# How to detect outlier in data of an unknown function $f(x)$

Suppose I have an unknown function $f(x)$ that is monotonic, convex and nonlinear by definition. I somehow evaluated it at 1000 nonuniform $x$ points. By ploting these data, I found a very obvious nonlinear curve with a dozen of outliers, which, e.g., contain sudden small or large peaks and valleys and violate monotonicity and convexity. How to carry out outlier recognition automatically? Sth. more systematic than inspecting by eyes...

Note that my ultimate purpose is not finding any fit of these data and I don't want any modification among the data except excluding outliers.

The data set is (updated, more reasonable, different from the one plotted in the comment below, apology for that I cannot retain the original one due to character limit)

data = {0,197.1278166},{3.356150145 10^-5,197.1004078},{6.745694819 10^-5,197.0729913},{0.0001016863402,197.045061},{0.0001362496775,197.01686},{0.0001711469601,196.9884324},{0.000206378188,196.9597409},{0.0002419433612,196.9307115},{0.0002778424797,196.9014819},{0.0003140755434,196.872016},{0.0003506425525,196.8422888},{0.0003875435068,196.8123176},{0.0004247784064,196.7820562},{0.0004623472513,196.7515632},{0.0005002500415,196.7208153},{0.000538486777,196.6896979},{0.0005770574578,196.6584585},{0.0006159620839,196.6269528},{0.0006552006552,196.5952411},{0.0006947731718,196.5632462},{0.0007346796338,196.5309877},{0.000774920041,196.4984332},{0.0008154943935,196.4656562},{0.0008564026913,196.4326006},{0.0008976449343,196.3992963},{0.0009392211227,196.365732},{0.0009811312564,196.3319839},{0.001023375335,196.2979535},{0.00106595336,196.2636788},{0.001108865329,196.2290179},{0.001152111244,196.1942448},{0.001195691104,196.159195},{0.001239604909,196.1239301},{0.00128385266,196.0884187},{0.001328434356,196.0526233},{0.001373349997,196.0165869},{0.001418599584,195.9803034},{0.001464183116,195.9438184},{0.001510100593,195.9069059},{0.001556352015,195.8698333},{0.001602937383,195.8325826},{0.001649856696,195.7950497},{0.001697109954,195.7573184},{0.001744697158,195.7193407},{0.001792618306,195.6811246},{0.001840873401,195.6426487},{0.00188946244,195.6039494},{0.001938385425,195.5650393},{0.001987642355,195.5259065},{0.00203723323,195.4864771},{0.00208715805,195.4468092},{0.002137416816,195.4069093},{0.002188009527,195.3667779},{0.002238936184,195.3263106},{0.002290196786,195.2857357},{0.002341791333,195.2448866},{0.002393719825,195.2037633},{0.002445982262,195.1624498},{0.002498578645,195.1209472},{0.002551508974,195.0791495},{0.002604773247,195.0372058},{0.002658371466,194.9949096},{0.00271230363,194.9524911},{0.002766569739,194.9098417},{0.002821169794,194.8672176},{0.002876103794,194.8240065},{0.002931371739,194.7805768},{0.002986973629,194.736972},{0.003042909465,194.702772},{0.003099179246,194.6586471},{0.003155782972,194.6141539},{0.003212720644,194.5696212},{0.003269992261,194.5248077},{0.003327597823,194.5436408},{0.00338553733,194.4342508},{0.003443810783,194.3887827},{0.003502418181,194.3432184},{0.003561359525,194.2972514},{0.003620634813,194.2511},{0.003680244047,194.2047559},{0.003740187226,194.1581676},{0.003800464351,194.1113705},{0.003861075421,194.064177},{0.003922020436,194.0169291},{0.003983299396,193.9695131},{0.004044912302,193.9217952},{0.004106859153,193.8738824},{0.004169139949,193.8257357},{0.00423175469,193.7773793},{0.004294703377,193.7288066},{0.004357986009,193.6800275},{0.004421602587,193.6307837},{0.004485553109,193.5822356},{0.004549837577,193.5328221},{0.004614455991,193.4831627},{0.004679408349,193.4332837},{0.004744694653,193.3832249},{0.004810314902,193.3329423},{0.004876269096,193.2824868},{0.004942557236,193.2308879},{0.005009179321,193.1799623},{0.005076135351,193.1282019},{0.005143425327,193.076852},{0.005211049248,193.0252946},{0.005279007114,192.9735373},{0.005347298925,192.9215745},{0.005415924682,192.8693928},{0.005484884384,192.8170039},{0.005554178031,192.7644067},{0.005623805624,192.7116614},{0.005693767162,192.6632774},{0.005764062645,192.610056},{0.005834692073,192.5566516},{0.005905655447,192.4989477},{0.005976952766,192.4451613},{0.00604858403,192.3911403},{0.00612054924,192.3368714},{0.006192848395,192.2823985},{0.006265481495,192.2279216},{0.00633844854,192.1732041},{0.006411749531,192.1188254},{0.006485384467,192.0635985},{0.006559353348,192.00814},{0.006633656175,191.9525783},{0.006708292947,191.896843},{0.006783263664,191.8408394},{0.006858568326,191.7846399},{0.006934206934,191.7282461},{0.007010179487,191.6716881},{0.007086485986,191.6149185},{0.007163126429,191.5579129},{0.007240100818,191.5007167},{0.007317409152,191.4433843},{0.007395051432,191.3858193},{0.007473027657,191.3280679},{0.007551337827,191.2700583},{0.007629981942,191.2117456},{0.007708960003,191.1534528},{0.007788272008,191.0949794},{0.00786791796,191.0363179},{0.007947897856,190.9769944},{0.008028211698,190.9179446},{0.008108859485,190.8586478},{0.008189841217,190.7992471},{0.008271156895,190.7396372},{0.008352806518,190.679838},{0.008434790086,190.619893},{0.0085171076,190.5597393},{0.008599759058,190.4999113},{0.008682744463,190.4394211},{0.008766063812,190.3788123},{0.008849717107,190.3179419},{0.008933704347,190.2569127},{0.009018025532,190.195623},{0.009102680662,190.134266},{0.009187669738,190.072727},{0.009272992759,190.0109841},{0.009358649726,189.9494614},{0.009444640637,189.8873756},{0.009530965494,189.8256654},{0.009617624297,189.7633164},{0.009704617044,189.7007075},{0.009791943737,189.637898},{0.009879604375,189.5750092},{0.009967598958,189.5118553},{0.01005592749,189.4485701},{0.01014458996,189.3851623},{0.01023358638,189.3216322},{0.01032291674,189.2578523},{0.01041258105,189.1938933},{0.01050257931,189.1297504},{0.01059291151,189.0654914},{0.01068357766,189.0011197},{0.01077457775,188.9365838},{0.01086591178,188.8719057},{0.01095757976,188.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• I don't think I would call the errors in your data outliers as such (i.sstatic.net/h6MwF.png). There seems to be a small systematic offset that comes and goes.
– user484
Nov 17, 2014 at 15:48
• @RahulNarain Yes, I just show the worst data set here. Other data sets look really like sudden outliers. Anyway, considering the proportion of such points and the overall smooth trend, I need to somehow identify and eliminate them. Nov 17, 2014 at 15:55

As you expect monotonic "smooth" behavior, a simple solution is to z-score the differences.

diff = data[[All, 2]] // Differences;
mn = Mean[diff]
std = StandardDeviation[diff]


(* 3 std is bad *)

bad = Position[diff, x_ /; Abs[x] > Abs[mn + 3 std]]

ListPlot[data, PlotRange -> All]
ListPlot[data[[bad // Flatten]], PlotStyle -> Red]
Show[%, %%]


This can also be changed to scan the data locally rather than calculate the statistics over the entire array.

• Thanks. This is good. But please allow me to ask more. Your method is not innately suitable for some smooth monotonic convex curve, at lease to some extent. Due to nonlinearity, the diff might vary extensively. The more it varies, the less justifiable your method becomes, right? It becomes harder to pick out outliers in the less varying region since overwhelmed by largely varying region. For instance, your method fails to pick out the last but 5 point, which even violates monotonicity. Nov 19, 2014 at 3:43
• Hope to hear comment from you or anybody else. I myself guess we'd better divide the data into a few pieces and then apply this z-score method accordingly. To avoid outliers pinpointed at division boundaries, we might as well try different random divisions a few times. Anyway, it sounds no more than a workaround. Nov 19, 2014 at 3:52
• This method is ideal only for uniformly sampled quadratic curves where the derivative should be constant (i.e., the values of "diff" are all equal if the data is perfect). For smooth curves such as yours I often add the additional step of detrending the "diff" list prior to the mean and standard deviation calculation with a low order polynomial or appropriate functional form. Since the pattern test uses ABS, it does not test for monotonicity. Nov 19, 2014 at 23:35

A common approach to removing outliers is to use an order statistic filter. The simplest of these is the MedianFilter:

x = data[[All, 1]];
ySmoothed = MedianFilter[data[[All, 2]], 5];
ListPlot[Transpose[{x, ySmoothed}]]


 n = 20 (*even*)
f = Interpolation[Transpose[{data[[All, 1]],
Join[ data[[;; (n/2 - 1), 2]] , MovingAverage[data[[All, 2]], n] ,
data[[-n/2 ;;, 2]] ]}]]

GraphicsColumn[{ListPlot[data],
ListPlot[ Select[  data , Abs[f[#[[1]]] - #[[2]]] < .2 &  ] ]}]