Compression of Spatio-Temporal Datamdmconferences.org/mdm2016/pdfs/Compressing.pdf · 2016. 6....

Compression of Spatio-Temporal Data

MDM 2016 Advanced Seminars

Goce TrajcevskiDept. of EECS

Northwestern University

Advanced Seminar MDM, Porto, June 2016

Pre-Introduction

Plethora of applications relying on some form of Location Based Service (LBS):

• transportation/routing• social networks, online/mobile marketing• traffic management, disaster response• environmental/structural health monitoring• ecology (flora and fauna)

Trajectories = location-in-time data

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Pre-Introduction

Novel trends

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Pre-Introduction

McKinsey 2011: Location data from GPS-equipped mobile phones = O(peta-bytes) (20% in 2010 → 70% in 2020)◦ 400-fold increase if cell-tower data included

◦ Coupled with other sensors data (US Express ~950 sensors)

Daily travel in the US averages 11 billion miles a day (approximately 40 miles per person)◦ 87% of them take place in personal vehicles – recording

location samples generated every 10 seconds ⇒ 275TB daily

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Pre-Introduction: the trends of Internet of Things

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Outline

Introduction Spatial data and Temporal data compression Compression of Spatio-temporal data◦ Trajectories/fundamentals◦ Constraints◦ Real-time/tracking◦ Mobile shapes

Alternative views and recent trends Concluding remarks

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Introduction – Basics

Broadly, data compression can be perceived as a science or an art – or a mix of both – aiming at development of efficient methodologies for a compact representation of information◦ take a dataset D1 with a size β bits as an input, and produce

a dataset D′1 as a representation of D1 having a size β′ bits, where β′ <β (hopefully << ).

Example:◦ Transmitting raw (un-compressed) HDTV signal would

require a channel resource enabling 884 Mbits/second which, in turn, means a bandwidth of 220 MHZ.◦ Compressed version of the respective mix of video-frames

and audio require 20Mbits per second – only 6MHz of a bandwidth (which is the amount of bandwidth allocated in the US).

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Introduction – History

Smoke signals

Polybius square

Heliographs

Chappe

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Introduction – History: the electricity era

Telegraph:◦ Some letters occur more frequently than others… a → . _ q → _ _ . _Exploit frequency to reduce average transmission time

Non-statistical approaches◦ Vocoder (exploit a compact description of voice-box)◦ Recover the original “voice” at the receiving-end

Claude Shannon (1940s)◦ Fano; Huffman – information-theoretic bounds

LZ77◦ Many variants (even more law suits…)

Images/Video – MPEG (multiple versions)9


Introduction – Taxonomies

Lossless vs. Lossy◦ Original data can/not be restored (i.e., distortion)

Entropy based vs. Dictionary based◦ Shannon/Huffman (frequency/probability of occurrence)◦ LZ variants (raw data has repetitions; collect a dictionary

and substitute repeated occurrences with its entry index)

Static vs. Dynamic/Adaptive◦ Properties (i.e., dictionary) known in advance◦ Properties vary in “real-time”…

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Outline



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Spatial and Temporal Data Compression

Cartography◦ Maps generalization Given a fixed-size window, represent larger area Equivalently, represent a given area in a smaller window

⇒ sacrifice the level of detail…

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Errors “from the get-go” Earth is not flat ⇒ projections◦ E.g., Mercator

Distorting the data at the benefit of “semantics”◦ 1:100,000 reduction scale will make any object (e.g., a

building) which has an edge smaller than 35 m - the vast majority of single family homes - to drop below 0.35mm; ◦ based on the physiology of the human vision, 0.35mm is the

limit of perceptibility. ◦ to keep a particular polygonal object with sides < 0.35mm

on the map, an “artificial enlargement” is needed13



Basic issues:◦ Philosophical objectives of why to generalize.◦ Cartometric evaluation of the conditions which indicates

when to generalize. ◦ The selection of appropriate spatial and attribute

transformations which provide the techniques on how to generalize.

Fundamental operators (“how”):◦ simplification, smoothing, aggregation, amalgamation, merging,

collapse, refinement, exaggeration, enhancement and displacement

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Canonical problem – polyline simplification Given a polyline PL1 with vertices {v1,v2, . . . ,vn}, and a

tolerance ε Construct another polyline PL′1 with vertices {v′1,v′2, . . . ,v′m} such that:◦ m ≤ n, and◦ for every point P ∈ PL1 its distance from PL′1 is smaller than

a given threshold: dist(P,PL′1) ≤ ε .

ASIDE: if {v′1,v′2, . . . ,v′m} ⊆ {v1,v2, . . . ,vn}, the simplification is strong; otherwise, it is a weak simplification

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By far the most popular heuristic approach: Douglas-Peucker (DP) or, Ramer-Douglas-Peucker (RDP)

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RDP algorithm

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Observations: Complexity: O(n2)◦ Hershberger and Snoeyink provided O(n logn) algorithm…

Is NOT optimal (in the sense of guaranteeing the minimal number of points for the output)

Still, widely popular due to “visual appeal”◦ Plus, first one to be implemented in FORTRAN◦ Part of many GIS implementations…

Solves only min-# variant of compression…◦ min-#: given ε, generate a sub-polyline with the smallest

number of vertices;◦ min-ε: given a “budget” m (< n) generate a sub-polyline with

the smallest distance from the original one18



Optimal algorithm:◦ Draw circles with radii ε centered at each vertex of the

polyline. ◦ Starting from the first vertex, draw the pair of tangents to each circle in

the sequence.◦ Let Ui and Li denote the upper and the lower ray emanating from v1

after drawing a pair of tangents to all the vertices up to vi

Essentially, the boundaries of a non-empty wedge◦ Repeat when wedge empty…◦ Repeat, starting at a different vertex (looking both backwards and

forward)

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Basic Properties:

◦ Note: a randomized algorithm yields O(n 4/3 + δ) for a small δ

Extensions to 3D:

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Life is not that simple… Errors may occur which have “topological nature”◦ Inside vs. outside (or, to-east vs. to-west)◦ Intersections (i.e., river outside (vs. inside) the city

boundary

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Other issues in spatial data compression: Application+device awareness:◦ Combining navigation with notifications on a limited display

(and during motion)

Distance measures:◦ Minimize area between original and reduced polyline◦ Minimize the notion of directionality-disturbance

Broader contexts:◦ Clustering◦ Point-sets compression Convex hull Non-convex hulls (α-shapes; χ-shapes)

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Other fields: Image processing Video processing◦ Subject to JPEG; MPEG families of compression◦ Either lossless, or rely in physiology when restoring

Spatial Data Warehousing◦ Multi-thematic layers◦ Context-based zoning E.g., soil types; plant/crop types

Spatial histograms for inter-molecular distances based on affinity…

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time in computer science has lead to development of numerous protocols and methodologies from systems as well as semantic-based perspectives: ◦ Synchronization among processes ◦ concurrency management ◦ distributed systems◦ events management ◦ collaboration among sensing and computing devices◦ …

Compression:◦ Temporal Databases◦ Time series (+ Data Streams)

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Temporal databases: majority of the applications do have certain time-related semantics◦ adding “fixes” to a non-temporal database to cater to this,

has proven to be either too much overhead, or simply infeasible.

Example: having a DATE attribute enables knowing when a particular tuple became valid in the database.

However, the natural expectation is to know when a particular tuple (or an attribute therein) ceased to be valid◦ ergo, adding another DATE column ◦ Problem becomes the one of how to express in SQL some

simple queries…25



Consider: “Who was Aaron’s manager when he worked in Capital account”?

Adding such features brought the concept of time-varying tables which, in turn, spurred the field of Temporal Databases )TDb◦ Temporal-SQL (TSQL), and the standardization of

incorporating and the temporal dimension in SQL3

Temporal Data Types:◦ Instant◦ Interval◦ Period

Kinds of time (user-defined; valid; transaction)

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Consider

and the query: “Retrieve the total number of employees per department, during January 1

and July 15 of 2015.”

Both Jack and James:◦ Increased salaries◦ Their intervals have been merged in the answer (coalescing) Lost the info about the salary increase…

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Granularity issue:◦ Time expressed in month/year◦ Flights expressed in minutes How to join such tables?

If interval-base temporal semantics is used, there are 13 possible relationships: Before Meet Overlap During Starts Finishes Plus 6 negations, plus Equal

Important in DW/BI – hierarchies along the time dimension…

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Time series and Data Streams◦ Two complementary classes of problems

Time series◦ Large datasets of sequences of (time, value) pairs.◦ Each sequence perceived as a point in N-dimensional space.◦ To decrease the “dimensionality curse”, often one resorts

to compressed representation, typically used for index

Data streams◦ Fast-arriving values, exceeding the memory capacity◦ Must be processed on-the-fly◦ Continuous queries (approximate answers)

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Time series◦ Index should be “build-able” in a reasonable time, and cater

to different distance functions Adaptive and non-adaptive representation methods Ensure lower-bounding (to be able to prune without false negatives

during searches)

Other desirable properties need to be maintained:◦ Quick answer to queries of interest (i.e., similarity in terms

of NN)◦ Maintain “perceptual similarity”

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Streaming Data◦ data model and query semantics must allow order-based

and time-based operations (e.g. queries over a five-minute moving window).

◦ The inability to store a complete stream suggests the use of approximate summary structures, referred to in the literature as synopses or digests.◦ Queries over the summaries may not return exact answers.◦ Streaming query plans may not use blocking operators that

must consume the entire input before any results are produced.

Best one can work towards is ensuring probabilistic guarantees on the bound of the error to a query-answer.

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Random sampling, histograms, wavelets… Example – consider the set:

C = {5.0,2.2,3.1,4.7,5.2,4.0,5.3,7.1,7.9,3.7,4.2,6.8,7.3,6.1}12 values in total

Histogram:

3 buckets (equi-width)◦ 5 numbers total ⇒ 58.6% savings

However, the query “how many members fall between 3 and 5.5 yields 4.5 (assuming uniformity)…

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Outline



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Spatio-Temporal Data Compression

Moving Object

Any real-world elementthat may be perceived as“unique” (car, person, animal)

Changes its spatial whereabouts over time(during its existence)

Trajectory: continuous mapping from Time to (some Geographical) 2D SpaceI(⊆) R → R2

or, even think of it as parameterization over timeα(t) = (αx(t), αy(t))

So, now Trajectory = {(αx(t), αy(t),t)| t ∈ I}

•Description•Representation•Manipulations

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Raw Data → Trajectory

Sample-points (location, time) need to be part of the trajectory, however, in-between?

Linear Interpolation: (x,y,t) = (xi,yi,ti)+ [(t −ti)/(ti+1 −ti)] (xi+1−xi,yi+1−yi,ti+1−ti)(assumption: constant velocity in—between samples…)

Interpolation via Bezier Curves

(time-slices)

-Spatiotemporal Data Model-Constraint Database Model-Moving Objects Database Model

Off the shelf – STER

Differential Geometry; Toplogy

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Road-Network Constrained

Distance → graph-distance⇒Dijkstra-like algorithmic approaches

The flow/speed in a given link may vary in time (A* based SP)…

Eco-routes

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Trajectory construction◦ (location, time) updates◦ Periodic (location, time, velocity) updates◦ Full-future trajectory

So, why simplification/compression?

As always:1. Save storage2. Save on communication/bandwidth

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T′ is called an ε –simplification of T with respect to a distance measure M (equivalently, T′ is a simplification of T with an M-tolerance ε ), denoted by

T′ = S(T,ε ,M) if DM(T,T′) ≤ε

T

T’

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Distance functions:◦ Hausdorff distance (ignoring time):

◦ From T to T’

◦ Symmetric: DH(T,T′) = max(˜DH(T,T’), ˜DH(T’, T))

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Hausdorff distance does not incorporate time Not appropriate for “mobile world” Consider “man walks the dog” example:

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Frechet distance: The most general way of incorporating time (i.e., all

the possible ways) Consider two curves:

Their Frechet distance is defined as:

where α and β range over all the possible continuous an monotonically increasing mappings[0,1] → [a1,b1] and [0,1] → [a2,b2]

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Restricted variants of Frechet distance: Eu – The three dimensional time uniform distance is

defined when tm is between ti and tj , as follows:◦ Eu(pm, pipj) = √(xm−xc)2 +(ym−yc)2 where pc = (xc,yc, tc) is the

unique point on pipj which has the same time value as pm(i.e., tc = tm).

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Et – The time distance is defined as: Et(pm, pipj) = |tm−tc|, where tc is the time of the point

on p′ip′j (which is the X-Y projection of pipj) that is closest in terms of the 2D Euclidean distance to p′m(the X-Y projection of pm);

if the closest point on p′ip′j has more than one time point, choose the one that maximizes |tm −tc|.

Intuitively:◦ project both on the X-Y plane, then find the point p′c on

the projected segment which is closest to p′m,◦ find the difference between their time-values.

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Compression is lossy – so, using the compressed trajectory to answer queries will generate errors (with respect to the answer applied to the original trajectory).

Errors depend on the distance function (and, of course, the type of query)

For a given query q, the error with respect to a trajectory T is bounded by δ if the difference between the answer of q on T and the answer of q on a ε -simplification of T is bounded by δ.◦ Clearly, semantics of the δ depends on the query type.

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Relationships among different distance-functions

ASIDE: when treating time as “almost-z” one needs to “relativize”

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Queries:◦ Where_at◦ When_at◦ Range (i.e., Intersect(T,P))◦ NN◦ Θ-join

ASIDE: heuristic (i.e., RDP) used;Optimal algorithm yields an increasein complexity

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So far, full-trajectories were considered◦ ASIDE: one may consider periodic re-simplification of the

compressed/simplified trajectories To enable further space-savings Applied to “old-enough” trajectories

◦ aka “Aging”

But what happens with the data that arrives to the MOD server in real-time?◦ One can store it all, and then apply simplification on the

completed motion◦ OR – one can attempt to compress the data “on the fly”…

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Real-time compression:◦ Motion model: sending (location, time, velocity) updates◦ When: event-triggered, whenever the actual location

deviates by > δ from the expected location (based on the previous update) – aka dead-reckoning

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With distance-based dead reckoning, the MOD server performs two tasks:◦ (1) corrects its own “knowledge” about the recent past and

approximates the actual trajectory between toldand tnow with a straight line-segment, which defines the actual simplification of the near-past trajectory;◦ (2) generates another infinite ray corresponding to the

future-expected trajectory, starting at the last update point, and using the newly received velocity vector for extrapolation.

As it turns out:◦ Using ε as a dead-reckoning threshold would generate a

simplified trajectory which would be a strong simplification with bound ≤ 2ε with respect to the entire trajectory…

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Generic real-time compression (tracking protocols)

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Tracking in sensor networks◦ Instantaneous location detected by trilateration (i.e.,

distances from 3 sensors)◦ Brute-force: transmit every location to the sink and let the

sink construct the trajectory

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Assign a buffer (transmitted to next principal, along with current location) When buffer exceeds certain capacity-threshold,

apply compression When buffer fills in, transmit the entire buffer-

simplified trajectory to the sink

ASIDE:The issue of “freshness”in the sink…

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An important context: what if the motion is constrained to an existing road network?

Often the case in practice – and one can capitalize on identifying “popular routes” and using them as “dictionary entries”.

Example:◦ If an object is known to move along an existing road

segment, than we need not store any GPS updates in-between (assuming uniformity of the motion) If the motion is non-uniform, then store the time-instants where the

speed changes, along with the distance from one of the end-points of that road-segment

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In other words, change the “traditional” network edge-oriented model

Into a route-based model

Generalize route:

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Key observation:

Compression is not based on each individual trajectory, but reduces the size of MOD as a whole (in terms of number of Bytes needed to represent the dataset)

While the issues of uncertainty (in terms of query-errors) are not alleviated, the overall compression ratio for the entire MOD is higher than the “sum of the individual savings”

Need more solid (ML-”ish”) prediction to be applicable for real-time compression

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What if the entities which move also have (deformable) extents?◦ Example: ◦ Spreading of toxic gasses or spills◦ Region affected by a hurricane (eye +tail)

To begin with, one can only get discrete spatial samples at discrete time-instants

Thus, in a sense:◦ Compact representation of point-sets◦ Those point-sets are close-enough in capturing a particular

continuous phenomenon

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Iso-contours:◦ Boundaries of region where a phenomenon has values

within certain bounds

Observation:◦ This is more similar to the min-ε variant (i.e., one is given a

budget of “size” for the polygons used) with respect to the area◦ One can dynamically vary the (sub)sizes allocated to

different values from the domain of the phenomenon…57



What to use when knowing that measurements were taken in discrete locations?

Using convex hull (while simple), need not be the best idea (i.e., one may end up with a lot of dead-space…). ◦ α-shapes need not generate a simple polygon

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Key observation when tracking:◦ Do not calculate the new shape from the scratch

(if/whenever possible)

◦ Similarly to trajectory-tracking – if the “freshness” of the sink is not “a must” – one can put a threshold based only on the pending queries and their answer-changes

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Outline



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Alternative Views/Trends

Behavioral classification of movement:◦ Classification relies on computing and analyzing movement

features jointly in both the spatial and temporal domains. focusing on the spatial domain, the underlying movement space is

partitioned into several zonings that correspond to different spatial scales, and features related to movement are computed for each partitioning level.

concentrating on the temporal domain, several movement parameters are computed from trajectories across a series of temporal windows of increasing sizes, yielding another set of input features for the classification.

For both the spatial and the temporal domains, ML techniques used to determine the “reliable scale”

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Zebra-fish motion in reaction to different pharmacological products

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Spatio-temporal Data warehousing

A wide range of materializationsat different levels of hierarchies

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Visualization of trajectories

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Visualization of trajectories Density map framework for expert users, who

explore distributions of attributes defined along trajectories.

In the exploration, the user mainly interacts with the distribution maps, though fine tuning for optimizing details is possible.

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Symbolic/Semantic Trajectories

Recall – MOD:◦ Collection of trajectories {Tr1, Tr2, …, Trk}◦ Each Tri a sequence:◦ [(xi1,yi1,ti1), (ki2,yi2,ti2), …, (xim,yim,tim)]

tij < ti(j+1)in-between location samples, interpolation assumed

Semantic Trajectories◦ Sequence of Semantic Episodes

𝑆𝑆𝑆𝑆 = [𝑠𝑠𝑠𝑠1 , 𝑠𝑠𝑠𝑠2 , 𝑠𝑠𝑠𝑠3 , … , 𝑠𝑠𝑠𝑠𝑛𝑛]◦ Each episode a tuple of the form𝑠𝑠𝑠𝑠𝑖𝑖 = (𝑑𝑑𝑑𝑑𝑖𝑖 , 𝑠𝑠𝑠𝑠𝑖𝑖 , 𝑥𝑥𝑖𝑖𝑖𝑖𝑛𝑛 ,𝑦𝑦𝑖𝑖𝑖𝑖𝑛𝑛 , 𝑡𝑡𝑖𝑖𝑖𝑖𝑛𝑛 , 𝑥𝑥𝑖𝑖𝑜𝑜𝑜𝑜𝑜𝑜 ,𝑦𝑦𝑖𝑖𝑜𝑜𝑜𝑜𝑜𝑜 , 𝑡𝑡𝑖𝑖𝑜𝑜𝑜𝑜𝑜𝑜 , 𝑡𝑡𝑑𝑑𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑡𝑡)

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Each semantic episode:𝑠𝑠𝑠𝑠𝑖𝑖 = (𝑑𝑑𝑑𝑑𝑖𝑖 , 𝑠𝑠𝑠𝑠𝑖𝑖 , 𝑥𝑥𝑖𝑖𝑖𝑖𝑛𝑛 ,𝑦𝑦𝑖𝑖𝑖𝑖𝑛𝑛 , 𝑡𝑡𝑖𝑖𝑖𝑖𝑛𝑛 , 𝑥𝑥𝑖𝑖𝑜𝑜𝑜𝑜𝑜𝑜 ,𝑦𝑦𝑖𝑖𝑜𝑜𝑜𝑜𝑜𝑜 , 𝑡𝑡𝑖𝑖𝑜𝑜𝑜𝑜𝑜𝑜 , 𝑡𝑡𝑑𝑑𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑡𝑡)

Consists of:da = defining annotation◦ typically expressing an activity (verb) such as “stop”, “move”; etc…

sp = semantic location/position of the activity ◦ e.g., POI (museum, restaurant, zoo), home, work, etc.

tin and tout

◦ entry/exit times of a semantic position.tagList◦ any additional semantic information (e.g., transportation mode)

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ST1 =[(drive, Adams St, 50, 10, 10:45, 10, 10, 11:00, drive, car, VW)(stop, “Roditis”, 10, 10, 11:00, 10, 10, 11:45, restaurant, eat,salad),(walk, parking lot, 10, 10, 11:45, 11, 10, 11:50, car, VW),(drive, Randolph St, 11, 10, 11:55, 25, 10, 12:00, car),(stop, traffic light, 25, 10, 12:00, 25, 10, 12:03, car),(. . .)(stop,“Starbucks”, 25, 40, 12:25, 25, 40, 1:30, coffee, eat, dessert)]

ST2 = [(move, Dearborn St, 60, 60, 11:30, 60, 40, 11:45, walk),(stop, “Arby’s” , 60, 40, 11:45, 60, 40, 12:30, fast-food, eat, beef),(move, Dearborn St, 60, 40, 12:30, 60, 35, 13:00, walk),(move, Chicago Ave, 50, 35, 13:00, 25, 35, 13:25, ride, bus 14),(stop, “Starbucks”, 25, 35, 13:25, 25, 35, 13:50, coffee, desert),. . .(move, Jackson St, 10, 20, 14:15, 50, 20, 14:40, ride, bus 151) ]

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From a certain perspective,

Clusters Flocks Convoys

Can be considered as compressed (more compact) representation of constituent trajectories…

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Instead of conclusions

The “saga” will continue

Internet of Things offers a plethora of cross-contexts compression approaches◦ Example: accounting for possible offline/indoor motion

Privacy assurances in the IoT context (while providing particular sampling-quality)

Analysis of collective motion in social settings is likely to generate novel compact representations◦ Interested only in groups who have exhibited a pattern of

sequentially visiting PoI’s within same time-delays and with above-threshold memberships

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