Import and interpolate data with pgfplots (and a solution for plotting the chi square distribution)

Question

I have to draw a chi-square distribution, and this is quite a challenge with pgfplots. I found some solutions but I discarded them all for different reasons and I choose the lazy solution:

1. I generated the curve using Octave/MATLAB, then I saved the points in a CSV file (downloadable here):

   x = .1:.1:8;
   pdf1 = chi2pdf(x,1);
   pdf2 = chi2pdf(x,2);
   m = [x' pdf1' pdf2' pdf3' pdf4' pdf5' pdf6' pdf7' pdf8'];
   csvwrite ('chisquare.csv', m);
   

2. I imported them in LaTeX using \pgfplotstableread[col sep=comma]{chisquare.csv}\dataChiSquare:

This is the plot of the chi-square distributions for K=1, ..., 8:

 \begin{tikzpicture}
  \begin{axis}[%
    no markers,
    height=5cm, width=10cm,
    smooth,
    axis x line=bottom, axis y line=left,
    xtick=\empty, ytick=\empty,
    clip=false,
    enlargelimits=upper,
    restrict x to domain = 0:7,
    restrict y to domain = 0:0.8,
   ]
   \addplot table[x = x, y = pdf1] from \dataChiSquare;
   \addplot table[x = x, y = pdf2] from \dataChiSquare;
   % .........
   \addplot table[x = x, y = pdf8] from \dataChiSquare;
  \end{axis}
 \end{tikzpicture}

My question is: how can I compute points of the plots, as I asked here? Is there any way to (easily) define a function that interpolates the points using tikz or pgfplots? I need to draw some vertical lines from x axis to the plot, but I also need the coordinates for other purposes.

The preamble of my other question can also be used to have a MWE for this one.

Thank you!

Answer 1

Here is an example using intersections and calc. The y value is stored in \yQ. I just made up some data but it should work with your data as well. (I could not use \pgfplotspointgetcoordinates because of expansion issues but it is straightforward to get the coordinate with `calc.)

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{filecontents}
\begin{filecontents*}{chisquare.csv}
x,pdf1,pdf2,pdf3,pdf4,pdf5,pdf6,pdf7,pdf8
0,1000,0.5,0,0,0,0,0,0
0.1,1.200038948430136,0.475614712250357,0.1200038948430136,0.02378073561251787,0.004000129828100455,0.0005945183903129469,8.000259656200907e-005,9.908639838549101e-006
0.2,0.8071711293576808,0.4524187090179798,0.1614342258715362,0.04524187090179798,0.01076228172476909,0.0022620935450899,0.0004304912689907634,7.540311816966337e-005
0.3,0.6269100992275207,0.4303539882125289,0.1880730297682562,0.06455309823187934,0.01880730297682562,0.004841482367390948,0.001128438178609537,0.0002420741183695475
0.4,0.5164415474672782,0.4093653765389909,0.2065766189869113,0.08187307530779819,0.02754354919825484,0.008187307530779821,0.002203483935860388,0.0005458205020519878
0.5,0.4393912894677224,0.3894003915357024,0.2196956447338612,0.09735009788392562,0.03661594078897687,0.0121687622354907,0.003661594078897689,0.001014063519624225
0.6000000000000001,0.381545289384093,0.3704091103408589,0.2289271736304558,0.1111227331022577,0.04578543472609117,0.01666840996533865,0.005494252167130943,0.001666840996533865
0.7000000000000001,0.3360144677267703,0.3523440448593567,0.2352101274087392,0.1233204157007749,0.05488236306203916,0.02158107274763561,0.007683530828685487,0.00251779182055749
0.8,0.298983539918205,0.3351600230178197,0.2391868319345639,0.1340640092071279,0.06378315518255041,0.02681280184142558,0.01020530482920807,0.003575040245523413
0.9,0.268136721052083,0.3188140758108867,0.2413230489468747,0.143466334114899,0.07239691468406238,0.03227992517585227,0.01303144464313124,0.004841988776377838
1,0.2419707245191433,0.3032653298563167,0.2419707245191434,0.1516326649281584,0.08065690817304777,0.0379081662320396,0.01613138163460956,0.006318027705339933
1.1,0.2194581724133437,0.2884749051902433,0.2414039896546781,0.1586611978546339,0.08851479620671532,0.04363182941002433,0.01947325516547737,0.007999168725171124
1.2,0.1998677639017332,0.2744058180470132,0.2398413166820799,0.1646434908282079,0.09593652667283198,0.04939304724846241,0.02302476640147969,0.009878609449692479
1.3,0.1826614817951091,0.261022888380508,0.2374599263336419,0.1696648774473302,0.1028993014112448,0.05514108517038233,0.02675381836692366,0.01194723512024951
1.4,0.1674325573450835,0.2482926518957047,0.2344055802831169,0.1738048563269934,0.1093892707987879,0.06083169971444766,0.03062899582366061,0.01419406326670446
1.5,0.1538663228054553,0.2361832763705073,0.2307994842081829,0.1771374572778805,0.1153997421040915,0.06642654647920521,0.03461992263122744,0.0166066366198013
1.6,0.1417145653062239,0.2246644820586108,0.2267433044899583,0.1797315856468887,0.1209297623946444,0.0718926342587555,0.03869752396628622,0.01917136913566813
1.7,0.1307781819238881,0.2137074659743633,0.2223229092706098,0.1816513460782088,0.1259829819200122,0.07720182208323877,0.04283421385280416,0.02187384959025099
1.8,0.1208951224732049,0.2032848298702996,0.2176112204517688,0.1829563468832696,0.1305667322710613,0.08233035609747134,0.04700402361758206,0.02469910682924141
1.9,0.1119318050861699,0.1933705117272506,0.2126704296637229,0.1837019861408881,0.1346912721203578,0.08725844341692184,0.05118268340573599,0.02763184041535858
2,0.1037768743551487,0.1839397205857212,0.2075537487102974,0.1839397205857211,0.1383691658068649,0.09196986029286061,0.05534766632274598,0.03065662009762019
2.1,0.09633657731357953,0.1749688745555777,0.2023068123585171,0.1837173182833566,0.1416147686509619,0.09645159209876222,0.059478202833404,0.03375805723456678
2.2,0.08953128037314291,0.1664355418490398,0.1969688168209144,0.1830790960339438,0.1444437990020039,0.1006935028186691,0.06355527156088171,0.03692095103351201
2.3,0.0832928061175755,0.1583183846895266,0.1915734540704237,0.1820661423929556,0.1468729814539915,0.1046880318759495,0.06756157146883611,0.04013041221911396
2.4,0.07756236924025954,0.150597105956101,0.1861496861766229,0.1807165271473213,0.1489197489412983,0.1084299162883928,0.07148147949182322,0.04337196651535711
2.5,0.07228895706727252,0.143252398430095,0.1807223926681813,0.1790654980376188,0.1506019938901511,0.1119159362735118,0.07530099694507554,0.04663164011396324
2.6,0.06742804459323157,0.1362658965170063,0.1753129159424021,0.1771456654721082,0.1519378604834151,0.1151446825568703,0.07900768745137589,0.04989602910797716
2.7,0.06294056442554451,0.1296201303229458,0.1699395239489702,0.1749871759359768,0.1529455715540732,0.1181163437567843,0.08259060863919951,0.05315235469055297
2.8,0.05879207325254447,0.1232984819708032,0.1646178051071245,0.1726178747591245,0.1536432847666496,0.1208325123313872,0.08604023946932379,0.05638850575464736
2.9,0.05495207145796054,0.1172851440468988,0.1593610072280856,0.1700634588680033,0.1540489736538161,0.1232960076793024,0.08934840471921335,0.0595930703783295
3,0.05139344326792309,0.1115650800742149,0.1541803298037693,0.1673476201113224,0.1541803298037693,0.1255107150834918,0.0925081978822616,0.0627553575417459
3.1,0.0480919926380412,0.1061239869133715,0.1490851771779277,0.1644921797157259,0.1540546830838587,0.1274814392796876,0.09551390351199233,0.06586541029450528
3.2,0.04502605584019208,0.1009482589973277,0.1440833786886146,0.1615172143957243,0.1536889372678556,0.1292137715165795,0.09836091985142761,0.06891401147550903
3.3,0.04217617598652614,0.09602495431037704,0.1391813807555363,0.1584411746121221,0.1530995188310899,0.1307139690550008,0.1010456824285194,0.07189268298025039
3.4,0.03952482794294565,0.0913417620263673,0.1343844150060152,0.1552809954448245,0.1523023370068173,0.1319888461281008,0.1035655891646358,0.0747936794725905
3.5,0.03705618452374813,0.08688697172522257,0.1296966458331184,0.1520522005191395,0.1513127534719715,0.1330456754542471,0.1059189274303801,0.07760997734831081
3.6,0.03475591672713835,0.08264944411079327,0.125121300217698,0.1487689993994279,0.1501455602612377,0.1338920994594851,0.1081048033880911,0.08033525967569108
3.7,0.03261102221401006,0.0786185831568138,0.1206607821918372,0.1454443788401056,0.1488149647032659,0.1345360504270977,0.1101230738804168,0.08296389776337691
3.8,0.03060967735558654,0.07478430961131752,0.1163167739512288,0.1420901882615033,0.1473345803382232,0.1349856788484282,0.1119742810570497,0.0854909299373378
3.9,0.02874110905657778,0.07113703579325677,0.1120903253206534,0.1387172197968507,0.1457174229168494,0.1352492893019295,0.1136595898751425,0.08791203804625421
4,0.02699548325659403,0.06766764161830635,0.1079819330263761,0.1353352832366127,0.1439759107018348,0.1353352832366127,0.1151807285614679,0.09022352215774178
4.100000000000001,0.02536380756620646,0.06436745179390209,0.1039916110214465,0.1319532761774993,0.1421218683959769,0.1352521080819368,0.1165399320847011,0.09242227385599021
4.2,0.023837845937227,0.06122821412649095,0.1001189529363534,0.128579249665631,0.1401665341108948,0.1350082121489126,0.1177398886531517,0.0945057485042388
4.3,0.0224100436236817,0.05824207888674848,0.09636318758183131,0.1252204696065092,0.1381205688672915,0.1346120048269975,0.1187836892258708,0.09647193679268147
4.4,0.02107346097903018,0.05540157918116694,0.09272322830773279,0.1218834741985672,0.1359940681846747,0.134071821618424,0.1196747800025138,0.09831933585351095
4.5,0.01982171487060489,0.05269961228093217,0.08919771691772202,0.1185741276320974,0.1337965753765831,0.1333958935861096,0.1204169178389248,0.1000469201895822
4.600000000000001,0.01864892668496918,0.05012942186140185,0.08578506275085825,0.1152976702812243,0.1315370962179827,0.1325923208234079,0.1210141285205441,0.1016541126312795
4.7,0.01754967605644785,0.04768458110777481,0.08248347746530486,0.1120587656032708,0.1292241146956443,0.1316690495838432,0.1214706678139056,0.1031407555073439
4.800000000000001,0.01651895958214552,0.04535897664470624,0.07929100599429849,0.108861543947295,0.1268656095908776,0.130633852736754,0.1217909852072426,0.1045070821894032
4.9,0.01555215389559439,0.04314679324968525,0.07620555408841254,0.1057096434617288,0.1244690716777404,0.1294943132406179,0.1219796902441857,0.1057536891465046
5,0.01464498256192649,0.0410424993119494,0.07322491280963242,0.1026062482798735,0.1220415213493874,0.1282578103498418,0.1220415213493874,0.1068815086248682
5.100000000000001,0.01379348633346337,0.03904083300057656,0.0703467803006632,0.09955412415147029,0.1195895265111275,0.1269315082931246,0.12198131704135,0.1078917820491559
5.2,0.012993996368508,0.03713678910716694,0.06756878111624158,0.09655565167863402,0.1171192206014854,0.1255223471822242,0.1218039894255449,0.1087860342245943
5.300000000000001,0.01224311007004337,0.03532560653021478,0.06488848337122986,0.09361285730506919,0.1146363206225061,0.1240370359292167,0.1215144998598565,0.1095660484041414
5.4,0.01153766924671384,0.03360275636987488,0.06230341393225473,0.09072744219866219,0.1121461450780586,0.122482046968194,0.1211178366843032,0.1102338422713746
5.5,0.01087474033728314,0.03196393060335379,0.05981107185505726,0.0879008091592229,0.1096536317342717,0.1208636125939315,0.1206189949076989,0.1107916448777706
5.600000000000001,0.01025159647287075,0.03040503131260898,0.05740894024807616,0.08513408767530517,0.1071633551297422,0.1191877227454272,0.1200229577453113,0.1112418745623988
5.7,0.009665701179594184,0.02892216043741923,0.05509449672368685,0.08242815724664482,0.1046795437750051,0.1174601240764689,0.1193346799035058,0.1115871178726455
5.800000000000001,0.009114693548557632,0.0275116100282036,0.05286522258163426,0.07978366908179048,0.1022060969911596,0.1156863201685962,0.1185590725097452,0.1118301094963097
5.9,0.008596374721056833,0.02616985297421619,0.0507186108542353,0.07720106627393776,0.09974660134666277,0.1138715727540582,0.1177009895890621,0.1119737132081573
6,0.008108695554940244,0.02489353418393197,0.04865217332964145,0.07468060255179593,0.09730434665928291,0.1120209038276939,0.1167652159911395,0.1120209038276939
6.100000000000001,0.00764974535371112,0.02367946219557045,0.04666344665763784,0.07222235969648989,0.09488234153719693,0.1101390985371471,0.1157564566753803,0.1119747501794329
6.2,0.0072177415535363,0.0225246011967789,0.04474999763192506,0.0698262637100146,0.09248332843931178,0.1082307087505227,0.1146793272647467,0.1118383990422068
6.300000000000001,0.006811020275148059,0.02142606343352008,0.04290942773343277,0.06749209981558829,0.09010979824020883,0.1063000572095515,0.1135383457826631,0.1116150600700292
6.4,0.006428027657939708,0.02038110198918311,0.04113937701081413,0.06521952636538594,0.0877640042897368,0.1043512421846175,0.1123379254908632,0.111307991663592
6.5,0.006067311902576734,0.019387103915861,0.03943752736674877,0.06300808772654828,0.085447975961289,0.1023881425556409,0.1110823687496757,0.110920487768611
6.600000000000001,0.005727515956354745,0.01844158370062,0.03780160531194131,0.060857226212046,0.08316353168627089,0.1004144232498759,0.1097758618258776,0.1104558655748635
6.7,0.005407370782485125,0.01754217705042251,0.03622938424265033,0.05876629311891542,0.08091229147525246,0.09843354097418335,0.1084224705768383,0.1099174540878381
6.800000000000001,0.005105689160609535,0.01668663498016303,0.03471868629214483,0.05673455893255431,0.07869568892886164,0.09644875018534235,0.1070261369432519,0.109308583543388
6.9,0.004821359971245364,0.01587281818903397,0.033267383801593,0.05476122275216719,0.07651498274366392,0.09446310924748844,0.1055906761862563,0.1086325756346117
7,0.004553342921640174,0.01509869171115925,0.0318734004514812,0.05284542098905737,0.07437126772012283,0.09247948673085041,0.1041197748081719,0.1078927345193254
7.100000000000001,0.004300663674745038,0.01436231982711971,0.03053471209068977,0.05098623538627498,0.07226548528129911,0.09050056781063809,0.1026169890994448,0.1070923385759218
7.2,0.004062409346773289,0.01366186122364628,0.02924934729676768,0.04918270040512661,0.07019843351224242,0.0885288607292279,0.1010857442576291,0.1062346328750735
7.300000000000001,0.00383772434215234,0.01299556438937767,0.02801538769771208,0.0474338100212285,0.06817077673109938,0.086566703288742,0.09952933402740517,0.1053228223346361
7.4,0.003625806497653241,0.01236176323516969,0.02683096808263397,0.04573852397012787,0.06618305460383049,0.08461626934473659,0.09795092081366917,0.1043600655251752
7.5,0.003425903510139483,0.01175887292800455,0.02569427632604612,0.04409577348001709,0.06423569081511527,0.08267957527503202,0.09635353622267297,0.10334946909379
7.600000000000001,0.003237309624752144,0.0111853859280828,0.02460355314811629,0.04250446652671462,0.0623290013085613,0.08075848640075782,0.09474008198901322,0.1022940827742932
7.7,0.003059362562475526,0.01063986821918858,0.02355709173106154,0.04096349264387606,0.06046320210972462,0.0788547233394614,0.09311333124897592,0.1011968949523088
7.800000000000001,0.002891440667935384,0.01012095572290219,0.022553237209896,0.03947172731931856,0.05863841674572961,0.0769698682726712,0.09147593012333818,0.1000608287544726
7.9,0.002732960259995687,0.00962735088769346,0.02159038605396593,0.03802803600638919,0.05685468327544363,0.07510537111261865,0.08983039957520095,0.09888873863161463
8,0.002583373169261507,0.009157819444367089,0.02066698535409205,0.03663127777746836,0.05511196094424543,0.0732625555549367,0.08817913751079275,0.09768340740658221
\end{filecontents*}
\usepackage{pgfplots} 
\pgfplotsset{compat=1.16}
\pgfplotstableread[col sep=comma,header=true]{chisquare.csv}\dataChiSquare
\usetikzlibrary{intersections,calc}
\begin{document}
 \begin{tikzpicture}
 \begin{axis}[%
    no markers,
    height=5cm, width=10cm,
    smooth,
    axis x line=bottom, axis y line=left,
    xtick=\empty, ytick=\empty,
    clip=false,
    enlargelimits=upper,
    restrict x to domain = 0:7,
    restrict y to domain = 0:0.8,
   ]
   \addplot[name path=plot1] table[x = x, y = pdf1]  \dataChiSquare;
   \path[name path=y0] (0,0) -- (4,0);
   \path[name path=p1] (1.2,0) -- (1.2,0.8);
   \coordinate[overlay] (X) at (1,1);
  \end{axis}
  \path [name intersections={of=plot1 and p1, by=Q}];
  \path let \p1=(X),\p2=(Q) in \pgfextra{\pgfmathsetmacro{\yQ}{\y2/\y1}
  \typeout{y\space coordinate\space of\space Q\space is\space \yQ}};
  \draw [name intersections={of=y0 and p1, by=P}]   (P) -- (Q);
 \end{tikzpicture}
\end{document}