Lab seminar 210219

1. Using Topological Analysis to Support
Event-Guided Exploration in Urban Data
IEEE transactions on visualization and computer graphics
Harish Doraiswamy, Nivan Ferreira, Theodoros Damoulas, Juliana Freire and Cl´audio T. Silva

2. Overview of the paper
• Motivation
 To examine prohibitively large number of spatio-temporal slices in efficient way
and discover interesting patterns from it
• Contribution
 Propose an efficient and scalable technique that automatically discovers events
 Guide users towards potentially interesting data slices
 Accomplish event detection through the application of topological analysis on a
time-varying scalar function
 Design an indexing scheme that groups similar patterns across time slices
 Design visual interface to aid in event-guided exploration of urban data
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3. Background
• A scalar function maps points in a spatial domain.
 The function value at each point on this graph is equal to the point’s y-coordinate.
• A super-level set of a real value a is the pre-image of the interval [a,+∞).
• A sub-level set of a is the pre-image of the interval (−∞,a].
• Critical points of a smooth real-valued function are exactly where the gradient becomes zero.
 Topological changes occur at critical points.
 A maximum captures a peak of the function, where the function value is higher than its
neighborhood.
 A minimum captures a valley of the function.
• Regular points are the points that are not critical.
 Topology of the super-level (sub-level) set is preserved across regular points.
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4. Background
• Topological Persistence
 The topology of the super-level sets change when the sweep in decreasing order
encounters a critical point.
 A creator is the critical point if a new component is created, a destroyer otherwise.
 The persistence value of 𝑣𝑐 is 𝜋𝑐 = 𝑓 𝑣𝑐 − 𝑓 𝑣𝑑 .
• Join tree and split tree
 The tree abstracts the topology of a scalar function f, and represent features of f.
 The join tree tracks the changes in the connectivity of super-level sets
 The split tree tracks the connectivity of the sub-level sets
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5. Data
• NYC Taxi Data
 Manhattan during 2011 and 2012
 Each trip consists of pickup and drop-off locations and times
 Average 500 thousand trips each day
 Identifying road closures and taxi hot spots
 Scalar function for an hourly interval at each node of this graph as the density of taxis
within a small circular region
 Minima and maxima are used to represent events
• MTA Subway Data
 Time stamps of all the stops for all the trips that happen each day.
 Delays in the schedule of the different trains ( scheduled – actual)
 The nodes of this path corresponds to the different stations along its route
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6. Managing Events
• Computing Events
1. Split tree to see the significant events
2. Persistence to capture the importance of a feature
3. Geometric size of a feature to consider the characteristic of hyper-volume
4. Remain top-k from the set of minima
• Event Group Index
 Define a notion of similarity between events based on their geometric and topological
properties
 Group similar events within a certain time interval into event groups
 Define a key to index these groups
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7. Event Group Index
• Similarity Between Events
 E is formally represented as a pair (R, τ)
• R is a subgraph of spatial region
• τ is a real number representing topological importance
 Graph distance metric δ, to measure the geometric similarity between R1 and R2
δ 𝐸1, 𝐸2 = 1 −
|𝑅1 ∩ 𝑅2|
max( 𝑅1 , 𝑅2 )
• 𝑅1 ∩ 𝑅2 denotes the maximum common subgraph between R1 and R2
• 𝑅 denotes the number of nodes in R
• Measures the amount of overlap between two regions, ensuring that similar regions have a significant
overlap
 Topological similarity between two events
T 𝐸1, 𝐸2 = |τ1 − τ2|
• Two events E1 and E2 are similar if δ 𝐸1, 𝐸2 ≤ εδ and τ 𝐸1, 𝐸2 ≤ ετ
• Ensures that the two events are topologically close with respect to the topological importance
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8. Event Group and Event Group Key
• Use a time period equal to one month
 not to miss periodic events
 not to create a computational bottleneck
• Given an event group Σ = 𝐸1, 𝐸2, … , 𝐸𝑘 , define the event group key of Σ as (𝑅Σ, τΣ)
𝑅Σ =
𝑖∈[1,𝑘]
𝑅𝑖 𝑎𝑛𝑑 τΣ =
𝑖=1
𝑘
τ𝑖/𝑘
• 𝑅𝛴 is the maximum common subgraph of the geometric regions  overlap for similarity condition
• τΣ captures average of the topological importance
• Follows definition of geometric and topological similarity measures
• The definition of event group key helps in using a consistent definition for the similarity between event
groups
• When two similar event groups are found, they are merged into a single group
• With given query, perform a linear search over the set to find events
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9. Visual Exploration Interface
• Map View and Query Interface
• Event Group Distribution View and Timeline View
 Range is the amount of time between the first and the last event
 Density is the number of events of group that happen per time unit
• Classification of event groups with two attributes
 Region I:
• Low range, but high density
• Rare occurrence (irregular pattern)
 Region II:
• High range and high density
• Occur over frequent periods, so can identify trend
 Region III:
• Small number of events that span a large range
• Potentially represent patterns that are regular over a large time interval
• Irregular with respect to the range of the input data
 Region IV:
• Low range as well as low density
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Filtering Interface
Event group size
Event size
Event time
Spatial region

10. Case Studies – NYC Taxi data
• To help them identify areas with high concentrations of taxis
• Minima events in NYC
 Regions where there are comparatively fewer taxis
• If this place is usually a high density of taxis, blockage of streets
• Hourly events
 Sixth avenue in Greenwich Village on October 31st
 This corresponds to the annual NYC Halloween Parade
• Daily events
 Fifth avenue on October 9th and 10th, 2011
 This corresponds to the Hispanic Day Parade on 9th October and the
Columbus Day Parade on October 10th
• Weekly events
 NYC Summer streets that happens on Park avenue
 Occurred on three consecutive Saturdays, 6th, 13th, and 20th
August respectively
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11. Case Studies – NYC Taxi data
• Querying events
 Search for events similar to a selected event that occurs in other
months
 find other parades that also occurred in the same location.
• Identifying trends
 Maxima events show high concentration of taxis
 If such concentrations are frequent, then it could imply taxi hot spots
 Optimize the amount of receiver place
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12. Case Studies – MTA data
• To identify events related to delays
• The amount of delay is applied as topological persistence for
importance measure
• Minimum event groups
 Find a station at which the delay is lower than that of its neighbors
 Signals where trains start to get delayed
 Frequent presence of such events are in Region II
 Wall Street station
• Events occur predominantly during the rush hour period on
weekdays
 14th street station
• 3 train sometimes waits for the 1 train
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13. Limitation and Future work
• The characteristic of the event is not explainable unless the user search the event
• The system should iterate the group to find the similar event
• No entire view of the system other than the graphs
• Scalar function should be assigned considerately
• Speed can be used for scalar function computation
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