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Semi-Markov model for simulating MOOC students Louis Faucon, Łukasz Kidzin´ski, Pierre Dillenbourg ComputerHumanInteractioninLearningandInstruction ÉcolePolytechniqueFédéraledeLausanne {louis.faucon,lukasz.kidzninski,pierre.dillenbourg}@epfl.ch ABSTRACT Weproposeaprobabilisticmodel, basedonextendedversionof Large-scaleexperimentsareoftenexpensiveandtimeconsuming. MarkovChains,calledsemi-MarkovChains.Inthemodel,wecan AlthoughMassiveOnlineOpenCourses(MOOCs)provideasolid balancethecomplexityofthestructureandthenumberofparame- andconsistentframeworkforlearninganalytics,MOOCpractition- terstoestimatebycross-validatingitsparameters. Wepresentan ersarestillreluctanttoriskresourcesinexperiments.Inthisstudy, algorithmforfittingthemodelaswellasillustrativeexamplesof wesuggestamethodologyforsimulatingMOOCstudents,which thefitonasetofMOOCs. allowestimationofdistributions,beforeimplementingalarge-scale experiment. Thecontributionsofthispaperarethreefold.First,weinvestigate towhatextentSemi-Markovchainscanbeusedtodescribebe- Tothisend,weemploygenerativemodelstodrawindependentsam- haviouralpatternsofstudents(RQ1). Second,sinceourmodel plesofartificialstudentsinMonteCarlosimulations.WeuseSemi- implicitlydividesusersintoclusters,weanalyseiftheseclusters MarkovChainsformodelingstudent’sactivitiesandExpectation- are interpretable (RQ2). Third, we analyse how these models Maximizationalgorithmforfittingthemodel.Fromthefittedmodel, canbeusedtoinferdistributionsofevents(RQ3). wegeneratesimulatedstudentswhoseprocessesofweeklyactivities aresimilartotheseoftherealstudents. 2. RELATEDWORK Modelingstudentsisakeyconceptinlearninganalyticsandedu- Keywords cationalresearchingeneral. Researchersbuildmodelspredicting MOOCs;simulationofstudents;generativemodels;Expectation- motivationandcognition,basedonstudent’sgoals[19]ortheypre- Maximization;Semi-Markovchains;Bayesianstatistics dictgoalsbymotivationaltraits[7].Largedatasetsallowresearchers tofindpredictivepowerofseeminglyslightlyrelatedsignalslike 1. INTRODUCTION thelengthofpausesinavideo[12]orpotentiallynoisysignalslike headmovementintheclassroom[16]. Vastamountsofdatawhichwegatherandanalyseinmodernlearn- ingenvironmentsallowustobuildmodelsofunprecedentedscale andaccuracy.Thisphenomenon,inparallelwithdevelopmentsin 2.1 GenerativemodelsinMOOCs computerscience,gaverisetonewpossibilitiesofinferencefrom Alltheaforementionedmodelsarefocusedonpredictionandbelong educationalenvironments.Inparticular,thegrowingfieldofSimu- totheclassofso-calleddiscriminativemodels. Inthisstudy,we latedLearners[8,11,14]providesuswithtoolsforinferencefrom suggestagenerativemodel,whichallowusnotonlytopredict,but educationalsimulations. alsotogenerateobservationsfromtheestimateddistribution.These modelscapturetheprobabilitystructureofinputvariablesandthe Inferencefromanysimulationsisboundedbythepredefinedlevel flowoftheprocesses.SeveralgenerativemodelsinMOOCshave of abstraction of the analysis. In the context of Massive Online beenapplied,e.g.toforums[3]. OpenCourses(MOOCs),ononehandasaneducationalinstitution wehaveaccesstoonlyahandfulofMOOCs,onanotherhand,we Amongmanygenerativemodelsthatcanbeencounteredineduca- havedataasgranularasstudent’sclickstreaminavideoplayer.We tionalresearch,Markovmodelswereemployedforvisualization[5], arethereforeobligedtomodelgranularityrobustly,dependingon for modeling engagement [17] and for modeling students reten- theavailabilityofthedata.Wearguethatunderstandingtheproper- tion[1]. tiesofthestatisticalmethodologyathandiscrucialforsuccessful inference. 2.2 Simulatedlearner Theareaofsimulatingstudents’behaviourlaysontheintersection ofcognitivescienceandartificialintelligence.Examplesofapplica- tionsofsimulationofstudentscanbefoundevenoutsidecomputer science,wheretheteachersimulatesstudent’sresponseinorderto self-improveinstructionalskills[18]. Anacknowledgedexample of the usage of simulating humans [9] for education deals with simulationsofpatientsbehaviourfortrainingmedicinestudents. EmergenceofInternetandnewdatastoragetechniquesallowre- Proceedings of the 9th International Conference on Educational Data Mining 358 searcherstocollectandanalysemassiveamountsofinformation ForumPost}.Weaddtothissetonespecialtypeofevent,EndOfDay. abouttheusers.Researchersemploysimulationsforclusteringstu- ThiseventcorrespondstotheendofinteractionswithMOOCsona dents[13]. Forareviewofearliertechniqueswereferto[2]. We givenday.Weusethesymbola Atodesignateanytypeofactivity. ∈ motivateourmethodologybytheadvancementsofusermodelingin Onecanextendthesetofactivitiestoothereventsifneededfor webcontext[4],aswefindthisenvironmentconceptuallycloseto certainapplication. theenvironmentofaMOOC. Note that we do not specify the regular ’end of a course’ event, 3. GENERALFRAMEWORK sinceweonlymodelthebehaviourwithinthelimitedtime-frame ofacourseandwetreatthelastdayofthecourseasthelastdayof 3.1 Dataset theprocess.Therefore,eachstudentwhowentthroughthewhole FromourinternalMOOCdatabase,aggregatingdatafromCours- coursewithoutdroppingouthasjustaEndOfDayeventonthelast eraandedX,weextractedeventsfor61EPFLcourses. Theraw dayofthecourse.NumberofEndOfDayeventsisthereforeequal datacontainedapproximately23millioneventsfor500,000stu- tothenumberofdaysofthecourse. dents,arrangedintuples: <StudentID,CourseID,EventType, Timestamp>.TheEventTypedescribesthetypeofanactivityand (s) (s) (s) takesoneoffourpossiblevaluespresentedinTable1.Wechoose TherandomsequentialvariableX1 ,X2 ,...,Xn representsthe (s) theseeventsasthemostdiscriminativeactionsfromthekeyareas: sequence of activities of one student s. Each Xi ∈A and the learning, validation and community engagement. Note that our sequencestopsafteranEndOfDaywhenthestudentreachestheend modellingtechniquecanbeeasilyextendedtocoverothertypesof ofthecourse.Wedenotethelengthofthesequenceforastudents events. asn(s).TheobservationofonestudentactivityalongoneMOOCis thusarealizationoftherandomsequenceX. Abbreviation Description Proportion VideoPlay watchingavideo 51% TheprobabilitydistributionP:Ingeneral,foreachstudents S Submission submittinganassignment 33% (s) ∈ wecanmodelthei-theventX withaprobabilitydistribution ForumView visitingtheforum 15% i ForumPost postingontheforum 1% P(s)(X(is)=a|X(is)1,,X(is)2,...,X(1s),Cs), − − Table1:Distributionofeventsinthedataset. wherea A,X(s),...,X(s) arethepreviouseventsofthatstudent ∈ 1 i 1 andCsarepersonalchar−acteristicsofthestudent. FortheanalysiswedevelopedourownPythonimplementationofthe algorithmfittingthemodel1.InSection5weexplainthealgorithm Thisdistributionrepresentsthestudent’sbehaviourprofileandal- indetail.Since23millioneventscanstillfitinmemoryofasingle lowstogeneratetypicalsequencesofactivities.Ourmainobjective computer,wedidnotrequireaspecificcomputingarchitectureto is to model this distribution as accurately as possible, given the performtheanalysis.However,giventheconsiderablesizeofthe limitedinformation. Theaccuratedistributionwouldallowusto dataset,thealgorithmtakesseveralminutestorun. drawsamplesofstudents. 3.2 Definitions 3.3 Assumptions Westartwithageneralframework,inwhichstudent’sactivityinany Asdiscussedintheprevioussection,assessingP(s) isunfeasible MOOCcanbeverypreciselydescribed.Next,weelevateabstraction duetodependenceontoomanyeventsinthepastandduetothe ofthemodelbyaddingassumptionssimplifyingtheanalysis.Our lack of information on personal student features. In order to fit goalistointroduceamodelwhosecomplexitycanbeadaptedto a probabilistic model we need to relax these dependencies. We thestructureofacourseandtheamountofavailabledata. introducefollowingassumptions: Weconsideramodelinwhichstudentsbehaviourisdescribedin A1 Students’behavioursfitintoasmallnumberofnaturalcate- asequentialmannerbythetypeofactivitytheyperformandthe goriesofbehaviour. timetheywaitbetweentwosessions.Furthermore,asmostofthe A2 Thetypeofactivitydependsonlyonhispreviousactivityand studentsperformatmost1MOOCsessionperday,wechoosea notonoldpastactivities. dailygranularityofactions. AssumptionA1mapsthespaceofallpossiblestudents’character- Asequenceofstudent’sdailyactivityisdescribedasalistof’active isticsintoalimitednumberofcategories,whicharemucheasier events’ (VideoPlay, Submission, ForumView and ForumPost) toattribute. ManystudiesonMOOCsexplicitlyclassifystudents followedbya’endofthedayevent’(EndOfDay)oronlyaEndOfDay intoasmallnumberofcategories[10],studentsaredividedbetween inthecasethestudentdidnotperformanyactivitythegivenday. ’Viewers’whoonlywatchvideos,’ForumActives’whosharewith Theformaldefinitionofthemodelisfollowing: theirpeersintheMOOCdiscussionforumand’Completers’who succeedintheassignments.Aswepresentinthenextsection,our ThesetofallstudentsS:Weusethesymbols S todesignate ∈ methodisbasedonunsupervisedclustering,wheregroupsemerge anindividualstudent. inthewayoptimalintermsofmaximumlikelihoodofthemodel. ThesetAofalltypesofactivities: Forthisstudywechoseaset AssumptionA2weimposethatonlythelastactivityhasanimpact offourtypesofevents: {VideoPlay,Submission,ForumView, onthecurrentactivity. Thisassumptionismoreconstraining,but 1Our implementation is available under https://github.com/ sincethecomplexityofhistorygrowsexponentiallywiththenumber lfaucon/edm2016-mooc-simulator ofstepsand,inordertobeabletoestimateparameters,wehaveto Proceedings of the 9th International Conference on Educational Data Mining 359 reducethesearchspace. Thissimplificationisusuallycalledthe whereristhenumberofrepetitionsandλkistheaveragenumber a ’Markovassumption’. ofrepetitionandneedstobeestimatedfromthedataforeachkand a. ApartfromtechnicalassumptionsrequiredforMarkovModels,we imposeotherassumptionsforconvenience.First,wedonotconsider ToillustratethatthePoissondistributionimprovesthemodel,let lengthofevents,sotheVideoPlayeventisonlythemomentwhen usconsideranexample. Supposeweexpectthatsomegroupof astudentstartswatchingavideo. Second,iftheseriesofevents studentsconnectstoaMOOCtwiceaweek,withapproximately happensduringmidnight,stillaneventEndOfDayisaddedtothe threedaysintervalbetweenconnections.Inthatcase,theaverage sequence. numberofrepetitionsoftheEndOfDayeventis3.SimpleMarkov Model,accuratelymodelstheaveragetobe3butimplicitlyassumes thatthemajorityofstudentsgetsonly1repetition. Semi-Markov 4. PROBABILISTICMODELING modelwithPoissondistributionalsogivestheaverageequalto3 4.1 Softclustering andthedistributionisconcentratedaround3. In Section 3 we proposed a simplified framework, in which we assumethatthereareonlyafewdifferentpossibleclassesofstu- 5. FITTINGTHEMODEL dents (A1). We enumerate clusters 1,2,...,K. For each student 5.1 Algorithm (s) s S weintroduceaprobabilitydistributionµ whichdescribes ∈ k TheExpectation-Maximisation(EM)algorithmhasbeenintroduced probabilitythatthestudentbelongstothebehaviourclassesk,for in1977in[6]. Thegoalofthisiterativetechniqueistocompute k 1,2,...,K . ∈{ } theparametersthatmaximizethelikelihoodofagivenprobabilistic model.TheEMalgorithmhasbeenproventoconvergeatleasttoa Thistechniqueisoftenreferredtoassoftclustering,weightedclus- localminimum.Thisminimumdependsontheinitializationpoint, teringorfuzzyclustering[15].Insteadofdiscretclusterassignment, thusmultiplerunswithdifferentrandominitialisationsareoften as for example in K-means, we obtain for each student a proba- usedinpracticeinordertoincreasethechancesoffindingtheglobal bilitydistributionamongtheclusters. Theseprobabilitiescanbe minimum. intuitivelyseenasourcertaintythatthestudentbelongstoagiven cluster. InthisstudyweusetheEMalgorithmforunsupervisedlearning. Neithertheparametersofthelatentclassesnortherepartitionof 4.2 Semi-MarkovChain thestudentsareknownatthebeginningandthealgorithmhasto Assumption(A2),i.e.dependenceonlyonthelaststate,allowsus estimatebothquantitiesatonce.Inoursettings,wedefineforeach tomodeltheprocessMarkovChains.Formally,inthedefinitionof k 1,2,...,K andstatesaandb: ∈{ } distributionofthenexteventwecandropdependenceoftheevents whichoccurredbeforethecurrentone,i.e.weidentify (k) - p ,theprobabilitythatastudentwiththebehaviourpro- b >a P(s)(X(s) X(s),...,X(s) )=P(s)(X(s) X(s) ) file−kperformstheactivityaaftertheactivityb: i | 1 i 1 i | i 1 − − AclapsrseicliMmianrakroyvaMnaoldyeslissirnevoeuarlceodnatenxti.mApotrratadnittiowneaalkMnaersksoovfmuosidnegl p(bk−)>a=P(Xi=a|Xi−1=b) considersthatastudentisequallylikelytostopwatchingvideos (k) whentheyhavewatchedone,aswhentheyhavealreadywatchedten - λa ,theaveragenumberofrepetitionsofaneventafroma studentofprofilek. videos.Inpractice,studentswatchvideossequentiallyandMarkov Modeldoesnotcaptureappropriatelythenumberofeventsinthe (s) - µ ,theprobabilitythatastudentsbelongstotheprofilek. sequence. k ToremedythisissueweemployedSemi-MarkovModels(alsocalled Wecanthuscomputethelikelihoodoftheobservedsequence,asa MarkovRenewalProcesses).Thekeyfeatureofthismodelisthat functionofclusterrepartitionandparametersofMarkovChainsby itallowstoreplacetheself-loops(transitionsfromoneeventtype toitself)intheMarkovChain,byaprobabilitydistributionofthe K numberofrepetitionofagivenstate. likelihood=s∏∈S[k∑=1µk(s)(a,b∏,r)∈Tsp(bk−)>aPλa(k)(r)], (1) InSemi-MarkovModels,westillneedtochooseaparametricdistri- whereTsisthesetoftuples(a,b,r) A A Ncorrespondingto bution,butwehavemorefreedomthanintraditionalMarkovChain. ∈ × × transitionsfromactivitybtoactivityawithrrepetitionsofactivity MarkovChainimplicitlyassumesthatprobabilityofstayinginthe a.Thegoalofthealgorithmistofindtheparametersthatmaximize samestateisthelargestfor1stepanddecreaseswithnumberof thelikelihood. steps. However,wewouldexpectthat1isnotthemostprobable numberofrepetitionatleastforaparticulargroupofstudents.This Inthefirststage,thealgorithminitializerandomlyKprofiles.Next, phenomenoncanbecapturedby,forexample,Poissondistribution, ititerativelyimprovesthe likelihood,byalternatingtwostepsas whichprovedtobemoreaccurateinourpreliminaryanalysis.Thus, described below. In each step it modifies the repartition or the foraneventa Aandaclasskwemodelthenumberofrepeated eventsRk by ∈ Markovchainparameters. a P(Rka=r)= e−λakr!λak r Ienitihteiarlitzhaetipob(nk):>aThaendinλiat(ika)liozrattihoenµck(osn).siIsntsoiunrcahlgooorsiitnhgmr,awndeosmtalryt (cid:0) (cid:1) − Proceedings of the 9th International Conference on Educational Data Mining 360 (s) withtheµk .Thiscanbedonebygeneratingarandomnumberk∗ from1toKforeachstudentsandbysetting µ(s)= 1 ifk=k∗ k 0 otherwise. (cid:26) Iterations: Theiterationphasehastwosteps. First,wecompute (k) (k) (s) the optimal values for pb >a and λa given that µk are fixed (equations(2)and(3)). − ∑ ∑ µ(s) k p(k) = s∈S(a,b,_)∈Ts (2) b−>a ∑ ∑ µ(s) k s∈S(_,b,_)∈Ts ∑ ∑ rµ(s) k λa(k)= s∈∑S(a,_,∑r)∈Ts µ(s) (3) k s∈S(a,_,_)∈Ts (s) Next, we compute the new values of µ according to the new k (k) (k) pb >aandλa (equations(4)). − ∏ p(k) P (r) µ(s)= (a,b,r)∈Ts b−>a λa(k) (4) k K ∑ ∏ p(c) P (r) c=1(a,b,r)∈Ts b−>a λa(c) Intuitively,theinthefirststepwecomputetheparametersofthe latentclassesgiventherepartitionofthestudentsandinthesecond stepwerecomputetherepartitionfromthenewclassesparameters. 5.2 Example: Interpretationclusters(K=3) Beforewepresenttheresultsforthechoiceofthenumberofclusters, inthissection,weillustratethebehaviourofthealgorithmandthe model when the number of clusters is small (K=3). Although in this case we may lose important variability among groups of students,smallnumberofclustersallowsustovisualisetheSemi- Markovmodelsandinterpreteachoftheclusters. The visualizations of the Semi-Markov models on Figure 1 can revealgeneralcharacteristicsofstudents’behaviours. Forexam- ple,Profiles1and3areingenerallessactiveastheyhavemore EndOfDayevents.Onthecontrary,Profile3hasaveryhighaverage numberofrepetitiononVideoPlayandconsiderableprobability togobacktoEndOfDayevents. Thismeansthatstudentsofthis Figure1:Threegraphicalrepresentationsofbehaviourprofiles clusterarenotfullyengagedinallMOOCactivities. extractedbytheEMalgorithm. Fromtoptobottom: profiles 1, 2 and 3 (thickness: transition probability; color: average Amoreinsightfulwaytoanalyseandinterpretthedifferencesisto numberofrepetitions) generatesequencesofeventsandcomparetheoutcomes. Wecan computetheexpectednumberofvideoswatchedortheexpected numberofpostontheforumdirectlyfromsimulatedsequences. Table2showstheaveragenumberofseveraltypesofeventsfor example, we can see that students of Profile 1 participate in the 100simulatedstudents(averagefrom10000simulations)overfour collaborativeactivitiesoftheMOOCmorerarely,butengageinthe weeksgeneratedwiththethreeMarkovmodelsfromFigure1.For assignmentsmorethaninwatchingthevideos.Thismightindicate Proceedings of the 9th International Conference on Educational Data Mining 361 thattheyalreadyhaveagoodunderstandingofthecontentofthe 6. SIMULATIONS courseanddonotneedtospendmoretimeonstudying. Tofully WithamodelfittedwiththeEMalgorithmathand,thealgorithm investigatethishypothesis,furtheranalysisshouldbeconducted. repartitioned students and chose parameters of a Semi-Markov Chainforeachoftheclusters. Sinceboththerepartitionandthe Profiles 1 2 3 Semi-MarkovChainsaregenerative,wecandrawsamplesfromthe WatchedVideos 1060 3133 2363 fitteddistribution,i.e. wecansimulatethestudents. Werunthe Submissions 1535 2423 442 simulationsandshowapossiblewaytomeasurethevalidityofthe ForumVisits 68 1711 255 results. Forumposts 3 96 15 Tovalidatepotentialvalueofsimulations,wefirstproposeasimple Table 2: Average number of events for 100 students over the accuracy measure. In equation (6), P (a >n) represents the real | | firstfourweeksoftheMOOC probabilitythatastudentperformsmorethanneventsoftypea duringthetimeoftheMOOC. a isthecountofeventsoftypea. | | 5.3 ChoiceoftheparameterK Psim(a >n)representsthesameprobabilitybutforasimulated | | Acommonchallengeofunsupervisedlearningandfittingaproba- student.InthemeasurewechosethevalueN=50becauseitcovers bilisticmodelisfindingthecorrectnumberofclasses.Inourcase, mostofthevariabilityinthestudentsactivitysequencesandisnot thesimilarityofthealgorithmwithotherclusteringtechniquessuch toolargeasstill19%ofthestudentshaveanactivitywithmorethan astheK-meansleadstothe"elbowheuristic",oftenusedinpractice. 50repetitions. Theideaistochoosethenumberofclusterslargeenoughtoexplain a large part of the variability, but such that a greater number of clusterswouldnotexplainsubstantiallymore. MSE= (A 11) N ∑ ∑ (Preal(|a|>n)−Psim(|a|>n))2 | |− ∗ a An<N ∈ (6) Inordertoprovethecorrectnessofthemodelingmethod,wedivided ourdatasetintoatrainingsetandatestsetforvalidatingtheresults. Thefirststepistorunthealgorithmonthetrainingsetwithseveral parameterK and then, usethe computed parametersto simulate anewpopulationofstudentsandfinallycomparethispopulation with the students from the testing set. In Figure 3 we can see thatthefitdoesnotimprovemuchafterK=15,becausetoohigh numberofclustersmakesthealgorithmlearnmostlythenoisefrom the random actions of the students instead of their real intrinsic behaviouralpatterns. Figure 2: Average distance of students from their model for differentnumberofclasses Inordertoconfirmtheresultofthisfirstmeasureofquality,we designedanothermeasuredescribedintheequation(5).Thegoalis toquantifyhowthestudentssequencesdivergefromtheirattributed cluster.Intheequation, A isthecardinalityofthesetofpossible | | activities, ps(a) is the probability of finding the activity a if we takeuniformlyatrandomanactivityofstudentsand p (a)isthe k probabilityoffindingtheactivityaifwetakeuniformlyatrandom anactivityfromasequencegeneratedbytheclassk. 1 Figure 3: Measure of accuracy of a simulation for different d2(s,k)= A ∑(ps(a)−pk(a))2 (5) numberofclasses | |a A ∈ Thesmallerrorprovesthatthedistributionobtainedfromsimula- Thisdistancemeasureshowsanelbowshapeforthesamevaluesof tionsisclosetotheoriginaldistribution.Thisimpliesthatthemodel Kbetween10and15asitcanbeseenonFigure2. 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