# I need to extract data from a multi-valued functions and interpolate them into separate single values functions

The graphed data look like this. I need to extract the data for each curve individually and use Interpolation to create a differentiable function for each curve. Can you show how to do this?

I will need to select data that falls between certain ranges. Ideally, I need to define a function xmin[t] and xmax[t] and select all data that falls between those limits.

Here is the data:

{{0.0001, {3.33812, 0.291642, 1.25777, 2.22443}}, {0.0002, {3.4381,
0.320099, 1.28631, 2.25285}}, {0.0003, {3.53812, 0.341598, 2.27456,
1.3081}}, {0.0004, {3.6381, 0.358445, 2.29152,
1.32505}}, {0.0005, {3.73812, 0.372028, 2.30516,
1.33859}}, {0.0006, {3.8381, 2.31621, 1.3496,
0.383104}}, {0.0007, {3.93812, 2.32534, 1.35872,
0.392211}}, {0.0008, {4.03811, 2.33303, 1.36642,
0.399914}}, {0.0009, {4.13811, 2.33961, 1.37303,
0.406549}}, {0.001, {4.23811, 2.34531, 1.37876,
0.412299}}, {0.0011, {4.33811, 2.35031, 1.38379,
0.417339}}, {0.0012, {4.43812, 2.35474, 0.421802,
1.38826}}, {0.0013, {4.5381, 1.39232, 2.3587,
0.425801}}, {0.0014, {4.63812, 1.39593, 2.36227,
0.429397}}, {0.0015, {4.7381, 1.39919, 2.3655,
0.432651}}, {0.0016, {4.83812, 1.40215, 2.36846,
0.435615}}, {0.0017, {4.9381, 1.40487, 2.37117,
0.438333}}, {0.0018, {5.03811, 1.40737, 2.37368,
0.440839}}, {0.0019, {5.1381, 1.40969, 2.37601,
0.443162}}, {0.002, {5.23811, 1.41184, 2.37819,
0.445325}}, {0.0021, {5.33811, 1.41385, 2.38022,
0.447348}}, {0.0022, {5.43811, 1.41574, 2.38214,
0.449249}}, {0.0023, {5.53812, 1.41752, 2.38395,
0.451041}}, {0.0024, {5.6381, 1.41921, 2.38566,
0.452738}}, {0.0025, {5.73812, 1.42081, 2.38729,
0.45435}}, {0.0026, {2.38885, 5.8381, 1.42233,
0.455887}}, {0.0027, {2.39031, 5.93812, 1.42379,
0.457356}}, {0.0028, {6.0381, 1.42518, 0.458765,
2.39172}}, {0.0029, {6.13811, 1.42652, 0.460121,
2.39308}}, {0.003, {6.23811, 1.42781, 0.461429,
2.39438}}, {0.0031, {6.33811, 1.42906, 0.462694,
2.39564}}, {0.0032, {6.43812, 1.43028, 0.46392,
2.39686}}, {0.0033, {6.5381, 1.43145, 0.465113,
2.39805}}, {0.0034, {6.63812, 1.4326, 0.466276,
2.39921}}, {0.0035, {6.7381, 0.467414, 1.43372,
2.40034}}, {0.0036, {6.83812, 0.468534, 1.43482,
2.40144}}, {0.0037, {6.9381, 0.469631, 1.43589,
2.40252}}, {0.0038, {7.03812, 0.470709, 1.43695,
2.40359}}, {0.0039, {7.1381, 0.471771, 1.43799,
2.40464}}, {0.004, {7.23811, 0.472818, 1.43902,
2.40567}}, {0.0041, {7.33811, 0.473854, 1.44004,
2.4067}}, {0.0042, {7.43811, 0.474881, 1.44105,
2.40771}}, {0.0043, {7.53812, 0.4759, 1.44206,
2.40872}}, {0.0044, {7.63811, 0.476915, 1.44306,
2.40973}}, {0.0045, {7.73812, 0.477927, 1.44406,
2.41074}}, {0.0046, {7.8381, 0.478939, 1.44506,
2.41174}}, {0.0047, {7.93812, 0.479952, 1.44606,
2.41274}}, {0.0048, {8.0381, 0.480969, 1.44707,
2.41375}}, {0.0049, {8.13812, 0.481992, 1.44809,
2.41477}}, {0.005, {8.2381, 0.483024, 1.44911,
2.41579}}, {0.0051, {8.33811, 0.484065, 1.45015,
2.41682}}, {0.0052, {8.43811, 0.48512, 1.45119,
2.41787}}, {0.0053, {8.53811, 0.486191, 1.45226,
2.41894}}, {0.0054, {8.63811, 0.487279, 1.45335,
2.42002}}, {0.0055, {8.7381, 0.488389, 1.45445,
2.42112}}, {0.0056, {0.489522, 1.45559, 2.42225,
8.83811}}, {0.0057, {0.490683, 1.45675, 2.42341,
8.93811}}, {0.0058, {0.491875, 1.45794,
2.42459}}, {0.0059, {0.493101, 1.45917,
2.42582}}, {0.006, {0.494366, 1.46044,
2.42708}}, {0.0061, {0.495675, 1.46175,
2.42839}}, {0.0062, {0.497032, 1.46312,
2.42974}}, {0.0063, {0.498444, 1.46454,
2.43115}}, {0.0064, {0.499918, 1.46603,
2.43263}}, {0.0065, {0.50146, 1.46759,
2.43419}}, {0.0066, {0.503079, 1.46923,
2.43582}}, {0.0067, {0.504785, 1.47096,
2.43755}}, {0.0068, {0.506588, 1.47279,
2.43937}}, {0.0069, {0.508502, 1.47473,
2.4413}}, {0.007, {0.510541, 1.47681,
2.44336}}, {0.0071, {0.512723, 1.47904,
2.44556}}, {0.0072, {1.48141, 0.515068,
2.44792}}, {0.0073, {0.517602, 2.45048,
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1.49312}}, {0.0077, {0.53032, 2.46327,
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1.50551}}, {0.008, {0.544197, 2.47719,
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1.52359}}, {0.0083, {2.49817, 1.53163,
0.56518}}, {0.0084, {2.50771, 1.54117,
0.574717}}, {0.0085, {2.5192, 1.55267,
0.586233}}, {0.0086, {2.53335, 0.600412,
1.56688}}, {0.0087, {1.58491, 2.55123,
0.61839}}, {0.0088, {1.60793, 0.641552,
2.57447}}, {0.0089, {0.672227, 1.63838,
2.60505}}, {0.009, {0.713472, 2.64647,
1.67997}}, {0.0091, {2.70345, 1.73691,
0.770497}}, {0.0092, {2.78061, 0.847133,
1.81329}}, {0.0093, {2.87386, 1.90741,
0.940996}}, {0.0094, {2.97355, 1.03865, 2.00498, 0.0873681,
0.0488886}}, {0.0095, {3.07352, 2.08944, 0.156519, 1.123,
0.0220564}}, {0.0096, {3.17355, 2.15397, 1.18737,
0.22091}}, {0.0097, {2.20105, 3.27352, 1.23452,
0.268062}}, {0.0098, {3.37354, 0.302724, 1.26878,
2.23542}}, {0.0099, {3.47353, 0.328366, 2.26121,
1.29471}}, {0.01, {3.57354, 0.348013, 2.28102,
1.31458}}, {0.0101, {3.67353, 0.363577, 2.29666,
1.33018}}, {0.0102, {3.77354, 0.376234, 2.30933,
1.34274}}, {0.0103, {3.87354, 2.31963, 1.35302,
0.386513}}, {0.0104, {3.97353, 2.32821, 1.36159,
0.395083}}, {0.0105, {4.07355, 2.33548, 1.36888,
0.402378}}, {0.0106, {4.17353, 2.34172, 1.37515,
0.408677}}, {0.0107, {4.27355, 2.34716, 1.38061,
0.414158}}, {0.0108, {4.37353, 1.38543, 2.35194,
0.41898}}, {0.0109, {4.47354, 2.35619, 0.423268,
1.38972}}, {0.011, {4.57353, 1.39364, 2.36001,
0.427118}}, {0.0111, {4.67354, 1.39712, 2.36345,
0.430586}}, {0.0112, {4.77353, 1.40027, 2.36658,
0.433732}}, {0.0113, {4.87354, 1.40314, 2.36945,
0.436605}}, {0.0114, {4.97354, 1.40578, 2.37209,
0.439244}}, {0.0115, {5.07353, 1.40821, 2.37453,
0.441682}}, {0.0116, {5.17355, 1.41047, 2.3768,
0.443945}}, {0.0117, {5.27353, 1.41257, 2.37892,
0.446057}}, {0.0118, {5.37355, 1.41454, 2.38091,
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2.22647}}, {0.0292, {3.44438, 0.321624, 1.28786,
2.2544}}, {0.0293, {3.5444, 0.342775, 2.27575,
1.30929}}, {0.0294, {3.64438, 0.359383, 2.29246,
1.32599}}, {0.0295, {3.7444, 0.372795, 2.30592,
1.33935}}, {0.0296, {3.84438, 2.31683, 1.35022,
0.383725}}, {0.0297, {3.9444, 2.32586, 1.35924,
0.392733}}, {0.0298, {4.04439, 2.33348, 1.36687,
0.40036}}, {0.0299, {4.14439, 2.33999, 1.37341,
0.406934}}, {0.03, {4.2444, 2.34565, 1.37909,
0.412635}}, {0.0301, {4.34439, 2.35061, 1.38409,
0.417635}}, {0.0302, {4.4444, 2.355, 0.422066,
1.38852}}, {0.0303, {4.54438, 1.39256, 2.35894,
0.426039}}, {0.0304, {4.6444, 1.39614, 2.36248,
0.429611}}, {0.0305, {4.74438, 1.39938, 2.3657,
0.432845}}, {0.0306, {4.8444, 1.40233, 2.36864,
0.435793}}, {0.0307, {4.94439, 1.40503, 2.37134,
0.438497}}, {0.0308, {5.04439, 1.40752, 2.37383,
0.44099}}, {0.0309, {5.14439, 1.40983, 2.37615,
0.443302}}, {0.031, {5.24439, 1.41197, 2.37832,
0.445456}}, {0.0311, {5.3444, 1.41398, 2.38035,
0.447471}}, {0.0312, {5.44439, 1.41586, 2.38225,
0.449364}}, {0.0313, {5.5444, 1.41763, 2.38406,
0.451151}}, {0.0314, {5.64438, 1.41931, 2.38577,
0.452842}}, {0.0315, {5.7444, 1.4209, 2.38739,
0.454449}}, {0.0316, {2.38894, 5.84438, 1.42242,
0.455981}}, {0.0317, {2.39041, 5.9444, 1.42387,
0.457446}}, {0.0318, {6.04439, 1.42527, 0.458852,
2.39181}}, {0.0319, {6.14439, 1.4266, 0.460205,
2.39316}}, {0.032, {6.24439, 1.42789, 0.461509,
2.39446}}, {0.0321, {6.34439, 1.42914, 0.462772,
2.39572}}, {0.0322, {6.4444, 1.43035, 0.463996,
2.39694}}, {0.0323, {6.54438, 1.43153, 0.465187,
2.39813}}, {0.0324, {6.6444, 1.43267, 0.466348,
2.39928}}, {0.0325, {6.74438, 0.467485, 1.43379,
2.40041}}, {0.0326, {6.8444, 0.468603, 1.43488,
2.40151}}, {0.0327, {6.94438, 0.4697, 1.43596,
2.40259}}, {0.0328, {7.0444, 0.470777, 1.43702,
2.40365}}, {0.0329, {7.14439, 0.471837, 1.43806,
2.4047}}, {0.033, {7.24439, 0.472884, 1.43909,
2.40574}}, {0.0331, {7.3444, 0.473919, 1.4401,
2.40676}}, {0.0332, {7.44439, 0.474945, 1.44112,
2.40778}}, {0.0333, {7.5444, 0.475964, 1.44212,
2.40879}}, {0.0334, {7.64439, 0.476978, 1.44312,
2.4098}}, {0.0335, {7.7444, 0.477991, 1.44412,
2.4108}}, {0.0336, {7.84438, 0.479002, 1.44512, 2.4118}}}


I need to find data that falls within certain ranges. Can you show me how to separate out the data for each curve in the graph individually?

• It sounds like you already know what you need to do for this one (i.e. use Interpolation). Have you tried using Interpolation on each part of your dataset? Nov 4, 2019 at 23:35

Here's a start:

data = CloudGet[
"https://www.wolframcloud.com/obj/55c838a4-7856-4dba-9e50-3317fdb0f33b"];

(curves =
Flatten[Thread /@ data, 1][[#]] & /@
FindCurvePath[
1]]) // ListLinePlot


fns = Interpolation /@ DeleteDuplicatesBy[First] /@ curves;
ListLinePlot[fns]


There are some problems with duplicated abscissae, but they can probably be handled by hand. Inspect:

curves[[{5, 6}]]
curves[[{8, 7}]]

• Thanks, this works. Can you help me - where do I find documentation on this Thread /@ data, 1][[#]] & /@ Nov 5, 2019 at 1:56
• @Klandgren You can look up Thread and Function and Map. You can also enter & and /@ into the search bar of the documentation, or search this Q&A for the shortcut symbols: mathematica.stackexchange.com/a/25616/4999 Nov 5, 2019 at 2:00
• Perfect, thanks. I am re-writing into those commands that I understand better. You did some slick Mathematica programming. Nov 5, 2019 at 2:43