The Lowess command creates a function whose values represent the result of the lowess data smoothing algorithm applied to the input data.

This command calls Statistics[Lowess]. See its help page for more examples and a detailed description.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{Sample}\left(\mathrm{Uniform}\left(0\,\mathrm{\pi }\right)\,200\right)$

$X≔\left[2.55952994787841\,2.84562929519763\,0.398940849170435\,2.86945487961263\,1.98661516237133\,0.306432219774318\,0.874927958430562\,1.71807896311840\,3.00809643996240\,3.03128673371506\,0.495156099507834\,3.04920715280643\,3.00702865285900\,1.52485257225906\,2.51415524187245\,0.445749079076013\,1.32500214706731\,2.87686799856516\,2.48879272666428\,3.01433435793099\,2.06007016313017\,0.112191547056034\,2.66761838926515\,2.93422632565759\,2.13230937623431\,2.38051082755506\,2.33461950250529\,1.23221752310796\,2.05924452437235\,0.537798840821172\,2.21810920321822\,0.100005836322157\,0.869979215163289\,0.145051701612858\,0.305148490360630\,2.58696906401403\,2.18286849744474\,0.996197397016342\,2.98521060790962\,0.108215553452901\,1.37835605710050\,1.19870124571845\,2.40494191784856\,2.49819416754753\,0.587077601625226\,1.53864022779815\,1.39985033469435\,2.03045220448505\,2.22853534133869\,2.37091813587784\,0.867158354105291\,2.13534893622882\,2.05805107666556\,0.510859832674940\,0.373842242178014\,1.56565684452034\,3.01512456940131\,1.06935329828115\,1.83867286686109\,0.703125944891039\,2.36017507439865\,0.801404940693480\,1.58951095654316\,2.19621429619387\,2.79885511322572\,3.01370289407711\,1.71912828886453\,0.435501531198557\,0.469020951089329\,0.808986039393864\,2.64119115514444\,0.798851025395767\,2.55815122750744\,0.765056252712111\,2.91936777185323\,1.09950642809356\,0.617622194485329\,0.788803203652482\,1.93536142886538\,1.48688077073878\,1.10477092395412\,2.61012511379108\,1.83866136917531\,1.72700764931232\,2.88144887620684\,0.897989761635400\,2.37881467707075\,2.36790978538221\,1.19520587794653\,1.78386429505166\,0.238303278834594\,0.169489296463311\,1.66754969307643\,2.44782604600639\,2.93428110394878\,0.408112390196775\,1.78701223418413\,1.47463418961226\,0.0373914541076151\,1.05910202300229\,\dots \,\text{⋯ 100 row vector entries not shown}\right]$

$\mathrm{Yerror}≔\mathrm{Sample}\left(\mathrm{Normal}\left(0\,0.1\right)\,200\right)$

$\mathrm{Yerror}≔\left[\mathrm{-0.0953810981766894}\,0.171103963707571\,0.0285687899013584\,\mathrm{-0.0606979079748253}\,0.128901014035489\,\mathrm{-0.0112726591574468}\,0.0959342847342146\,\mathrm{-0.00314499457928086}\,\mathrm{-0.00771108360917845}\,\mathrm{-0.0433343735980858}\,0.00482823379489400\,\mathrm{-0.0965862973322655}\,0.235332740922138\,\mathrm{-0.0694524292206791}\,\mathrm{-0.105716425581988}\,0.167202282817078\,0.143312197000282\,0.0904681118110068\,0.00420740059943495\,0.151051878859773\,\mathrm{-0.0386596747220680}\,0.0440691237914762\,\mathrm{-0.00670225429357946}\,0.0933890481068025\,0.0478184461625122\,0.0278017403383086\,0.109443084167184\,0.0397625283920667\,\mathrm{-0.121323537606438}\,0.261193618963694\,0.0993582186147168\,0.0255303351071521\,\mathrm{-0.0241543939453960}\,0.0962199411551923\,\mathrm{-0.0157960144948491}\,\mathrm{-0.0342292986942833}\,\mathrm{-0.0525952037444790}\,0.150651932610134\,0.192128739829213\,\mathrm{-0.197094892675294}\,0.0716624529444702\,0.0210781638490448\,0.0218981113573270\,0.157603381498783\,0.0716964674343195\,0.0693803864165980\,0.0616715533493450\,0.0143206648485258\,\mathrm{-0.181484887394651}\,\mathrm{-0.0337656194550488}\,0.0276303588503054\,0.0951330555516539\,\mathrm{-0.0429790945303218}\,0.0455379098359191\,0.0366036903524287\,\mathrm{-0.0553457126157233}\,0.106723361618346\,0.140201886256280\,\mathrm{-0.0720866213397492}\,\mathrm{-0.137249480688211}\,0.0549362085338733\,\mathrm{-0.00167268056838582}\,0.129099704821460\,0.0254442515639488\,\mathrm{-0.0372671496029909}\,0.00869660121897113\,0.0640425662408623\,\mathrm{-0.0292282538277321}\,0.0962534001496196\,0.0583098777724654\,0.282818705363368\,0.135753946910954\,\mathrm{-0.166917470877993}\,0.0935508850482316\,\mathrm{-0.00462483722461739}\,0.105955669223280\,0.0633363321550364\,\mathrm{-0.234756705334042}\,\mathrm{-0.146622567958132}\,0.131133388816615\,\mathrm{-0.167983185691991}\,0.0833409806086111\,0.139678949514575\,0.0948898999584073\,0.201911097408961\,0.0463137263598911\,\mathrm{-0.131204554889707}\,0.104905900745445\,0.142937189262709\,0.126189572096261\,0.0812593501961863\,\mathrm{-0.00426991039536866}\,\mathrm{-0.0284706376734075}\,\mathrm{-0.0256090970343518}\,\mathrm{-0.0420351675532923}\,\mathrm{-0.00415469312979765}\,\mathrm{-0.0337334452865951}\,\mathrm{-0.135343885108397}\,\mathrm{-0.146335429275237}\,0.00740581624632347\,\dots \,\text{⋯ 100 row vector entries not shown}\right]$

$Y≔\mathrm{map}\left(\sin \,X\right)+\mathrm{Yerror}$

$Y≔\left[0.454367047823680\,0.462765422150169\,0.417011371451704\,0.208093241995450\,1.04368688151902\,0.290386326063255\,0.863431606558844\,0.986028510037196\,0.125388971861870\,0.0667479930224107\,0.479997241960240\,\mathrm{-0.00433216009474287}\,0.369491006894417\,0.929492342125722\,0.481355758387552\,0.598336169048849\,1.11325658273403\,0.352111636664697\,0.611620407680281\,0.277966968110887\,0.844014702644611\,0.156025460819454\,0.449723773184358\,0.299272415266748\,0.894268876685337\,0.717506928624505\,0.831639936250323\,0.982990192299385\,0.761738576735361\,0.773440413284528\,0.897065351753078\,0.125369558917136\,0.740161140300520\,0.240763529723844\,0.284638794481224\,0.492394052801364\,0.765863976074385\,0.990062283469757\,0.347874168316888\,\mathrm{-0.0890904272044914}\,1.05320289800621\,0.952645850199230\,0.693708919326861\,0.757521280138683\,0.625626778338910\,1.06886342361315\,1.04709583411662\,0.910525881763713\,0.609891571965787\,0.662853704236079\,0.790123843673663\,0.939961111426311\,0.840642423433526\,0.534465390413217\,0.401798671725056\,0.944641080274323\,0.232854589833541\,1.01709170995567\,0.892248495259157\,0.509355909689420\,0.759222697004252\,0.716661533636967\,1.12892459124919\,0.836162747483787\,0.298799518668049\,0.136238022700087\,1.05306153698097\,0.392636918139674\,0.548266579652729\,0.781897555687775\,0.762596553350432\,0.852309066164837\,0.383981839378259\,0.786128462229965\,0.215775500347837\,0.996939038436453\,0.642434607259717\,0.474753699054593\,0.787656341372445\,1.12761454419245\,0.725378096076094\,0.590139980696884\,1.10401710930675\,1.08271370183219\,0.459130591665432\,0.828389469888631\,0.559727806794417\,0.803680382760712\,1.07322838379324\,1.10357633664839\,0.317313545556865\,0.164409075004430\,0.966852405590904\,0.613828544030345\,0.163794594547286\,0.392722759490367\,0.942982816073745\,0.860036098392529\,\mathrm{-0.108952687520426}\,0.879321951048542\,\dots \,\text{⋯ 100 row vector entries not shown}\right]$

$L≔\mathrm{CurveFitting}:-\mathrm{Lowess}\left(X\,Y\,\mathrm{fitorder}=1\,\mathrm{bandwidth}=0.3\right)\:$

$P≔\mathrm{ScatterPlot}\left(X\,Y\right)$

$Q≔\mathrm{plot}\left(L\left(x\right)\,x=0..\mathrm{\pi }\right)$

$R≔\mathrm{plots}:-\mathrm{shadebetween}\left(L\left(x\right)\,0\,x=\frac{\mathrm{\pi }}{8}..\frac{3\mathrm{\pi }}{8}\,\mathrm{showboundary}=\mathrm{false}\,\mathrm{positiveonly}\right)$

$\mathrm{plots}:-\mathrm{display}\left(P\,Q\,R\right)$

$\mathrm{int}\left(L\,\frac{\mathrm{\pi }}{8}..\frac{3\mathrm{\pi }}{8}\,\mathrm{numeric}\,\mathrm{\epsilon }=0.01\right)$

$0.538403465288975$

$\mathrm{Optimization}:-\mathrm{Maximize}\left(L\,\mathrm{map}\left(\mathrm{unapply}\,\left\{-x\,x-\mathrm{\pi }\right\}\,x\right)\,\mathrm{optimalitytolerance}=0.001\right)$

$\left[1.00180213049401101\,\left[\begin{array}{c}1.58152459006664\end{array}\right]\right]$

The CurveFitting[Lowess] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.