Time geography and context. Basic time geographic …...Time geography and space–time prism Harvey J. Miller The Ohio State University, USA Time geography is a constraints-oriented

Time geography andspace–time prism

Harvey J. MillerThe Ohio State University, USA

Time geography is a constraints-orientedapproach to understanding human activities inspace and time. Time geography recognizes thathumans have fundamental spatial and temporallimitations: people can physically only be in oneplace at a time and activities occur at a sparse setof places for limited durations. Participating in anactivity requires allocating scarce available timeto access and conduct the activity. Constraintson activity participation include the location andtiming of anchors that compel presence (such ashome and work), the time budget for access andactivity, and the ability to trade time for space inusing mobility or information and communicationtechnologies (ICTs). Time geography is a phys-ical not a behavioral theory; it highlights thenecessary spatiotemporal conditions for humanactivities, but does not explain the sufficientevents that lead to specific activities. But sincethese necessary conditions vary by individual andsituation, time geography supports an approachto understanding human and environmentalsystems that recognizes individual histories andthe importance of geographic context.

Time geography is consistent with some coreideas in fields such as geography, transportation,urban science, social sciences, and environmentalsciences. These include an integrated perspectiveon human and physical phenomena, the need tobuild macro-level explanations from micro-levelprocessing, and situating human activities within

The International Encyclopedia of Geography.Edited by Douglas Richardson, Noel Castree, Michael F. Goodchild, Audrey Kobayashi, Weidong Liu, and Richard A. Marston.© 2017 John Wiley & Sons, Ltd. Published 2017 by John Wiley & Sons, Ltd.DOI: 10.1002/9781118786352.wbieg0431

context. Basic time geographic concepts, suchas events being sparsely distributed in time andspace, limited time availability, and trading timefor space to access activities, seem mundane,since they are common and correspond witheveryday experience. But this is why time geog-raphy is needed: these seemingly banal but utterlycrucial factors in our scientific explanations ofhuman behavior should not be neglected. Timegeography provides a framework that demandsrecognition of the fundamental constraintsunderlying human experience and also providesan effective conceptual system for keeping trackof these conditions.

Time geography originates from ProfessorTorsten Hägerstrand (1916–2004), a Swedishgeographer who spent his career at the Univer-sity of Lund. He nurtured the ideas for a longtime, but time geography emerged dramaticallyto the international scientific community witha now-famous 1969 presidential address to theRegional Science Association (Hägerstrand1970). Hägerstrand was concerned that humangeography and regional science were neglect-ing much of what comprises a livable world.He also wanted to provide a counterbalanceto increasing specialization and fragmentationin science, technology, and administration byoffering a more holistic view of human activities.Hägerstrand believed in the integrative power ofthe regional approach in geography, but felt thatit needed to be more inclusive and rich. Timegeography upgrades the regional approach bydescribing the “bare skeleton” of spatiotemporalconditions and constraints that emerge from theinterplay of historical and geographical factors.

Time geography is an active and flourishingresearch domain one half-century after its initial

TIME GEOGRAPHY AND SPACE–TIME PRISM

conceptualization in the 1960s. This ranks timegeography among elite and enduring scientificideas. Time geography has endured for goodreasons. It is beautiful: the geometry of thespace–time path and prism (discussed below) andtheir relationships are intuitive and appealing. Itis elegant: it can help to explain much with fewbasic principles. It is robust: it can help us under-stand a wide range of phenomena in humanand linked human-environmental systems. Itis sensitive: it treats people as individuals andrecognizes social differences across a wide rangeof factors (gender, age, socioeconomic status,culture) as well as geographic context. It is ecolog-ical: it connects the individual to the aggregate,balancing nomothetic law seeking with contextand situation, as well as balancing agency andstructure. Finally, it is practical: Hägerstrand wasprescient when thinking about how to keep trackof the basic spatial and temporal existential factsof human activities. It is now possible to collect,store, manage, and analyze individual-level dataon mobile objects and human activities withease and power that would seem magical fromthe perspective of the 1960s.

Classical time geography

Time geography recognizes three major types ofconstraints on human activities. Capability con-straints limit the activities of individuals throughtheir own physical capabilities and/or availableresources. People need to conduct maintenanceactivities such as eating and sleeping; theserequire time and place. Also, individuals withprivate automobiles can generally travel fasterthan individuals who walk or rely on publictransportation. Coupling constraints define where,when, and for how long an individual has tojoin with other individuals for shared activitiessuch as work, meetings, and classes. Authority

constraints are fiat restrictions over particularspace–time domains. For example, a shoppingmall or gated community can make it difficultand illegal to enter at designated times, while apublic street cannot.

Space–time path

Activities such as personal and domestic mainte-nance, work, shopping, health care, education,and recreation are sparsely distributed in timeand space; they are available for limited durationat relatively few locations. Participating in activi-ties requires trading time for space to access theselocations at their available times. The space–timepath highlights these requirements. Figure 1 illus-trates a space–time path between activity stations.Stations are places where activities can occur;classical time geography treats these as tubes des-ignating their locations in space and availabilityin time (e.g., work hours, operating hours fora store, appointments, scheduled lectures). Theclassical space–time path focuses on physicalmobility and interaction. Virtual interaction viaICTs is possible; for example, a telephone callcan be represented as a connection between twospace–time paths. However, virtual interactionis muted relative to physical interaction in theclassic theory.

Time geography also classifies activities basedon their flexibility for an individual. Fixed activ-ities are those that cannot be easily rescheduledor relocated (e.g., work, meetings), while flexibleactivities can be more easily rescheduled and/oroccur at more than one location (e.g., shopping,recreation). These categories can be arbitrary;for example, a software developer can code inan office or a cafe. Nevertheless, the dichotomyprovides an effective means for understandinghow the location and timing of some activitiescondition accessibility to other activities. Fixedactivities act as space–time anchors because other

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Time

WorkHome Child’s

school

Market

Space

Figure 1 A space–time path among activitystations.

activities must occur at the temporal gapsbetween fixed activities.

Space–time prism

The space–time prism (STP) highlights the influ-ence of space–time anchors on the ability toparticipate in flexible activities. The STP is theenvelope of all possible space–time paths betweenknown locations and times. Figure 2 illustratesa planar STP. In this case, two space–time anchorsframe a prism. Anchors correspond to knownlocations and times for the mobile object. Theseare often (but not always) the locations andtimes of fixed activities that compel presence.A maximum travel speed represents the object’smobility capabilities; in classic time geographythis is a uniform across space and time. (Timegeography uses the term “velocity,” but this isincorrect since velocity implies magnitude and

direction. “Speed” implies magnitude only andis, therefore, the more appropriate term.) Giventhese anchors and the speed limit, the prismdefines the envelope all possible space–timepaths between the anchors. The spatial footprintof the STP is the potential path area (PPA); thisis the region in space that is accessible to themoving object.

The prism in Figure 2 is general since itaccounts for stationary activity time and hastwo anchors that are spatially separate. A prismwithout stationary activity time consists only ofits time-forward cone rooted at the first anchor andits time-backward cone rooted at the second anchor.The STP will have a larger volume and PPA,as no stationary activity means more time to bemobile. The two anchors may also be coincidentspatially; this prism consists of two right conesinstead of oblique cones as in Figure 2. A prismmay also have only one anchor. An STP withits first anchor only is a time-forward conedelimiting all destinations that can be reachedfrom that origin within a specific time limit.Conversely, a prism with its second anchor onlyis a time-backward cone delimiting all originsthat can reach that destination within a specifictime limit.

The STP measures accessibility: the ability foran individual to travel and participate in activi-ties and the amount of time available for activityparticipation at locations. An activity at a sta-tion is not feasible unless that station intersectswith the prism spatially and temporally, the latterfor at least as long as the minimum activity timerequired. This delimits the subset of opportuni-ties in an environment that is available to a personbased on their STP constraints.

The STP provides a dramatically differentimage of accessibility than place-based accessi-bility measures, such as those based on spatialinteraction (“gravity”) models or simply count-ing the number of opportunities near a person’s

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home and work location. The locations and tim-ings of fixed activities vary by age/life cycle stage,socioeconomic status and culture. For example,STP-based accessibility measures capture genderdifferences in accessibility due to householdorganization that are missed by place-basedaccessibility measures such as home–work com-mute length. Place-based accessibility measuresassume everyone at a place, such as home andwork, has the same space–time scheduling con-straints, while people-based measures derivedfrom the space–time prims recognize individual

capability constraints that are masked throughhomogenization by place. Time geographyalso facilitates understanding how the temporalorganization of service and trading hours canhave differential impacts beyond the locations ofservices and businesses.

Bundling and intersections

Time geography recognizes two types of rela-tionships between paths and prisms. Bundlingrefers to the convergence in space and time of

Space

Potential patharea (PPA)

Stationaryactivity time

Maximum travel speed

Prism anchors

Time

Tim

e b

udget

Figure 2 A planar space–time prism.

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two or more paths. Path bundling is necessary(although not sufficient) evidence of shared activ-ities and individuals meshing their space–timeactivities to participate in projects. Bundlingcan occur when objects are in motion or sta-tionary; examples of the former include publictransportation and ride-sharing. Intersection isthe condition of two or more time geographicfeatures sharing some locations in space withrespect to time. Two or more people cannotphysically meet unless their STPs intersect, or apath is within an STP.

Bundling and intersections are necessary con-ditions for the emergence of broader space–timeactivity systems, such as a university or a city.Time geography is an ecological theory; itconsiders interactions between individuals andbetween the individual and the aggregate.Bundling and intersections require individualsto synchronize (coordinate over time) as well assynchrorize (coordinate over space). Coordinationalso occurs at multiple scales. Individuals mustconduct projects consisting of sequenced activitieslinked by mobility events to meet larger goalssuch as hosting a dinner party. Cities and regionsare time systems balancing the supply of availabletime and demands on that time. This is a richand intricate conceptualization of cities andsocieties; Alan Pred memorably referred to a“ballet of adjustments” as disruptions propagatethrough a time system due to activities andprojects changing to meet the new space–timerequirements for participation and interaction.

Analytical time geography

Classical time geography is conceptually rich butlimited analytically. The rise of geographic infor-mation systems (GIS) motivated renewed interestin time geography, particularly in relaxingstrict assumptions such as the maximum speed

constraining an STP being uniform in spaceand time. The development and deploymentof location-aware technologies (LATs), such as theglobal positioning system (GPS), mobile phones,and radiofrequency identification (RFID) chips,have greatly expanded capabilities for collectingdata on mobile objects and have led to the devel-opment of mobile objects databases and mobilitymining or exploratory analysis of mobile objectsdata. These scientific and technological develop-ments led to the development of analytical timegeography and supporting GIS software.

Path analytics

Location-aware technologies do not generatespace–time paths directly; they generate a tem-poral sequence of spatial locations that are usedto construct the path. There are several waysto generate this temporal sequence. Event-basedrecording captures the time and location when aspecified event occurs; for example, a person tex-ting or calling using a mobile phone. Time-basedrecording captures mobile object positions atregular time intervals; this typifies GPS receivers.In change-based recording a capture occurs whenthe position of the object is sufficiently dif-ferent from a previous location; this includesdead-reckoning methods as well as some mobileobjects data technologies that avoid recordinglocations to manage data volume. Location-basedrecording occur when a mobile object comes closeto locations where sensors are located; examplesinclude stationary radiofrequency identificationand Bluetooth sensors.

The simplest and most common way togenerate a path is linear interpolation: assumethe object followed the straight-line segmentbetween recorded locations. This works well fortime-based and change-based recording withhigh capture frequencies. Event-based recordingcreates more issues since events are often not

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very frequent, and the spatial resolution can becoarse and variable; this is often the case withmobile phone data resolved only to membershipin a service cell. Even coarser are paths derivedfrom location-based recording; these are oftensimply sequences of visits at the sensor locations.Space–time paths can also be matched to otherspatially referenced data such as transportationnetworks.

The ease of collecting space–time path datafrom location-aware technologies often comesat the expense of path semantics: details aboutthe moving objects such as the reasons formobility behavior. Semantics can be recoveredby overlaying paths with other georeferenceddata. This method can produce errors relatedto data inaccuracies and intrinsic ambiguities.For example, GPS positional error means thatit can be hard to tell if a person is inside oroutside of an activity location, for example, acoffee house. If the person is inside the coffeehouse, what was she doing – dining, working,socializing, or some combination of the above?Often, there is no unambiguous link betweenlocations and activities, especially in an erawith near-ubiquitous access to information andcommunication. Methods for recovering pathsemantics include decomposing the trajectory

into a sequence of moves and stops, and anno-tating these sequences based on map matchingwith background geographic information. Alsoavailable are advanced data mining techniquessuch as machine learning algorithms.

Table 1 summarizes the fundamental mea-surements available from space–time paths. Theprimitive position allows the derivation of dis-tance, direction, and spatial extent in the object’smovement; an example is spatial range meth-ods. From these primary derivatives the spatialdistribution of the object, changes in direc-tion, and the path’s sinuosity can be derived.From the temporal primitives instant and intervalthe duration of the object’s movement and itsspeed, as well as the temporal distribution of theobject’s movement and changes in duration, canbe derived. Primary derivatives from combiningspatial and temporal intervals include the object’sspeed (a scalar) and velocity (a vector), and corre-sponding secondary derivatives acceleration andan approaching rate with respect to a location,boundary or region.

Prism analytics

It is difficult to analytically describe the entireSTP. However, it is easy to describe its spatialextent at a moment in time; this can serve as

Table 1 Measurable primitives and their derivations from movement data.Dimension Primitive Derivatives

Primary Secondary

Spatial Position DistanceDirectionSpatial extent

Spatial distributionChange of directionSinuosity

Temporal InstanceInterval

DurationTravel time

Temporal distributionChange of duration

Spatiotemporal SpeedVelocity

AccelerationApproaching rate

Dodge, Weibel, and Lautenschütz 2008.

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the basis for a wide range of prism analytics. STPparameters are {xi, xj, ti, tj, sij, aij} where xi, xj arethe first and second anchor locations with associ-ated departure and arrival times ti, tj respectively,sij is the maximum travel speed, and aij is thestationary activity time. At a moment in timet ∈ (ti, tj), the spatial extent of an STP (denotedbyZij(t)) is the intersection of three convex spatialsets: (i) the future disc fi(t) comprising all loca-tions that can be reached from the first anchorby time ti + t; (ii) the past disc pj(t) encompassingall locations at time t that can reach the secondanchor by time tj − t; and (iii) the geo-ellipse gijthat constrains the prism locations to account forany stationary activity time:

Zij(t) = {fi(t) ∩ pj (t) ∩ gij} (1)

fi(t) = {x} |‖x − xi‖ ≤ (t − ti)sij (2)

pj(t) = {x} |‖x − xj‖ ≤ (tj − t)sij (3)

gij ={x} |‖x−xi‖+‖x−xj‖

≤ (tj− ti−aij)sij (4)

Figure 3 provides an illustration. This defini-tion of the STP is not limited to 2-D space. In1-D space, the sets described by equations (2)–(4)are line segments. In 2-D space, the discs arecircles and the geo-ellipse is an ellipse. In 3-Dspace, the discs are spheres and the geo-ellipseis a spheroid. There are scalable methods forcalculating these objects and their intersections.Also, only one or two of the sets are relevant atany time, since the future and past disc changein size and may be enclosed by the other twoobjects. For example, with a general prism as inFigure 2, it is only necessary to solve (in the fol-lowing order) a disc, a disc–ellipse intersection,an ellipse, a disc–ellipse intersection, and finallya disc. Similarly, finding path–prism intersectionsonly requires testing if a point lies within a disc,ellipse, or a disc–ellipse intersection. Findinga prism–prism intersection requires solving for

the intersection of two, three, or four of thesets based on the prisms’ morphologies at thatmoment in time. The worst case is a four-setintersection involving two discs and two ellipses(Miller 2005).

Over time, the future disc traces the time-forward cone with an apex at the first anchor, thepast disc traces the time-backward with an apex atthe second anchor and the geo-ellipse is a cylin-der. Over time, the prism in space is a lens-likeset that traces the PPA footprint. These dynamicscan be computed at discrete moments in timeusing the static construction described above andreconstruct the corresponding spatiotemporalregion for the prism.

Error and uncertainty in pathsand prisms

Error in space–time paths results from twosources. Measurement error occurs when therecording of a mobile object’s location has noiseor uncertainty. The left-hand side of Figure 4illustrates this for one segment of a space–timepath; the captured locations have some degreeof spatial error. This is equivalent to the prob-lem of error in line segments and polylines, awell-established topic in the GIS literature. Sam-pling error occurs when the captured locationsundersample an object’s movement pattern. Thiscreates a spatial uncertainty region equivalentto the STP over time. Therefore, a space–timepath with sampling error can be viewed as asequence of linked STPs. The right-hand sideof Figure 4 illustrates this; the black curve is theactual path of the mobile object, the dashed redpolyline indicates the interpolated movement,and the blue ellipses indicate the spatial uncer-tainty region surrounding each paired samplelocations.

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pj (t)

fi (t)

gij

zij(t)

Figure 3 Analytical construction of a planar STP.

Combined measurement and sampling error ina space–time path is equivalent to an STP withmeasurement error. Therefore, if it is possible tosolve the measurement error problem for STPs itis also possible to solve the problem of combinedmeasurement and sampling errors in space–timepaths. One strategy is Monte Carlo simulation,which involves generating multiple prism real-izations based on sampling from the stochasticerror distributions. Monte Carlo simulation iseffective for theoretical investigations but it iscumbersome for applications: it is awkward torun prism error simulations when executingSTP-based mobile objects database queries ormeasuring accessibility in a model.

Spatial error propagation theory providesan analytical strategy for analyzing prismerror. Given a direct geographic measurementx = μx + εx, where μx is the true location vectorand εx is an error vector, it must be determinedhow that error propagates through a geographicoperation y = f (x). If f(x) is nonlinear, the errorpropagation is approximated using the first-orderpartial derivatives of the function. In the case

Figure 4 Measurement and sampling error inspace–time paths.

of the analytical prism, the relevant functionsare the intersections of the boundaries of spatialobjects described by equations (2)–(4). Theseboundaries are:

fi ∶ ‖x − xi‖ − (t − ti)sij = 0 (5)

pj ∶ ‖x − xj‖ − (tj − t)sij = 0 (6)

gij ∶ ‖x−xi‖+‖x−xj‖

−(tj− ti−aij)sij = 0 (7)

A problem is that equations (5)–(7) define theseboundaries implicitly rather than explicitly (thatis, in the form f (x, y) = 0 rather than f (x) = y),meaning that solving for the intersection pointsrequires finding the roots of high order poly-nomials. This difficulty can be resolved usingimplicit function methods that allow calculationof the required first order partial derivatives with-out having their explicit functions. This allowsthe error propagation from STP parameters tothe constructed STP and to STP intersections tobe estimated analytically. However, some of therequired circle and ellipse intersection cases are

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still unsolved and the techniques are not tractablebeyond two STP intersections. Still required arescalable approximations and heuristics for STPerror based on these analytical techniques. Theanalytical techniques can also serve as boundsfor improving the efficiency of simulation-basedtechniques.

The methods discussed above estimate errorpropagation from measured STP parameters tothe constructed object. Another strategy is toconstruct objects that encompass the possibleSTP consistent with the prism parameters andtheir errors. A rough STP consists of the upperand lower bounds on an STP. A reliable STP is theset of space–time locations where an individualcan conduct an activity and meet the secondanchor constraints with a specified probability.

Properties of the prism interior

The STP is traditionally a binary concept; alllocations within the prism interior are consid-ered to be equally accessible. However, this isa simple characterization that masks intricateproperties of the prism interior. Intuitively, itwould be expected that the distribution of visitprobabilities would be unequal: locations thatare near the space–time axis connecting theprism anchors are more likely to be visited thanlocations near the prism boundary since thereare more possible paths through the former.

One way to model visit probabilities within aplanar STP is to use the theory of random walks(RWs). An RW is a stochastic process in discretespace and time; given a spatial lattice of discretelocations, an RW involves, at each time step, arandom choice of direction. Results from RWtheory confirm intuition that locations near aprism anchor in a forward-time or backward-time cone are more likely to be visited; visit

probabilities follow a bivariate multinomialdistribution centered on the prism anchor.

However, movement within an STP with twoanchors is not completely random; the objectneeds to move from the first prism anchor tosecond anchor within the time budget. A directedrandom walk (DRW) is an RW process with direc-tional biases. A DRW process constrained by anSTP requires updating the RW direction biasesat each step to account for the remaining timeto travel to the second anchor given the speedlimit. This suggests a visit probability distributionsimilar to a bivariate normal distribution that iscentered on the axis connecting the anchors andmoves along that axis with respect to time.

A technique for modeling directed movementin continuous time and space is Brownian bridges(BBs). A BB is a continuous stochastic processbetween two known values; it has been appliedwidely to model animal movement betweenrecorded locations. While BB can capture direc-tionally biased random movement betweenprism anchors, is it not constrained by a maxi-mum speed and, therefore, does not capture theSTP boundary; it is still possible for the objectto travel outside the prism boundary. A truncatedBrownian bridge (TBB) imposes a maximumspeed on a BB process, fully representing STPconstraints. Figure 5 illustrates visit probabili-ties at a moment in time within the PPA of aplanar STP based on a TBB process. Figure 5shows the two prism anchors with the spatialprojection of the space–time axis connecting theanchors. The gray area is the PPA for the STP:these locations have zero probability of beingoccupied at this moment in time. The blue,yellow, orange and red colors show increasingprobability that the locations may be occupiedby the object at that moment in time. Theseresults are also consistent with the assumptionof a bivariate random distribution whose centermoves with time along the axis connecting the

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anchor points; however, rather than assumed it isderived from fundamental movement principles.

Another strategy for estimating visit prob-abilities within planar STPs is to use spatialinterpolation to infer unknown object locationsfrom the known locations at the prism anchors.This assumes that the most likely locations arealong the axis between the anchors, but asseen above this assumption is consistent withtheory. Time-geographic density estimation (TDE)integrates kernel density estimation with theconstraints imposed by an STP. This method cangenerate the fine-grained movement patternsof objects between anchors derived from sparsetracking data.

Other types of space–time prisms

A planar STP has interesting properties and isuseful for applications where it can be assumedthat objects move in constrained space with auniform maximum speed. However, in someapplications the assumptions of unconstrainedspace and uniform maximum speed are unre-alistic; examples include vehicles within atransportation network or movement acrossterrain. In addition, the STP assumes unrealisticmotions such as infinite rates of acceleration anddeceleration.

A network time prism (NTP) is a prism subjectto network routes and allowable speeds that varyby network arc and, possibly, time. Figure 6illustrates an NTP for a network embedded in2-D space with time along the third orthog-onal dimension (Kuijpers and Othman 2009).The green subportion of the network arcs isthe potential network area (PNA), the networkanalog of the planar PPA; these are the accessiblelocations within the network. The red polygoncomprises the full NTP; this is the envelope ofpossible space–time paths between the anchorsconstrained by the network.

Figure 5 Visit probabilities within a planarspace–time prism’s potential path area at a momentin time.

Calculating an NTP involves three steps.A precomputation step uses the planar PPAas an upper bound on the NPA; this speedscomputation by eliminating network nodes thatcannot possibly be in the NPA. It then solvesfor the shortest path tree in this subnetworktwice: once for travel from the first anchor anda second time for travel to the second anchor.The procedure assigns the earliest arrival timeand latest departure time at each node. Thesecond step builds the spatial footprint of theNTP by testing if only one of an arc’s end nodesis included in the NPA and (if so) solving forthe NPA boundaries within the edge. The thirdstep computes the full spatiotemporal region ofthe NTP based on the earliest arrival times andlatest departure times at network vertices. Thesearrival and departure times dictate, for each nodeincident to an arc, whether the mobile objectcan traverse the full arc to an adjacent nodeor can only traverse partway and return to theoriginal node. These cases correspond to the

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Figure 6 A network time prism.

rectangular and triangular regions in Figure 6,respectively. The computational bottleneckin the algorithm is the shortest path trees inthe precomputation step; although this is onlyapplied to a subportion of the network, thissubnetwork can be large, as can the number ofprisms to be processed. However, the Booleanquery about whether two NTPs intersect can besolved analytically.

Similar to an STP, the parameters of an NTPcan also be measured with uncertainty. An anchorregion is the set of all possible locations andtimes for a prism anchor within a network arc.These regions are not necessarily continuousin space or time and can have nonuniformprobabilities. There are two ways to calculatean NTP with anchor regions. The first way isto calculate the envelope of all NTPs having ananchor point within a given anchor region. Thesecond way is to calculate, for any space–timepoint, the probability that an NTP with givenanchor regions contains that point. As with thestandard NTP, the computational bottleneckis the precomputation phase involving shortestpath calculations.

A field time prism (FTP) is a planar STPwhere travel speeds (magnitude) or velocities(magnitude and direction) vary continuouslyacross space. Fields are useful for describingmovement across terrain or through water andair that are subject to currents. Fields can extendthe NTP by treating edges as 2-D regionswith continuously varying speeds or velocitiesdue to factors such as traffic. In these cases,space–time paths have unobserved componentscorresponding to minimum cost curves throughan inverse speed or velocity field rather thanstraight line segments through a uniform plane.The FTP generalizes the prism concept: theSTP and the long standing concept of isochrones(curves of equal travel time) are special casesof the FTP. It also links time geography tothe continuous transportation or urban fieldstradition in quantitative geography and regionalscience.

Space–time prisms as described above are phys-ically impossible; they assume that the mobileobject can instantly accelerate and decelerateat locations such as the prism anchors and theprism’s sharp corners. Physical limitations onacceleration and deceleration mean that an STPis an overestimate of the true space–time regionaccessible to the object. While this may not mat-ter for some STP applications, kinetics can makea difference for applications such as microscalemovement (e.g., pedestrians, athletes), activetransport modes such as bicycling, movementthrough media such as airplanes and ships, andanimal behavior. It can also make a differencefor applications in sustainable transportation, asvehicle fuel consumption and emissions are oftendominated by acceleration events.

A kinetic time prism (KTP) is the set of allpossible kinetic paths between two locationsand times, where a kinetic path is a space–timepath that obeys physical limits on accelerationand deceleration. Figure 7 illustrates a KTP in

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t

x

Figure 7 A kinetic (solid line) and classic (dottedline) space–time prism in 1-D space.

1-D space and time, overlaid with a classic STP.Solving a KTP is scalable in 1-D space and timebut complex in 2-D space. In 2-D space, onlyone-quarter of the prism must be solved; theremainder can be obtained through point andreflection symmetries. But calculating the firstquarter of the KTP requires solving parametricfunctions describing the 1-D prism rotatingaround the line connecting the prism anchors.This can only be accomplished if it is assumedthat the object’s initial heading is unknown; stillopen is the case where the object’s initial headingis known.

Path and prisms collections

With the proliferation of location-aware tech-nologies generating mobile objects data, it isoften the case that there are large collections ofspace–time paths and prisms to analyze. Withthese collections, it is often useful to summarizethe paths or prisms through clustering (findinggroups of similar paths or prism) or aggregation(forming a composite representative paths orprisms). It may also be desired to search througha collection of paths and prisms for other casesthat resemble a reference path or prism. This

requires methods for calculating path and prismsimilarity.Path similarity is the degree of correspon-

dence between two space–time paths. Geometricsimilarity measures focus only on the geometry,ignoring sequence and time; these includeEuclidean and Hausdorff distances. Euclideandistances, such as the average, minimum andmaximum distance are intuitive and scalable;however these measures are sensitive to noiseand outliers. The Hausdorff distance is the maxi-mum distance from a location in one path to theclosest location in the other path. However, theHausdorff distance can be misleading, as it doesnot take into account the temporal sequence ofthe path locations, only their geometry.Dynamic similarity measures include Fréchet

distance, dynamic time warping, longest com-mon subsequences, and edit-distance functions.Unlike Hausdorff, the Fréchet distance takesinto account the sequence of locations in thepath and is, therefore, better suited to mobileobjects. One way to think about the Fréchetdistance is to imagine two space–time pathsrepresenting a person walking a dog: the Fréchetdistance is the shortest leash that connects thetwo. Dynamic time warping measures the similaritybetween two sequences or trajectories based onthe effort involved to stretch or compress timeto get the sequences to match. Least commonsubsequence (LCSS) measures similarity basedon the length of least common subsequencebetween two sequences. Edit-distance functions area generalization of LCSS; these measure simi-larities between sequential patterns based on thecost of the insertion, deletion, and substitutionoperations required to transform one sequenceinto the other.

Other methods for analyzing collections ofspace–time paths include path clustering meth-ods and spatial field methods. Path clusteringmethods find groupings of similar space–time

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paths in collections of mobile objects; these areoften based on path similarity measures such asthe ones already discussed. Spatial field methodstranslate movement patterns of objects into fieldsor surfaces that summarize mobility and activityfrequency by geographic location, allowing theidentification of “hot spots” or locations withhigh mobility activity.

Similar to space–time paths, it may desired toanalyze a collection of space–time prisms forpatterns. This may involve clustering or aggre-gating prisms based on morphology, findingprisms similar to reference prism, and sum-marizing and aggregating prisms to discoversynoptic patterns. However, to date there hasbeen little research on these questions; it is amuch less developed research topic than pathcomparison and summarization. A fundamentalproblem is to develop efficient numeric measuresthat summarize meaningful prism properties.It is easy to calculate basic properties such asprism volume and PPA size for classic prisms,as these have an elegant geometry consisting ofcones and cylinders. More generally, measuresare required that can capture a wide range ofmorphological properties. These properties maybe straightforward for classic prisms, but morecomplex prisms such as NTPs, FTPs, and KTPs,or prisms with fuzzy or probabilistic boundarieshave more complex shapes and, consequently,are more difficult to describe analytically.

One possible strategy for measuring planarspace–time prism morphology is shape analysisor quantitative measures summarizing geomet-ric form. Shape analytical techniques typicallyreduce geometric forms to a single numeric indi-cator or a small set of indicators; for example,roughness, perforation, and elongation. How-ever, most shape measures are specific to 2-Dspace: required are scalable shape measures thatcan handle 3- or 4-D space (treating time as

static) and the evolution of shape (treating timeas dynamic).

A possible way to summarize NTP structure isthrough network indices based on graph theory.Network indices summarize network propertiessuch as connectivity, the distribution of arclengths/costs, and properties of shortest pathsand tours within the network. Graph theory andnetwork analysis is a very well established fieldthat has been newly reinvigorated due to the riseof network science to analyze phenomena suchas social network and cities. A research frontieris determining which existing network indicesdescribe relevant properties of NTPs, and thedevelopment of new network indices tailoredfor NTPs.

Joint accessibility and time ecology

As mentioned, time geography recognizesbundling and intersections among paths andprisms as indicators of shared activities. The nextstep beyond calculating bundles and intersectionsare the higher-level properties associated withspace–time coordination and dynamics.Collective motion methods focus on individual

object movement patterns within the context ofa larger group of mobile objects. Distance-basedmeasures search for collective patterns, such asflocks, by searching for moving objects that aredensely connected in space given user-defineddistance and time thresholds. Relative motionmethods consider the individuals’ directionsto detect collective patterns such as flock-ing, leadership and convergence in a group ofmoving objects. Joint accessibility measures useprism–prism intersections as a basis for analyzingthe potential benefits to coordinated activityparticipation.

Static measures of collective motion and jointaccessibility do not directly address the ultimate

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goal of understanding the dynamic processesthough which shared and interlinked activitiesform and intertwine in space and time. Thishighlights a weakness of time geography: itsinability to specify an explanation of humanactivity that goes beyond the implications ofspace–time constraints limiting possible activi-ties. Time geography can explain why infeasibleactivities did not occur, but it cannot explainwhy some feasible activities occurred whileothers did not. However, it would be difficult tobuild a dynamic, processes-based time ecologytheory based only on the negative space impliedby prisms and stations. A more promising wayforward is to link time geography with modelingapproaches such as activity-based travel demandmodels and agent-based models that naturallycapture the emergent properties associated withindividuals’ interactions over space and time.The natural fit between activity-based analysisand time geography is well recognized and wellutilized in many models. However, the linkagesbetween time geography and agent-based mod-eling are less developed, somewhat surprisinglygiven their complementarity.

Time geography and virtual interaction

Classical time geography recognizes the pos-sibility of virtual interaction; for example,Hägerstrand (1970) shows a telephone callbetween two space–time paths. However, virtualinteraction is neglected relative to travel andphysical interaction in classical time geography.This is understandable given the era when timegeography was initially formulated. But in thecontemporary world virtual interaction is morepervasive and cannot be ignored when discussingphysical mobility and activities.

Table 2 is a typology of possible communi-cation modes based on space–time constraints.

Synchronous presence (SP) includes face-to-faceconversations; this requires people involved to bephysically present or proximal at the same time.Asynchronous presence (AP) requires co-locationin space but not coincidence in time; examplesinclude notes left on an office door or in ageocache. Synchronous telepresence (ST) requirescoincidence in time but not in space; it includesmedia such as telephones, television, radio, andteleconferencing. Asynchronous telepresence (AT)does not require co-location in space or coinci-dence in time; it includes mail, e-mail, messages,and webpages. Classical time geography focuseson SP, recognizes but does not develop ST, andmostly ignores AP and AT.

There are at least two strategies for incorpo-rating the wider range of communication modesinto time geography. One strategy is to treat theseas spatial and temporal relationships betweenspace–time paths and prisms. SP requires theprism to intersect both spatially and temporally,while AP interactions only require spatial coinci-dence between the prisms (in other words, theyshare the same locations but at different times).ST requires temporal but not spatial coincidence

Table 2 Communication modes based on spatialand temporal constraints.

Temporal Spatial

Presence Telepresence

Synchronous SPFace-to-face

conversation

STTelephoneTelevisionRadioTeleconferencing

Asynchronous APTrail signsNote left on

office doorGeocaches

ATMailE-mailNewspapersWebpages

Janelle 2004.

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(the prisms share some interval in time but notlocations in space), while AT only requires oneprism to precede the other in time.

Another strategy for extending time geog-raphy is to make communication technologyexplicit using time geographic entities and derivespace–time constraints based on those entities. Aportal is a type of space–time station (Figure 1)where a person can access communication ser-vices. It includes a point location, and an accessrange and time intervals when the service isavailable. Portals correspond to real world enti-ties such as wired Internet connections (a pointlocation with zero range), wireless access points,and cellular telephone base stations (both charac-terized as point locations with different ranges).An individual can access a communication ser-vice only if his or her paths or prism intersectwith the service footprint of an appropriateportal. When this occurs, it generates a messagewindow or an interval of time when a messagecould be sent or received. Message windowscan be compareed using temporal predicatessuch as before, during, and after to determinewhich communication events are feasible. Thisapproach is consistent with the perspective thattemporal constraints dominate spatial constraintson telepresence.

Discovering time geographic knowledge

As the size of mobility datasets become richerand voluminous, the challenges move beyonddeveloping scalable techniques for comput-ing low-level time geographic properties andrelationships to discovering higher-level timegeographic knowledge. Mobility mining is a set ofactivities and techniques for discovering novelknowledge hidden in moving objects data. Thisinvolves three major activities: (i) trajectoryreconstruction and management; (ii) space–time

knowledge discovery; and (iii) space–timeknowledge delivery.Trajectory reconstruction and management requires

processing the raw mobility data to obtain thespace–time paths of the mobile objects as well asdata structures and access methods for processingthese paths efficiently. These include specializeddata warehouse designs, mobile object indexingmethods, and semantic trajectory compressionmethods for reducing storage requirements.Preserving locational privacy through security,anonymization, and other protocols is also aconcern.Space–time knowledge discovery involves process-

ing the trajectories to discover patterns such asclusters and behavioral rules. As noted, path andprism similarity methods can be used in conjunc-tion with clustering methods to provide synopticsummaries of large mobile objects databases.Collection motion methods can also scale tolarge mobile databases. Another technique isspace–time association rule mining that discoversrules describing how objects move among a setof regions over time. Sequence mining techniquessearch for temporal patterns in mobile objectsdata. Periodic pattern mining search for recurrentpatterns in sequential data includes finding loca-tions that are repeatedly visited by mobile objectsand the recurrent movement patterns betweenthese locations. Under specific conditions, causalrelationships can also be inferred from sequentialpattern in movement data.Space–time knowledge delivery from movement

data requires methods for transparent manage-ment of the knowledge discovery process andlinking discovered knowledge to real-worldsemantics. A strategy for managing mobilitymining is through visual analytics. Visual analyticsis an extension of scientific visualization, butrather than seeking insights into data, visualanalytics seek insights into how data are pro-cessed during exploration and analysis. Visual

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analytics for moving objects include techniquesfor analyzing and comparing entire trajectories,the distribution of properties within trajecto-ries, synoptic visualization, and investigatingmovement within geographic context.

Connecting movement patterns to real-worldmovement semantics requires a conceptual sys-tem for describing low-level patterns and theirrelationships with higher-level behavior. Systemdimensions can include movement parameters(direct measures and derivatives; Table 1), thenumber of objects (one, group, or a cohort of simi-lar objects such as people aged 20–30 years), pathtype (semantically continuous or discontinuous),influencing factors (the object’s intrinsic properties,spatial constraints on movement, environmentalfactors, and the influence of other agents), andscale/granularity of the data (spatial and temporalscales, temporal granularity). Generic patternsare low level and shared with many differenttypes of mobile objects. These can be primi-tive (based on a single mobility parameter) orcompound (based on multiple primitives and/orinter-object relations). Behavioral patterns are highlevel and specific to the object type. Examplesinclude pursuit/evasion, fighting, flocking,and leader/follower. Unlike generic patterns,behavioral patterns are open-ended and canbe added to the framework as more empiricalmovement patterns are discovered in differentdomains (Dodge, Weibel and Lautenschütz2008).

SEE ALSO: Accessibility, in transportationplanning; Agent-based modeling; Behavioralgeography; Big data; Data model, movingobjects; Geographic data mining; Geographicinformation science; Geographic informationsystem; Geolocation services; GIS fortransportation; Graph theory; Information andcommunications technology; Informationtechnology and mobility; Network analysis;

Regional geography; Representation: time;Representation: trajectories; Transportgeography; Transport networks; Transporttechnology; Uncertainty

References

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Further reading

Andrienko, G., N. Andrienko, P. Bak, D. Keim, and S.Wrobel. 2013. Visual Analytics of Movement. Berlin:Springer.

Andrienko, N., G. Andienko, N. Pelekis, and S.Spaccapietra. 2008. “Basic Concepts of MovementData.” In Mobility, Data Mining and Privacy, editedby F. Giannotti and D. Pedreschi, 15–38. Berlin:Springer.

Bleisch, S., M. Duckham, A. Galton, P. Laube, andJ. Lyon. 2014. “Mining Candidate Casual Rela-tionships in Movement Patterns.” International Jour-nal of Geographical Information Science, 28: 363–382.

Boman, M., and E. Holm. 2004. “Multi-agent Sys-tems, Time Geography and Microsimulations.”In Systems Approaches and their Applications, editedby M.-O. Olsson. and G. Sjöstedt, 95–118.Dordrecht, The Netherlands: Kluwer Academic.

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Burns, L.D. 1979. Transportation, Temporal and SpatialComponents of Accessibility. Lexington, MA: Lexing-ton Books.

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Dodge, S., P. Laube, and R. Weibel. 2012. “Move-ment Similarity Assessment Using Symbolic Rep-resentation of Trajectories.” International Journal ofGeographic Information Science, 26: 1563–1588.

Downs, J.A., 2010. “Time-Geographic Density Esti-mation for Moving Point Objects.” In Geo-graphic Information Science, edited by S.I. Fabrikant,T. Reichenbacher, M. van Kreveld, and C.Schliederet, 16–26. Lecture Notes in ComputerScience 6292. Berlin: Springer.

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Farber, S., T. Neutens, H.J. Miller, and X. Li. 2013.“The Social Interaction Potential Of Metropoli-tan Regions: A Time-Geographic MeasurementApproach Using Joint Accessibility.” Annals of theAssociation of American Geographers, 103: 483–504.

Giannotti, F., and D. Pedreschi. 2008. “Mobility, DataMining and Privacy: A Vision of Convergence.”In Mobility, Data Mining and Privacy, edited by F.Giannotti and D. Pedreschi, 1–11. Berlin: Springer.

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Kuijpers, B., R. Grimson, and W. Othman. 2011.“An Analytic Solution to the Alibi Query in theSpace–Time Prisms Model for Moving ObjectData.” International Journal of Geographical Informa-tion Science, 25: 293–322.

Kuijpers, B., H.J. Miller, and W. Othman 2011.“Kinetic Space–Time Prisms.” In GIS ’11: Pro-ceedings of the 19th ACM SIGSPATIAL InternationalConference on Advances in Geographic InformationSystems, November 1–4, Chicago, IL, 162–170.New York: Association for Computing Machinery.

Kuijpers, B., H.J. Miller, T. Neutens, and W. Oth-man. 2010. “Anchor Uncertainty and Space–TimePrisms on Road Networks.” International Journal ofGeographical Information Science, 24: 1223–1248.

Kwan, M.-P. 1998. “Space–Time and Integral Mea-sures of Individual Accessibility: A ComparativeAnalysis Using a Point-Based Network.” Geograph-ical Analysis, 30: 191–216.

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