Introduction and Applications of Temporal Point Processes · Introduction and Applications of...
Transcript of Introduction and Applications of Temporal Point Processes · Introduction and Applications of...
ETHZLECTURE,14DECEMBER2018
IntroductionandApplicationsofTemporalPointProcesses
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IsabelValera
MPIforIntelligentSystems
OutlineoftheLecture
1.Intensityfunction2.Basicbuildingblocks3.Superposition4.MarksandSDEswithjumps
INTROTOTEMPORALPOINTPROCESSES(TPPS)
APPLICATION:CLUSTERINGEVENTSEQUENCES1.ProblemStatement2.IntroductiontoDPMM3.CRP+HP(a.k.a.HDHP)4.Generativeprocess
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1.ProblemStatement2.IntroductiontoDPs
3.CRP+HP(a.k.a.HDHP)4.GenerativeProcess
APPLICATION:CLUSTERINGEVENTSEQUENCES
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IntroductiontoprogrammingDiscretemathProjectpresentation
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For/do-whileloops
If…else
Howtowriteswitch Private
functions
Definefunctions
Classinheritance
Classdestructor
Plotlibrary
Logic
Settheory Geometry
GraphTheoryPowerpointvs.Keynote
Exportpptxtopdf
PPtemplates
1styearcomputersciencestudent
CLUSTERINGEVENTSEQUENCES
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1styearcomputersciencestudent
CLUSTERINGEVENTSEQUENCES
Content+Dynamics=Cluster(learningpattern)E.g.,programing+semester
math+semester presentation+week
CLUSTERINGEVENTSEQUENCES
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Several people share same clusters
IntroductiontoprogrammingDiscretemathProjectpresentation
CLUSTERINGEVENTSEQUENCES
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Event cluster (topic) is hidden è Clustering of events
Unknown number of clusters è Dirichlet Process
OutlineoftheLecture
1.Intensityfunction2.Basicbuildingblocks3.Superposition4.MarksandSDEswithjumps
INTROTOTEMPORALPOINTPROCESSES(TPPS)
APPLICATION:CLUSTERINGEVENTSEQUENCES1.ProblemStatement2.IntroductiontoDPMM3.CRP+HP(a.k.a.HDHP)4.Generativeprocess
1.ProblemStatement2.IntroductiontoDPs
3.CRP+HP(a.k.a.HDHP)4.GenerativeProcess
APPLICATION:CLUSTERINGEVENTSEQUENCES
DirichletProcess(DP)
G0 =1X
`=1
⇡`�✓`
G0 ⇠ DP (�, H) ⇡ = (⇡`)1`=1 ⇠ GEM(�)
✓` ⇠ H(✓)Stick breaking
⇥
G0
H
Cluster parameters
DirichletProcess:Randomprocesswhoserealizationconsistsofprobabilitydistributions
Concentration parameter
Base distribution
DirichletProcess(DP)
Properties:§ Infinite-dimensionalgeneralizationofDirichletdistribution
§ Priorforclustering§ Infinitemodelcomplexity=infinite#ofgroups
§ Partitiondataintogroups
G0 =1X
`=1
⇡`�✓`
Conjugacytothemultinomial
ChineseRestaurantProcess(CRP)
Exchangeability:
ü Clusters
· · ·b1 b5
b7b10b12b14
�1 �4 �6 ü Observations
1. Table assignment:
p(bn+1 = `|b1, . . . , bn) =⇢ m`
n+� k K+
↵n+� k +K+ + 1
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2. Cluster (dish) assignment:
�j(K+1) =
(✓` w.p. m`
K+�1for ` = 1, . . . , L
✓L+1 w.p. �1
K+�1
ChineseRestaurantProcess(CRP)
Exchangeability:
ü Clusters
· · ·b1 b5
b7b10b12b14
�1 �4 �6 ü Observations
CLUSTERINGEVENTSEQUENCES:
- Eachuserperformasequenceofevents- Eventsarenotexchangeable
t t
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1.ProblemStatement2.IntroductiontoDPs
3.CRP+HP(a.k.a.HDHP)4.GenerativeProcess
APPLICATION:CLUSTERINGEVENTSEQUENCES
CRP+HP(I)
Foreachuserandcluster,werepresenteventsasacountingprocess:
t = 0 t = T
Task Task
t
(tn, pn, qn)Event:
Cluster (hidden)
Time Content
pnNu,`(T ) = 9
CRP+HP(II)
IntensityforeachuserandclusterfromaHawkesprocess:
t = 0 t = T
Task Task
t
�⇤
u,`(t) = µu⇡` +X
j:tj2Hu,`(t)
k✓`(t� tj)
Exogeneousevent Endogenousevent
Clusterprobability(popularity)
Newtaskrate
Memory Hawkesprocess
Intensityorrate
(events/hour)
CRP+HP(III)
t
(tn, pn) ⇠ Hawkes
0
B@�⇤u,1(t)...
�⇤u,1(t)
1
CA
ClusterTime
Mark(content/words)
Users adopt more than one learning pattern
User’seventssampledforaninfinitemultivariateHawkes:
CRP+HP(IV)
Different users adopt same learning patterns
- Infinite#oflearningpatterns.- Sharedparametersacrossusers.
LearningpatterndistributionfromaDirichletprocess:
t t
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1.ProblemStatement2.IntroductiontoDPs
3.CRP+HP(a.k.a.HDHP)4.GenerativeProcess
APPLICATION:CLUSTERINGEVENTSEQUENCES
GenerativeProcess(I)
1. Time and user sampling:
Exchangeability: X Events ü Clusters
· · ·
· · ·
b1
b2b3
b4
b5
b6
b7
b8b9
b10
b11
b12
b13
b14 User 1
User 2
�2
�1
�3
�4
�5
�6
�7
(tn, un) ⇠ Hawkes
0
B@
�0 +P
i:ti2Hu(tn)bi(tn, ti)
.
.
.
�0 +P
i:ti2HU(tn)bi(tn, ti)
1
CA
GenerativeProcess(I)
Exchangeability: X Events ü Clusters
· · ·
· · ·
b1
b2b3
b4
b5
b6
b7
b8b9
b10
b11
b12
b13
b14 User 1
User 2
�2
�1
�3
�4
�5
�6
�7
2. Task (table) assignment :
bn =
(k w. p. �un,k(tn)
�un (tn), for k = 1, . . . ,K
Knew w. p. �0�un (tn)
GenerativeProcess(I)
Exchangeability: X Events ü Clusters
· · ·
· · ·
b1
b2b3
b4
b5
b6
b7
b8b9
b10
b11
b12
b13
b14 User 1
User 2
�2
�1
�3
�4
�5
�6
�7
3. Learning pattern (dish) assignment:
�j(K+1) =
(✓` w.p. m`
K+�1for ` = 1, . . . , L
✓L+1 w.p. �1
K+�1
GenerativeProcess(II)
General approach to sample from BNP+HP processes:
GenerativeProcess(II)
Suitable for online inference (Sequential Monte Carlo):
Example:OnlineLearningActivity
t t
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16kusers
1.6millionquestionsusing31.4ktags
Wegathereddatafrom..Over4years(09/2010–08/2014):learningevent:question(andanswers)
time:timewhenthequestionwasasked
content:questiontags
user:userwhoaskedthequestion
Experiments
Example:OnlineLearningActivity
SEP MAR SEP MAR SEP MAR SEP MAR
SEP MAR SEP MAR SEP MAR SEP MAR
Intensity
Versioncontroltaskstendtobespecific,quicklysolvedafterperformingfewquestions
IntensitiesContent
Example:OnlineLearningActivity
IntensitiesContent
SEP MAR SEP MAR SEP MAR SEP MAR
SEP MAR SEP MAR SEP MAR SEP MAR
Machinelearningtaskstendtobemorecomplexandrequireaskingmorequestions
OutlineoftheLecture
1.Intensityfunction2.Basicbuildingblocks3.Superposition4.MarksandSDEswithjumps
INTROTOTEMPORALPOINTPROCESSES(TPPS)
APPLICATION:CLUSTERINGEVENTSEQUENCES1.ProblemStatement2.IntroductiontoDPMM3.CRP+HP(a.k.a.HDHP)4.Generativeprocess
MoreaboutTTPs
1.Intensityfunction2.Basicbuildingblocks3.Superposition4.MarksandSDEswithjumps
TEMPORALPOINTPROCESSES(TPPS):INTRO
MODELS&INFERENCE1.Modelingeventsequences2.Clusteringeventsequences3.Capturingcomplexdynamics4.Causalreasoningoneventsequences
Slides/references: learning.mpi-sws.org/tpp-icml18
1.MarkedTPPs:anewsetting2.Stochasticoptimalcontrol3.Reinforcementlearning
RL&CONTROL