TIME GEOGRAPHY and SPACE–TIME PRISM Conceptualization in the 1960S

Home , Time geography, Transport geography

context. Basic time geographic concepts, such Time geography and as events being sparsely distributed in time and space–time prism space, limited time availability, and trading time for space to access activities, seem mundane, Harvey J. Miller since they are common and correspond with The Ohio State University, USA everyday experience. But this is why time geography is needed: these seemingly banal but utterly Time geography is a constraints-oriented crucial factors in our scientific explanations of approach to understanding human activities in human behavior should not be neglected. Time space and time. Time geography recognizes that geography provides a framework that demands humans have fundamental spatial and temporal recognition of the fundamental constraints limitations: people can physically only be in one underlying human experience and also provides place at a time and activities occur at a sparse set an effective conceptual system for keeping track of places for limited durations. Participating in an of these conditions. activity requires allocating scarce available time Time geography originates from Professor to access and conduct the activity. Constraints Torsten Hägerstrand (1916–2004), a Swedish on activity participation include the location and geographer who spent his career at the Univer- timing of anchors that compel presence (such as sity of Lund. He nurtured the ideas for a long home and work), the time budget for access and time, but time geography emerged dramatically activity, and the ability to trade time for space in to the international scientific community with using mobility or information and communication a now-famous 1969 presidential address to the technologies (ICTs). Time geography is a phys- Regional Science Association (Hägerstrand ical not a behavioral theory; it highlights the 1970). Hägerstrand was concerned that human necessary spatiotemporal conditions for human geography and regional science were neglect- activities, but does not explain the sufficient ing much of what comprises a livable world. events that lead to specific activities. But since He also wanted to provide a counterbalance these necessary conditions vary by individual and to increasing specialization and fragmentation situation, time geography supports an approach in science, technology, and administration by to understanding human and environmental offering a more holistic view of human activities. systems that recognizes individual histories and Hägerstrand believed in the integrative power of the importance of geographic context. the regional approach in geography, but felt that Time geography is consistent with some core it needed to be more inclusive and rich. Time ideas in fields such as geography, transportation, geography upgrades the regional approach by urban science, social sciences, and environmental describing the “bare skeleton” of spatiotemporal sciences. These include an integrated perspective conditions and constraints that emerge from the on human and physical phenomena, the need to interplay of historical and geographical factors. build macro-level explanations from micro-level Time geography is an active and flourishing processing, and situating human activities within research domain one half-century after its initial

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 TIME GEOGRAPHY AND SPACE–TIME PRISM conceptualization in the 1960s. This ranks time constraints are fiat restrictions over particular geography among elite and enduring scientific space–time domains. For example, a shopping ideas. Time geography has endured for good mall or gated community can make it difficult reasons. It is beautiful: the geometry of the and illegal to enter at designated times, while a space–time path and prism (discussed below) and public street cannot. their relationships are intuitive and appealing. It is elegant:itcanhelptoexplainmuchwithfew Space–time path basic principles. It is robust: it can help us under- stand a wide range of phenomena in human Activities such as personal and domestic mainte- and linked human-environmental systems. It nance, work, shopping, health care, education, is sensitive: it treats people as individuals and and recreation are sparsely distributed in time recognizes social differences across a wide range and space; they are available for limited duration of factors (gender, age, socioeconomic status, at relatively few locations. Participating in activi- culture) as well as geographic context. It is ecolog- ties requires trading time for space to access these ical: it connects the individual to the aggregate, locations at their available times. The space–time balancing nomothetic law seeking with context path highlights these requirements. Figure 1 illus- and situation, as well as balancing agency and trates a space–time path between activity stations. structure. Finally, it is practical: Hägerstrand was Stations are places where activities can occur; prescient when thinking about how to keep track classical time geography treats these as tubes des- of the basic spatial and temporal existential facts ignating their locations in space and availability of human activities. It is now possible to collect, in time (e.g., work hours, operating hours for store, manage, and analyze individual-level data a store, appointments, scheduled lectures). The on mobile objects and human activities with classical space–time path focuses on physical ease and power that would seem magical from mobility and interaction. Virtual interaction via the perspective of the 1960s. ICTs is possible; for example, a telephone call can be represented as a connection between two space–time paths. However, virtual interaction Classical time geography is muted relative to physical interaction in the classic theory. Time geography recognizes three major types of Time geography also classifies activities based constraints on human activities. Capability con- on their flexibility for an individual. Fixed activ- straints limit the activities of individuals through ities are those that cannot be easily rescheduled their own physical capabilities and/or available or relocated (e.g., work, meetings), while flexible resources. People need to conduct maintenance activities can be more easily rescheduled and/or activities such as eating and sleeping; these occur at more than one location (e.g., shopping, require time and place. Also, individuals with recreation). These categories can be arbitrary; private automobiles can generally travel faster for example, a software developer can code in than individuals who walk or rely on public an office or a cafe. Nevertheless, the dichotomy transportation. Coupling constraints define where, provides an effective means for understanding when, and for how long an individual has to how the location and timing of some activities join with other individuals for shared activities condition accessibility to other activities. Fixed such as work, meetings, and classes. Authority activities act as space–time anchors because other

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Time direction. “Speed” implies magnitude only and is, therefore, the more appropriate term.) Given these anchors and the speed limit, the prism Market defines the envelope all possible space–time paths between the anchors. The spatial footprint of the STP is the potential path area (PPA); this istheregioninspacethatisaccessibletothe moving object. The prism in Figure 2 is general since it Home Child’s accounts for stationary activity time and has Work school two anchors that are spatially separate. A prism without stationary activity time consists only of its time-forward cone rooted at the first anchor and its 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 be mobile. The two anchors may also be coincident

Space spatially; this prism consists of two right cones instead of oblique cones as in Figure 2. A prism Figure 1 A space–time path among activity may also have only one anchor. An STP with stations. its first anchor only is a time-forward cone delimiting all destinations that can be reached activities must occur at the temporal gaps from that origin within a specific time limit. between fixed activities. Conversely, a prism with its second anchor only is a time-backward cone delimiting all origins that can reach that destination within a specific Space–time prism time limit. The space–time prism (STP) highlights the influ- The STP measures accessibility: the ability for ence of space–time anchors on the ability to an individual to travel and participate in activi- participate in flexible activities. The STP is the ties and the amount of time available for activity envelope of all possible space–time paths between participation at locations. An activity at a sta- known locations and times. Figure 2 illustrates tion is not feasible unless that station intersects a planar STP. In this case, two space–time anchors with the prism spatially and temporally, the latter frame a prism. Anchors correspond to known for at least as long as the minimum activity time locations and times for the mobile object. These required. This delimits the subset of opportuni- are often (but not always) the locations and ties in an environment that is available to a person times of fixed activities that compel presence. based on their STP constraints. A maximum travel speed represents the object’s The STP provides a dramatically different mobility capabilities; in classic time geography image of accessibility than place-based accessi- this is a uniform across space and time. (Time bility measures, such as those based on spatial geography uses the term “velocity,” but this is interaction (“gravity”) models or simply count- incorrect since velocity implies magnitude and ing the number of opportunities near a person’s

3 TIME GEOGRAPHY AND SPACE–TIME PRISM home and work location. The locations and tim- capability constraints that are masked through ings of fixed activities vary by age/life cycle stage, homogenization by place. Time geography socioeconomic status and culture. For example, also facilitates understanding how the temporal STP-based accessibility measures capture gender organization of service and trading hours can differences in accessibility due to household have differential impacts beyond the locations of organization that are missed by place-based services and businesses. accessibility measures such as home–work com- mute length. Place-based accessibility measures Bundling and intersections assume everyone at a place, such as home and work, has the same space–time scheduling con- Time geography recognizes two types of rela- straints, while people-based measures derived tionships between paths and prisms. Bundling from the space–time prims recognize individual refers to the convergence in space and time of

Time

Maximum travel speed

Stationary activity time Time budget

Prism anchors Potential path area (PPA)

Space

Figure 2 A planar space–time prism.

4 TIME GEOGRAPHY AND SPACE–TIME PRISM two or more paths. Path bundling is necessary constraining an STP being uniform in space (although not sufficient) evidence of shared activ- and time. The development and deployment ities and individuals meshing their space–time of location-aware technologies (LATs), such as the activities to participate in projects. Bundling global positioning system (GPS), mobile phones, can occur when objects are in motion or sta- and radiofrequency identification (RFID) chips, tionary; examples of the former include public have greatly expanded capabilities for collecting transportation and ride-sharing. Intersection is data on mobile objects and have led to the devel- the condition of two or more time geographic opment of mobile objects databases and mobility features sharing some locations in space with mining or exploratory analysis of mobile objects respect to time. Two or more people cannot data. These scientific and technological develop- physically meet unless their STPs intersect, or a ments led to the development of analytical time path is within an STP. geography and supporting GIS software. Bundling and intersections are necessary conditions for the emergence of broader space–time Path analytics activity systems, such as a university or a city. Time geography is an ecological theory; it Location-aware technologies do not generate considers interactions between individuals and space–time paths directly; they generate a tem- between the individual and the aggregate. poral sequence of spatial locations that are used Bundling and intersections require individuals to construct the path. There are several ways to synchronize (coordinate over time) as well as to generate this temporal sequence. Event-based synchrorize (coordinate over space). Coordination recording captures the time and location when a also occurs at multiple scales. Individuals must specified event occurs; for example, a person tex- conduct projects consisting of sequenced activities ting or calling using a mobile phone. Time-based linked by mobility events to meet larger goals recording captures mobile object positions at such as hosting a dinner party. Cities and regions regular time intervals; this typifies GPS receivers. are time systems balancing the supply of available In change-based recording a capture occurs when time and demands on that time. This is a rich the position of the object is sufficiently dif- and intricate conceptualization of cities and ferent from a previous location; this includes societies; Alan Pred memorably referred to a dead-reckoning methods as well as some mobile “ballet of adjustments” as disruptions propagate objects data technologies that avoid recording through a time system due to activities and locations to manage data volume. Location-based projects changing to meet the new space–time recording occur when a mobile object comes close requirements for participation and interaction. to locations where sensors are located; examples include stationary radiofrequency identification and Bluetooth sensors. Analytical time geography The simplest and most common way to generate a path is linear interpolation: assume Classical time geography is conceptually rich but the object followed the straight-line segment limited analytically. The rise of geographic infor- between recorded locations. This works well for mation systems (GIS) motivated renewed interest time-based and change-based recording with in time geography, particularly in relaxing high capture frequencies. Event-based recording strict assumptions such as the maximum speed creates more issues since events are often not

5 TIME GEOGRAPHY AND SPACE–TIME PRISM very frequent, and the spatial resolution can be into a sequence of moves and stops, and anno- coarse and variable; this is often the case with tating these sequences based on map matching mobile phone data resolved only to membership with background geographic information. Also in a service cell. Even coarser are paths derived available are advanced data mining techniques from location-based recording; these are often such as machine learning algorithms. simply sequences of visits at the sensor locations. Table 1 summarizes the fundamental mea- Space–time paths can also be matched to other surements available from space–time paths. The spatially referenced data such as transportation primitive position allows the derivation of dis- networks. tance, direction, and spatial extent in the object’s The ease of collecting space–time path data movement; an example is spatial range meth- from location-aware technologies often comes ods. From these primary derivatives the spatial distribution of the object, changes in direc- attheexpenseofpath semantics: details about tion, and the path’s sinuosity can be derived. the moving objects such as the reasons for From the temporal primitives instant and interval mobility behavior. Semantics can be recovered the duration of the object’s movement and its by overlaying paths with other georeferenced speed, as well as the temporal distribution of the data. This method can produce errors related object’s movement and changes in duration, can to data inaccuracies and intrinsic ambiguities. be derived. Primary derivatives from combining For example, GPS positional error means that spatial and temporal intervals include the object’s it can be hard to tell if a person is inside or speed (a scalar) and velocity (a vector), and corre- outside of an activity location, for example, a sponding secondary derivatives acceleration and coffee house. If the person is inside the coffee an approaching rate with respect to a location, house, what was she doing – dining, working, boundary or region. socializing, or some combination of the above? Often, there is no unambiguous link between locations and activities, especially in an era Prism analytics with near-ubiquitous access to information and It is difficult to analytically describe the entire communication. Methods for recovering path STP. However, it is easy to describe its spatial semantics include decomposing the trajectory extent 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 Distance Spatial distribution Direction Change of direction Spatial extent Sinuosity Temporal Instance Duration Temporal distribution Interval Travel time Change of duration Spatiotemporal Speed Acceleration Velocity Approaching rate Dodge, Weibel, and Lautenschütz 2008.

6 TIME GEOGRAPHY AND SPACE–TIME PRISM the basis for a wide range of prism analytics. STP the intersection of two, three, or four of the parameters are {xi, xj, ti, tj, sij, aij} where xi, xj are sets based on the prisms’ morphologies at that the first and second anchor locations with associ- moment in time. The worst case is a four-set ated departure and arrival times ti, tj respectively, intersection involving two discs and two ellipses sij is the maximum travel speed, and aij is the (Miller 2005). stationary activity time. At a moment in time Over time, the future disc traces the time- , t ∈(ti tj), the spatial extent of an STP (denoted forward cone with an apex at the first anchor, the by Zij(t)) is the intersection of three convex spatial past disc traces the time-backward with an apex at sets: (i) the future disc fi(t) comprising all loca- the second anchor and the geo-ellipse is a cylin- tions that can be reached from the first anchor der. Over time, the prism in space is a lens-like by time ti + t; (ii) the past disc pj(t) encompassing set that traces the PPA footprint. These dynamics all locations at time t that can reach the second can be computed at discrete moments in time anchor by time t − t; and (iii) the geo-ellipse g j ij using the static construction described above and that constrains the prism locations to account for reconstruct the corresponding spatiotemporal any stationary activity time: region for the prism. Zij(t)={fi(t)∩pj (t)∩gij} (1) |‖ ‖ ≤ fi(t)={x} x − xi (t − ti)sij (2) Error and uncertainty in paths |‖ ‖ ≤ and prisms pj(t)={x} x − xj (tj − t)sij (3) |‖ ‖ ‖ ‖ gij ={x} x−xi + x−xj Error in space–time paths results from two ≤ sources. Measurement error occurs when the (tj −ti −aij)sij (4) recording of a mobile object’s location has noise Figure 3 provides an illustration. This defini- or uncertainty. The left-hand side of Figure 4 tion of the STP is not limited to 2-D space. In illustrates this for one segment of a space–time 1-D space, the sets described by equations (2)–(4) path; the captured locations have some degree are line segments. In 2-D space, the discs are of spatial error. This is equivalent to the prob- circles and the geo-ellipse is an ellipse. In 3-D lem of error in line segments and polylines, a space, the discs are spheres and the geo-ellipse well-established topic in the GIS literature. Sam- is a spheroid. There are scalable methods for calculating these objects and their intersections. pling error occurs when the captured locations Also, only one or two of the sets are relevant at undersample an object’s movement pattern. This any time, since the future and past disc change creates a spatial uncertainty region equivalent in size and may be enclosed by the other two to the STP over time. Therefore, a space–time objects. For example, with a general prism as in path with sampling error can be viewed as a Figure 2, it is only necessary to solve (in the fol- sequence of linked STPs. The right-hand side lowing order) a disc, a disc–ellipse intersection, of Figure 4 illustrates this; the black curve is the an ellipse, a disc–ellipse intersection, and finally actual path of the mobile object, the dashed red a disc. Similarly, finding path–prism intersections polyline indicates the interpolated movement, only requires testing if a point lies within a disc, and the blue ellipses indicate the spatial uncer- ellipse, or a disc–ellipse intersection. Finding tainty region surrounding each paired sample a prism–prism intersection requires solving for locations.

7 TIME GEOGRAPHY AND SPACE–TIME PRISM

pj(t)

zij(t)

fi(t)

gij

Figure 4 Measurement and sampling error in space–time paths. Figure 3 Analytical construction of a planar STP.

of the analytical prism, the relevant functions Combined measurement and sampling error in are the intersections of the boundaries of spatial a space–time path is equivalent to an STP with objects described by equations (2)–(4). These measurement error. Therefore, if it is possible to boundaries are: solve the measurement error problem for STPs it ‖ ‖ is also possible to solve the problem of combined fi ∶ x − xi −(t − ti)sij = 0(5) measurement and sampling errors in space–time ‖ ‖ paths. One strategy is Monte Carlo simulation, pj ∶ x − xj −(tj − t)sij = 0(6) which involves generating multiple prism real- ‖ ‖ ‖ ‖ izations based on sampling from the stochastic gij ∶ x−xi + x−xj error distributions. Monte Carlo simulation is −(tj −ti −aij)sij = 0(7) effective for theoretical investigations but it is cumbersome for applications: it is awkward to A problem is that equations (5)–(7) define these run prism error simulations when executing boundaries implicitly rather than explicitly (that STP-based mobile objects database queries or is, in the form f (x, y)=0 rather than f (x)=y), measuring accessibility in a model. meaning that solving for the intersection points Spatial error propagation theory provides requires finding the roots of high order poly- an analytical strategy for analyzing prism nomials. This difficulty can be resolved using error. Given a direct geographic measurement implicit function methods that allow calculation x =μx +εx, where μx is the true location vector of the required first order partial derivatives with- and εx is an error vector, it must be determined out having their explicit functions. This allows how that error propagates through a geographic the error propagation from STP parameters to operation y = f (x).Iff(x) is nonlinear, the error the constructed STP and to STP intersections to propagation is approximated using the first-order be estimated analytically. However, some of the partial derivatives of the function. In the case required circle and ellipse intersection cases are

8 TIME GEOGRAPHY AND SPACE–TIME PRISM still unsolved and the techniques are not tractable probabilities follow a bivariate multinomial beyond two STP intersections. Still required are distribution centered on the prism anchor. scalable approximations and heuristics for STP However, movement within an STP with two error based on these analytical techniques. The anchors is not completely random; the object analytical techniques can also serve as bounds needs to move from the first prism anchor to for improving the efficiency of simulation-based second anchor within the time budget. A directed techniques. random walk (DRW)isanRWprocesswithdirec- The methods discussed above estimate error tional biases. A DRW process constrained by an propagation from measured STP parameters to STP requires updating the RW direction biases the constructed object. Another strategy is to at each step to account for the remaining time construct objects that encompass the possible to travel to the second anchor given the speed STP consistent with the prism parameters and limit. This suggests a visit probability distribution their errors. A rough STP consists of the upper similar to a bivariate normal distribution that is and lower bounds on an STP.A reliable STP is the centered on the axis connecting the anchors and set of space–time locations where an individual moves along that axis with respect to time. can conduct an activity and meet the second A technique for modeling directed movement anchor constraints with a specified probability. in continuous time and space is Brownian bridges (BBs). A BB is a continuous stochastic process between two known values; it has been applied Properties of the prism interior widely to model animal movement between recorded locations. While BB can capture direc- tionally biased random movement between The STP is traditionally a binary concept; all prism anchors, is it not constrained by a maxi- locations within the prism interior are consid- mum speed and, therefore, does not capture the ered to be equally accessible. However, this is STP boundary; it is still possible for the object a simple characterization that masks intricate to travel outside the prism boundary. A truncated properties of the prism interior. Intuitively, it Brownian bridge (TBB) imposes a maximum would be expected that the distribution of visit speed on a BB process, fully representing STP probabilities would be unequal: locations that constraints. Figure 5 illustrates visit probabili- are near the space–time axis connecting the ties at a moment in time within the PPA of a prism anchors are more likely to be visited than planar STP based on a TBB process. Figure 5 locations near the prism boundary since there shows the two prism anchors with the spatial are more possible paths through the former. projection of the space–time axis connecting the One way to model visit probabilities within a anchors. The gray area is the PPA for the STP: planar STP is to use the theory of random walks these locations have zero probability of being (RWs). An RW is a stochastic process in discrete occupied at this moment in time. The blue, space and time; given a spatial lattice of discrete yellow, orange and red colors show increasing locations, an RW involves, at each time step, a probability that the locations may be occupied random choice of direction. Results from RW by the object at that moment in time. These theory confirm intuition that locations near a results are also consistent with the assumption prism anchor in a forward-time or backward- of a bivariate random distribution whose center time cone are more likely to be visited; visit moves with time along the axis connecting the

9 TIME GEOGRAPHY AND SPACE–TIME PRISM anchor points; however, rather than assumed it is derived from fundamental movement principles. Another strategy for estimating visit probabilities within planar STPs is to use spatial interpolation to infer unknown object locations from the known locations at the prism anchors. This assumes that the most likely locations are along the axis between the anchors, but as seen above this assumption is consistent with theory. Time-geographic density estimation (TDE) integrates kernel density estimation with the constraints imposed by an STP. This method can generate the fine-grained movement patterns of objects between anchors derived from sparse tracking data.

Other types of space–time prisms Figure 5 Visit probabilities within a planar space–time prism’s potential path area at a moment A planar STP has interesting properties and is in time. useful for applications where it can be assumed that objects move in constrained space with a Calculating an NTP involves three steps. uniform maximum speed. However, in some A precomputation step uses the planar PPA applications the assumptions of unconstrained as an upper bound on the NPA; this speeds space and uniform maximum speed are unre- computation by eliminating network nodes that alistic; examples include vehicles within a cannot possibly be in the NPA. It then solves transportation network or movement across for the shortest path tree in this subnetwork terrain. In addition, the STP assumes unrealistic twice: once for travel from the first anchor and motions such as infinite rates of acceleration and a second time for travel to the second anchor. deceleration. The procedure assigns the earliest arrival time A network time prism (NTP) is a prism subject and latest departure time at each node. The to network routes and allowable speeds that vary second step builds the spatial footprint of the by network arc and, possibly, time. Figure 6 NTP by testing if only one of an arc’s end nodes illustrates an NTP for a network embedded in is included in the NPA and (if so) solving for 2-D space with time along the third orthog- the NPA boundaries within the edge. The third onal dimension (Kuijpers and Othman 2009). step computes the full spatiotemporal region of The green subportion of the network arcs is the NTP based on the earliest arrival times and the potential network area (PNA), the network latest departure times at network vertices. These analog of the planar PPA; these are the accessible arrival and departure times dictate, for each node locations within the network. The red polygon incident to an arc, whether the mobile object comprises the full NTP; this is the envelope of can traverse the full arc to an adjacent node possible space–time paths between the anchors or can only traverse partway and return to the constrained by the network. original node. These cases correspond to the

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A field time prism (FTP) is a planar STP where travel speeds (magnitude) or velocities (magnitude and direction) vary continuously across space. Fields are useful for describing movement across terrain or through water and air that are subject to currents. Fields can extend the NTP by treating edges as 2-D regions with continuously varying speeds or velocities due to factors such as traffic. In these cases, space–time paths have unobserved components corresponding to minimum cost curves through an inverse speed or velocity field rather than straight line segments through a uniform plane. The FTP generalizes the prism concept: the STP and the long standing concept of isochrones Figure 6 A network time prism. (curves of equal travel time) are special cases of the FTP. It also links time geography to rectangular and triangular regions in Figure 6, the continuous transportation or urban fields respectively. The computational bottleneck tradition in quantitative geography and regional in the algorithm is the shortest path trees in science. the precomputation step; although this is only Space–time prisms as described above are phys- applied to a subportion of the network, this ically impossible; they assume that the mobile subnetwork can be large, as can the number of object can instantly accelerate and decelerate prisms to be processed. However, the Boolean at locations such as the prism anchors and the query about whether two NTPs intersect can be prism’s sharp corners. Physical limitations on solved analytically. acceleration and deceleration mean that an STP Similar to an STP, the parameters of an NTP is an overestimate of the true space–time region can also be measured with uncertainty. An anchor accessible to the object. While this may not mat- region is the set of all possible locations and ter for some STP applications, kinetics can make times for a prism anchor within a network arc. a difference for applications such as microscale These regions are not necessarily continuous movement (e.g., pedestrians, athletes), active in space or time and can have nonuniform transport modes such as bicycling, movement probabilities. There are two ways to calculate through media such as airplanes and ships, and an NTP with anchor regions. The first way is animal behavior. It can also make a difference to calculate the envelope of all NTPs having an for applications in sustainable transportation, as anchor point within a given anchor region. The vehicle fuel consumption and emissions are often second way is to calculate, for any space–time dominated by acceleration events. point, the probability that an NTP with given A kinetic time prism (KTP) is the set of all anchor regions contains that point. As with the possible kinetic paths between two locations standard NTP, the computational bottleneck and times, where a kinetic path is a space–time is the precomputation phase involving shortest path that obeys physical limits on acceleration path calculations. and deceleration. Figure 7 illustrates a KTP in

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t requires methods for calculating path and prism similarity. Path similarity is the degree of correspon- dence between two space–time paths. Geometric similarity measures focus only on the geometry, ignoring sequence and time; these include Euclidean and Hausdorff distances. Euclidean distances, such as the average, minimum and x maximum distance are intuitive and scalable; Figure 7 A kinetic (solid line) and classic (dotted however these measures are sensitive to noise line) space–time prism in 1-D space. and outliers. The Hausdorff distance is the maximum distance from a location in one path to the closest location in the other path. However, the Hausdorff distance can be misleading, as it does 1-D space and time, overlaid with a classic STP. not take into account the temporal sequence of Solving a KTP is scalable in 1-D space and time the path locations, only their geometry. but complex in 2-D space. In 2-D space, only Dynamic similarity measures include Fréchet one-quarter of the prism must be solved; the distance, dynamic time warping, longest com- remainder can be obtained through point and mon subsequences, and edit-distance functions. reflection symmetries. But calculating the first Unlike Hausdorff, the Fréchet distance takes quarter of the KTP requires solving parametric into account the sequence of locations in the functions describing the 1-D prism rotating path and is, therefore, better suited to mobile around the line connecting the prism anchors. objects. One way to think about the Fréchet This can only be accomplished if it is assumed distance is to imagine two space–time paths that the object’s initial heading is unknown; still representing a person walking a dog: the Fréchet open is the case where the object’s initial heading distance is the shortest leash that connects the is known. two. Dynamic time warping measures the similarity between two sequences or trajectories based on the effort involved to stretch or compress time Pathandprismscollections to get the sequences to match. Least common subsequence (LCSS) measures similarity based With the proliferation of location-aware tech- on the length of least common subsequence nologies generating mobile objects data, it is between two sequences. Edit-distance functions are often the case that there are large collections of a generalization of LCSS; these measure simi- space–time paths and prisms to analyze. With larities between sequential patterns based on the these collections, it is often useful to summarize cost of the insertion, deletion, and substitution the paths or prisms through clustering (finding operations required to transform one sequence groups of similar paths or prism) or aggregation into the other. (forming a composite representative paths or Other methods for analyzing collections of prisms). It may also be desired to search through space–time paths include path clustering meth- a collection of paths and prisms for other cases ods and spatial field methods. Path clustering that resemble a reference path or prism. This methods find groupings of similar space–time

12 TIME GEOGRAPHY AND SPACE–TIME PRISM paths in collections of mobile objects; these are static) and the evolution of shape (treating time often based on path similarity measures such as as dynamic). the ones already discussed. Spatial field methods A possible way to summarize NTP structure is translate movement patterns of objects into fields through network indices based on graph theory. or surfaces that summarize mobility and activity Network indices summarize network properties frequency by geographic location, allowing the such as connectivity, the distribution of arc identification of “hot spots” or locations with lengths/costs, and properties of shortest paths high mobility activity. and tours within the network. Graph theory and Similar to space–time paths, it may desired to network analysis is a very well established field analyze a collection of space–time prisms for that has been newly reinvigorated due to the rise patterns. This may involve clustering or aggre- of network science to analyze phenomena such gating prisms based on morphology, finding as social network and cities. A research frontier prisms similar to reference prism, and sum- is determining which existing network indices marizing and aggregating prisms to discover describe relevant properties of NTPs, and the synoptic patterns. However, to date there has development of new network indices tailored been little research on these questions; it is a for NTPs. much less developed research topic than path comparison and summarization. A fundamental Joint accessibility and time ecology problem is to develop efficient numeric measures that summarize meaningful prism properties. As mentioned, time geography recognizes It is easy to calculate basic properties such as bundling and intersections among paths and prism volume and PPA size for classic prisms, prisms as indicators of shared activities. The next as these have an elegant geometry consisting of step beyond calculating bundles and intersections cones and cylinders. More generally, measures are the higher-level properties associated with are required that can capture a wide range of space–time coordination and dynamics. morphological properties. These properties may Collective motion methods focus on individual be straightforward for classic prisms, but more object movement patterns within the context of complex prisms such as NTPs, FTPs, and KTPs, a larger group of mobile objects. Distance-based or prisms with fuzzy or probabilistic boundaries measures search for collective patterns, such as have more complex shapes and, consequently, flocks, by searching for moving objects that are are more difficult to describe analytically. densely connected in space given user-defined One possible strategy for measuring planar distance and time thresholds. Relative motion space–time prism morphology is shape analysis methods consider the individuals’ directions or quantitative measures summarizing geomet- to detect collective patterns such as flock- ric form. Shape analytical techniques typically ing, leadership and convergence in a group of reduce geometric forms to a single numeric indi- moving objects. Joint accessibility measures use cator or a small set of indicators; for example, prism–prism intersections as a basis for analyzing roughness, perforation, and elongation. How- the potential benefits to coordinated activity ever, most shape measures are specific to 2-D participation. space: required are scalable shape measures that Static measures of collective motion and joint can handle 3- or 4-D space (treating time as accessibility do not directly address the ultimate

13 TIME GEOGRAPHY AND SPACE–TIME PRISM goal of understanding the dynamic processes Synchronous presence (SP) includes face-to-face though which shared and interlinked activities conversations; this requires people involved to be form and intertwine in space and time. This physically present or proximal at the same time. highlights a weakness of time geography: its Asynchronous presence (AP) requires co-location inability to specify an explanation of human in space but not coincidence in time; examples activity that goes beyond the implications of include notes left on an office door or ina space–time constraints limiting possible activi- geocache. Synchronous telepresence (ST) requires ties. Time geography can explain why infeasible coincidence in time but not in space; it includes activities did not occur, but it cannot explain media such as telephones, television, radio, and why some feasible activities occurred while teleconferencing. Asynchronous telepresence (AT) others did not. However, it would be difficult to does not require co-location in space or coinci- build a dynamic, processes-based time ecology dence in time; it includes mail, e-mail, messages, theory based only on the negative space implied and webpages. Classical time geography focuses by prisms and stations. A more promising way on SP, recognizes but does not develop ST, and forward is to link time geography with modeling mostly ignores AP and AT. approaches such as activity-based travel demand There are at least two strategies for incorpo- models and agent-based models that naturally rating the wider range of communication modes capture the emergent properties associated with into time geography. One strategy is to treat these individuals’ interactions over space and time. as spatial and temporal relationships between The natural fit between activity-based analysis space–time paths and prisms. SP requires the and time geography is well recognized and well prism to intersect both spatially and temporally, utilized in many models. However, the linkages while AP interactions only require spatial coinci- between time geography and agent-based mod- dence between the prisms (in other words, they eling are less developed, somewhat surprisingly share the same locations but at different times). given their complementarity. ST requires temporal but not spatial coincidence

Time geography and virtual interaction Table 2 Communication modes based on spatial and temporal constraints. Temporal Spatial Classical time geography recognizes the pos- sibility of virtual interaction; for example, Presence Telepresence Hägerstrand (1970) shows a telephone call Synchronous SP ST between two space–time paths. However, virtual Face-to-face Telephone interaction is neglected relative to travel and conversation Television physical interaction in classical time geography. Radio This is understandable given the era when time Teleconferencing geography was initially formulated. But in the Asynchronous AP AT contemporary world virtual interaction is more Trail signs Mail pervasive and cannot be ignored when discussing Note left on E-mail Newspapers physical mobility and activities. office door Geocaches Webpages Table 2 is a typology of possible communication modes based on space–time constraints. Janelle 2004.

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(the prisms share some interval in time but not knowledge discovery; and (iii) space–time locations in space), while AT only requires one knowledge delivery. prism to precede the other in time. Trajectory reconstruction and management requires Another strategy for extending time geog- processing the raw mobility data to obtain the raphy is to make communication technology space–time paths of the mobile objects as well as explicit using time geographic entities and derive data structures and access methods for processing space–time constraints based on those entities. A these paths efficiently. These include specialized portal is a type of space–time station (Figure 1) data warehouse designs, mobile object indexing where a person can access communication ser- methods, and semantic trajectory compression vices. It includes a point location, and an access methods for reducing storage requirements. range and time intervals when the service is Preserving locational privacy through security, available. Portals correspond to real world enti- anonymization, and other protocols is also a ties such as wired Internet connections (a point concern. location with zero range), wireless access points, Space–time knowledge discovery involves process- and cellular telephone base stations (both charac- ing the trajectories to discover patterns such as terized as point locations with different ranges). clusters and behavioral rules. As noted, path and An individual can access a communication ser- prism similarity methods can be used in conjunc- vice only if his or her paths or prism intersect tion with clustering methods to provide synoptic with the service footprint of an appropriate summaries of large mobile objects databases. portal. When this occurs, it generates a message Collection motion methods can also scale to window or an interval of time when a message large mobile databases. Another technique is could be sent or received. Message windows space–time association rule mining that discovers can be compareed using temporal predicates rules describing how objects move among a set such as before, during,andafter to determine of regions over time. Sequence mining techniques which communication events are feasible. This search for temporal patterns in mobile objects approach is consistent with the perspective that data. Periodic pattern mining search for recurrent temporal constraints dominate spatial constraints patterns in sequential data includes finding loca- on telepresence. tions that are repeatedly visited by mobile objects and the recurrent movement patterns between these locations. Under specific conditions, causal Discovering time geographic knowledge relationships can also be inferred from sequential pattern in movement data. As the size of mobility datasets become richer Space–time knowledge delivery from movement and voluminous, the challenges move beyond data requires methods for transparent manage- developing scalable techniques for comput- ment of the knowledge discovery process and ing low-level time geographic properties and linking discovered knowledge to real-world relationships to discovering higher-level time semantics. A strategy for managing mobility geographic knowledge. Mobility mining is a set of mining is through visual analytics. Visual analytics activities and techniques for discovering novel is an extension of scientific visualization, but knowledge hidden in moving objects data. This rather than seeking insights into data, visual involves three major activities: (i) trajectory analytics seek insights into how data are pro- reconstruction and management; (ii) space–time cessed during exploration and analysis. Visual

15 TIME GEOGRAPHY AND SPACE–TIME PRISM analytics for moving objects include techniques Regional geography; Representation: time; for analyzing and comparing entire trajectories, Representation: trajectories; Transport the distribution of properties within trajecto- geography; Transport networks; Transport ries, synoptic visualization, and investigating technology; Uncertainty movement within geographic context. Connecting movement patterns to real-world References movement semantics requires a conceptual system for describing low-level patterns and their Dodge, S., R. Weibel, and A.-K. Lautenschütz. 2008. relationships with higher-level behavior. System “Towards a Taxonomy of Movement Patterns.” dimensions can include movement parameters Information Visualization, 7: 240–252. (direct measures and derivatives; Table 1), the Hägerstrand, T. 1970. “What About People in number of objects (one, group, or a cohort of simi- Regional Science?” Papers of the Regional Science lar objects such as people aged 20–30 years), path Association, 24: 1–12. type (semantically continuous or discontinuous), Janelle, D.G. 2004. “Impact of Information Technolo- influencing factors (the object’s intrinsic properties, gies.” In The Geography of Urban Transportation,3rd spatial constraints on movement, environmental edn, edited by S. Hanson and G. Giuliano, 86–112. factors, and the influence of other agents), and New York: Guilford. scale/granularity of the data (spatial and temporal Kuijpers, B., and W. Othman. 2009. “Modeling Uncertainty of Moving Objects on Road Net- scales, temporal granularity). Generic patterns works via Space–Time Prisms.” International Journal are low level and shared with many different of Geographical Information Science, 23: 1095–1117. types of mobile objects. These can be primi- Miller, H.J. 2005. “A Measurement Theory for Time tive (based on a single mobility parameter) or Geography.” Geographical Analysis, 37: 17–45. compound (based on multiple primitives and/or inter-object relations). Behavioral patterns are high level and specific to the object type. Examples include pursuit/evasion, fighting, flocking, Further reading and leader/follower. Unlike generic patterns, behavioral patterns are open-ended and can Andrienko, G., N. Andrienko, P.Bak, D. Keim, and S. be added to the framework as more empirical Wrobel. 2013. Visual Analytics of Movement. Berlin: Springer. movement patterns are discovered in different Andrienko, N., G. Andienko, N. Pelekis, and S. domains (Dodge, Weibel and Lautenschütz Spaccapietra. 2008. “Basic Concepts of Movement 2008). Data.” In Mobility, Data Mining and Privacy,edited by F. Giannotti and D. Pedreschi, 15–38. Berlin: SEE ALSO: Accessibility, in transportation Springer. planning; Agent-based modeling; Behavioral Bleisch, S., M. Duckham, A. Galton, P. Laube, and geography; Big data; Data model, moving J. Lyon. 2014. “Mining Candidate Casual Rela- tionships in Movement Patterns.” International Jour- objects; Geographic data mining; Geographic nal of Geographical Information Science, 28: 363–382. information science; Geographic information Boman, M., and E. Holm. 2004. “Multi-agent Sys- system; Geolocation services; GIS for tems, Time Geography and Microsimulations.” transportation; Graph theory; Information and In Systems Approaches and their Applications,edited communications technology; Information by M.-O. Olsson. and G. Sjöstedt, 95–118. technology and mobility; Network analysis; Dordrecht, The Netherlands: Kluwer Academic.

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