9
$\begingroup$

I want to trace the curve for the given data set.

This is my code

     pts = {{465.302, 247.796}, {100.732, 247.01}, {119.224, 42.4657}, {325.255, 
  17.7339}, {318.756, 283.584}, {499.567, 63.6977}, {310.721, 
  287.444}, {26.9471, 180.403}, {395.927, 273.221}, {368.045, 
  271.637}, {394.756, 24.373}, {187.579, 19.9764}, {379.001, 
  17.2288}, {545.629, 105.618}, {550.008, 137.879}, {388.283, 
  271.924}, {283.586, 11.0365}, {145.416, 258.876}, {9.03499, 
  134.851}, {534.812, 102.146}, {178.031, 268.363}, {447.15, 
  257.755}, {349.99, 276.643}, {73.4742, 56.8668}, {544.916, 
  101.512}, {288.193, 280.948}, {42.0569, 213.622}, {186.703, 
  269.109}, {378.495, 274.047}, {228.13, 281.019}, {51.1296, 
  72.1907}, {147.055, 268.347}, {45.4008, 206.851}, {208.656, 
  275.126}, {469.567, 248.686}, {308.452, 279.432}, {40.7974, 
  208.963}, {337.657, 282.141}, {443.21, 42.952}, {449.687, 
  258.946}, {311.266, 276.839}, {152.425, 269.8}, {419.123, 
  33.8306}, {79.9482, 53.6496}, {276.149, 9.86281}, {272.033, 
  7.11489}, {97.5072, 49.8366}, {262.92, 281.142}, {25.6081, 
  106.191}, {379.908, 274.569}, {543.667, 98.5575}, {160.077, 
  27.2338}, {398.438, 19.9315}, {265.113, 277.998}, {352.039, 
  11.3923}, {223.75, 279.167}, {550.373, 146.266}, {361.002, 
  11.6061}, {381.973, 17.7216}, {524.548, 206.913}, {47.1688, 
  82.1359}, {400.572, 265.738}, {302.766, 284.17}, {24.9839, 
  113.105}, {386.87, 267.605}, {543.044, 110.881}, {544.145, 
  118.1}, {72.178, 228.569}, {201.287, 17.4099}, {537.593, 
  96.732}, {245.844, 282.46}, {125.95, 35.7945}, {126.276, 
  35.1486}, {283.323, 12.9579}, {432.03, 32.0569}, {490.156, 
  229.188}, {187.804, 270.442}, {442.752, 40.183}, {410.278, 
  263.768}, {21.8946, 179.793}, {154.48, 23.3589}, {69.2665, 
  225.007}, {376.301, 276.076}, {466.387, 52.4701}, {520.09, 
  77.7554}, {189.943, 271.459}, {124.29, 250.907}, {281.935, 
  11.6622}, {247.094, 8.95544}, {389.127, 17.8674}, {68.6922, 
  229.083}, {30.9424, 191.098}, {161.086, 31.3124}, {216.199, 
  17.6549}, {475.856, 238.632}, {247.733, 17.1029}, {498.519, 
  224.687}, {408.9, 269.554}, {20.7042, 188.348}, {451.979, 
  44.9689}, {30.7126, 98.6237}, {373.218, 273.415}, {492.192, 
  58.8905}, {308.128, 286.02}, {443.24, 38.9625}, {60.6258, 
  222.16}, {41.5267, 80.9147}, {236.23, 284.485}, {544.81, 
  193.11}, {416.114, 29.3403}, {54.0816, 224.732}, {142.162, 
  256.687}, {165.4, 28.8549}, {15.773, 134.739}, {538.801, 
  202.256}, {332.078, 284.059}, {353.132, 14.7648}, {278.279, 
  282.172}, {526.683, 204.325}, {61.4494, 219.364}, {475.606, 
  237.591}, {178.634, 276.961}, {124.782, 40.4963}, {258.218, 
  279.259}, {289.924, 280.441}, {177.697, 276.02}, {546.535, 
  106.27}, {305.448, 12.5493}, {417.535, 28.6531}, {15.9444, 
  129.405}, {226.386, 282.084}, {107.532, 47.8901}, {184.42, 
  20.8464}, {303.016, 14.5519}, {92.1222, 47.4099}, {553.302, 
  154.452}, {17.0436, 124.738}, {15.8854, 135.147}, {18.5364, 
  176.525}, {247.784, 11.4122}, {229.798, 278.079}, {484.061, 
  56.0596}, {205.851, 13.477}, {214.129, 10.7242}, {85.6134, 
  242.559}, {123.336, 256.978}, {96.8295, 55.244}, {333.557, 
  285.583}, {230.976, 280.373}, {253.041, 16.7594}, {19.1676, 
  109.802}, {554.128, 159.326}, {455.361, 40.7296}, {517.993, 
  75.6124}, {128.245, 38.9897}, {252.325, 8.19425}, {151.285, 
  262.697}, {441.277, 37.0819}, {15.4735, 172.528}, {370.261, 
  273.103}, {20.3036, 167.98}, {306.387, 8.86591}, {10.7912, 
  157.721}, {464.234, 253.389}, {248.294, 282.49}, {504.109, 
  221.291}, {77.1577, 240.759}, {489.979, 236.638}, {476.421, 
  237.917}, {105.401, 40.3942}, {479.418, 50.3713}, {351.19, 
  18.4029}, {536.189, 189.566}, {56.21, 75.3998}, {20.4849, 
  122.668}, {122.235, 42.563}, {422.536, 260.39}, {266.302, 
  284.858}, {464.772, 40.2306}, {387.285, 278.219}, {107.814, 
  41.0823}, {442.766, 259.048}, {454.204, 44.0058}, {528.349, 
  88.016}, {269.989, 11.8865}, {34.5577, 100.702}, {436.206, 
  260.15}, {538.176, 195.335}, {228.174, 12.8474}, {27.2441, 112.313},
  {221.57, 18.0464}, {231.513, 279.885}, {401.089, 270.572}, {439.9, 
  36.1922}, {188.533, 15.7943}, {360.085, 16.9698}, {22.98, 
  187.373}, {75.8475, 57.0842}, {364.183, 278.657}, {407.266, 
  27.5398}, {255.345, 278.215}, {524.146, 211.964}, {167.79, 
  21.6328}, {325.392, 278.029}, {210.571, 20.5786}, {134.737, 
  34.2591}, {89.602, 54.9735}, {551.263, 117.395}, {317.205, 
  10.4819}, {329.518, 16.9544}, {558.44, 145.311}, {125.7, 
  39.5867}, {481.071, 237.54}, {243.18, 282.807}, {496.341, 
  225.601}, {455.069, 37.2108}, {282.097, 287.254}, {282.487, 
  13.6979}, {162.886, 265.388}, {219.283, 20.5863}, {452.148, 
  249.101}, {493.321, 59.9754}, {512.818, 216.94}, {39.1416, 
  91.0125}, {251.071, 284.862}, {556.502, 143.996}, {541.606, 
  193.46}, {548.117, 174.13}, {550.339, 167.337}, {148.208, 
  31.7878}, {535.255, 103.158}, {96.8475, 240.15}, {109.504, 
  256.249}, {18.8648, 167.318}, {231.479, 15.4998}, {422.7, 
  265.448}, {239.009, 284.592}, {59.9833, 72.9196}, {396.71, 264.941},
  {178.13, 273.65}, {163.102, 272.844}, {471.25, 51.5322}, {291.099, 
  5.8935}, {312.323, 16.8829}, {209.496, 281.543}, {30.554, 
  99.8663}, {99.138, 47.9444}, {549.832, 146.061}, {499.16, 
  59.3727}, {475.705, 51.8307}, {344.309, 276.729}, {443.58, 
  252.146}, {230.523, 276.124}, {226.899, 278.722}, {262.362, 
  17.402}, {363.855, 11.8984}, {12.1561, 146.561}, {133.904, 
  39.2059}, {497.374, 64.7685}, {90.1903, 57.6868}, {137.696, 
  26.9482}, {62.0312, 75.8927}, {285.149, 14.6837}, {448.484, 
  38.2635}, {132.798, 257.957}, {50.4869, 73.3527}, {426.976, 
  267.593}, {8.66257, 158.774}, {178.69, 276.365}, {137.446, 
  31.084}, {425.264, 36.2503}, {282.6, 14.8515}, {375.604, 
  18.6047}, {549.023, 164.189}, {540.398, 198.418}, {530.539, 
  89.5846}, {88.3028, 49.2858}, {458.625, 48.4402}, {187.563, 
  270.068}, {416.335, 269.791}, {484.147, 232.806}, {29.2683, 
  102.471}, {238.71, 281.789}, {96.4815, 250.023}, {17.1013, 
  115.931}, {207.999, 273.452}, {210.403, 17.5497}, {509.899, 
  66.3078}, {42.2069, 85.9688}, {47.9787, 212.531}, {420.237, 
  30.7798}, {15.0482, 143.138}, {375.78, 279.715}, {421.687, 
  29.3802}, {230.66, 17.2373}, {352.842, 278.854}, {16.633, 
  114.365}, {429.668, 255.528}, {201.304, 20.4621}, {256.545, 
  281.11}, {64.7614, 68.6319}, {237.643, 9.94991}, {559.796, 
  148.963}, {319.699, 277.684}, {20.9321, 117.817}, {468.063, 
  47.8955}, {499.741, 69.329}, {417.194, 25.0924}, {102.675, 
  39.9647}, {108.555, 256.231}, {198.2, 15.5882}, {497.102, 
  233.738}, {394.98, 23.1966}, {460.588, 43.9541}, {14.8324, 
  140.164}, {233.131, 11.3155}, {113.474, 248.203}, {28.4194, 
  186.293}, {14.7, 134.36}, {494.66, 60.3701}, {350.545, 
  277.697}, {219.984, 19.9239}, {484.729, 244.139}, {51.4992, 
  80.0979}, {537.291, 95.0876}, {128.789, 252.439}, {122.28, 
  38.9316}, {186.056, 277.165}, {65.7549, 224.443}, {173.935, 
  26.7053}, {541.124, 180.923}, {306.021, 279.904}, {15.3222, 
  138.105}, {185.841, 278.291}, {127.048, 37.8384}, {468.983, 
  250.595}, {530.001, 211.434}, {55.0502, 78.4593}, {37.0245, 
  196.488}, {477.551, 244.24}, {296.013, 286.34}, {529.772, 
  93.176}, {184.913, 271.522}, {507.766, 68.6946}, {140.926, 
  30.4062}, {431.863, 33.0485}, {230.038, 9.12564}, {309.197, 
  282.328}, {554.096, 121.979}, {171.371, 24.9378}, {308.336, 
  9.18316}, {79.4051, 239.439}, {545.299, 100.732}, {514.833, 
  214.542}, {356.856, 13.0061}, {517.816, 84.9898}, {284.289, 
  284.336}, {171.784, 273.039}, {366.461, 15.6313}, {352.427, 
  281.581}, {225.607, 18.817}, {25.3003, 190.18}, {323.462, 
  284.304}, {75.2392, 57.8409}, {527.443, 82.8886}, {440.453, 
  41.8566}, {29.3609, 193.342}, {74.9634, 61.9223}, {558.773, 
  159.527}, {26.9852, 188.329}, {559.62, 147.743}, {129.347, 
  36.4634}, {409.779, 30.896}, {73.1597, 59.8505}, {465.424, 
  249.259}, {41.9467, 89.3623}, {47.2739, 86.4977}, {220.584, 
  275.888}, {413.35, 28.2137}, {235.712, 283.017}, {421.992, 
  263.72}, {93.6369, 238.182}, {547.273, 129.125}, {209.45, 
  272.191}, {321.587, 17.437}, {238.685, 9.76702}, {45.1402, 
  88.351}, {196.799, 21.6793}, {338.091, 16.8859}, {311.161, 
  15.1921}, {367.131, 277.12}, {190.915, 17.0876}, {29.3558, 
  196.305}, {475.513, 247.84}, {76.8984, 59.2354}, {475.917, 
  243.902}, {226.69, 284.785}, {299.745, 15.4357}, {102.394, 
  252.598}, {538.07, 105.176}, {269.752, 8.81818}, {480.457, 
  242.972}, {539.959, 193.981}, {149.172, 28.4105}, {90.7731, 
  242.446}, {418.281, 29.6794}, {86.0118, 60.0169}, {549.199, 
  120.943}, {26.0357, 190.247}, {380.708, 17.8098}, {189.625, 
  23.5462}, {489.574, 229.305}, {101.304, 251.321}, {11.753, 
  155.419}, {169.47, 267.773}, {289.964, 15.879}, {179.18, 
  273.113}, {404.775, 26.465}, {479.374, 239.821}, {443.412, 
  255.279}, {559.408, 141.156}, {503.839, 227.549}, {477.721, 
  56.7715}, {203.543, 16.9369}, {554.106, 116.562}, {530.594, 
  83.6009}, {558.078, 162.95}, {359.572, 13.0296}, {33.1991, 
  203.17}, {189.232, 24.4352}, {364.847, 271.466}, {64.3696, 
  220.982}, {550.47, 169.292}, {183.707, 269.145}, {547.798, 
  124.055}, {475.595, 237.944}, {254.887, 10.2161}, {30.1837, 
  93.8457}, {407.629, 28.9718}, {530.981, 200.279}, {18.9379, 
  114.71}, {15.0161, 121.044}, {144.415, 35.8059}, {33.5156, 
  101.305}, {133.144, 40.1251}, {74.4527, 57.735}, {169.355, 
  272.777}, {237.579, 276.271}, {41.7276, 89.4028}, {310.732, 
  285.575}, {24.4119, 104.173}}

ListPlot[{pts[[All, {1, 2}]]}, PlotStyle -> {Red}, PlotRange -> Full]

enter image description here

I am using LeastSquares to fit the data, however, it's not working as desired. I am unable to find the problem with the following code:

{x, y} = {pts[[All, 1]], pts[[All, 2]]};
m = Map[x \[Function] {1, x, x^2, x^3, x^4, x^5, x^6, x^7}, x];
{a, b, c, d, e, f, g, h} = 
 LeastSquares[m, y, Tolerance -> 10^-8]
ff = Fit[pts[[All, {1, 2}]], {1, s, s^2, s^3, s^4, s^5, s^6, s^7}, s]

Show[Plot[
  a x^0 + b x^1 + c x^2 + d x^3 + e x^4 + f x^5 + g x^6 + 
   h x^7, {x, -600, 600}, 
  PlotRange -> {{0, 600}, {0, 300}}], 
 ListPlot[{pts[[All, {1, 2}]]}, PlotStyle -> {Red}], ImageSize -> 600]

![![enter image description here

Do I need to try other ways of fitting the curve? Please help me solve this problem.

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Please provide some points of your dataset in copyable form (can't access your link). Thanks! $\endgroup$
    – Ulrich Neumann
    Jan 5 at 8:40
  • $\begingroup$ I have edited it. $\endgroup$
    – user84456
    Jan 5 at 8:48

2 Answers 2

Reset to default
16
$\begingroup$

You have to fit implicitly, for example fit a conic section (or ellipse )

nlm = NonlinearModelFit[Map[{#[[1]], #[[2]], 0} &,pts], {1, a, b, c, d, e} . {1, x, y , x^2, x y, y^2} , {a, b, c, d,e}, {x, y}]

nlm[x, y] (*1 - 0.00601378 x + 0.0000104909 x^2 - 0.0123825 y + 2.96388*10^-7 x y + 0.0000417392 y^2*)

Show[{ListPlot[pts], ImplicitRegion[nlm[x, y] == 0, {x, y}] // Region}]

enter image description here

Improve this answer
$\endgroup$
10
  • $\begingroup$ What if it is not a closed loop! Can I use ImplicitRegion in that case? $\endgroup$
    – user84456
    Jan 5 at 9:03
  • 1
    $\begingroup$ Why not? Alternatively try ContourPlot[nlm[x,y]==0,...] $\endgroup$
    – Ulrich Neumann
    Jan 5 at 9:07
  • $\begingroup$ Doesn't really affect the main point of the answer, but one could use Append[0] /@ pts as a more expressive alternative to Map[{#[[1]], #[[2]], 0} &,pts] $\endgroup$
    – Lukas Lang
    Jan 5 at 9:15
  • $\begingroup$ ContourPlot[nlm[x,y]==0,...] this works. Thank you. Is there a way to increase the accuracy? $\endgroup$
    – user84456
    Jan 5 at 9:18
  • 1
    $\begingroup$ NonlinearModelFit[...,WorkingPrecision -> 100] works fine $\endgroup$
    – Ulrich Neumann
    Jan 5 at 9:37
9
$\begingroup$

NB: see comment by @JimB. Original post was wrong.

Updated Post

I have upvoted @UlrichNeumann answer. For what it is worth you can use LinearModelFit. For example:

d = {#1^2, #1, #2, #1 #2, #2^2} & @@@ pts;
lm = LinearModelFit[d, {a, b, c, e}, {a, b, c, e}];
v = {1, x^2, x, y, x y}
exp = y^2 - lm["BestFitParameters"] . v;
{u1, u2} = 
  y^2 - lm["SinglePredictionBands"] /. Thread[{1, a, b, c, e} -> v];
ContourPlot[{exp == 0, u1 == 0, u2 == 0}, {x, 0, 600}, {y, 0, 300}, 
 Epilog -> Point[pts], AspectRatio -> Automatic]

enter image description here

I leave the original post for transparency.

Original Post

I have upvoted @UlrichNeumann answer. For what it is worth you can use LinearModelFit. For example:

d = {#1^2, #1, #2, #1 #2, #2^2} & @@@ pts;
lm = LinearModelFit[d, {a, b, c, e}, {a, b, c, e}];
exp = y^2 - lm["BestFitParameters"] . {1, x^2, x, y, x y}
ci = lm["ParameterConfidenceIntervalTableEntries"];
exp2 = y^2 - ci[[All, 3, 1]] . {1, x^2, x, y, x y}
exp3 = y^2 - ci[[All, 3, 2]] . {1, x^2, x, y, x y}
ContourPlot[{exp == 0, exp2 == 0, exp3 == 0}, {x, 0, 600}, {y, 0, 
  300}, Epilog -> Point[pts], PlotLabel -> Row[{exp, "=0"}], 
 AspectRatio -> Automatic]

enter image description here

Improve this answer
$\endgroup$
3
  • $\begingroup$ +1 But your approach for obtaining a 95% confidence interval is a bit "unusual" because of two reasons: (1) the parameter estimators are not independent as shown with lm["CorrelationMatrix"] // MatrixForm, and (2) even if the parameter estimators were independent, using 95% confidence intervals for each parameter doesn't necessarily result in a 95% confidence interval for the ellipse. (And as of this moment, I have to confess I don't know the correct way of doing so.) $\endgroup$
    – JimB
    Jan 6 at 18:58
  • $\begingroup$ @JimB yes I had a lot of reservation about ci and have to admit did not give it enough thought and on reflection there are other ways. Thank you for your comment. I think my dumb “ plug and play” was wrong as you allude to and needed to be called out. I think the only relevance of answer as you can use ordinary least squares in a linear model. $\endgroup$
    – ubpdqn
    Jan 7 at 0:31
  • $\begingroup$ The good thing is that you thought that a confidence region (or some measure of precision) was necessary for a curve fit. That doesn't always happen in this forum. $\endgroup$
    – JimB
    Jan 7 at 2:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.