359,    0.0211358, -0.0435434, -0.249447, -0.185737, 0.0660567,    0.159894}, {0, 0, 0, 0, 0, 0, 0.0519615, 0. + 0.0519615 I,    0, (0.288 - 0.498831 I) Sin[y], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    0, -0.0422717, -0.195326, 0, 0, 0, 0, 0.576 Cos[y], -0.0366083, 0,    0, 0.0211358, -0.195326, 0.276945, 0.143353, -0.248295, 0.159894,    0.293945}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -0.576 Sin[y],    0. - 0.03 I, -0.03, 0, 0, 0, 0, 0, 0, 0, -0.0906385, 0,    0, -0.265868, 0.0366083, -0.0906385, 0, 0, 0.265868, -0.0366083,    0.576 Cos[y], 0, 0, 0. - 0.06 I, 0, 0.0911964, 0, 0.356683, 0,    0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. + 0.03 I, -0.576 Sin[y], 0,    0.03, -0.0519615, 0, 0, 0, 0, 0, 0, 0.234664, 0.234164, 0, 0, 0,    0.234664, -0.234164, 0, 0, 0, 0.576 Cos[y], 0. + 0.03 I, 0, 0,    0, -0.208851, 0, 0.0722586, -0.286706}, {0, 0, 0, 0, 0, 0, 0, 0, 0,    0, 0.03, 0, -0.576 Sin[y], 0. - 0.03 I, 0. - 0.0519615 I, 0, 0, 0,    0, 0, 0, 0.234164, -0.0357261, 0, 0, 0, -0.234164, -0.0357261, 0, 0,    0, 0. - 0.03 I, 0.576 Cos[y], 0, 0, 0.356683, 0, 0.343409, 0,    0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -0.03,    0. + 0.03 I, -0.576 Sin[y], 0, 0, 0, 0, 0, 0, -0.265868, 0, 0,    0.216359, 0.0211358, 0.265868, 0, 0, 0.216359, 0.0211358,    0. + 0.06 I, 0, 0, 0.576 Cos[y], 0, 0, 0.0722586,    0, -0.00936269, -0.319789}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    0.0519615, 0. + 0.0519615 I, 0, -0.576 Sin[y], 0, 0, 0, 0, 0,    0.0366083, 0, 0, 0.0211358, -0.195326, -0.0366083, 0, 0,    0.0211358, -0.195326, 0, 0, 0, 0, 0.576 Cos[y], 0, -0.286706,    0, -0.319789, 0.293945}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,    0, 0, 0. - 0.265 I, -0.265, 0, 0, 0.0157771, -0.185737, 0.0349767,    0.0435434, -0.276945, 0.0157771, 0.185737, 0.0349767, -0.0435434,    0.276945, 0.0911964, 0, 0.356683, 0, 0, 1.02645, 0, 0, 0. - 0.53 I,    0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. + 0.265 I, 0,    0, 0.265, -0.458993, -0.185737, 0.205344, -0.239136, -0.249447,    0.143353, 0.185737, 0.205344, 0.239136, -0.249447, 0.143353,    0, -0.208851, 0, 0.0722586, -0.286706, 0, 1.02645, 0. + 0.265 I, 0,    0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.265, 0, 0,    0. - 0.265 I, 0. - 0.458993 I, 0.0349767, -0.239136, -0.0707859,    0.185737, 0.248295, 0.0349767,    0.239136, -0.0707859, -0.185737, -0.248295, 0.356683, 0, 0.343409,    0, 0, 0, 0. - 0.265 I, 1.02645, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0,    0, 0, 0, 0, 0, 0, 0, -0.265, 0. + 0.265 I, 0, 0,    0.0435434, -0.249447, 0.185737, 0.0660567,    0.159894, -0.0435434, -0.249447, -0.185737, 0.0660567, 0.159894, 0,    0.0722586, 0, -0.00936269, -0.319789, 0. + 0.53 I, 0, 0, 1.02645,    0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.458993,    0. + 0.458993 I, 0, 0, -0.276945, 0.143353, 0.248295, 0.159894,    0.293945, 0.276945, 0.143353, -0.248295, 0.159894, 0.293945,    0, -0.286706, 0, -0.319789, 0.293945, 0, 0, 0, 0, 1.02645}}
(*empty line*)

 ListPlot[Transpose@Table[Eigenvalues[m], {y, -Pi, Pi, .01}]]

enter image description here


Original code (m is a 3x3 matrix):


alle = Table[Eigenvalues[m], {y, -Pi, Pi, .01}];
original = Show[

             MapIndexed[ListPlot[Flatten[Take[alle, All, {#}] , 2],
                                 Joined -> True, PlotRange -> All, 
                                 PlotStyle -> Hue[First@#2/3]] &, Range[3]]];
v = Table[V[i], {i, Length[alle]}];
SetOptions[NMinimize, MaxIterations -> 500];
Do[
  alle = MapThread[ Prepend[  Drop[ #2 , {#1}]  , #2[[#1]] ]  &,
       { (v /. Last@Minimize[{Total[((#[[1]] + #[[3]] - 2 #[[2]])^2 & /@
    Partition[ Indexed @@@ Transpose[{alle, v}], 3, 1, {1, 3}, {}])],
          Table[k <= V[i] <= 3, {i, Length[alle]}]}, v, Integers]) , 

    alle}] , {k, 2}];
Show[original, 
     ListPlot[Flatten@Take[alle, All, {#}], Joined -> True, 
              PlotStyle -> {Thick, Black, Dashed}] & /@ Range[3]]

My adoption (m is my matrix):


alle = Table[Eigenvalues[m], {y, -Pi, Pi, .01}];
original = Show[
             MapIndexed[ListPlot[Flatten[Take[alle, All, {#}] , 2],
                                 Joined -> True, PlotRange -> All, 

                                 PlotStyle -> Hue[First@#2/40]] &, Range[40]]];
v = Table[V[i], {i, Length[alle]}];
SetOptions[NMinimize, MaxIterations -> 500];
Do[
  alle = MapThread[ Prepend[  Drop[ #2 , {#1}]  , #2[[#1]] ]  &,
       { (v /. Last@Minimize[{Total[((#[[1]] + #[[3]] - 2 #[[2]])^2 & /@
    Partition[ Indexed @@@ Transpose[{alle, v}], 40, 1, {1, 40}, {}])],
          Table[k <= V[i] <= 40, {i, Length[alle]}]}, v, Integers]) , 
    alle}] , {k, 40-1}];
Show[original, 

     ListPlot[Flatten@Take[alle, All, {#}], Joined -> True, 
              PlotStyle -> {Thick, Black, Dashed}] & /@ Range[40]]

My Questions:




  1. Can someone explain george2079's code, because I still have trouble to figure out exactly what is going on?




    • What is the v table for?





    • I understand the idea of the cost function but how is it used to reorder the eigenvalues?





  2. Is my adoption of the 3x3 example correct for my 40x40 example?

  3. How do I add my eigenvectors so that I still know at the end which eigenvectors belongs to which eigenvalue?

  4. Does the above code work on your system and if it does how long does the job take to finish?

  5. If it does not finish within reasonable time (for me that is < 40min): Is there a way to increase performance?



Note If you have a better idea how to solve my problem, this is of course also great. I am not bound in any way to use the above code.



Answer



Apologies for the procedural approach, but this seems to work. Trace out one line at a time, using Interpolation to extrapolate where the next point is likely to be:


c = {};
frames = {};
xvals = Pi Range[-1, 1, 1/100];
alle = Table[({y, #} & /@ Eigenvalues[m]), {y, xvals}] ;
colors = Table[Hue[RandomReal[{0, 2/3}]], {Length@m}];
Clear[g];

Monitor[
 Do[
  line =
    First@Position[alle, First@SortBy[#, #[[2]] &]] & /@ alle[[;; 2]];
  MapIndexed[(nextx = #;
    proj = {nextx, 
      Quiet[ Interpolation[alle[[Sequence @@ #]] & /@ line]@nextx ]};
  AppendTo[line, Position[alle, First@Nearest[alle[[First@#2 + 2]],
      proj]][[1]]] ) &, xvals[[3 ;;]]];
 AppendTo[c, alle[[Sequence @@ #]] & /@ line];

 alle = Delete[alle, line];
 AppendTo[frames, 
     g = Show[ { 
      If[Length@First@alle > 0, ListPlot[Flatten[alle, 1]], 
       Graphics[]] ,
      Graphics[ 
        MapIndexed[{Thick, colors[[First@#2]], Line[#]} &, c]]}, 
        PlotRange -> {-2, 2}, AxesOrigin -> {0, 0}]],
         {nrem, Length@m, 1, -1}], g]


enter image description here


enter image description here


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plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]
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plotting - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1.
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