review article an overview of moving object trajectory...

Review ArticleAn Overview of Moving Object TrajectoryCompression Algorithms

Penghui Sun Shixiong Xia Guan Yuan and Daxing Li

School of Computer Science and Technology China University of Mining and Technology No 1 College RoadXuzhou Jiangsu 221116 China

Correspondence should be addressed to Shixiong Xia xiasxcumteducn and Guan Yuan yuanguancumteducn

Received 28 December 2015 Revised 23 February 2016 Accepted 3 April 2016

Academic Editor Vladan Papic

Copyright copy 2016 Penghui Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Compression technology is an efficient way to reserve useful and valuable data as well as remove redundant and inessential datafrom datasets With the development of RFID and GPS devices more and more moving objects can be traced and their trajectoriescan be recorded However the exponential increase in the amount of such trajectory data has caused a series of problems inthe storage processing and analysis of data Therefore moving object trajectory compression undoubtedly becomes one of thehotspots in moving object data mining To provide an overview we survey and summarize the development and trend of movingobject compression and analyze typical moving object compression algorithms presented in recent years In this paper we firstlysummarize the strategies and implementation processes of classical moving object compression algorithms Secondly the relateddefinitions about moving objects and their trajectories are discussed Thirdly the validation criteria are introduced for evaluatingthe performance and efficiency of compression algorithms Finally some application scenarios are also summarized to point outthe potential application in the future It is hoped that this research will serve as the steppingstone for those interested in advancingmoving objects mining

1 Introduction

In recent years with the rapid development and extensiveuse of GPS devices RFID sensors satellites and wirelesscommunication technologies it is possible to track variouskinds of moving objects all over the world and collect amyriad of trajectory data with respect to the mobility ofvarious moving objects (such as people vehicles and ani-mals) containing a great deal of knowledge These data needan urgent and effective analysis A moving object spatial-temporal trajectory is a sequence of position attribute andtime [1] which are three basic characteristics of geographicphenomena and three basis data of GIS database [2] Movingobjects move continuously while their locations can onlybe updated at discrete times leaving the location of amoving object between two updates uncertain for the limitof acquisition storage and processing technologies [3] Thesimplest description of a trajectory is a finite sequence ofgeolocations with timestamps

As time goes on it will lead to a series of difficultiesin storing transmitting and analyzing data for the size oftrajectory data is sharply increasing and the scale of data isgrowing huge and complex First of all the shear volumesof data can quickly overwhelm available data storage whichwill make it difficult to store the data For instance if datais collected at 2 second intervals 1 GB of storage capacityis required to store just over 800 objects for a single dayTherefore the storage of data will result in an enormouscost The cost of transmitting a large amount of trajectorydata which may be expensive and problematic is the secondmajor problem highlighting the need of compressing dataAccording to [4] the cost of sending a volume of data overremote networks can be prohibitively expensive typicallyranging from $5 to $7 perMb Thus tracking a fleet of 800vehicles for a single daywould incur a cost of $5000 to $7000or approximately $1825000 to $2555000 annually Finallyalong with the increasing of the data scale it is difficult for usto extract the valuable and useful patterns

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6587309 13 pageshttpdxdoiorg10115520166587309

2 Mathematical Problems in Engineering

P1P2

P3

P4

P6

P8

P9

P7P5

S1

S2

S3

S4

X

Y

Figure 1 A schematic of the trajectory compression

To address these issues two categories of trajectorycompression strategies have been proposed aiming to reducethe size of a trajectory while not compromising muchprecision in its new data representation [5] One is theoffline compression which reduces the size of trajectory afterthe trajectory has been fully generated The other is onlinecompression compressing a trajectory instantly as an objecttravels On the one hand it can reduce the memory space bycompressing which will make the storage of data easier Onthe other hand it can cut down the size of data which willbe convenient for the transmission of data What is more itcan reserve the useful information in trajectories and removeredundant data from trajectories which have the potential tomake the thorough analysis of trajectory data easierThe datacompression is a method that reduces the size of data to cutdown the memory space and improve the efficiency of trans-mission storage and processing without losing informationor reorganizes data to reduce the redundancy and memoryspace according to certain algorithms Data compression canbe classified into two categories namely lossless and lossycompression Moving object trajectory compression aims toreduce the size and memory space of a trajectory on thepremise that the information contained in trajectory datais reserved that is to say in order to cut down the size ofdata it removes redundant location points while ensuringthe accuracy of the trajectory [6] Figure 1 is a schematicdiagram of a trajectory compression where the originaltrajectory is represented by black lines and the compressedtrajectory consists of red lines (namely 119878

1 1198782 1198783 and 119878

4)

There are 9 points in the original trajectory but only 5points are retained to approximately represent the originaltrajectory after compressing whose compression ratio is closeto 50 Thus it can be seen that trajectory compressionsplay an important role in the storage and analysis of dataBut trajectory compression tends to cause a certain lossof information while compressing trajectories Thereforevarious trajectory algorithms existing in literature balance thetradeoff between accuracy and storage size

The trajectory compression technology derives from thetopographic cartography and computer graphics The mostnative and simplest compressionmethod is uniform samplingalgorithmwhich simply takes every 119894th point in the trajectory[7] In 1961 Bellman put forward a new algorithm called

Bellman algorithm which solves linear generalization prob-lems by dynamic programming methods [8] This methodwill guarantee that the segments connecting the specificnumber of points selected from the curve are closest to theoriginal curve but the time overhead of it is giant which isup to119874(1198993) One of the most classical trajectory compressionalgorithms called Douglas-Peucker algorithm was presentedin 1973 by Douglas and Peucker [9] In 2001 Keogh et alput forward the Opening Window algorithm to compresstrajectory data online However traditional error metrics(such as perpendicular distance) are not suitable for movingobject trajectories whose spatial characteristics and temporalcharacteristics need to be simultaneously considered dueto their internal features In 2004 Meratnia and Rolf putforward a top-down speed-based algorithm and a top-downtime-ratio algorithm [3] The former improves the existingcompression techniques by exploiting the spatiotemporalinformation hiding in the time series while the latter is atransformation of DP algorithm which took a full consid-eration of spatiotemporal characteristics by replacing spatialerror with SED In 2006 Potamias et al put forward STTracealgorithm estimating the safe zone of successor point bylocation velocity and direction Meanwhile it is suitable forsmallmemory devices [10] Gudmundsson et al developed animplementation of the Douglas-Peucker path-simplificationalgorithm in 2009 which works efficiently even in the casewhere the polygonal path given as input is allowed to self-intersect [11] In 2009 Schmid et al proposed that trajectoriesstored in the form of trajectory points can be instead ofsemantic information of road networks [12] Since thenmanyresearchers have been doing a great deal of studies aboutthe semantic information of road networks [13ndash18] With theincreasing of the data traditional compression algorithmsare quite limited for online trajectory data Therefore onlinetrajectory compression becomes one of the hot topics [19ndash21] For example Opening Window Time-Ratio algorithmwas put forward by Meratnia which is an extension toOpening Window using SED instead of spatial error to taketemporal features into account [2] And Trajcevski et al putforward another online algorithm called Dead Reckoningalgorithm which estimates the successor point throughthe current point and its velocity [22] Out of traditionalposition preserved trajectory compression algorithms manyscholars have focused on different perspectives For exampleBirnbaum et al proposed a trajectory simplicity algorithmbased on subtrajectories and their similarity [23] Long et alproposed a polynomial-time algorithm for optimal direction-preserving simplification which supports a border appli-cation range than position-preserving simplification [24]Nibali and He proposed an effective compression systemfor trajectory data called Trajic which can fill the gap ofgood compression ratio and small error margin [25] Similarto STTrace Muckell et al put forward the Spatial QUalItySimplificationHeuristicmethod [26] In 2012 Chen et al pro-posed aMultiresolution PolygonalApproximation algorithmwhich compressed trajectories by a joint optimization onboth the LSSD and the ISSD criteria [27] In 2014 Muckell etal proposed a new algorithm SQUISH-E which compresses

Mathematical Problems in Engineering 3

Table 1 Summary of trajectory compression algorithms (119899 is the size of trajectories)

Algorithm Computationalcomplexity Batch or online Error criteria

Uniform sampling algorithm 119874(119899) Batchonline Target compressionratio (120582)

Bellman algorithm 119874(1198993) Batch Target compression

ratio (120582)Douglas-Peucker algorithm 119874(119899 log 119899) Batch Spatial distance (120576)Opening Window algorithm 119874(119899

2) Online Spatial distance (120576)

Top-down time-ratio algorithm 119874(119899 log 119899) Batch SED distance (120576)Top-down speed-based algorithm 119874(119899

2) Batchonline Speed difference (119904

119889)

Opening Window Time-Ratio algorithm 119874(119899 log 119899) Online SED distance (120576)

Dead Reckoning 119874(119899) Online SED distance (120576)velocity (v)

Semantic trajectory compression 119874(1198992) Batch Spatial distance (120576)

Paralleled road-network-based trajectorycompression 119874(119899) Batch TSND NSTD

Similarity based compression of GPStrajectory data 119874(119899 log 119899) Batch Similarity (120583)

Spatial QUalIty Simplification Heuristicalgorithm 119874(log 119899) Batchonline SED distance (120576)

Spatial QUalIty SimplificationHeuristic-Extended algorithm 119874(log 119899) Batchonline SED distance (120576)

compression ratio (120582)

trajectories with provable guarantees on errors [28] Thisalgorithm has the flexibility of tuning compression withrespect to compression ratio and error Algorithms involvedin this paper are summarized in Table 1 from the complexityapplication scope and error metric of them

In literature [29] the traditional trajectory compressionalgorithms were classified into the following 4 categorieswhich are now unable to contain all of the compressionalgorithms

(1) Top-Down The data series is recursively partitioneduntil some halting condition is met The popular top-downcompression methods include Douglas-Peucker algorithmtop-down speed-based algorithm and top-down time-ratioalgorithm

(2) Bottom-Up Starting from the finest possible representa-tion successive data points are merged until some haltingcondition is met The algorithm may not visit all data pointsin sequence

(3) Sliding Window Starting from one to the end of the dataseries a window of fixed size is moved over the data pointsand compression takes place only on the data points insidethe window Spatial QUalIty Simplification Heuristic methodand SQUISH-E algorithm are the popular sliding windowmethods

(4) OpeningWindow Starting from one to the end of the dataseries a compression takes place on the data points insidethe window whose size is decided by the number of pointsto be processed Its process will not end until some halting

condition is met The window size is not constant whilecompressing The famous Opening Window methods areOpening Window algorithm and Opening Window Time-Ratio algorithm

The organization of this paper is as follows the basic idealof compression and typical algorithms of compression areintroduced and discussed in Section 1The related definitionsabout moving objects and their trajectories are summarizedin Section 2 The survey of moving object compressionalgorithms is given in Section 3 Some validation criteria ofcompression performance are discussed in Section 4 to revealtheir benefit for moving object compression In Section 5some public trajectory datasets are described Some typicalapplication scenarios are listed to show the application ofmoving object compression in Section 6 In Section 7 somedisadvantages and future works are summarized

2 Related Definitions about Moving Objectsand Their Trajectories

21 Trajectory Data A spatial-temporal trajectory of a mov-ing object is defined as a sequence of position attribute andtime in literature [1] It is necessary for a formal descriptionof a trajectory and its correlation attributes to describe themethods in this paper A trajectory formally defined inliterature [30] is also suitable in this paper Giving TD asTrajectory Database denotes trajectory sets and TD = TR

1

TR2 TR

119899 A trajectory (TR) is a chronological sequence

consisting of multidimensional locations which is denotedby TR

119894= 1198751 1198752 119875

119898 (1 le 119894 le 119899) 119875

119895(1 le 119895 le 119898)

a sampling point in TR119894 is represented as ⟨Location

119895 119879119895⟩


which means that the position of the moving object isLocation

119895at time119879

119895 Location

119895is amultidimensional location

point A trajectory 1198751198881

1198751198882

119875119888119894

(1 le 1198881lt 1198882lt sdot sdot sdot le

119898) represents a trajectory segment or subtrajectory of atrajectory TR

119894 denoted as TS (Trajectory Segment) TS

119894=

1198711198941

1198711198942

119871119894num

In this section we classify the derivation of trajectories

into 4 major categories briefly introducing a few applicationscenarios in each category [31]

(1) Mobility of People Real-world movements of people arerecorded in the formof spatial-temporal trajectories passivelyand actively Such records can be translated into a greatamount of spatial-temporal trajectories that can be used inhuman behavior analysis and inferring social ties

Active Recording Travelers actively log their travel routes forthe purpose of memorizing a journey and sharing experiencewith friends In Flickr a series of geotagged photos canformulate a spatial-temporal trajectory as each photo has alocation tag and a time stamp corresponding to where andwhen the photo was taken Likewise the ldquocheck-insrdquo of a userin a location-based social network can be also regarded as aspatial-temporal trajectory when sorted chronologically

Passive Recording A user carrying a mobile phone uninten-tionally generates many spatial-temporal trajectories repre-sented by a sequence of cell tower IDs with correspondingtransition times In addition transaction records of a creditcard also indicate the spatial-temporal trajectory of thecardholder as each transaction contains a time stamp anda merchant ID denoting the location where the transactionoccurred

(2) Mobility of Transportation Vehicles A great number ofvehicles (such as taxis buses vessels and aircrafts) havebeen equipped with a GPS device For instance many taxishave been equipped with a GPS sensor which enables themto report a time-stamped location with a certain frequencySuch reports formulate a large amount of spatial-temporaltrajectories that can be used for resource allocation trafficanalysis and improving transportation networks

(3) Mobility of Animals Biologists are collecting the movingtrajectories of animals like tigers and birds for the purposeof studying animalsrsquo migratory traces behavior and livingsituations

(4) Mobility of Natural Phenomena Meteorologists envi-ronmentalists climatologists and oceanographers are busycollecting the trajectories of natural phenomena such ashurricanes tornados and ocean currents These trajectoriescapture the change of the environment and climate helpingscientists deal with natural disasters and protect the naturalenvironment we live in

22 Road Network A road network is defined as a directedgraph 119866 = (119881 119864) where 119881 is a finite vertex set in whichevery vertex denotes a location point and 119864 is a finite edge

Figure 2 A schematic of road network

set in which every edge denotes a segment connecting 2vertexes A road network can also be regard as a constrained2-dimensional space often referred to as 15-dimensionalspace The road network contains 29 vertexes (119881) and 36edges (119864) in Figure 2

In 2-dimensional space a point (119901) is a two-tuple inthe form of (119909 119910) and a polyline (pl) is a set of pointsThe distance (119889) of the points (119901) in the polyline (pl) isthe distance along the polyline (pl) from its starting pointto point 119901 The definitions of road network point distancepolyline segment and measurement in road network spaceare described as follow

(1) Road Network Point The point 119901 in road network spacecan be denoted in the form of 119901 = (119890 119889) where 119890 is the pointin a road and 119889 is the measurement of 119901 along the road

(2) Road Network Distance The distance (119889) between 2random points (119901

1and 119901

2) in road network space is the

shortest path length along the road from 1199011to 1199012

(3) Road Network Polyline The road network polyline isdenoted as pl = (119901

1 1199012 119901

119899) where 119899 gt 1 The length of

polyline is the summation of the distance between 2 adjacentvertexes

(4) RoadNetwork SegmentThe roadnetwork segment in roadnetwork space is a road network polyline which owns andonly owns 2 vertexes

(5) Edge of Road Network The edge (119890) in road network is thepath between 2 adjacent intersections

(6) A TrajectoryModel Based on RoadNetworkThe trajectorymodel based on road network is a new representation ofmoving object trajectories which matches GPS points withroad network to more accurately describe the spatial motioninformation of moving objects by their motion laws inroad network Meanwhile the model introduces a nonlinearinterpolation function among sampling points to preferablydescribe variable motions The trajectory model based onroad network separates the locations from time stamps Inother words a trajectory is represented by a spatial pathand a temporal sequence The spatial path of a trajectory


1

3

2

1

5

9

2 3 4

8

12

6

7

1110

t1

t4 t5

t2 t3e15 e16

e6

e3

e13Δ

Δ

Δ

Figure 3 A trajectory in road network

in a road network is a sequence of consecutive edges Asshown in Figure 3 a trajectory sequentially passes edges11989015 11989016 11989013 1198906 and 119890

3 Consequently it can be represented

by a spatial path in the format of ⟨11989015 11989016 11989013 1198906 1198903⟩ Note

that a trajectory can start from or end at any point of anedge not necessarily an endpoint The temporal informationof a trajectory is captured by a two-tuple (119889

119894 119905119894) where 119889

119894

represents the road network distance the object has traveledat the time stamp 119905

119894from the start of the trajectory and 119905

119894

represents the time the object has traveled at the location 119889119894

from the start of the trajectory

23 Perpendicular Distance and Synchronized Euclidean Dis-tance The perpendicular distance of point 119901 is the shortest

distance between the current point and the segment con-necting the first and last points of the trajectory whileSynchronized Euclidean Distance of point 119901 is the distancebetween the currently real point and the synchronized pointacquired by interpolating between the precursor point andthe successor point of the current point In Figure 4 theperpendicular distance of point 119901

119894+1is denoted as 119889

perpand the

Synchronized Euclidean Distance of point 119901119894+1

is denoted asSED

As shown in Figure 4 119901119894+1

is the current sampling point119901119894119901119894+2

is the segment connecting the first and last points of thetrajectory 1199011015840

119894+1is the synchronized point of 119901

119894+1in segment

119901119894119901119894+2

and the coordinate of 1199011015840119894+1

is calculated by

1199051015840

119894+1= 119905119894+1

1199091015840

119894+1= 119909119894+1

+119905119894+1

minus 119905119894

119905119894+2

minus 119905119894

(119909119894+2

minus 119909119894)

1199101015840

119894+1= 119910119894+1

+119905119894+1

minus 119905119894

119905119894+2

minus 119905119894

(119910119894+2

minus 119910119894)

(1)

The perpendicular distance and Synchronized EuclideanDistance between 119901

119894+1and 119901

119894119901119894+2

are calculated by formula(2) according to formula (1)

119889perp=

1003816100381610038161003816119909119894 lowast 119910119894+2 + 119909119894+2 lowast 119910119894+1 + 119909119894+1 lowast 119910119894 minus 119909119894+2 lowast 119910119894 minus 119909119894+1 lowast 119910119894+2 minus 119909119894 lowast 119910119894+11003816100381610038161003816

4 lowast radic(119909119894minus 119909119894+2)2

+ (119910119894minus 119910119894+2)2

SED = radic(119909119894minus 1199091015840

119894+2)2

+ (119910119894minus 1199101015840

119894+2)2

(2)

24 Trajectory Similarity The trajectory similarity is calcu-lated by measuring the similarity between two trajectoriesor subtrajectories utilizing Euclidean distance PCA PlusEuclidean distance Hausdorff distance Frechet distance andso on In this section we introduce 4 classical trajectorysimilarity measurements

241 Euclidean Distance Let 119871119894and 119871

119895be 119901-dimensional

trajectory segmentswith length of 119899Their Euclidean distancedenoted as119863

119864is given in

119863119864(119871119894 119871119895) =

1

119899

119899

sum

119896minus1

radic

119901

sum

119898minus1

(119886119898

119896minus 119887119898

119896)2

(3)

242 PCA Plus Euclidean Distance When computing PCA(Principal Components Analysis) Plus Euclidean distancetrajectory is firstly represented as a 1D signal by concatenatingthe 119909 and the 119910 projectionsThen location signal is convertedinto the first few PCA coefficientsThe trajectory similarity is

the Euclidean distance computed with the PCA coefficientsas shown in

119863PCA119864

(119871119894 119871119895) = radic

119870

sum

119896minus1

(119886119888

119896minus 119887119888

119896)2

(4)

Here 119886119888119896and 119887119888119896are respectively the 119896th PCA coefficient

in two-dimensional space trajectory segments 119871119894and 119871

119895

whose length is 119899 and119870 ≪ 2119899

243HausdorffDistance Given 2 trajectory segments119871119894and

119871119895 their Hausdorff distance denoted as119863

119867(119871119894 119871119895) is given in

119863119867(119871119894 119871119895) = max (ℎ (119871

119894 119871119895) ℎ (119871

119895 119871119894))

where ℎ (119871119894 119871119895) = max119886isin119871119894

(min119887isin119871119895

(dist (119886 119887))) (5)

In the formula ℎ(119871119894 119871119895) is the direct Hausdorff distance

of 119871119894and 119871

119895 and dist(119886 119887) is the Euclidean distance between

sampling points 119886 and 119887 in 119871119894and 119871

119895 respectively


pi+1(ti+1 xi+1 yi+1)

pi(ti xi yi)

SED dperp

p998400i+1(t

998400i+1 x

998400i+1 y

998400i+1)


Figure 4 A schematic of perpendicular distance and SynchronizedEuclidean Distance

244 Discrete Frechet Distance Discrete Frechet distancefully considers the location and sequential relationship of thepoint in trajectories while measuring their similarity It scansthe points on two trajectories and calculates its Euclideandistance point by point The maximum Euclidean distance isthe Discrete Frechet distance between two trajectories Thecalculating formula is shown as

119863119865(119871119894 119871119895)

= min 119862 119862 is the coupling between 119871119894and 119871

119895

where 119862 =119870max119896minus1

dist (119886119896119894 119887119896

119894)

(6)

Here 119871119894and 119871

119895are the trajectory segments whose lengths

are 119898 and 119899 respectively Consider 119870 = min(119898 119899) 119886119896119894

and 119887119896119895are the 119896th points on trajectory segments 119871

119894and 119871

119895

respectively dist(119886119896119894 119887119896

119895) is the Euclidean distance between 119886119896

119894

and 119887119896119895

245 Others In addition to the 4 trajectory similarity mea-sures discussed above Vlachos et al put forward longestcommon subsequence which is different from distance calcu-lation and is used to obtain the longest common subsequenceexisting in two trajectory sequences [32] Chen et al proposedDynamic Time Warping method which is a well-knowntechnique to find an optimal alignment between two given(time-dependent) sequences under certain restrictions [33]Lee et al put forward a comprehensive distance functionwhich is composed of three components the angle distancethe parallel distance and the perpendicular distance [1] Themethod overcomes the limitations of the trajectory similaritymeasure by the length of trajectory segments It can morecomprehensively measure the similarity between trajectorysegments Yuan et al extract trajectory structure and proposea structure similarity measurement for comparing trajecto-ries in microlevel [34]

25 Compressive Sensing Compressive sensing (CS) is anefficient signal processing technique to acquire and recon-struct a signal by finding solutions to underdeterminedlinear systems It is also known as compressed sensing

compressive sampling or sparse sampling CS is with theprinciple that through optimization the sparsity of a signalcan be exploited to recovery from far fewer samples thanrequired by the Shannon-Nyquist sampling theorem Thereare two conditions under which recovery is possible Thefirst one is sparsity which requires the signal to be sparse insomedomainThe second one is incoherencewhich is appliedthrough the isometric property which is sufficient for sparsesignals [35]

In this section we will discuss CS given in literature [16]briefly Given a vector 119909 isin R119899 the representation 120579 isin R119899

can be computed on a basis 120595 isin R119899times119899 by solving the linearequation 119909 = 120595120579 which is said to be compressible if 120579 hasa large number of elements with small magnitude If thereis a basis on which a given vector 119909 has a compressiblerepresentation then 119909 is also compressible Compressivesensing considers the problem of recovering an unknowncompressible vector 119909 from its projections LetΦ be an119898times119899

projection matrix with119898 lt 119899 Consider the equation

119910 = Φ119909 + 119911 (7)

where 119911 isin R119899 is a noise vector whose norm is bounded by120598 Compressive sensing aims to reconstruct 119909 from 119910 and Φgiven the knowledge that 119909 is compressible on the basis 120595Compressive sensing shows that under certain conditions it ispossible to recover 119909 by solving the following ℓ

1optimization

problem

minisinR119899

10038171003817100381710038171003817100381710038171003817100381710038171

subject to 10038171003817100381710038171003817119910 minus Φ120595

100381710038171003817100381710038172le 120598

(8)

Given 119909 can be estimated from = 120595In the context of trajectory compression 119909 is the tra-

jectory measured by a Mobile Sensor Networks node Thedimension of 119909 is large The MSN node computes 119910 = Φ119909

and transmits 119910 to the server The server can compute anestimated trajectory by using 119910 Φ and 120595 to solve theaforementioned ℓ

1optimization problem as shown in (8)

Note that the compression is lossy with 1 minus119898119899 representingboth space savings and reduction in wireless transmissionrequirement

3 Trajectory Compression Algorithms

In this section we comprehensively analyze moving objecttrajectory compression algorithms which have been one ofthe research hotspots in the moving object data miningfield Existing trajectory compression algorithms include 2categories single trajectory compression (STC) and multipletrajectory compression (MTC) The former compresses eachtrajectory individually ignoring the commonalities amongtrajectories and the latter compresses several trajectories orsubtrajectories at the same time by the commonalities amongtrajectories (such as similarity)There are some different clas-sification strategies about compression algorithms but theyare not unable to contain all of the compression algorithms asthe rapid development of compression technologyTherefore


in this paper we present a new classification strategy to dividetrajectory compression algorithms into 5 categories on thebasis of compression theories

31 Distance Based Trajectory Compression Distance (suchas perpendicular distance and Synchronized Euclidean Dis-tance) information is one of the most classic and commoncompression metrics in trajectory compression algorithmsMany researchers have devoted their talent to compress tra-jectories by deciding whether the sampling point is reservedbased on distance since 1973 The earliest distance based tra-jectory compression algorithm is Douglas-Peucker algorithmproposed by Douglas and Peucker [9] which recursivelyselects the point whose perpendicular distance is greater thangiven threshold until all points reserved meet the conditionKeogh et al put forward Opening Window algorithm thatonline compresses trajectory data based on perpendiculardistance A transformation of Douglas-Peucker algorithmcalled top-down time-ratio algorithm which takes a fullconsideration of spatial-temporal characteristics by replacingperpendicular distance with SED is proposed by Meratniaand Rolf [3] Then an extension to Opening Window calledOpening Window Time-Ratio algorithm using SED insteadof perpendicular distance to take temporal features intoaccount is proposed by Wu and Cao [2] Gudmundssonet al developed an implementation of the Douglas-Peuckeralgorithm which works efficiently even in the case where thepolygonal path given as input is allowed to self-intersect [11]

Perpendicular distance based trajectory compression issimple and efficient but it just considers the spatial featuresand ignores the temporal features of trajectories Synchro-nized Euclidean Distance based trajectory compression notonly is simple and efficient but also has a better compressioneffect than perpendicular distance based trajectory compres-sion for it takes the spatial-temporal features of trajectoriesinto account Distance based trajectory compression providesan effective way to compress trajectory data and a satisfac-tory compression result which has been applied to manyfields such as animal migration hurricane prediction andaerospace field But there are some obvious shortcomingsin processing limited trajectories such as the trajectoryof human activities in urban and taxi motion track andkeeping the internal features in trajectories for distance basedtrajectory compression pays more attention to keeping theholistic geometrical characteristics of trajectories

32 Velocity Based Trajectory Compression Velocity is oneof the most basic features of moving objects and it canreflect the motion features of moving objects as well asthe internal features in trajectories The researches on com-pressing trajectory data based on velocity are not perfectby now A famous velocity based trajectory compression istop-down speed-based algorithm proposed by Meratnia andRolf [3] improving the existing compression techniques byexploiting the spatiotemporal information hiding in the timeseries which can be made by analyzing the derived speedssubsequent to the trajectory A large difference between thetravel speeds of two subsequent segments is a criterion that

can be applied to retain the data point in the middle Anonline algorithm called Dead Reckoning algorithm proposedby Trajcevski et al [22] compresses trajectory by estimat-ing the successor point through the current point and itsvelocity A polynomial-time algorithm for optimal direction-preserving trajectory simplification which supports broaderapplication range than position-preserving simplificationproposed by Long et al [24] can be also regarded as avelocity based trajectory compression This method uses themaximum angular difference between the direction of themovement during each time period in original trajectory andthe direction of the movement during the same time periodin a simplification of original trajectory

Velocity based trajectory compression not only is simpleand efficient but also can keep the internal features intrajectories however it is not popular for the existing velocitybased trajectory compression methods only take speed intoaccount which may lead to greater errors and break theholistic geometrical characteristics of trajectories In thefuture study we hope that researchers will pay their attentionto compressing trajectory data by various features of velocity(such as velocity direction and accelerated velocity) except forthe magnitude of velocity

33 Semantic Trajectory Compression Semantic informationin road network has more practical significance in repre-senting moving object trajectories that are collected fromlimited moving objects Semantic trajectory compressionstores trajectories in the form of semantic information inroad network instead of trajectory points compresses spa-tial information in trajectory data by spatial compressionmethods and compresses temporal information in trajectorydata by temporal compression techniques until some haltingcondition is met The new and novel representation for tra-jectories that replaces trajectory data by the form of semanticinformation in road network was proposed by Schmid etal [12] in 2009 Many researchers have paid their attentionto semantic trajectory compression since then Semantictrajectory compression was applied to humanmotion datasetin urban area by Richter et al [13] which identifies therelevant reference points along the trajectory determinesall possible descriptions of how movement continues fromhere and exploits motion feature description of referencepoints to compress trajectory data Song et al [14] proposeda new framework namely paralleled road-network-basedtrajectory compression to effectively compress trajectorydata under road network constraints Different from existingworks PRESS proposed a novel representation for trajecto-ries to separate the spatial representation of a trajectory fromthe temporal representation and proposes a Hybrid SpatialCompression (HSC) algorithm and error Bounded TemporalCompression (BTC) algorithm to compress the spatial andtemporal information of trajectories respectively

Semantic trajectory compression is only suitable forlimited moving objects such as movement in road networkurban movement and orbital trajectory which will get amore realistic significance result in compressing trajectoriesof limited moving objects


34 Similarity Based Trajectory Compression Similaritybased trajectory compression splits original trajectories intosubtrajectories and then clusters subtrajectories with highsimilarity into the same group and clusters subtrajectorieswith low similarity into the different groups And then itunifies spatial information of trajectory data in the samegroup by a certain strategy which will keep a set of spatialinformation and all temporal information in every groupuntil some halting condition is met A famous similaritybased compression is similarity based compression of GPStrajectory data proposed by Birnbaum et al [23] which splitstrajectories into subtrajectories according to the similaritiesamong them For each collection of similar subtrajectoriesthis technique stores only one subtrajectoryrsquos spatial dataEach subtrajectory is then expressed as a mapping betweenitself and a previous subtrajectory

Similarity based trajectory compression has great advan-tages in retaining the commonalities among trajectories It issuitable for trajectory set and may be not suitable for a singletrajectory for the error may be large

35 Priority Queue Based Trajectory Compression Priorityqueue based trajectory compression selects the best subsetof trajectory points and permanently removes redundantand inessential trajectory points from original trajectory byutilizing local optimization strategies until some haltingcondition ismetThe Spatial QUalIty SimplificationHeuristic(SQUISH)method based on the priority queue data structureproposed byMuckell et al [26] prioritizes themost importantpoints in a trajectory stream It uses local optimization toselect the best subset of points and permanently removesredundant or insignificant points from the original GPStrajectory Three years later Muckell et al [28] presented anew version of SQUISH called SQUISH-E (Spatial QUalItySimplification Heuristic-Extended) which has the flexibilityof tuning compression with respect to compression ratio anderror

Priority queue based trajectory compression is not onlyan online trajectory compression algorithm but also a tra-jectory compression algorithm that requires presetting thememory buffer Hence it can be well applied to real-timeapplications and small memory devices It is suitable for allkinds of trajectories but the compression effect andmatchingeffect may be a little worse than the other compressionmethods

36 Others Considering that if the movement pattern andinternal features are neglected applications such as trajec-tory clustering outlier detection and activity discovery maybe not so accurate as we expectedTherefore we expect that anew algorithm called structure features based trajectory com-pression which compresses trajectories based on movementpattern and structure features in trajectories such as movingdirection of objects internal fluctuation in trajectories andtrajectory velocity or acceleration will attract more attentionof researchers for instance a polynomial-time algorithmfor optimal direction-preserving simplification proposed byLong et al which supports border application range than

position-preserving simplification [24] At present most ofthe portable equipment used for data collection is inexpen-sive power saving and of lower computational capabilitywhile the data processing procedure is often performed insupercomputers which have a higher computational capa-bility In order to effectively reduce the transport cost weexpect that compressive sensing based trajectory compres-sion which reduces the data scale in the process of acquiringdata by combining compressive sensing with trajectory fea-tures will attract more attention of researchers for instanceRana et al present an adaptive algorithm for compressiveapproximation of trajectory in 2011 which performs trajec-tory compression so as to maximize the information aboutthe trajectory subject to limited bandwidth [36] Four yearslater another compression method called adaptive trajectory(lossy) compression algorithm based on compressive sensinghas been proposed by Rana et al which has two innovativeelements [16] First they propose a method to computea deterministic projection matrix from a learnt dictionarySecond they propose a method for the mobile nodes toadaptively predict the number of projections needed basedon the speed of the mobile nodes

4 Validation Criteria ofCompression Performance

Compression result validation is very important for com-pression algorithms and it can measure the level of successand correctness reached by the algorithms There are manysolutions to validate the result mainly including AnalysisExperience Evaluation and Example The Analysis solutionincludes rigorous derivation and proof or carefully designedexperiment with statistically significant results Experiencesolution is applied in real-world scenarios or projects andthe evidence of approachrsquos correctness (usefulness or effec-tiveness) can be obtained from the process of executionEvaluation uses a set of examples to illustrate the pro-posed approach with a nonsystemic analysis of gatheredinformation from the execution of examples Example usesonly one or several small-scale examples to illustrate theproposed approach without any evaluation or comparison ofthe execution result In this section wemainly discuss 2 kindsof compression validation solutions The first compressionvalidation solution is performancemetrics which are used forcomparing the efficiency and performance of trajectory com-pression algorithms And the other compression validationsolution is accuracy metrics which are used for comparingthe accuracy and information loss of trajectory compressionalgorithms This section respectively denotes the originaltrajectory as OT whose length is 119898 and the compressedtrajectory as RT whose length is 119899 in order to facilitate thevalidation of trajectory compression

41 Performance Metrics

411 Compression Ratio Compression ratio (R) is an impor-tant index to measure the advantages and disadvantages of


SplE

Originaltrajectory

Compressedtrajectory

P9984003

P9984002

P9984004

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P4

P5

P7

P8

P6

P1P3

Figure 5 A schematic of spatial error

trajectory compression performance which is defined asin

R = (1 minus119899

119898) lowast 100 (9)

Compression ratio is the most common compressionindex which can accurately reflect the change of the sizeof trajectory data But R is influenced by the originalsignal data sampling rate and quantization accuracy and soon it is difficult to make an objective measurement Forinstance a compression ratio of 70 indicates that 30 of theoriginal points remained in the compressed representationof the trajectory namely if there are 100 points in originaltrajectory only 30 points will be reserved in the compressedrepresentation of the trajectory after compressing

412 Compression Time Compression time (119879) is an impor-tant index tomeasure the efficiency of trajectory compressionperformance which reflects the total time required by thecompression For example a compression time of 24 indicatesthat the total time of compressing original trajectory is 24ms

42 Accuracy Metrics

421 Spatial Error Given an original trajectory OT and itscompressed representation RT the spatial error (SplE) of RTwith respect to a point 119901

119894in OT is defined as the distance

between 119901119894(119909119894 119910119894 119905119894) and its estimation 119901

1015840

119894(1199091015840

119894 1199101015840

119894 1199051015840

119894) If RT

contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016

and 11990110158408= 1199018in Figure 5 where there is a trajectory containing

1199011 1199012 119901

8) Otherwise 1199011015840

119894is defined as the closest point to

119901119894along the line between precursor point and successor point

of 119901119894in trajectory RTThe precursor point of 119901

2is 1199011and the

successor point of 1199012is 1199014 Therefore the spatial error of RT

with respect to 1199012is the perpendicular distance from 119901

2to

line 11990111199014

422 SED Error Temporal characteristics of trajectory dataare not considered in spatial error so Synchronized EuclideanDistance (SED) is introduced to overcome this limitationSED is also the distance between 119901

119894(119909119894 119910119894 119905119894) and its estima-

tion 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840

119894) which is obtained by linear interpolation

method owning the same time coordinate with 119901119894 If RT

contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840

119894is defined as the location point

owning the same time coordinate with 119901119894in trajectory RT

Originaltrajectory

SEDE


P9984002

P9984003 P998400

4

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P1P3

P4

P5

P6

P7

P8

Figure 6 A schematic of SED error

HE2 HE3 HE4

HE5HE6

HE7

HE1

Originaltrajectory


P1P3

P5

P7

P8

P6

P2

P4

Figure 7 A schematic of heading error

The estimation point of 1199012is 11990110158402 Therefore the SED error of

RT with respect to 1199012is the distance between 119901

2and 1199011015840

2

423 Heading Error Heading error (HE) is the angulardeflection betweenmoving direction from the actual locationpoint 119901

119894minus1(119909119894minus1 119910119894minus1 119905119894minus1) to 119901119894(119909119894 119910119894 119905119894) along original trajec-

tory andmoving direction from the estimation location point1199011015840

119894minus1(1199091015840

119894minus1 1199101015840

119894minus1 1199051015840

119894minus1) to 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840

119894) along compressed trajec-

tory The estimation 1199011015840

119894owning the same time coordinate

with 119901119894is obtained by linear interpolation method As shown

in Figure 7 we specify clockwise direction is positive valueand anticlockwise direction is negative value to facilitate thecalculation

424 Speed Error Speed error (SpdE) is an importantmetricfor various kinds of transit applications For instance velocitymeasurement system gets overspeed hotspots by velocityinformation [37] as well as acceleration and deceleration datahelp to identify all kinds of irregular driving behaviors whichwill help police to find vehiclersquos illegal activities [38] Thecomputing method of speed error is similar to heading errorIt calculates the difference value between actual velocity andestimated velocity instead of calculating angular deflection

425 Information Loss Degree Information Loss Degree(ILD) that can comprehensively analyze the accuracy anderror of trajectory compression results is a comprehensiveindex to measure the advantages and disadvantages of trajec-tory compression effectiveness Information Loss Degree canbe calculated by the SED distance Dynamic Time Warpingdistance and Speed Corner between original trajectory andcompressed trajectory

Information Loss Degree based on SED (ILDSED) is themean value of maximum SED distance error (SEDEmax(OTRT)) average SED distance error (SEDEavg(OTRT)) andminimum SED distance error (SEDEmin(OTRT)) between


original trajectory OT and compressed trajectory RT whichcan be calculated as

ILDSED (OTRT) =SEDEmax (OTRT) + SEDEavg (OTRT) + SEDEmin (OTRT)

3 (10)

Information Loss Degree based on DTW (ILDdtw) ismeasured by the time warping distance between original

trajectory OT and compressed trajectory RT which can becalculated as

ILDdtw (OTRT) =

0 119898 = 119899 = 0

infin 119898 = 0 119899 = 0

SEDE (op1 rp1) +min

ILDdtw (Rest (OT) Rest (RT))

ILDdtw (Rest (OT) RT)

ILDdtw (OTRest (RT))

others

(11)

Here SEDE(op1 rp1) is the SED error between point

op1and rp

1 which respectively are the first point of OT

and RT Rest(OT) and Rest(RT) are the remaining trajectoryafter removing the first sampling point ILDdtw calculates theInformation Loss Degree by DTW error

Information Loss Degree based on Speed Corner (ILD-corner) is measured by the original and compressed SpeedCorner of moving objects which can be calculated as

ILDcorner (OTRT)

=

summin(119898119899)1

((10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816) (

10038161003816100381610038161205791198941003816100381610038161003816 +

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816))

119898 + 119899

(12)

5 Public Trajectory Datasets

There are quite a few real trajectory datasets that are publiclyavailable In this section a detailed description of realtrajectory datasets is given from their sources characteristicssampling rate and so on

51 GeoLife Trajectory Dataset A GPS trajectory datasetfrom Microsoft Research GeoLife project was collected by182 users in a period of over 5 years from April 2007 toAugust 2012 A GPS trajectory of this dataset is representedby a sequence of time-stamped points each of which containsthe information of latitude longitude and altitude Thisdataset whose size is 155GB contains 17621 trajectories witha total distance of 1292951 kilometers and a total durationof 50176 hours These trajectories were recorded by differentGPS loggers and GPS-phones and have a variety of samplingrates 915 percent of the trajectories are logged in a denserepresentation for example every 1sim5 seconds or every 5sim10meters per point

52 T-Drive Taxi Trajectories A sample of trajectories fromMicrosoft Research T-Drive project was generated by over30000 taxicabs in a period of 6 months from March 2009 toAugust 2009 The total distance traveled by the taxis is morethan 800 million kilometers and the total number of GPSpoints is nearly 15 billion The size of the dataset is 756Mband the average sampling interval and average distancebetween two consecutive points are around 31 minutes and300 meters respectively

53 GPS Trajectory with Transportation Labels This is aportion of GPS trajectory dataset collected in (MicrosoftResearch Asia) GeoLife project Each trajectory has a set oftransportation mode labels such as driving taking a busriding a bike and walkingThere is a label file associated witheach folder storing the trajectories of a user A GPS trajectoryof this dataset is represented by a sequence of time-stampedpoints each of which contains the information of latitudelongitude height speed heading direction and so forthThese trajectories were recorded by different GPS loggers orGPS-phones and have a variety of sampling rates 95 percentof the trajectories are logged in a dense representationfor example every 2sim5 seconds or every 5sim10 meters perpoint while a few of them do not have such a high densitybeing constrained by the devices The size of the dataset is560Mb

54 Check-in Data from Location-Based Social NetworksThe dataset from a LBSN in China whose size is 1068Mbconsists of 2756710 check-in data generated by 10049 usersexcluding the timestamp and relationships between usersEach check-in includes the information of ID latitudelongitude and timestamp


55 Hurricane Trajectories This dataset is provided by theNational Hurricane Service (NHS) containing 1740 trajec-tories of Atlantic Hurricanes from 1851 to 2012 NHS alsoprovides annotations of typical hurricane tracks for eachmonth throughout the annual hurricane season that spansfrom June to November The data were collected every 6hours

56 Movebank Animal Tracking Data Movebank is a freeonline database of animal tracking data helping animaltracking researchers to manage share protect analyze andarchive their data Movebank is an international projectwith over 11000 users including people from research andconservation groups around the world A lot of datasets arecollected in this database such as Continental black-tailedgodwits (data from Senner et al 2015) whose size is 5161Mb[39] andNavigation experiments in lesser black-backed gulls(data fromWikelski et al 2015) whose size is 2909Mb [40]

6 Application Scenarios ofTrajectory Compression

(1) The unrestricted movement of moving objects is a typicalapplication scenario of trajectory compression such as a birdflying in the sky a fish swimming in the sea and a horserunning on the grasslandDistance based trajectory compres-sion and velocity based trajectory compression have a highefficiency in this application scenario and a good applicationprospect in many fields such as studying animalsrsquo migratorytraces behavior and living situations as well as animalmigration research and hurricanes tornados and oceancurrents prediction For instance animal tracking data helpsbiologists understand how individuals and populations movewithin local areas migrate across oceans and continents andevolve through millennia This information is being used toaddress environmental challenges such as climate and landuse change biodiversity loss invasive species and the spreadof infectious diseases However the data that need to beanalyzed always have a large scale which will make themdifficult to be analyzed and find the useful information in thedata so it is necessary to compress the data by removing theredundant data and only keeping the valuable data

(2)The restrictedmovement of moving objects is anothervery important application scenario of trajectory compres-sion such as themotion track of taxis in urban area Semantictrajectory compression can effectively and efficiently com-press the trajectory data in this application scenario withrespect to transport analysis smart city plan and smarttransportation management For instance vast amounts oftrajectory data can be collected by vehicle positioning equip-ment and other devices which can be used to help police tofind dangerous driving predict the stream of people in majorfestivals in important places of a city and trace escaping routeof criminals But the large scale of the data will lead to thedifficulty of finding dangerous driving predict the stream ofpeople in major festivals in important places of a city andtrace escaping route of criminals for police so the data are in

urgent need of compression which can remove the redundantdata and only reserve the valuable data in the dataset

(3) Priority queue based trajectory compression is widelyapplied to small memory devices and has a high efficiencyin this scenario For instance most of the portable mobiledevices have a small memory If the data have to be analyzedon portable mobile devices it is easy to meet the breakdown(out of memory) that will lead to the device not workingTherefore it is necessary for portable mobile devices witha compress application that may compress the data byremoving the redundant data before analyzing them

7 Conclusion and Future Work

Trajectory compression is an efficient way to reduce thesize of trajectory data and reserve the useful and valuableinformation in large scale dataset which is one of the impor-tant components of data mining technology In this paperthe research status and new development of moving objecttrajectory compression algorithms in recent years have beensurveyed and summarized Firstly the representative com-pression algorithms proposed in recent years are analyzedand summarized from algorithmic thinking key technologyand the advantages and disadvantages Then the existingalgorithms are classified into several categories according tocompression theories Thirdly some typical valid criteria ofcompression result are summarized Lastly some applicationscenarios are pointed out and discussed

On the basis of summarizing and surveying on the mov-ing object trajectory compression and its theories methodsand techniques we also summarize the problems and thechallenges existing in moving object trajectory compressionwhich mainly includes the following aspects (1) Most of thecurrent trajectory compression algorithms pay more atten-tion to the holistic outline geometrical characters of trajectoryand ignore the movement patterns and the internal featuresin trajectories (2) Most of the current trajectory compres-sion algorithms cannot fully combine time dimension withspace dimensions and they just regard time dimension asthe additional dimension of space dimension of trajectoryobject (3)The general applicability of trajectory compressionalgorithm is low (4) Few researchers have paid their attentionto compressing trajectories by compressive sensing whichreduces the data scale in the process of acquiring data

Competing Interests

No potential conflict of interests was reported by the authors

Authorsrsquo Contributions

Penghui Sun and Shixiong Xia contributed equally to thiswork

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities China (with Grant2015XKMS085)


References

[1] J-G Lee J Han and K-Y Whang ldquoTrajectory clustering apartition-and-group frameworkrdquo in Proceedings of the ACMSIGMOD International Conference on Management of Data(SIGMOD rsquo07) pp 593ndash604 ACM Beijing China June 2007

[2] X-C Wu and Z-Y Cao ldquoBasic conception function andimplementation of temporal GISrdquo Earth Science-Journal ofChina University of Geosciences vol 27 no 3 pp 241ndash245 2002

[3] N Meratnia and A Rolf ldquoSpatiotemporal compression tech-niques for moving point objectsrdquo in Advances in DatabaseTechnologymdashEDBT2004 vol 2992 ofLectureNotes in ComputerScience pp 765ndash782 Springer Berlin Germany 2004

[4] M Prior-Jones Satellite Communications Systems Buyersrsquo GuideBritish Antarctic Survey Cambridge UK 2008

[5] H Wang H Su K Zheng S Sadiq and X Zhou ldquoAn effective-ness study on trajectory similarity measuresrdquo in Proceedings ofthe Twenty-Fourth AustralasianDatabase Conference (ADC rsquo13)vol 137 pp 13ndash22 2013

[6] D Zhang and X Zhang ldquoA spatiotemporal compression algo-rithm for GPS trajectory datardquo Journal of Transport Informationand Safety vol 31 no 3 pp 32ndash48 2013

[7] W R Tobler ldquoNumerical map generalizationrdquo DiscussionPaper Michigan Inter-University Community of MathematicalGeographers Ann Arbor Mich USA 1966

[8] R Bellman ldquoOn the approximation of curves by line segmentsusing dynamic programmingrdquo Communications of the ACMvol 4 no 6 p 284 1961

[9] D H Douglas and T K Peucker ldquoAlgorithms for the reductionof the number of points required to represent a digitized line orits caricaturerdquo Cartographica vol 10 no 2 pp 112ndash122 1973

[10] M Potamias K Patroumpas and T Sellis ldquoSampling trajectorystreams with spatiotemporal criteriardquo in Proceedings of the 18thInternational Conference on Scientific and Statistical DatabaseManagement (SSDBM rsquo06) pp 275ndash284 Vienna Austria July2006

[11] J Gudmundsson J Katajainen D Merrick C Ong and TWolle ldquoCompressing spatio-temporal trajectoriesrdquo Computa-tional Geometry vol 42 no 9 pp 825ndash841 2009

[12] F Schmid F K Richter and P Laube ldquoSemantic trajectorycompressionrdquo in Advances in Spatial and Temporal Databasespp 411ndash416 Springer Berlin Germany 2009

[13] K F Richter F Schmid and P Laube ldquoSemantic trajectorycompression representing urban movement in a nutshellrdquoJournal of Spatial Information Science no 4 pp 3ndash30 2015

[14] R Song W Sun B Zheng and Y Zheng ldquoPRESS a novelframework of trajectory compression in road networksrdquo Pro-ceedings of the VLDBEndowment vol 7 no 9 pp 661ndash672 2014

[15] G Kellaris N Pelekis and Y Theodoridis ldquoMap-matchedtrajectory compressionrdquo Journal of Systems and Software vol86 no 6 pp 1566ndash1579 2013

[16] R Rana M Yang T Wark C T Chou and W Hu ldquoSim-pleTrack adaptive trajectory compression with deterministicprojection matrix for mobile sensor networksrdquo IEEE SensorsJournal vol 15 no 1 pp 365ndash373 2015

[17] K Liu Y Li J Dai et al ldquoCompressing large scale urbantrajectory datardquo inProceedings of the 4th InternationalWorkshopon Cloud Data and Platforms (CloudDP rsquo14) p 3 ACMAmsterdam The Netherlands 2014

[18] I S Popa K Zeitouni V Oria and A Kharrat ldquoSpatio-temporal compression of trajectories in road networksrdquo GeoIn-formatica vol 19 no 1 pp 117ndash145 2014

[19] G Liu M Iwai and K Sezaki ldquoAn online method for trajectorysimplification under uncertainty of GPSrdquo Information andMedia Technologies vol 8 no 3 pp 665ndash674 2013

[20] W Pan C Yao X Li and L Shen ldquoAn online compressionalgorithm for positioning data acquisitionrdquo Informatica vol 38no 4 pp 339ndash346 2014

[21] G Yuan S Xia and Y Zhang ldquoInteresting activities discoveryformoving objects based on collaborative filteringrdquoMathemati-cal Problems in Engineering vol 2013 Article ID 380871 9 pages2013

[22] G Trajcevski H Cao P Scheuermanny O Wolfsonz and DVaccaro ldquoOn-line data reduction and the quality of historyin moving objects databasesrdquo in Proceedings of the 5th ACMInternational Workshop on Data Engineering for Wireless andMobile Access (MobiDE rsquo06) pp 19ndash26 ACM 2006

[23] J Birnbaum H-C Meng J-H Hwang and C LawsonldquoSimilarity-based compression of GPS trajectory datardquo in Pro-ceedings of the 4th International Conference on Computing forGeospatial Research and Application (COMGeo rsquo13) pp 92ndash95San Jose Calif USA July 2013

[24] C Long R C W Wong and H V Jagadish ldquoDirection-preserving trajectory simplificationrdquo Proceedings of the VLDBEndowment vol 6 no 10 pp 949ndash960 2013

[25] A Nibali and Z He ldquoTrajic an effective compression systemfor trajectory datardquo IEEE Transactions on Knowledge and DataEngineering vol 27 no 11 pp 3138ndash3151 2015

[26] J Muckell J H Hwang V Patil C T Lawson F Ping andS S Ravi ldquoSQUISH an online approach for GPS trajectorycompressionrdquo inProceedings of the 2nd International Conferenceon Computing for Geospatial Research amp Applications article 13Washington DC USA May 2011

[27] M ChenM Xu and P Franti ldquoA fastmultiresolution polygonalapproximation algorithm for gps trajectory simplificationrdquoIEEE Transactions on Image Processing vol 21 no 5 pp 2770ndash2785 2012

[28] J Muckell P W Olsen Jr J-H Hwang C T Lawson andS S Ravi ldquoCompression of trajectory data a comprehensiveevaluation and new approachrdquo GeoInformatica vol 18 no 3pp 435ndash460 2014

[29] E Keogh S Chu DHart andM Pazzani ldquoAn online algorithmfor segmenting time seriesrdquo in Proceedings of the 1st IEEEInternational Conference on Data Mining (ICDM rsquo01) pp 289ndash296 San Jose Calif USA December 2001

[30] J G Lee J Han and X Li ldquoTrajectory outlier detection apartition-and-detect frameworkrdquo in Proceedings of the IEEE24th International Conference on Data Engineering (ICDE rsquo08)pp 140ndash149 IEEE Washington DC USA 2008

[31] Y Zheng ldquoTrajectory data mining an overviewrdquoACMTransac-tions on Intelligent Systems and Technology vol 6 no 3 article29 2015

[32] M Vlachos M Hadjieleftheriou D Gunopulos and E KeoghldquoIndexing multidimensional time-seriesrdquo The VLDB Journalvol 15 no 1 pp 1ndash20 2006

[33] L Chen M T Ozsu and V Oria ldquoRobust and fast similaritysearch for moving object trajectoriesrdquo in Proceedings of theACM SIGMOD International Conference on Management ofData (SIGMOD rsquo05) pp 491ndash502 ACM June 2005

[34] G Yuan S Xia L Zhang Y Zhou and C Ji ldquoAn efficienttrajectory-clustering algorithm based on an index treerdquo Trans-actions of the Institute of Measurement and Control vol 34 no7 pp 850ndash861 2012


[35] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation Theory vol 52 no 4 pp 1289ndash1306 2006

[36] R Rana W Hu T Wark and C T Chou ldquoAn adaptive algo-rithm for compressive approximation of trajectory (AACAT)for delay tolerant networksrdquo in Wireless Sensor Networks vol6567 of Lecture Notes in Computer Science pp 33ndash48 SpringerBerlin Germany 2011

[37] J G Harper ldquoTraffic violation detection and deterrence impli-cations for automatic policingrdquo Applied Ergonomics vol 22 no3 pp 189ndash197 1991

[38] M Karpiriski A Senart and V Cahill ldquoSensor networks forsmart roadsrdquo in Proceedings of the 4th Annual IEEE Interna-tional Conference on Pervasive Computing and CommunicationsWorkshops p 310 IEEE Pisa Italy 2006

[39] N R SennerM A Verhoeven JM Abad-Gomez et al ldquoWhenSiberia came to the Netherlands the response of continentalblack-tailed godwits to a rare spring weather eventrdquo Journal ofAnimal Ecology vol 84 no 5 pp 1164ndash1176 2015

[40] M Wikelski E Arriero A Gagliardo et al ldquoTrue navigationin migrating gulls requires intact olfactory nervesrdquo ScientificReports vol 5 Article ID 17061 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


P1P2

P3

P4

P6

P8

P9

P7P5

S1

S2

S3

S4

X

Y

Figure 1 A schematic of the trajectory compression

To address these issues two categories of trajectorycompression strategies have been proposed aiming to reducethe size of a trajectory while not compromising muchprecision in its new data representation [5] One is theoffline compression which reduces the size of trajectory afterthe trajectory has been fully generated The other is onlinecompression compressing a trajectory instantly as an objecttravels On the one hand it can reduce the memory space bycompressing which will make the storage of data easier Onthe other hand it can cut down the size of data which willbe convenient for the transmission of data What is more itcan reserve the useful information in trajectories and removeredundant data from trajectories which have the potential tomake the thorough analysis of trajectory data easierThe datacompression is a method that reduces the size of data to cutdown the memory space and improve the efficiency of trans-mission storage and processing without losing informationor reorganizes data to reduce the redundancy and memoryspace according to certain algorithms Data compression canbe classified into two categories namely lossless and lossycompression Moving object trajectory compression aims toreduce the size and memory space of a trajectory on thepremise that the information contained in trajectory datais reserved that is to say in order to cut down the size ofdata it removes redundant location points while ensuringthe accuracy of the trajectory [6] Figure 1 is a schematicdiagram of a trajectory compression where the originaltrajectory is represented by black lines and the compressedtrajectory consists of red lines (namely 119878

1 1198782 1198783 and 119878

4)

There are 9 points in the original trajectory but only 5points are retained to approximately represent the originaltrajectory after compressing whose compression ratio is closeto 50 Thus it can be seen that trajectory compressionsplay an important role in the storage and analysis of dataBut trajectory compression tends to cause a certain lossof information while compressing trajectories Thereforevarious trajectory algorithms existing in literature balance thetradeoff between accuracy and storage size

The trajectory compression technology derives from thetopographic cartography and computer graphics The mostnative and simplest compressionmethod is uniform samplingalgorithmwhich simply takes every 119894th point in the trajectory[7] In 1961 Bellman put forward a new algorithm called

Bellman algorithm which solves linear generalization prob-lems by dynamic programming methods [8] This methodwill guarantee that the segments connecting the specificnumber of points selected from the curve are closest to theoriginal curve but the time overhead of it is giant which isup to119874(1198993) One of the most classical trajectory compressionalgorithms called Douglas-Peucker algorithm was presentedin 1973 by Douglas and Peucker [9] In 2001 Keogh et alput forward the Opening Window algorithm to compresstrajectory data online However traditional error metrics(such as perpendicular distance) are not suitable for movingobject trajectories whose spatial characteristics and temporalcharacteristics need to be simultaneously considered dueto their internal features In 2004 Meratnia and Rolf putforward a top-down speed-based algorithm and a top-downtime-ratio algorithm [3] The former improves the existingcompression techniques by exploiting the spatiotemporalinformation hiding in the time series while the latter is atransformation of DP algorithm which took a full consid-eration of spatiotemporal characteristics by replacing spatialerror with SED In 2006 Potamias et al put forward STTracealgorithm estimating the safe zone of successor point bylocation velocity and direction Meanwhile it is suitable forsmallmemory devices [10] Gudmundsson et al developed animplementation of the Douglas-Peucker path-simplificationalgorithm in 2009 which works efficiently even in the casewhere the polygonal path given as input is allowed to self-intersect [11] In 2009 Schmid et al proposed that trajectoriesstored in the form of trajectory points can be instead ofsemantic information of road networks [12] Since thenmanyresearchers have been doing a great deal of studies aboutthe semantic information of road networks [13ndash18] With theincreasing of the data traditional compression algorithmsare quite limited for online trajectory data Therefore onlinetrajectory compression becomes one of the hot topics [19ndash21] For example Opening Window Time-Ratio algorithmwas put forward by Meratnia which is an extension toOpening Window using SED instead of spatial error to taketemporal features into account [2] And Trajcevski et al putforward another online algorithm called Dead Reckoningalgorithm which estimates the successor point throughthe current point and its velocity [22] Out of traditionalposition preserved trajectory compression algorithms manyscholars have focused on different perspectives For exampleBirnbaum et al proposed a trajectory simplicity algorithmbased on subtrajectories and their similarity [23] Long et alproposed a polynomial-time algorithm for optimal direction-preserving simplification which supports a border appli-cation range than position-preserving simplification [24]Nibali and He proposed an effective compression systemfor trajectory data called Trajic which can fill the gap ofgood compression ratio and small error margin [25] Similarto STTrace Muckell et al put forward the Spatial QUalItySimplificationHeuristicmethod [26] In 2012 Chen et al pro-posed aMultiresolution PolygonalApproximation algorithmwhich compressed trajectories by a joint optimization onboth the LSSD and the ISSD criteria [27] In 2014 Muckell etal proposed a new algorithm SQUISH-E which compresses










119889)



















1

TR2 TR



119894= 1198751 1198752 119875

119898 (1 le 119894 le 119899) 119875

119895(1 le 119895 le 119898)


119895 119879119895⟩



119895at time119879

119895 Location



1198751198882

119875119888119894




119894=

1198711198941

1198711198942

119871119894num















1and 119901




1 1199012 119901







1

3

2

1

5

9

2 3 4

8

12

6

7

1110

t1

t4 t5

t2 t3e15 e16

e6

e3

e13Δ

Δ

Δ






119894 119905119894) where 119889

119894



119894





119894+1is denoted as 119889

perpand the


is denoted asSED





119894+1in segment

119901119894119901119894+2


is calculated by

1199051015840

119894+1= 119905119894+1

1199091015840

119894+1= 119909119894+1

+119905119894+1

minus 119905119894

119905119894+2

minus 119905119894

(119909119894+2

minus 119909119894)

1199101015840

119894+1= 119910119894+1

+119905119894+1

minus 119905119894

119905119894+2

minus 119905119894

(119910119894+2

minus 119910119894)

(1)


119894+1and 119901

119894119901119894+2


119889perp=



+ (119910119894minus 119910119894+2)2

SED = radic(119909119894minus 1199091015840

119894+2)2

+ (119910119894minus 1199101015840

119894+2)2

(2)





119864is given in

119863119864(119871119894 119871119895) =

1

119899

119899

sum

119896minus1

radic

119901

sum

119898minus1

(119886119898

119896minus 119887119898

119896)2

(3)



119863PCA119864

(119871119894 119871119895) = radic

119870

sum

119896minus1

(119886119888

119896minus 119887119888

119896)2

(4)



119895




119867(119871119894 119871119895) is given in

119863119867(119871119894 119871119895) = max (ℎ (119871

119894 119871119895) ℎ (119871

119895 119871119894))

where ℎ (119871119894 119871119895) = max119886isin119871119894

(min119887isin119871119895

(dist (119886 119887))) (5)


of 119871119894and 119871



119895 respectively



pi(ti xi yi)

SED dperp

p998400i+1(t

998400i+1 x

998400i+1 y

998400i+1)




119863119865(119871119894 119871119895)


119895

where 119862 =119870max119896minus1

dist (119886119896119894 119887119896

119894)

(6)

Here 119871119894and 119871




119894and 119871

119895



119894

and 119887119896119895







119910 = Φ119909 + 119911 (7)


1optimization

problem

minisinR119899

10038171003817100381710038171003817100381710038171003817100381710038171


100381710038171003817100381710038172le 120598

(8)





























SplE

Originaltrajectory


P9984003

P9984002

P9984004

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P4

P5

P7

P8

P6

P1P3



R = (1 minus119899

119898) lowast 100 (9)



42 Accuracy Metrics




1015840

119894(1199091015840

119894 1199101015840

119894 1199051015840

119894) If RT

contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840




2is 1199011and the



2to

line 11990111199014


119894(119909119894 119910119894 119905119894) and its estima-

tion 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840



contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840



Originaltrajectory

SEDE


P9984002

P9984003 P998400

4

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P1P3

P4

P5

P6

P7

P8


HE2 HE3 HE4

HE5HE6

HE7

HE1

Originaltrajectory


P1P3

P5

P7

P8

P6

P2

P4




2and 1199011015840

2




119894minus1(1199091015840

119894minus1 1199101015840

119894minus1 1199051015840

119894minus1) to 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840












3 (10)



ILDdtw (OTRT) =

0 119898 = 119899 = 0

infin 119898 = 0 119899 = 0

SEDE (op1 rp1) +min




others

(11)


op1and rp




ILDcorner (OTRT)

=

summin(119898119899)1

((10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816) (

10038161003816100381610038161205791198941003816100381610038161003816 +

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816))

119898 + 119899

(12)


















Competing Interests




Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119889)



















1

TR2 TR



119894= 1198751 1198752 119875

119898 (1 le 119894 le 119899) 119875

119895(1 le 119895 le 119898)


119895 119879119895⟩



119895at time119879

119895 Location



1198751198882

119875119888119894




119894=

1198711198941

1198711198942

119871119894num















1and 119901




1 1199012 119901







1

3

2

1

5

9

2 3 4

8

12

6

7

1110

t1

t4 t5

t2 t3e15 e16

e6

e3

e13Δ

Δ

Δ






119894 119905119894) where 119889

119894



119894





119894+1is denoted as 119889

perpand the


is denoted asSED





119894+1in segment

119901119894119901119894+2


is calculated by

1199051015840

119894+1= 119905119894+1

1199091015840

119894+1= 119909119894+1

+119905119894+1

minus 119905119894

119905119894+2

minus 119905119894

(119909119894+2

minus 119909119894)

1199101015840

119894+1= 119910119894+1

+119905119894+1

minus 119905119894

119905119894+2

minus 119905119894

(119910119894+2

minus 119910119894)

(1)


119894+1and 119901

119894119901119894+2


119889perp=



+ (119910119894minus 119910119894+2)2

SED = radic(119909119894minus 1199091015840

119894+2)2

+ (119910119894minus 1199101015840

119894+2)2

(2)





119864is given in

119863119864(119871119894 119871119895) =

1

119899

119899

sum

119896minus1

radic

119901

sum

119898minus1

(119886119898

119896minus 119887119898

119896)2

(3)



119863PCA119864

(119871119894 119871119895) = radic

119870

sum

119896minus1

(119886119888

119896minus 119887119888

119896)2

(4)



119895




119867(119871119894 119871119895) is given in

119863119867(119871119894 119871119895) = max (ℎ (119871

119894 119871119895) ℎ (119871

119895 119871119894))

where ℎ (119871119894 119871119895) = max119886isin119871119894

(min119887isin119871119895

(dist (119886 119887))) (5)


of 119871119894and 119871



119895 respectively



pi(ti xi yi)

SED dperp

p998400i+1(t

998400i+1 x

998400i+1 y

998400i+1)




119863119865(119871119894 119871119895)


119895

where 119862 =119870max119896minus1

dist (119886119896119894 119887119896

119894)

(6)

Here 119871119894and 119871




119894and 119871

119895



119894

and 119887119896119895







119910 = Φ119909 + 119911 (7)


1optimization

problem

minisinR119899

10038171003817100381710038171003817100381710038171003817100381710038171


100381710038171003817100381710038172le 120598

(8)





























SplE

Originaltrajectory


P9984003

P9984002

P9984004

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P4

P5

P7

P8

P6

P1P3



R = (1 minus119899

119898) lowast 100 (9)



42 Accuracy Metrics




1015840

119894(1199091015840

119894 1199101015840

119894 1199051015840

119894) If RT

contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840




2is 1199011and the



2to

line 11990111199014


119894(119909119894 119910119894 119905119894) and its estima-

tion 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840



contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840



Originaltrajectory

SEDE


P9984002

P9984003 P998400

4

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P1P3

P4

P5

P6

P7

P8


HE2 HE3 HE4

HE5HE6

HE7

HE1

Originaltrajectory


P1P3

P5

P7

P8

P6

P2

P4




2and 1199011015840

2




119894minus1(1199091015840

119894minus1 1199101015840

119894minus1 1199051015840

119894minus1) to 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840












3 (10)



ILDdtw (OTRT) =

0 119898 = 119899 = 0

infin 119898 = 0 119899 = 0

SEDE (op1 rp1) +min




others

(11)


op1and rp




ILDcorner (OTRT)

=

summin(119898119899)1

((10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816) (

10038161003816100381610038161205791198941003816100381610038161003816 +

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816))

119898 + 119899

(12)


















Competing Interests




Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119895at time119879

119895 Location



1198751198882

119875119888119894




119894=

1198711198941

1198711198942

119871119894num















1and 119901




1 1199012 119901







1

3

2

1

5

9

2 3 4

8

12

6

7

1110

t1

t4 t5

t2 t3e15 e16

e6

e3

e13Δ

Δ

Δ






119894 119905119894) where 119889

119894



119894





119894+1is denoted as 119889

perpand the


is denoted asSED





119894+1in segment

119901119894119901119894+2


is calculated by

1199051015840

119894+1= 119905119894+1

1199091015840

119894+1= 119909119894+1

+119905119894+1

minus 119905119894

119905119894+2

minus 119905119894

(119909119894+2

minus 119909119894)

1199101015840

119894+1= 119910119894+1

+119905119894+1

minus 119905119894

119905119894+2

minus 119905119894

(119910119894+2

minus 119910119894)

(1)


119894+1and 119901

119894119901119894+2


119889perp=



+ (119910119894minus 119910119894+2)2

SED = radic(119909119894minus 1199091015840

119894+2)2

+ (119910119894minus 1199101015840

119894+2)2

(2)





119864is given in

119863119864(119871119894 119871119895) =

1

119899

119899

sum

119896minus1

radic

119901

sum

119898minus1

(119886119898

119896minus 119887119898

119896)2

(3)



119863PCA119864

(119871119894 119871119895) = radic

119870

sum

119896minus1

(119886119888

119896minus 119887119888

119896)2

(4)



119895




119867(119871119894 119871119895) is given in

119863119867(119871119894 119871119895) = max (ℎ (119871

119894 119871119895) ℎ (119871

119895 119871119894))

where ℎ (119871119894 119871119895) = max119886isin119871119894

(min119887isin119871119895

(dist (119886 119887))) (5)


of 119871119894and 119871



119895 respectively



pi(ti xi yi)

SED dperp

p998400i+1(t

998400i+1 x

998400i+1 y

998400i+1)




119863119865(119871119894 119871119895)


119895

where 119862 =119870max119896minus1

dist (119886119896119894 119887119896

119894)

(6)

Here 119871119894and 119871




119894and 119871

119895



119894

and 119887119896119895







119910 = Φ119909 + 119911 (7)


1optimization

problem

minisinR119899

10038171003817100381710038171003817100381710038171003817100381710038171


100381710038171003817100381710038172le 120598

(8)





























SplE

Originaltrajectory


P9984003

P9984002

P9984004

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P4

P5

P7

P8

P6

P1P3



R = (1 minus119899

119898) lowast 100 (9)



42 Accuracy Metrics




1015840

119894(1199091015840

119894 1199101015840

119894 1199051015840

119894) If RT

contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840




2is 1199011and the



2to

line 11990111199014


119894(119909119894 119910119894 119905119894) and its estima-

tion 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840



contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840



Originaltrajectory

SEDE


P9984002

P9984003 P998400

4

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P1P3

P4

P5

P6

P7

P8


HE2 HE3 HE4

HE5HE6

HE7

HE1

Originaltrajectory


P1P3

P5

P7

P8

P6

P2

P4




2and 1199011015840

2




119894minus1(1199091015840

119894minus1 1199101015840

119894minus1 1199051015840

119894minus1) to 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840












3 (10)



ILDdtw (OTRT) =

0 119898 = 119899 = 0

infin 119898 = 0 119899 = 0

SEDE (op1 rp1) +min




others

(11)


op1and rp




ILDcorner (OTRT)

=

summin(119898119899)1

((10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816) (

10038161003816100381610038161205791198941003816100381610038161003816 +

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816))

119898 + 119899

(12)


















Competing Interests




Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

3

2

1

5

9

2 3 4

8

12

6

7

1110

t1

t4 t5

t2 t3e15 e16

e6

e3

e13Δ

Δ

Δ






119894 119905119894) where 119889

119894



119894





119894+1is denoted as 119889

perpand the


is denoted asSED





119894+1in segment

119901119894119901119894+2


is calculated by

1199051015840

119894+1= 119905119894+1

1199091015840

119894+1= 119909119894+1

+119905119894+1

minus 119905119894

119905119894+2

minus 119905119894

(119909119894+2

minus 119909119894)

1199101015840

119894+1= 119910119894+1

+119905119894+1

minus 119905119894

119905119894+2

minus 119905119894

(119910119894+2

minus 119910119894)

(1)


119894+1and 119901

119894119901119894+2


119889perp=



+ (119910119894minus 119910119894+2)2

SED = radic(119909119894minus 1199091015840

119894+2)2

+ (119910119894minus 1199101015840

119894+2)2

(2)





119864is given in

119863119864(119871119894 119871119895) =

1

119899

119899

sum

119896minus1

radic

119901

sum

119898minus1

(119886119898

119896minus 119887119898

119896)2

(3)



119863PCA119864

(119871119894 119871119895) = radic

119870

sum

119896minus1

(119886119888

119896minus 119887119888

119896)2

(4)



119895




119867(119871119894 119871119895) is given in

119863119867(119871119894 119871119895) = max (ℎ (119871

119894 119871119895) ℎ (119871

119895 119871119894))

where ℎ (119871119894 119871119895) = max119886isin119871119894

(min119887isin119871119895

(dist (119886 119887))) (5)


of 119871119894and 119871



119895 respectively



pi(ti xi yi)

SED dperp

p998400i+1(t

998400i+1 x

998400i+1 y

998400i+1)




119863119865(119871119894 119871119895)


119895

where 119862 =119870max119896minus1

dist (119886119896119894 119887119896

119894)

(6)

Here 119871119894and 119871




119894and 119871

119895



119894

and 119887119896119895







119910 = Φ119909 + 119911 (7)


1optimization

problem

minisinR119899

10038171003817100381710038171003817100381710038171003817100381710038171


100381710038171003817100381710038172le 120598

(8)





























SplE

Originaltrajectory


P9984003

P9984002

P9984004

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P4

P5

P7

P8

P6

P1P3



R = (1 minus119899

119898) lowast 100 (9)



42 Accuracy Metrics




1015840

119894(1199091015840

119894 1199101015840

119894 1199051015840

119894) If RT

contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840




2is 1199011and the



2to

line 11990111199014


119894(119909119894 119910119894 119905119894) and its estima-

tion 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840



contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840



Originaltrajectory

SEDE


P9984002

P9984003 P998400

4

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P1P3

P4

P5

P6

P7

P8


HE2 HE3 HE4

HE5HE6

HE7

HE1

Originaltrajectory


P1P3

P5

P7

P8

P6

P2

P4




2and 1199011015840

2




119894minus1(1199091015840

119894minus1 1199101015840

119894minus1 1199051015840

119894minus1) to 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840












3 (10)



ILDdtw (OTRT) =

0 119898 = 119899 = 0

infin 119898 = 0 119899 = 0

SEDE (op1 rp1) +min




others

(11)


op1and rp




ILDcorner (OTRT)

=

summin(119898119899)1

((10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816) (

10038161003816100381610038161205791198941003816100381610038161003816 +

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816))

119898 + 119899

(12)


















Competing Interests




Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











pi(ti xi yi)

SED dperp

p998400i+1(t

998400i+1 x

998400i+1 y

998400i+1)




119863119865(119871119894 119871119895)


119895

where 119862 =119870max119896minus1

dist (119886119896119894 119887119896

119894)

(6)

Here 119871119894and 119871




119894and 119871

119895



119894

and 119887119896119895







119910 = Φ119909 + 119911 (7)


1optimization

problem

minisinR119899

10038171003817100381710038171003817100381710038171003817100381710038171


100381710038171003817100381710038172le 120598

(8)





























SplE

Originaltrajectory


P9984003

P9984002

P9984004

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P4

P5

P7

P8

P6

P1P3



R = (1 minus119899

119898) lowast 100 (9)



42 Accuracy Metrics




1015840

119894(1199091015840

119894 1199101015840

119894 1199051015840

119894) If RT

contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840




2is 1199011and the



2to

line 11990111199014


119894(119909119894 119910119894 119905119894) and its estima-

tion 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840



contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840



Originaltrajectory

SEDE


P9984002

P9984003 P998400

4

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P1P3

P4

P5

P6

P7

P8


HE2 HE3 HE4

HE5HE6

HE7

HE1

Originaltrajectory


P1P3

P5

P7

P8

P6

P2

P4




2and 1199011015840

2




119894minus1(1199091015840

119894minus1 1199101015840

119894minus1 1199051015840

119894minus1) to 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840












3 (10)



ILDdtw (OTRT) =

0 119898 = 119899 = 0

infin 119898 = 0 119899 = 0

SEDE (op1 rp1) +min




others

(11)


op1and rp




ILDcorner (OTRT)

=

summin(119898119899)1

((10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816) (

10038161003816100381610038161205791198941003816100381610038161003816 +

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816))

119898 + 119899

(12)


















Competing Interests




Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























SplE

Originaltrajectory


P9984003

P9984002

P9984004

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P4

P5

P7

P8

P6

P1P3



R = (1 minus119899

119898) lowast 100 (9)



42 Accuracy Metrics




1015840

119894(1199091015840

119894 1199101015840

119894 1199051015840

119894) If RT

contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840




2is 1199011and the



2to

line 11990111199014


119894(119909119894 119910119894 119905119894) and its estima-

tion 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840



contains 119901119894 then 1199011015840

119894is 119901119894(eg 1199011015840

1= 1199011 11990110158404= 1199014 11990110158406= 1199016


1199011 1199012 119901

8) Otherwise 1199011015840



Originaltrajectory

SEDE


P9984002

P9984003 P998400

4

P9984005

P9984006

P9984007

P9984008

P9984001

P2

P1P3

P4

P5

P6

P7

P8


HE2 HE3 HE4

HE5HE6

HE7

HE1

Originaltrajectory


P1P3

P5

P7

P8

P6

P2

P4




2and 1199011015840

2




119894minus1(1199091015840

119894minus1 1199101015840

119894minus1 1199051015840

119894minus1) to 1199011015840

119894(1199091015840

119894 1199101015840

119894 1199051015840












3 (10)



ILDdtw (OTRT) =

0 119898 = 119899 = 0

infin 119898 = 0 119899 = 0

SEDE (op1 rp1) +min




others

(11)


op1and rp




ILDcorner (OTRT)

=

summin(119898119899)1

((10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816) (

10038161003816100381610038161205791198941003816100381610038161003816 +

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816))

119898 + 119899

(12)


















Competing Interests




Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












3 (10)



ILDdtw (OTRT) =

0 119898 = 119899 = 0

infin 119898 = 0 119899 = 0

SEDE (op1 rp1) +min




others

(11)


op1and rp




ILDcorner (OTRT)

=

summin(119898119899)1

((10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816) (

10038161003816100381610038161205791198941003816100381610038161003816 +

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816))

119898 + 119899

(12)


















Competing Interests




Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Competing Interests




Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









review article an overview of moving object trajectory...

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