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Uncertain Observation Times. Shaunak Chatterjee & Stuart Russell Computer Science Division University of California, Berkeley. Overview. Why uncertain observation times matter Scenarios considered: Each event is observed: Efficient DP algorithm Missing and false events: - PowerPoint PPT Presentation
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Uncertain Observation Times
Shaunak Chatterjee & Stuart RussellComputer Science Division
University of California, Berkeley
Overview
• Why uncertain observation times matter• Scenarios considered:1. Each event is observed: – Efficient DP algorithm
2. Missing and false events: – Practical approximation algorithm
3. Multiple asynchronous observation streams
Motivation
• Two types of data streams– Automatically time-stamped data traces– Human annotations for temporal events• Many essential facts cannot be recorded automatically• Human-generated timestamps often wrong
• Assuming the correctness of timestamps can lead to nonsense results
Example: at 16.30, nurse enters “gave phenylephrine at 16.00”
Data entry time
Event timestamp
Example: at 16.30, nurse enters “gave phenylephrine at 16.00”
Data entry time
Actual event time
Ubiquity of uncertain observation times
• Nurse monitoring a patient in the ICU– Hundreds of events recorded by the nurse • Usually recorded after event, sometimes before
• Manual recording of events– Science experiments• Biology, chemistry, physics
– Industrial plants• Multiple observation traces– Various historians’ accounts of a period– Only one underlying truth
Sample trace generated from model
• Correct chronological ordering of time stamps
Actual time of event (ai)
Recording time of event (di)
Time
Nurse gives medicine at 10:23 a.m.
Nurse records event at 11:00 a.m.
Nurse records time of event as 10:30 a.m.
Previous event’s time stamp (10:15 a.m.)
Recorded time of event (mi)
Dynamic Bayesian networks
• DBNs are discrete-time multivariate stochastic process models (include HMMs and KFs)
• DBNs facilitate modeling of complex systems with sensor noise etc.
Large-scale physiological models pursued since the 1960s, but little attention paid to nature of real data
Simple DBN representationY1 Y2 Y3 Y7
X1 X2 X3 X7
a1 a2 a3
m1 m2 m3
d1 d2 d3
Y4 Y5 Y6
X4 X5 X6
Y8
X8
false true true truefalse false false false
2 5 7
2 4 8
6 6 8
Objective
• To design a graphical model that allows for uncertainty in observation times
• Derive efficient inference algorithms – Naïve algorithm has O(MT) complexity– Reduce to O(MT)• Ordering constraints• Dynamic programming
Key constraint assumption
• Person recording events gets the order right• Valid association
• Invalid association
• For all i, j: mi > mj => ai > aj
Time
Recorded time of event (mi)
Actual time of event (ai)
Time
Recorded time of event (mi)
Actual time of event (ai)
Pre-computation step
• Likelihood of the data segment between the current event time stamp (ak) and the next hypothesized event time stamp (ak+1)
• Pre-compute for all k, and all possible values of ak and ak+1
Modified Baum-Welch algorithm
Complexity
• Modified time complexity O(MS2T)– M: maximum size of the time window of
uncertainty– S: # states in system– T: number of time steps
• Space complexity– O(KM2) – storing – O(KM) – storing α, β and γ
Simulation results – Increased likelihood of evidence
Window of uncertainty
Simulation results – General accuracy of inference
Simulation results – Computation time vs size of uncertainty window
Unreported events, false reports
• Not all events are reported– Unobserved– Negligence
• Not all reports are true– Double entry of a single data point– Misinterpretation of information– Intended actions reported but not carried out
Missing and false reports
a1 a2 a4
m1 m2 m3
θ1θ2 θ4
Φ1 Φ2 Φ3
a3
θ3
Actual time of event (ai)
Recorded time of event (mi)
Event i reported? (θi)
Index of event corresponding to report j (φi)
Modified DP and complexity
• The previous algorithm was compact because of the one-to-one correspondence between events and reports– Now have to consider all possibilities• Unless there are constraints (more on this later)
• Chronological mapping of events’ time stamps still holds– This again leads to an efficient dynamic program
Computational complexity
• In the general case, uncertainty windows are no longer limited, since event i can be associated with any report j
• O(IJT2) – I is the number of hypothesized events– J is the number of reports– T is the length of the temporal sequence
Practical assumptions – I
• Data entries are made in blocks– All reports in a given block (e.g., the night shift) must be
for events that occurred (really or otherwise) in that block
– Computational complexity is linear in T if blocks are of constant size
Practical assumptions – II
• When unobserved events and false reports are both rare events– We can perform approximate inference by NOT
considering all possible ai mj associations– The posterior distribution is highly concentrated along
the “skewed diagonal” corresponding to a small number of errors
– Assuming a bounded number of errors gives time complexity proportional to T
Simulation results – Posterior is peaked around the skewed diagonal
Simulation results – Hypothesizing more events leads to better recall
Effect of varying c
Multiple observation sequences
• Formulation– Several “sources” reporting on the same events– Key assumption• Individual report sequences are independent given the actual
truth (the X chain)
ai ai+1 aI
θiθi+1 θI
Φj(1) ΦJ
(1)
mj+1(1) mJ
(1)
Φj+1(1)
mj(1)
mj(R) mj+1
(R) mJ(R)
Φj(R) Φj+1
(R) ΦJ(R)
Latent trajectory
Evidence trajectory 1
Evidence trajectory R
Multiple observation sequences
• Formulation– Several “sources” reporting on the same events– Key assumption
• Individual report sequences are independent given the actual truth (the X chain)
• Inference– Similar DP algorithms apply, given the assumptions of
ordering constraints, blocks, etc.– Complexity increases linearly with the number of report
sequences
Conclusions• Handling uncertainty in observation times is critical for correct
modeling and inference• Assumptions about qualitative accuracy (e.g., order of events)
can be very helpful
• Given such assumptions, the computational complexity of inference remains unchanged (modulo some constant factors) while handling the following cases– Noisy observation times– Missing and false reports– Multiple report sequences
QUESTIONS?Thank You!