A Spatiotemporal Uncertainty Model of Degree 1.5 for
Continuously Changing Data Objects
Byunggu Yu
University of Wyoming Department of Computer ScienceLaramie, WY 82071-3315, USA +1-307-766-2440
[email protected]
ABSTRACT
To support emerging database applications that deal with
continuously changing (or moving) data objects (CCDO), one
requires an efficient data management system that can store, update, and retrieve large sets of CCDOs. Although actual CCDOs can continuously change, computer systems cannot deal with continuously occurring infinitesimal changes. Thus, in the data management system, each object’s spatiotemporal values are always associated with a certain degree of uncertainty at every point in time, and the queries are mostly processed over estimates characterizing the uncertainty. Unfortunately, there is a marked lack of formal explication of the uncertainty of multidimensional CCDOs in space-time. This paper presents our logical and mathematical bases for capturing, representing, and processing CCDOs of varying dimensionality.
Categories and Subject Descriptors
E.2 [Data]: Data Storage Representations – object representations; H.2.1 [Database Management]: Logical Design
– data model; H.2.8 [Database Management]: Database Applications – spatial databases and GIS; G.1.2 [Numerical
Analysis]: Approximation – nonlinear approximation, spline and piecewise polynomial approximation
General Terms
Management, Measurement, Design
Keywords
trajectory, spatiotemporal databases, continuously changing data, spatiotemporal model, spatiotemporal uncertainty
1.
INTRODUCTION
An increasing number of emerging applications deal with a large number of continuously changing (or moving) data objects (CCDOs), such as vehicles, humans, animals, sensors, mobile
computers, nano-robots, orbital objects, economic indicators, temporal geographic objects, RFIDs, and bank portfolios (or assets). For example, in earth science applications, temperature, wind speed and direction, radio or microwave image, and various other measures (e.g., CO2) associated with a certain geographic region can change continuously. Several models of watches, handheld devices, and monitoring devices equipped with a GPS or RFID are already available to consumers. Accordingly, new services and applications dealing with large sets of CCDOs are appearing. In the future, more complex and larger applications that deal with higher dimensional CCDOs (e.g., a moving sensor platform capturing multiple stimuli) will become commonplace – increasingly complex sensor devices will continue proliferate alongside potential applications associated therewith. Efficient support for these CCDO applications will offer significant benefit in many broader challenging areas including mobile databases, satellite image analysis, sensor networks, homeland security, internet security, environmental control, and disease surveillance. To support large-scale CCDO applications, one requires a data management system that can store, update, and retrieve CCDOs. Each CCDO has both multidimensional-temporal (i.e.., 2 or 3D geographic space or other information dimensions that vary with time) properties representing its continuous trajectory in an information space-time continuum as well as non-temporal properties such as identification, phone number, and address (step functions of time).
Importantly, although CCDOs can continuously move or change (thus drawing continuous trajectories in a space-time), computer systems cannot deal with continuously occurring infinitesimal changes – this would effectively require infinite computational speed and sensor resolution. Thus, each object’s continuously changing attribute values (e.g., location, velocity, and acceleration) can only be discretely updated. Hence, they are always associated with a degree of uncertainty at every point in time.
Unfortunately, the related techniques for storing, updating, and retrieving CCDOs have been developed on an ad hoc basis, and there is a marked lack of formal explication of the concepts of the uncertainty of (low to high) multidimensional CCDOs. Moreover, most related uncertainty models are not spatiotemporal but spatial, aggravating the query performance by not differentiating the uncertainty regions along the time dimension (degraded selectivity for queries). Developing a CCDO application system by simply adopting the known techniques may result in various anomalies,
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.
SAC’06, April, 23-27, 2006, Dijon, France.
data redundancy, data inconsistency, and inefficient query systems. This paper presents our framework for capturing and managing the uncertainty of low- to high-dimensional CCDOs using geographic spatiotemporal objects as the exemplar. Section 2 summarizes related techniques and models. Section 3 characterizes CCDOs. Sections 4 and 5 present and verify a novel “spatiotemporal” uncertainty model called the Funnel Uncertainty
Model (FUM) capable of efficiently supporting CCDOs of varying
dimensionality. Sections 6 and 7 conclude the paper.
2.
RELATED WORK
Several application-specific models of uncertainty have been proposed. One popular uncertainty model is that, at any point in time, the location of the object is within a certain distance d, of its last reported location. If the object moves further than d, it reports its new location and possibly changes the distance threshold d for future updates [12]. Given a point in time, the uncertainty is a circle with radius d, bounding all possible locations of the object. Another model assumes that the object always moves along straight lines (linear routes). The location of the object at any point in time is within a certain interval, centered at its last reported location, along the line of movement [12]. Different CCDO trajectory models that have no uncertainty consideration are found in the literature [7]. These models make sure that the exact velocity is always known by requiring updates at the data management system whenever the object’s speed or direction changes. Other models assume that the object travels with known velocity along a straight line, but can deviate from this path by a certain distance [9, 11].
An important study on the issues of uncertainty in the recorded past trajectories (history) of CCDOs is found in [6]. Assuming the maximum velocity of an object is known, they prove that all possible locations of the object during the time interval between two consecutive observations (states) lie on an error ellipse. A complete trajectory of any object is obtained by using linear interpolation between two adjacent samples. That is, a trajectory is approximated by a sequence of connected straight lines each of which connects two consecutively reported CCDO observations. By using the error ellipse, the authors demonstrate how to process uncertainty range queries for trajectories. Unfortunately, the error ellipse defined and proved in [6] is, in fact, the projection of a three-dimensional spatiotemporal uncertainty region onto the two-dimensional data space and is thus inefficient for spatiotemporal range queries: a 3-dimensional spatiotemporal query window whose extent is 0.1 along every dimension occupies 0.13 in the original space-time but 0.12 (a much larger portion) in the projected space. The first spatiotemporal uncertainty model explicitly generalized to high-dimensional CCDOs appears in [14]. However, this model strictly assumes that each sensed (reported) CCDO location is exactly correct (no support for instrument or measurement errors) and does not take into account higher order derivatives, such as velocity and acceleration. Our approach, a mechanism that explicitly leverages an understanding and characterization of uncertainty for a generalized case of the CCDO, offers an alterative construct that are suitable for higher dimensional data and that can produce minimally bounding, spatiotemporal uncertainty regions for both past and future trajectories of CCDOs by taking into account the temporally-varying higher order derivatives, such as velocity and
acceleration. We call this uncertainty model the funnel uncertainty model (FUM); because, for each reported CCDO state, the model produces a funnel-shaped uncertainty solid in the space-time.
3.
THE CCDO CONCEPT
Before presenting our uncertainty model, we explicate “CCDO” through a series of ontological abstractions. This represents a conceptual CCDO model. The formalization of the CCDO concept begins with the definition of the ontological terms [5] of a generic CCDO: As explicated in Table 1, a CCDO trajectory is a sequence of connected segments in space-time, and each segment has two endpoints that are consecutively reported (factual) states of the CCDO.
Table 1. A basic ontological abstraction of CCDO
Abstraction Definition
CCDO A CCDO is a data object consisting of one or more
trajectories and zero, one, or more non-temporal
properties.
trajectory A trajectory consists of dynamics and f:time snapshot, where time is a past, current, or future
point in time.
snapshot A snapshot is a probability distribution that represents the probability of every possible state at a specific point in time.
state A state is a point in a multidimensional information space-time of which time is one dimension. Optional properties include velocity, acceleration, and even higher order derivatives.
dynamics The dynamics of a trajectory is the domain of the properties of all states of the trajectory.
Examination of Table 1, one may observe the following: only a subset of states, called reported states, are given to the database system. This is due to the fact that a database cannot be continuously updated. The reported states are the factual known states of the CCDOs, and these known factual states can be committed to the database. All in-between states and future states of the CCDOs are then interpolated and extrapolated on the fly [13]. Because these “estimated” states are not exact, each of them is associated with a degree of uncertainty (a snapshot). Theoretically, an infinite number of possible states may exist between two reported states.
Given the theoretic possibility of an infinite number of states between two factual states, a mathematical model and computational approach is required to efficiently manage the “in-between” and “future” states.
4.
THE FUM
4.1
The 1
stDegree FUM
Consider an object moving through one dimension of space over time. Figure 1 shows an example of a trajectory segment connecting two known (reported) states of a CCDO. Let Mv be the
maximum rate of change (i.e., the norm of the maximum possible velocity) of the CCDO, A be the reported state (value) of this CCDO at ti, and B be the state at time tj. Then all possible states of
the CCDO between ti and tj are bounded by the lines where | cot
θ
| = Mv. The shaded region covers all possible locations (i.e., more
generically, states) of the CCDO between ti and tj. We call this
region the “uncertainty region” of the trajectory segment <A B>. The snapshot of the CCDO at tk that is between ti and tj is the
cross section of this uncertainty region, produced by the cutting line time = tk. Note that the uncertainty region in Figure 1 is the
overlapping region of two isosceles trapezoids – for each trapezoid, the shorter one of the two parallel sides represent the error of the corresponding reported state. This type of error (also known as the instrument and measurement error) exists due to various reasons including limited sensor resolution, GPS error, and measurement normalization. The other side, which is the longer one of the two parallel sides, has a length of 2×Mv×(tj-ti)+e,
where e is the length of the shorter parallel side.
Figure 1. An example of a 2-dimensional spatiotemporal uncertainty region representing a trajectory segment
Similarly, when a CCDO continuously changes in a two-dimensional information space (e.g., geographic space or X-Y plane), the uncertainty region of each trajectory segment connecting three-dimensional spatiotemporal points A and B is represented by the overlapping region of the two funnels (see Figure 2). Analogous to the isosceles trapezoid in a single dimension information space, each funnel is a right circular funnel with height = tj-ti, and the diameter of the base of each funnel is
2×Mv×(tj-ti)+e. Note that, when Mv is not known, the uncertainty
region is bounded only by the boundaries of the space1.
Figure 2. Overlapping uncertainty regions in a 3-dimensional space-time
It is important to note that, as shown in Figure 2, the projection of the 3-dimensional uncertainty regions onto the 2-dimensional data
1
For indexing, data values are normalized to a certain range by an order-preserving domain transformation.
space is, in fact, the uncertainty ellipse that can be defined by the ellipse model [6] with a simple modification taking into account the instrument and measurement errors. However, the FUM and the ellipse model are inherently different in that, while the ellipse model present uncertainty regions in space, the FUM produces uncertainty regions in the space-time. This difference makes the FUM more selective in processing spatiotemporal selection queries with reference objects that have a limited range along the time dimension. In keeping with the intended generic nature of the CCDO, Theorem 1 offers a logical basis for extending the FUM to higher dimensional data.
Theorem 1. The FUM of degree 1: Given the norm of the
maximum possible velocity, Mv, and two d-dimensional
location-time points (reported states) P1 and P2, where P1.time < P2.time,
the uncertainty region between P1.time and P2.time is the
overlapping region of Funnel1 and Funnel2, where Funnel1 (resp., Funnel2) is a hyper-funnel defined as follows: (1) the top
(directrix) is a hyper-circle, centred at P1 (resp., P2) and
perpendicular to the time dimension, with a diameter of e, where ±(e/2≥0) is the instrument and measurement error; (2) the base is a (d-1)-dimensional hyper-circle that is perpendicular to the time dimension at P2.time (resp., P1.time); (3) the funnel is right
circular; (4) any lateral surface line that meets both a top (resp., base) boundary point L1 and a base (resp., top) boundary point L2
at right angles (i.e., the generatrix) is a linear parametric function
P(u)=a0+a1u, where the coefficients can be derived by solving the
following constraints for a0 and a1: P(u=0) = L1; P(u=P2
.time-P1.time) = L2. Proof:
=> Suppose that there is a possible state P that is in the exterior of
Funnel1. Then to travel from P1 to P, the required velocity must
exceed Mv. However Mv is given as the norm of the maximum
possible velocity. By contradiction, P cannot be exterior to
Funnel1.
<= Suppose that there is a possible state P that is in the exterior of
Funnel2. Then to travel from P to P2, the required velocity must
exceed Mv. However Mv is given as the norm of the maximum
possible velocity. By contradiction, P cannot be exterior to
Funnel2 □
The uncertainty region and snapshots of the future trajectory of a CCDO can be computed in a similar manner (e.g., by extending the uncertainty boundaries of B in Figure 1 in the opposite (future) direction along the time dimension, the patterned trapezoidal region in Figure 1). The bolded line in Figure 1 represents all possible locations of the CCDO at the current point
c in time.
4.2
The FUM of Degree 1.5
The FUM presented in Section 4.1 assumes that the acceleration is infinite. That is, any CCDO can instantly reach its maximum velocities regardless of the current velocity. Thus, the lateral surface of each spatiotemporal uncertainty region is generated by a 1st degree parametric function (1st degree generatrix). This is the reason why we call the model the FUM of degree 1.
Considering real mobile objects and many types of other CCDOs, the maximum acceleration and deceleration are limited. This section presents an improved FUM that can produce smaller
ti tj time Data Space A B θ θ θ θ ti tj space time A B θ θ c
uncertainty regions, given the following additional information: (1) each reported state includes not only a location-time but the corresponding velocity as well; (2) the maximum acceleration and deceleration are known. The rest of this section will show an improved version of the FUM called the FUM of degree 1.5 that can reduce the size of the spatiotemporal uncertainty regions: the uncertainty regions are more reduced as the maximum acceleration, maximum deceleration and the reported velocities become smaller.
When each reported state includes a velocity vector v, we can find both the direction and speed. Considering both pieces of information in defining the uncertainty regions results in a highly complicated and computationally expensive model (further discussed in Section 6). In this paper, we present the FUM of degree 1.5, which produces symmetric uncertainty regions as in the 1st degree FUM. The FUM of degree 1.5 takes only the speed component of v (i.e., norm of v). This is the reason why we call this model the FUM of degree 1.5, not 2.
When the norm of the maximum acceleration, Ma, is defined, the
trajectory of a CCDO draw a non-linear, 2nd degree curve until it
reaches the maximum velocity Mv. Figure 3 shows the difference
between the 1st degree FUM with an infinite acceleration and the FUM of degree 1.5 with a reported velocity and Ma defined. In the
figure, the shaded region represents the uncertainty region of the FUM of degree 1.5 and the larger region of the trapezoid represents the uncertainty region of the 1st degree FUM. This
difference exists because it takes the object tacc > 0 to reach the
maximum velocity. Because of the constant maximum acceleration, the velocity (the first derivative of the trajectory) can change only linearly until it reaches the maximum. Then the velocity is fixed at the maximum. That is, the boundaries are
line-curves: In Figure 3, f1 and f2 are a second degree function and a
first degree function, respectively.
Figure 3. The difference between the 1st degree FUM and the FUM of degree 1.5
Therefore, the base of the shaded region at tj has an extent of
, ) ( ) ) ( ) ( ( 2 min vt t t M dt t tmin M e v j t ti i i a + + − ⋅ + − ⋅ ⋅
∫
where v(ti) is the reported velocity at time ti, Mv is the maximum
velocity constant, Ma is the maximum acceleration constant, e is
the report error, and tmin =min{ti+
(
Mv− v(ti))
Ma,tj}is the earliest point in time at which the velocity reaches the maximum (i.e.,v
(
t
min)
=
M
v) or tj (if it cannot reach themaximum velocity before tj). One can find that, unlike the FUM
of degree 1, this FUM’s uncertainty region size is affected not only by Mv but also by the reported velocity and Ma. Theorem 2
defines this improved model, the FUM of degree 1.5, generalized to multidimensional space-time.
Theorem 2. The FUM of degree 1.5: Given the norm Mv of the
maximum possible velocity, the norm Ma of the maximum
possible acceleration, the norm Md of the maximum possible
deceleration, and two pairs of d-dimensional location-time point and velocity (two reported states) <P1, P1’> and <P2, P2’>, where P1.time < P2.time, the uncertainty region between P1.time and P2.time is the overlapping region of Funnel1 and Funnel2, where Funnel1 (resp., Funnel2) is a hyperbolic hyper-funnel defined as
follows: (1) the top (directrix) is a (d-1)-dimensional hyper-circle, centred at P1 (resp., P2) and perpendicular to the time dimension,
with a diameter of e, where ±(e/2≥0) is the instrument and measurement error; (2) the base is a (d-1)-dimensional hyper-circle that is perpendicular to the time dimension at P2.time (resp., P1.time); (3) the centre of the top circle projected onto the base is the centre of the base; (4) any lateral surface line-curve that meets both a top (resp., base) boundary point L1 and a base (resp., top)
boundary point L2 at right angles (i.e., the generatrix) is
represented by the following parametric function:
Funnel1: ≤ ∆ + ∆ ≤ + + = u t u b b t u u a u a a u P acc acc if if ) ( 1 0 2 2 1 0 ,
where 0 ≤ u ≤ (L2.time - L1.time) and
(
( ))
, .time .time}min{M vt M L2 L1
tacc = v− i a −
∆ .
The coefficients are derived by solving the following constraints for a0, a1, and a2: P(u=0) = L1, P’(u=0) = P1’, and P(u= tacc) =
LM; and by solving the following constraints for b0 and b1: P(u= tacc) = LM' and P(u=L2.time-L1.time) = L2, where LM = LM’.
Funnel2: ≥ ∆ − − + ∆ − − ≥ + + = u t L L u b b t L L u u a u a a u P dec dec ) time . time . ( if ) time . time . ( if ) ( 1 2 1 0 1 2 2 2 1 0 ,
where 0≤ u ≤ (L2.time - L1.time) and
(
( ))
, .time .time}min{M v t M L2 L1
tdec = v− j d −
∆ .
The coefficients are derived by solving the following constraints for a0, a1, and a2: P(u= L2.time-L1.time) = L2, P’(u=L2 .time-L1.time) = P2’, and P(u=L2.time-L1.time- tdec) = LM; and by
solving the following constraints for b0 and b1: P(u= L2 .time-L1.time- tdec) = LM’ and P(u=0) = L1, where LM = LM’.
Our proof of Theorem 2 is similar to the proof of Theorem 1. Because of the limitation of paper length, we do not present the full proof in this paper.
ti tj tacc time space reaches Mv f1 f2
5.
EXPERIMENTS
To verify the proposed uncertainty models, we have conducted the following experiment. Using a portable GPS device (Trimgle Navigation’s ProXRS Receiver with GPS logger), which can record a pair <location-time, velocity> every second, we collected real GPS data. Every report was 4-dimensional (i.e., longitude, latitude, altitude, and time). We placed the GPS device in a car and drove from a location in the north of Denver, Colorado, to Loveland, Colorado, USA along the interstate highway 25 in such a way that the resulting trajectory includes both relatively straight movement and some winding movement. Every second, we logged a spatiotemporal data from the GPS device. For the comparison between the two versions of FUM, we created trajectories based on a subset of logged records: We randomly selected logged spatiotemporal records with a sampling ratio of about 5%.
In the first experiment, we removed the sampled altitude values (3-dimensional case). The maximum velocity Mv was set to
180km/hour (50meters/sec); the report (instrument and measurement) error was fixed at ±2 meters along every spatial dimension for every sample; for the FUM of degree 1.5, Both constants Ma and Md were set to (5/3 meters/sec) per sec (i.e., 0 –
50meters/sec takes 30 seconds with the maximum acceleration). Both FUMs of degree 1 and 1.5 were used to define the uncertainty regions of the sampled trajectory states. For illustration sake, all 3-dimensional spatiotemporal uncertainty regions were projected onto the longitude-latitude plane. Figure 4 shows these uncertainty regions. The results in Figure 4 show that the uncertainty regions produced by the FUM of degree 1.5 are smaller than their counterparts produced by the FUM of degree 1. In addition, Figure 5 confirms that this difference is increased when the object is moving slowly.
In the second experiment, we took all sampled state values, including the altitude values, to test the FUMs in a 4-dimensional space-time. All other parameters were set as in the first experiment. The results in Figure 6 conform with the results of the first experiment – the uncertainty regions produced by the FUM of degree 1.5 are smaller than their counterparts produced by the FUM of degree 1.
6.
FUTURE ISSUES
Based on the spatiotemporal uncertainty modelling framework introduced in this paper, one might be able to devise even higher degree uncertainty models. For example, the second degree FUM takes into account not only reported speeds but also reported directions, fully utilizing reported velocity vectors. The uncertainty region is not symmetric anymore and the computation overhead for defining individual uncertainty regions increases substantially. However, it is also expected that the higher degree models can substantially reduce the uncertainty regions for many applications, resulting in reduced false drops or alarms.
Processing a query with uncertainty means that each result data item is associated with the probability (or likelihood) that the item really satisfies the query predicate [4]. To support probabilistic query processing, one needs to calculate the probability density of each snapshot or uncertainty region: An appropriate application-specific distribution (e.g., skewed-normal random distribution) can be used to estimate the probability density [1,2,10]. [13]
provides a series of methods that can significantly reduce the errors associated with the peak point of the probability density.
Figure 4. The results of the first experiment: dark ellipses represent the projected spatiotemporal uncertainty regions produced by the FUM of degree 1; the lightly coloured ellipses represent the projected spatiotemporal uncertainty regions of the FUM of degree 1.5; a ‘+’ sign represents a trajectory sample; the curve passing through all the samples represents the actual trajectory
Figure 5. A magnified view of the northern part of Figure 4
Loveland, Colorado Greely, Colorado U-turn ending point North
(a)
(b)
Figure 6. The results of the second experiment: (a) the 4-dimensional spatiotemporal uncertainty regions that were produced by the FUM of degree 1 and that were projected into the 3-dimensional (longitude-latitude-altitude) data space; (b) the FUM of degree 1.5
If the uncertainty regions can be further minimized, the spatiotemporal regions requiring indexing can also be commensurately limited and the query results will be associated with more probable likelihoods. By taking into account for how the environment may be variably constraining movement and thus variably affecting the set of possible positions of the moving object, one can further reduce the uncertainty regions and contextualize (modify) the probability distributions [8].
7.
SUMMARY
In this paper, we proposed a novel and practical framework for managing multidimensional CCDOs (Spatiotemporal Data Representation and Processing). The proposed spatiotemporal uncertainty models can more efficiently support both conventional CCDOs that move in a 2- or 3-dimensional geographic space and emerging high-dimensional CCDOs, such as combined sensor streams and satellite data.
8.
REFERENCES
[1] Azzalini and Capitanio, A. Statistical applications of the multivariate skew-normal distribution. Journal of the Royal
Statistical Society, Series B(61), 1999, 579-602.
[2] Azzalini and Valle, A.D. The multivariate skew-normal distribution. Biometrika, 83, 1996, 715-726.
[3] Catmull, E. and Rom, R. A Class of Local Interpolation Splines. In: R.E. Barnhill and R.F. Riesenfled (eds.):
Computer Aided Geometric Design, Academic Press, New
York, 1974.
[4] Cheng, R., Kalashnikov, D., and Prabhakar, S. Evaluating Probabilistic Queries over Imprecise Data. IEEE
Transactions on Knowledge and Data Engineering, 16(9),
2004, 1112-1127.
[5] Guarino, N. Formal Ontology and Information Systems. In:
N. Guarino (eds.): FOIS'98, IOS Press, Trento, Italy, 1998,
3-17.
[6] Pfoser, D. and Jensen, C.S. Capturing the Uncertainty of Moving-Objects Representations. In Proc. SSDBM Int. Conf.
on Scientific and Statistical Database Management, 1999,
123-132.
[7] Pfoser, D. and Jensen, C.S. Querying the Trajectories of On-Line Mobile Objects. In Proc. ACM MobiDE International
Workshop on Data Engineering for Wireless and Mobile Access, 2001, 66-73.
[8] Prager, S.D. and Yu, B. Contextualized Probability for Approximation of Spatiotemporal Data Distributions. In
Proc. ISAS/CITSA International Conference on Information Systems Analysis and Synthesis, 2005, 318-322.
[9] Sistla, P.A., Wolfson, O., Chamberlain, S., and Dao S. Querying the Uncertain Position of Moving Objects.
Temporal Databases: Research and Practice, 1399, 1998.
[10]R Development Core Team, R: A language and environment
for statistical computing, R Foundation for Statistical
Computing, Vienna, Austria, 2004.
[11]Trajcevski, G., Wolfson, O., Zhang, F., and Chamberlain, S. The Geometry of Uncertainty in Moving Object Databases.
In Proc. Int’l Conf. on Extending Database Technology,
May 2002.
[12]Wolfson, O., Sistla, P.A., Chamberlain, S., and Yesha, Y. Updating and Querying Databases that Track Mobile Units.
Distributed and Parallel Databases, 7(3), 1999, 257-387.
[13]Yu, B., Kim, S.H., Bailey, T., and Gamboa, R. Curve-Based Representation of Moving Object Trajectories. IEEE IDEAS
International Database Engineering and Applications Symposium, 2004, 419-425.
[14]Yu, B., Prager, S.D., and Bailey, T., 2005. The Isosceles Triangle Uncertainty Model: A Spatiotemporal Uncertainty Model for Continuously Changing Data, In: Christopher Gold (eds.): ISPRS DMGIS Dynamic and Multi-Dimensional GIS, Vol. XXXVI, Part 2/W29. ISPRS
ICWG II/IV, 2005, 179-183.
BIOGRAPHY
Dr. Byunggu Yu, Ph.D. is currently an Assistant Professor of Computer Science at the University of Wyoming. He has published many refereed computer science research papers in the area of database and has served as a reviewer or program committee member for a number of journals and conferences. His major research interests include spatial databases and spatiotemporal databases.
