Learning Hawkes Processes Under Synchronization Noise11-14-00)-11-14-20-4830... · Learning Hawkes...
Transcript of Learning Hawkes Processes Under Synchronization Noise11-14-00)-11-14-20-4830... · Learning Hawkes...
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Learning Hawkes Processes Under Synchronization Noise
William Trouleau
Presented at ICML’19on Tue Jun 11th 2019
Jalal Etesami Negar KiyavashMatthias Grossglauser Patrick Thiran
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Question of interest
Learning the causal structure of networks of multivariate time series
in continuous time
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?
Don’t listen to @Bob, it’s FAKE NEWS!
Charly@TruthSeeker
This candidate will stop global warming! Vote for him!
Bob@Bob
Example1: Information Diffusion
• Consider a network of users
• We observe a a sequence of discrete events in continuous time:tweets, Facebook posts…
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Don’t listen to @Bob, it’s FAKE NEWS!
Charly@TruthSeeker
This candidate will stop global warming! Vote for him!
Bob@Bob
Example1: Information Diffusion
?
• Consider a network of users
• We observe a a sequence of discrete events in continuous time:tweets, Facebook posts…
• Questions of interest:Who influences whom? How does fake news spread?
![Page 5: Learning Hawkes Processes Under Synchronization Noise11-14-00)-11-14-20-4830... · Learning Hawkes Processes Under Synchronization Noise William Trouleau Presented at ICML’19 on](https://reader033.fdocuments.net/reader033/viewer/2022060505/5f1e3a85483a6e168c4fcc16/html5/thumbnails/5.jpg)
Don’t listen to @Bob, it’s FAKE NEWS!
Charly@TruthSeeker
This candidate will stop global warming! Vote for him!
Bob@Bob
Example1: Information Diffusion
?
• Consider a network of users
• We observe a a sequence of discrete events in continuous time:tweets, Facebook posts…
• Questions of interest:Who influences whom? How does fake news spread?
![Page 6: Learning Hawkes Processes Under Synchronization Noise11-14-00)-11-14-20-4830... · Learning Hawkes Processes Under Synchronization Noise William Trouleau Presented at ICML’19 on](https://reader033.fdocuments.net/reader033/viewer/2022060505/5f1e3a85483a6e168c4fcc16/html5/thumbnails/6.jpg)
Example 2: Disease Dynamics
?
• Consider a network of hospitals
• We observe a a sequence of discrete events in continuous time: interactions, infections, recoveries…
• Questions of interest:Who infected whom?How does the disease spread?How to control it?
![Page 7: Learning Hawkes Processes Under Synchronization Noise11-14-00)-11-14-20-4830... · Learning Hawkes Processes Under Synchronization Noise William Trouleau Presented at ICML’19 on](https://reader033.fdocuments.net/reader033/viewer/2022060505/5f1e3a85483a6e168c4fcc16/html5/thumbnails/7.jpg)
Example 2: Disease Dynamics
?
• Consider a network of hospitals
• We observe a a sequence of discrete events in continuous time: interactions, infections, recoveries…
• Questions of interest:Who infected whom?How does the disease spread?How to control it?
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How do we usually solve it?
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Method: Multivariate Hawkes Process (MHP)• Temporal Point Process
• Widely used model to learn causal structure between time series
• Captures mutually exciting patterns of influence between dimensions
λi(t|Ht) = µi +d!
j=1
!
τ∈Hjt
κij(t− τ)
αij
λi(t|Ht)
λj(t|Ht)
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Method: Multivariate Hawkes Process (MHP)• Temporal Point Process
• Widely used model to learn causal structure between time series
• Captures mutually exciting patterns of influence between dimensions
λi(t|Ht) = µi +d!
j=1
!
τ∈Hjt
κij(t− τ)
Exogenous intensity:constant, independent
of the pastEndogenous intensity:
due to excitation from past events, with excitation kernel
κij(t) = αije−βt1{t > 0}
αij
λi(t|Ht)
λj(t|Ht)
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Method: Multivariate Hawkes Process (MHP)
• Prior work assume perfect traces without noise
• What if the observed stream of events is subject to a random and unknown time shift?
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How to learn MHPs under noisy observations?
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• What it events have systematic measurement errors?
T
NA
NB
tA1
zA
t̃A1 t̃A2 t̃A3tA2 tA3
tB1 t̃B1 tB2 t̃B2
zA zA
zB zB
t0
Multivariate Hawkes Process under Synchronization Noise
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• What it events have systematic measurement errors?
T
NA
NB
tA1
zA
t̃A1 t̃A2 t̃A3tA2 tA3
tB1 t̃B1 tB2 t̃B2
zA zA
zB zB
t0
Order of events can be switched
Multivariate Hawkes Process under Synchronization Noise
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• What it events have systematic measurement errors?
T
NA
NB
tA1
zA
t̃A1 t̃A2 t̃A3tA2 tA3
tB1 t̃B1 tB2 t̃B2
zA zA
zB zB
t0
Events can enter the observation window…
…or escape it
Multivariate Hawkes Process under Synchronization Noise
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• What it events have systematic measurement errors?
• Edges learnt by maximum likelihood estimation can be significantly affected by even small delays
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ffici
ents
NA
NB
NA
NB
NA
NB
NA
NB
NA
NB
NA
NB
Lear
ntN
etw
ork
AB
BA
AB
AB
))
AB T
NA
NB
tA1
zA
t̃A1 t̃A2 t̃A3tA2 tA3
tB1 t̃B1 tB2 t̃B2
zA zA
zB zB
t0
Ground truthNetwork
Multivariate Hawkes Process under Synchronization Noise
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New approach DESYNC-MHP• Idea:
• Consider the noise as parameters • Maximize the joint log-likelihood over
both MHP parameters and noise
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New approach DESYNC-MHP• Idea:
• Consider the noise as parameters • Maximize the joint log-likelihood over
both MHP parameters and noise
• Challenges: resulting objective is • Non-smooth • Non-convex
![Page 19: Learning Hawkes Processes Under Synchronization Noise11-14-00)-11-14-20-4830... · Learning Hawkes Processes Under Synchronization Noise William Trouleau Presented at ICML’19 on](https://reader033.fdocuments.net/reader033/viewer/2022060505/5f1e3a85483a6e168c4fcc16/html5/thumbnails/19.jpg)
New approach DESYNC-MHP• Idea:
• Consider the noise as parameters • Maximize the joint log-likelihood over
both MHP parameters and noise
• Challenges: resulting objective is • Non-smooth • Non-convex
• Solution: • Approximate the objective with a
smooth approximation • Use SGD to escape local minima
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Experimental Results
φЈɘ φЈɞ φЈȯ φЈɱ φЈȯ φЈɞ φЈɘNoise variance ᇐɞ
ΚЈΘЈϩЈϨЈ΅ЈνЈφЈЈAv
erag
e ac
cura
cy (std)
Classic MLEDESYNC-MHP MLEφ�ᇀ � φӳЈ
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Learning Hawkes Processes Under Synchronization Noise
William Trouleau Jalal Etesami Negar KiyavashMatthias Grossglauser Patrick Thiran
Come check out our poster tonight !