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Probabilistic Measurement of Uncertainty in Moving Objects Databases Abstract


Probabilistic Measurement of Uncertainty in Moving Objects Databases
Talel Abdessalem, Laurent Decreusefond
Ecole Nationale Sup? erieure des T? el? ecommunications D? epartement Informatique et R? eseaux, LTCI – UMR CNRS 5141 46, rue Barrault – 75013 Paris – France {Talel.Abdessalem, Laurent.Decreusefond}@enst.fr

Jos? e Moreira
Universidade de Aveiro – Departamento de Electr? onica e Telecomunica? c? oes, IEETA Campus Universit? ario de Santiago – 3810 - 193 Aveiro – Portugal jmoreira@det.ua.pt

July 11, 2005

Abstract
The representation of moving objects in spatial database systems has become an important research topic in recent years. As it is not realistic to track and store the location of objects at every time instant, one of the issues that has been raised in this domain has to do with handling uncertainty in the location of moving objects. There are several works proposing to enrich spatiotemporal languages with estimates about the validity of the answers to queries about the object’s movement. Although, research in developing statistical tools to compute those estimations is nearly non-existent. In this paper, we propose three statistical tools for computing probabilistic estimates about the location of a moving object at a certain time and show how to use them for evaluating probabilistic range queries. We also compare the three proposals in terms of ex-

pressivity and ability for dealing with the different semantics proposed in the spatiotemporal query languages literature. The focus is on applications dealing with the history about the spatiotemporal behavior of non-network constrained moving objects, for monitoring or data-mining purposes, for instance.

1

Introduction

The spatiotemporal databases research community is giving particular attention to moving objects applications. Real-time systems, using recent information about object’s movement for anticipation of events in near future, or historical systems, recording information about object’s movement during large periods of time for monitoring or data-mining purposes, are noteworthy examples. Both cases require functionality allowing answering questions of the kind where and when. Although, as it is not

possible to continuously monitor and record the location of moving objects, the knowledge that may be captured and stored by computer systems is only a partial representation of the actual spatiotemporal behavior of real-world objects. For that reason, there are numerous proposals for extending spatiotemporal query languages with adequate semantics for dealing with uncertainty in the location of moving objects. The goal is to incorporate functionalities allowing answering questions such as “Which were the moving objects that have been within a certain area during some period of time, with a probability of at least 80%”. Answering such kind of queries requires methods for computing probabilistic estimates about the location of moving objects at a given time. Although, research in this domain is almost non-existent. This paper deals with the computation of probabilistic estimates for the validity of answers to spatiotemporal queries. The focus is on systems representing the history about the movement of non-network constrained moving objects. We propose three statistical tools that allow obtaining realistic estimates for the location of a moving object at a certain time. The methods explore di?erent features and are applicable under di?erent conditions, according to the information that may be disclosed by real applications. We also explain how we can use these methods to implement di?erent semantics proposed in the literature for dealing with uncertainty in the location of moving objects. The remaining part of this paper is organized as follows. Section 2 presents an overview of current proposals for dealing with uncertainty in the location of non-network constrained moving objects. Section 3 depicts the two main approaches that we have delineated to solve the proposed problem and presents three methods for computing probabilistic estimates about the location of moving objects.

Section 4 presents the tools that have been developed to test the proposed methods and puts in evidence the domain of application covered by each one. Finally, section 5 concludes the paper.

2

Uncertainty in objects movement representation

The representation of the objects’ movement is inherently imprecise and therefore, answers to user queries based on such information may be inaccurate [8, 13]. Imprecision may be introduced by the measurement process – it may depend on the accuracy of the GPS, for instance –, or by the sampling approach, as depicted below. Notice that the imprecision refers to the spatial dimension only, as it is commonly assumed that measurement instruments are able to determine precisely the time a position sample is taken. As an example, consider the case of a port authority dealing with a spread of toxic waste in the sea and querying a nautical surveillance system to know which ships have crossed the polluted zone during a speci?ed time interval. Imagine that the ship responsible for the waste has actually followed the trajectory represented in Figure 1. The black dots represent four observations made during the speci?ed time interval, the shaded region represents the polluted area and the hatched line a trajectory that might have been inferred from the observations.

Figure 1: Uncertainty about moving objects trajectories

The hatched line does not cross the shaded region and, thus, an answer to a query based on this estimation of the trajectory would not include the guilty ship. On the contrary an answer may also include false candidates whose inferred trajectory crosses the area even though they have not actually been there.

2.2

Bounding uncertainty

There are physical constraints on the movement of objects allowing limiting uncertainty of their position at a certain time. For instance, the uncertainty zone for a train moving on a railway is a section of the railway and for a ship moving freely in the ocean is an area. When it comes to future movements, considering a two-dimensional space (Figure 2), the uncertainty area is a circle centered on 2.1 Uncertainty of past, present and the expected location of the object. The circle bounds the maximum deviation allowed for future positions an object at a given time instant. Objects are committed to send a location update when the The preceding example focuses on the history deviation reaches the bound. of the objects’ movement. In general, the focus may be put on the past movement or on the future movement of objects, depending on applications requirements. Two main approaches were raised. The ?rst approach [8, 6], focusing on past movements, addresses the needs of mining applications of spatiotemporal data: tra?c mining, environment monitoring, etc. In this case, uncertainty is bounded using a sequence of positions of objects and some known physical constraints on their movement. Figure 2: Expected location of a moving object In the second approach [10, 11, 19, 13], the focus is put on the uncertainty about the fuFor past movements, since the positions beture movement of objects. This approach ad- tween two consecutive samples are not meadresses the needs of real-time applications and sured, the best to do is to limit the poslocation-based services: real-time tra?c con- sibilities of where the moving object could trol, real-time mobile workforce management, have been [9, 8, 7]. Let us consider a twodigital battle?elds, etc. These systems make dimensional space and a moving object m, an use of speed patterns information in the con- instant t belonging to a time interval t1 , t2 struction of future movements and uncertainty and two consecutive observations (p1 , t1 ) and is ?xed in advance. The latter allows avoid- (p2 , t2 ), (Figure 3). We denote by p1 and p2 ing frequent updates of the database. In fact, the positions of the moving object at observathe database is not updated as long as the ac- tion time instants t1 and t2 , respectively. d detual object’s movement deviation from its ex- notes the distance between p1 and p2 . At time pected location, as inferred from the informa- t, the distance d1 between m and p1 is inferior tion stored in the database, is less than the to r1 = Vmax × ?t1 , where ?t1 = t ? t1 and threshold previously ?xed. Vmax is a user-de?ned value standing for the

maximum velocity of moving object m. The distance d2 between m and p2 , at instant t, is inferior to r2 = Vmax × ?t2 , where ?t2 = t2 ? t. So, at time t, the moving object might be at any location within the area de?ned by the intersection of the two circles of radius r1 and r2 . This is a so-called lens area [8] representing the set of all possible locations for a moving object at a certain time instant.

intersection of the two circles with the ellipsis. Lens areas for time intervals comprising one or more observations are the union of several lens areas computed using the method described above.

2.3

Using probabilistic methods

Consider now that we have methods that allow estimating the location of a moving object at any time instant. The evaluation of spatiotemporal query expressions could than be augmented with probabilistic estimates of the validity of answers to users queries. Thus, it would be possible to answer queries such as Figure 3: A lens area for a time instant “Which are the planes for which the probability of being inside Area C within 5 minutes is The set of all locations where a moving obat least 40%?”, or “Which were the ships that ject might have been between two consecutive were in a certain area during a given time inobservations corresponds to an ellipsis (Figterval, with a probably of at least 60%?”. ure 4). This means that the ellipsis covers all Figure 5 illustrates this kind of query [8]. possible lens areas between the two consecutive It considers the case of the anticipation of observations [7, 2]. the location of an object moving on a twodimensional space. If we assume that the distribution of probability in this lens area is uniform, then, the object is said to be within the given area with a probability of 30%, if at least 30% of its lens area is within that area. Figure 4: A lens area for a time interval between consecutive observations Figure 4 shows how to compute the lens area for a time interval ta , tb , between two consecutive observations. The circle with radius rb = Vmax × (tb ? t1 ) corresponds to the maximum distance from p1 that could be reached by the moving object at tb . The same reasoning applies to ra . In addition, the moving Figure 5: Probability of intersection between a object could not have been outside the ellipsis lens area and a query window just de?ned. So, the lens area is de?ned by the

2.4

Related works

In recent years, uncertainty handling emerged as an important issue in moving object database research. Several aspects were investigated and two complementary models were proposed: [8] focusing on past objects’ movement and [19, 13] dealing with future objects’ movement. Moreover, [18, 17] investigated the communication cost for updating the database in the case of real-time applications. [5] discusses how the uncertainty of network constrained moving objects can be reduced by using reasonable modeling methods and location update policies. Finally, [9] added fuzziness in object location and considered the case of moving objects that may change their geometry in time. An important issue of the current research activity in this domain is the design of a probabilistic model of uncertainty. The goal is to handle more realistic (non-uniform) distributions of probability on the location of moving objects, and measure the validity of the answers to user queries. Recent results [4, 3, 12] are going toward this goal, even if they just brie?y touch upon the possibility of a nonuniform distribution. Besides, the evaluation of user queries may require handling temporal types such as time instants, time intervals or a combination of them. Evidently, the probability of presence of a moving object within a given area at a certain time, should be less or equal than the probability of presence of the moving object within the same area during any time that includes the previous one. However, the methods presented in section 2.3 are suitable for dealing only with time instants and are not adequate to deal with time intervals. For instance, consider ?gure 6 where A is an area of interest, L is the lens area calculated for the location of an object at a time instant t (1st case) and L is a lens area for the loca-

tion of the same object during a time interval ta , tb ? t (2nd case).

(1st case) Lens area for t

(2nd case) Lens area for ta , tb ? t

Figure 6: Intersection of a lens area (L) with an area of interest (A) Considering that the distribution of probability in the lens area is uniform, then the probability of presence of the moving object within (A∩L ) A at time t is P (t) = area area(L ) and the probability of being within A during time ta , tb is (A∩L ) P ( ta , tb ) = area area(L ) . As A ∩ L is equal to A ∩ L and the area of L is less than the area of L then, against evidence, P (t) is greater than P ( ta , tb ).

3

Probabilistic reasoning

This section presents the statistical tools that we propose for the evaluation of probabilistic estimates about the location of moving objects. We will denote the probability of presence of a moving object within a given region during a certain time, simply as P (t). We only consider the past objects’ movement and we assume that it is represented as an ordered sequence of observations, denoted {(t, p)}, where p is a two-dimensional value denoting the location of the object at time instant t. We also consider that the objects move freely in space with no obstacles or networks constraining their movements. It is also important to notice that the movement of real-world objects is not random. Indeed, it is reasonable to consider that, most often, their movement is smooth, i.e., that the

movement between two locations is approximately linear and uniform. This means that the locations in the neighborhood of the position expected for an object1 , denoted p ?, should have a greater weight in the computation of the probabilities, than those locations that are far from p ?. As we are interested in obtaining realist estimates, it is desirable that the density functions for distribution of the probabilities within lens areas take this feature into account. Moreover, the location of a moving object at the time instants corresponding to the observations is precisely known, and thus, we must have P (t) = 1 for those time instants or any time interval containing them. Finally, we also assume that there are systems for which it is not reasonable to estimate in advance the maximum velocity of a moving object with an acceptable accuracy. We propose a speci?c method for those cases. We have investigated two main guidelines for the implementation of the proposed statistical tools, which were designated by pointbased and trajectory-based approaches. We have developed methods based on each of these approaches and we will put in evidence the strengths and weaknesses of each one.

P (t) =

WLensArea(t) ∩ Region WLensArea(t)

(1)

The main issue is the de?nition of a density function over a complex form, such as a lens area. Since we can easily de?ne a density function over a circle, we propose to perform an anamorphosis of the lens area into a circle. The method that we propose consists in four steps: ? First, we de?ne a local coordinates system for the lens area, to make the formulation of the lens area equation easier. ? Second, we de?ne the anamorphosis of the lens area into a circle of radius 1, to which the chosen density of probability will be associated. ? Third, we de?ne how to transform the density of probability over the circle of radius 1 into a density of probability over the lens area. ? Finally, we complete the process by the evaluation of P (t). 3.1.1 Lens area equations

3.1

Point-based approach

As referred in section 2.4, there are several authors suggesting using a density function to estimate the probabilities of presence of a moving object at each point inside the lens area. Then, assuming that t denotes a time instant, the values of P (t) may be calculated using the weight of the intersection of the lens area with the region considered, as shown in formula (1).
The position expected for a moving object at a certain time instant is estimated assuming that the movement between consecutive observations is linear and uniform.
1

The lens area is given by the intersection of two circles. If one circle contains the other, the result of the intersection is the smaller circle. This situation may arise for time instants in the neighborhood of the instants of observations. Otherwise, the result of the intersection is an area similar to the one depicted in ?gure 7. The lens area in ?gure 7 delimits the set of all possible locations for a moving object at time instant t, between two consecutive locations: p0 , observed at instant t0 , and p1 , observed at t1 . As presented in section 2, the left centered circle delimits the lens area of the object during the time interval [t0 , t]. The right centered circle delimits the lens area of this object during

Applying the Al-Kashi’s theorem (law of cosines) [1, 16], we obtain the following abscises for p0 and p1 in the local coordinates system:
x0 = ? d x1 =
2

+r0 2 ?r1 2 2d

(4)

d2 ?r0 2 +r1 2 2d

= d + x0

Figure 7: Lens area parameters the time interval [t, t1 ]. The radiuses of the circles depend on the maximum velocity (Vmax ) previously estimated:
r0 = Vmax × (t ? t0 ) (2) r1 = Vmax × (t1 ? t)

In the general case, the lens area is de?ned by two arcs located in the half-plans x > 0 and x < 0. Otherwise, it is a circle centered on p0 or p1 . So, we use the following explicit equations to represent the lens area:

fL (y ) =

x1 ? x0 ? x0 + x1 +

r1 2 ? y 2 r0 ?
2

, in the general case , if x1 ? r1 2 ? y 2 > 0

y2

fR (y ) =

r0 2 ? y 2 r1 2 ? y 2

, in the general case , if x0 + r0 2 ? y 2 < 0 (5)

To make the formulation of the lens area parameters easier, we use a local coordinates system based on the lens area axes (see ?gure 7). Let us consider d as the distance between p0 and p1 . The origin o of the coordinates system corresponds to the projection of the lens area summits over the x-axis. Formally, o is considered as the center of mass, i.e. the barycenter2 , x1 , and p1 with of points p0 with mass d=(x 1 ?x0 ) x0 mass d=(x1 ?x0 ) , and is de?ned as:
o 0 0 = bar p0 x0 0 , x1 d , p1 x1 0 , x0 d (3)

Now, we can easily change from the global coordinates system to the local one, by a translation of vector
?o.X ?o.Y

, that places the ori-

gin at o, followed by a rotation of an angle θ, such as:
cos θ = sin θ =
p1 .X ?p0 .X d p1 .Y ?p0 .Y d

(6)

3.1.2

Lens area anamorphosis

2 Consider two points A1 and A2 de?ned by their cartesian coordinates (x1 , y1 ) and (x2 , y2 ). The mass, also referred to as the weighting coe?cients, for each point is m1 and m2 , respectively. The barycenter of ((A1 , m1 ), (A2 , m2 )) is a point d with cartesian coorx1 +m2 x2 and yg = dinates (xg , yg ) such as: xg = m1m 1 +m2 m1 y1 +m2 y2 . m1 +m2

As referred above, de?ning a density function over complex objects such as lens areas would not be a simple task. To cope with this problem, we propose using an anamorphosis, to transform the lens area into a circle of radius 1 (?gure 8). The density function will be then de?ned over the circle. To achieve such transformation, we de?ne an a?ne bijection [15, 14] between the lens area and the circle. This one-to-one transformation preserves collinearity (i.e., all points lying on

(a) 1st case

- Let us denote the maximum height of a lens area by r. Depending on the shape of the lens area, r may be equal to the length of the line segment between the summits of the lens area (?gure 8(a)), or it may be equal to the radius of the smaller of the two circles that de?ne the lens area. The latter occurs for time instants near to the instants of observations, when the lens area is a circle or when it looks like the one presented in ?gure 8(b). Equation 8 shows how to calculate r.

(b) 2nd case Figure 8: Lens area anamorphosis a line initially still lie on a line after transformation), as well as the ratios of masses and distances (i.e., the barycenter of a line segment stills the barycenter of the corresponding line segment after transformation). So, determining the point in the circle that corresponds to a point in the lens area, is formally de?ned as follows: - Consider a point
M x y

r=

r0 2 ? x0 2 r0 r1

, in the general case , if x0 > 0 , if x1 < 0

(8)

- The points Lc and Rc in the boundary of the unity disk (?gure 8), that correspond to the points Ll and Rl in the boundary of the lens area, are de?ned as follows:

Lc

?

1?
y r

y2 r

and Rc

1?
y r

y2 r

(9)

that belongs the

lens area (?gure 8). - Using the explicit equations (5), we can de?ne the endpoints of the line segment containing M , as Ll xL =yfL (y) and
Rl xR = fR (y ) y

- Point P in the unity disk is the image of M , obtained by an a?ne one-to-one transformation of line segment Ll Rl in the lens area into the line segment Lc Rc in the unity disk:

.
P = bar Lc ,

- Supposing that M is the barycenter of the line segment Ll Rl , then M must verify equation (7). This means that mass coe?cients of Ll and Rl are proportional to their distance from M .
xR ? x Ll , xR ? xL x ? xL , Rl , xR ? xL

xR ? x xR ? xL

, Rl ,

x ? xL xR ? xL

(10)

- So, P is the image of M obtained by an anamorphosis α such as:

M = bar

(7)

M

x y

→P

α

2x?xL ?xR xR ?xL y r

1?

y 2 r

(11)

3.1.3

De?ning the density function

3.1.4

The density of probability over the lens area is then de?ned throw the density of probability over the unity disk. However, we cannot simply apply the transformation and keep the probability corresponding to each point, since the di?erential surface element has been changed by the anamorphosis:
f (α (x, y )) dx dy = f (x, y ) dx dy = 1 (12)

Probability of presence of an object within a region

Considering a region Z and a lens area L, then P (t) is proportional to the weight of the intersection zone L ∩ Z . As the space has been decomposed into granules, the value of P (t) may be easily calculated by summing the weight of the points within L that intersect Z :
1 Wtot

L

D

P (t) =

f (α (m g, n g ))
(m g,n g )∈L ∩ Z

(15)

where, L stands for the lens area and
D stands for the unity disk.

Hence, the density of probability must be normalized to guarantee that the whole probability is equal to 1. As in computation it is not possible to deal with in?nite sets, we have introduced the notion of granularity g in this model. We de?ne g as the distance between two consecutive points (granules). So, the coordinates of each point become multiples of g . We de?ne the weight of a point as its corresponding density of probability over the unity disk. Considering all the points in a lens area, the sum of their weights gives the total weight of the lens area Wtot .
Wtot =

The computation of this probability may be performed simultaneously with the computation of the total weight of the lens area. The weight of each point in the intersection zone is added simultaneously to the weight of the lens area Wtot and to the weight of the intersection zone WL∩Z . Then, P (t) is obtained as follows:
P (t) = WL∩Z Wtot (16)

3.2

Trajectory-based approach

f (α (m g, n g ))
m>0|m g ?x0 <r0 m<0|x1 ?m g<r1 n|(m g ?x0 )2 +(n g )2 <r0 2

The Trajectory-based approach consists in the generation of a large number of trajectories between each two consecutive observations recorded for the object. The probability of presence of a moving object within a given region, during a time interval ?t, is then estimated by the number of trajectories intersecting that region, over the total number of trajectories generated (17).
P (?t) = #trajectories(?t) ∩ region #trajectories generated (17)

(13)

Let us now consider a motion section de?ned The probability associated to each point is then de?ned as the ratio of its weight over the by three consecutive observations, as shown in ?gure 9. Let T0,1 (respectively T1,2 ) be the set total weight of the lens area. of the N0,1 (respectively N1,2 ) trajectories generated for the ?rst (respectively second) step f (α (mM g, nM g )) mM g (14) of the motion section. Let Z0,1 (respectively P M = nM g Wtot Z1,2 ) be the subset of the K0,1 (respectively

Figure 9: Trajectory-based approach K1,2 ) trajectories that do not intersect the forbidden region. The obtained set of trajectories over the two steps of the motion section is then T0,2 = T0,1 × T1,2 , and the subset of the trajectories that do not intersect the forbidden region is Z0,2 = Z0,1 × Z1,2 . Thus, the probability of presence of the moving object in the given region during the query window ?t may be calculated as follows:
P0,2 (?t) = #(T0,2 ? Z0,2 (?t)) #T0,2 ? #Z0,2 (?t) = #T0,2 #T0,2

ceive a random number of impacts of random strength, from random directions, during a certain period of time. The movement of the particles between two impacts is linear and uniform. No other interaction with the particles exists. This theory has been ?rstly set by Robert Brown in 1827, when observing pollen particles ?oating on water. It is applied today to many di?erent domains, like ?nancial assets modeling or signal processing. In the later the Brownian motion theory is used to simulate noise associated to processed signals. A brown noise (or ”Brownian”), is a noise in which each successive sample is a small random increment or decrement above the previous sample. We follow a similar process to generate our Brownian movements. Brownian movements In our case, Brownian movements are generated between consecutive observations. So, the origin and the destination of each generated movement must coincide with the given observations. The Brownian movement generation is then done in a suite of constant steps between the origin and the destination. The number of steps is ?xed in advance. We propose two generators of Brownian movements. The ?rst one generates onedimensional Brownian movements. This kind of movement don’t enable backward steps. The moving object executes a suite of constant steps following the same direction: from the origin to the destination. Time is also decomposed on a regular (uniform) basis. Within each step the projection of the movement over the line de?ned by the origin and destination is uniform. However, the movement over the perpendicular directions is directed by a Brownian law. Figure 10 shows a representation of the so-called one-dimensional Brownian motion. The second generator we propose combines two one-dimensional Brownian motions

#(Z0,1 (?t) × Z1,2 (?t)) #(T0,1 × T1,2 ) #Z0,1 (?t)#Z1,2 (?t) K0,1 K1,2 P0,2 (?t) = 1 ? =1? #T0,1 #T1,2 N0,1 N1,2 Hence: P0,2 (?t) = 1 ? P0,2 (?t) = 1 ? P 0,1 (?t)P 1,2 (?t) For the general case, where 0 < i < j , we obtain: Pi,j (?t) = 1 ?
k∈[i,j ?1]

P k,k+1 (?t)

(18)

The key issue for this approach is the development of a generator for movement data. The generated movements should be random, but they must comply with some physical constraints on the movement of real world objects. We have implemented two kinds of generators that we designate by the Brownian motion generator and the vector-oriented motion generator. Only the latter requires knowing the maximum velocity of the object. 3.2.1 Brownian motion generator

The Brownian motion is originally used to describe the movement of particles that re-

termediate points between the origin and the destination), p is the ?xed step length and N (0, 1) is a generator of random values accordingly to a Gaussian law, with an average null and a variance equal to 1. To constraint the movement origin and destination, the Brownian generator must ful?ll ?0 = B ?N ?1 = 0. This is the following criteria: B obtained in the following equation: Figure 10: One-dimensional Brownian motion to obtain a random movement, called twodimensional Brownian movement. There is no longer regularity between the generated movement steps. In particular, we can observe backward movement steps. Thus, the movement is more realistic and the set of trajectories generated covers an area that is closer to the shape of a lens area, than that one covered by the onedimensional generator. This kind of movement is illustrated in ?gure 11.
?k = Bk ? k Bn?1 B n?1 ?k ∈ [0, n ? 1] (20)

The advantage of the one-dimensional Brownian motion is that it is simple to implement. This movement is generated easily using a Brownian value to which we apply a rotation and a translation. The obtained vector is then added to an initial movement vector obtained on the basis of uniform movement following the axis origin-destination. Let τ be the path generated between two points A(xa , ya ) and B (xb , yb ) and τd the path from A to B following a straight line:
? ? τd (0) = A ? τ (k) = A + k AB d n?1
τd (n ? 1) = B (21) ?k ∈ [0, n ? 1]

The one-dimensional movement is then calculated for each step using the following formula, where σ is a user de?ned parameter that enables amplifying the Brownian values: Figure 11: Two-dimensional Brownian motion
?k )), ?k ∈ [0, n ? 1] (22) τ (k) = τd (k) + trans(rot(σ . B

Implementation

The generation of two-dimensional Brownian movement requires using four parameters To implement the generators described above, σ[2,2] instead of a single σ , and two indepenwe de?ne the Brownian as follows: dent Brownians.
B0 = 0 √ Bk = Bk?1 + p . N (0, 1) ?k ∈ [1, n ? 1] (19)

?
τ (k) = τd (k) + σ11 σ21 σ12 σ22 .?

?1 B k ?2 B k

? ?
(23)

where n is the number of points of the Brownian (which corresponds to the number of in-

?k ∈ [0, n ? 1]

The shape of the movements generated is in?uenced by the four parameters σ[2,2] . The parameters in the ?rst line have an in?uence on the shifting of positions in the direction of the line connecting the origin and destination, and the other parameters have an in?uence on the shifting of positions in the perpendicular direc- Figure 12: Generation of vector-oriented movetion. ments Implementation 3.2.2 Vector-oriented motion generator The time interval between two observations is decomposed into steps representing the smallest unit for the simulation. All steps have the same duration. New orientation and speed values are generated at each step accordingly to the following criteria: ? for the cape angle, the distribution is centered towards the closest point of the circumference. Considering that αavg (n) is the mean value of the angles generated for steps 1 through (n ? 1), the new orientation α(n) generated at step n is bounded by ?max , as follows:
α(n) ∈ [αavg (n) ? ?max , αavg (n) + ?max ] (24)

This movement generator may be suitable for applications for which the Brownian movements doesn’t represent a realistic solution. Since there is no restrictions for the changing of cape between two successive steps of a Brownian movement, instantly U-turns are made possible. This is not adequate for moving objects like ?shing ships, for instance.

Vector-oriented movements A vector-oriented movement is a movement composed of a suite of steps, each one is calculated using a random speed and orientation values. The main issue for this approach is the convergence of the generated movements towards the chosen destination, without interfering with the desirable random features of the generated values. To deal with this problem, we have de?ned a circumference with a center at the origin and radius equal to the distance between the origin and the destination. Then the movement generated must converge towards the circumference. For each step, the cape angle is generated accordingly to a distribution centered towards the closest point of the circumference. Once a point of the circumference is attained, it is enough to perform a rotation to make the last point generated coincide with the destination intended (?gure 12).

? for the speed value at step n, the distribution is centered on the average speed vavg (n), which should be equal to the average speed required to reach the closest point of the circumference. Our experiments revealed that the trajectories generated converge quickly towards the neighborhood of the circumference, but after that, they turned around during a certain time before reaching it. To cope with this situation, we tried to decrease the variance of the random variables over time accordingly to a chosen law (linear, quadratic or exponential). In this way, convergence is imposed arti?cially but the results obtained were satisfactory for our model.

Finally, it may arise that, during the generation of a movement, we obtain a average speed value greater than the maximum speed allowed. In this case, that generated part of the movement is simply discarded and a new one is generated.

4

Tools and applications

This section compares the proposed methods and introduces some issues related with their application for the implementation of spatiotemporal operations. As an example, we show how to use them for evaluating probabilistic range queries.

4.1

Comparison of the methods

The point-based and the vector-oriented methods are applicable to systems where it is possible to estimate the maximum velocity of the objects in advance. Otherwise, the Brownian motion method should be used. The point-based method allows estimating values for P (t), only when t is a time instant. Extending this method to cope with time intervals is not trivial. We could conceive a method based on the notion of a temporal granularity, as we did for space in section 3.1.3. The method would consist in the speci?cation of a sequence of lens areas, one for each temporal granule within the desired time interval, and building a density function for each lens area, using the point-based method proposed in this paper. Then, the density function for a time interval would result from a combination of density functions de?ned for each lens area. The problem is that the probability for the presence of a moving object at a certain location depends on its location at previous instant. As the location of the moving object at the previous instant may be one within a possibly large set of points, we were not able to ?nd

a solution for such a complex problem, allowing maintaining the desirable properties from the statistical point of view. The vector-oriented motion method allows estimating values for P (t), where t is a time instant or a time interval. The method is also simpler than the previous one, but a definite choice should only be made after benchmarking for performance and reliability evaluation purposes. The former, benchmarking for performance evaluation, would indicate which method is the most e?cient from response times point of view. The latter concerns the quality of the estimations. This is an interesting issue, also raised in [3], as we are interested in comparing the reliability of the results of probabilistic queries, but in fact we do not know which are the true results. The Brownian motion generator should be used for systems where the maximum velocity of the moving objects cannot be estimated accurately in advance. Under these circumstances, it is not possible to de?ne lens areas. Consequently, it is not possible to guaranty that when P (t) = 0, the moving object has not been within the region of interest during the speci?ed time instant or time interval t. As we will see below this feature restricts the applications of this method.

4.2

Tools and functionality

We have developed two tools for the evaluation of the proposed methods: one implementing the Brownian motion generator and the other implementing the methods based on the knowledge of the maximum velocity. The Brownian motion generator tool has been implemented in C language and runs on Windows systems (?gure 13). The tool allows drawing polygons and paths. Polygons are shown in its triangulated form. The paths are line segments de?ned by a sequence of points. The parameters for each sim-

icy for controlling these parameters in order to apply this solution for cases where the maximum speed is known as well. Although, we were not succeeded in ?nding a solution that would allow preserving the desirable statistical properties of the generated values and giving realistic answers to user queries. The second tool implements the point-based and the vector-oriented methods. It has been developed in C language and runs on Linux systems (?gure 14).

Figure 13: Brownian motion tool ulation are entered in the simulation window, where Brownian motion type allows choosing between one or two-dimensional Brownian motions, Paths stands for the number of trajectories that should be generated, Step de?nes the number of motion sections between each pair of consecutive points, and Sigma are the coef?cients that direct the spatial distribution of the generated values. The color of the pixels in the neighborhood of the paths that could have been followed by a moving object are darker for smaller probabilities and lighter for higher probabilities. The results are displayed in the property window, where Proba main denotes the value estimated for the probability of the presence of the object within the region de?ned by the polygon during the whole simulation, and proba denotes the values estimated for each individual segment. Notice that the Sigma parameters may be adjusted to mimic the features of real-world applications. We have tried to establish a pol-

Figure 14: Vector-oriented motion tool The functionalities o?ered here are approximately the same as those o?ered by the previous tool. In addition, as this tool implements the methods based on the maximum velocity, it also allows drawing lens areas, for time intervals de?ned between two consecutive observations (the ellipsis) or for arbitrary time intervals (the area de?ned by the intersection of two circles). Another di?erence is the displaying method. This tool does not use a colored scale for representing the probabilities. Hence, the zones of higher or lower probabilities should be inferred visually, considering the density of the trajectories displayed. Previous ?gure shows that there are locations within a lens area, which are not crossed by any trajectory. As the probabilities inferred for those locations would be zero, this is not

adequate to implement some operations proposed in the spatiotemporal query languages literature. To overcome this problem it is required to consider that every location within a lens area has an initial probability equal to a very small value.

4.3

Application to spatial-temporal query languages

The application of the proposed methods to answer probabilistic range queries, such as “Which were the ships that were in a certain region during a time interval, with a probability of at least 30%” [8, 4] could be expressed as follows in a SQL-like language:
select objId from movingObjects where probWithin(objMotion, region, time) > 0.3;

Consequently, these methods may also be used to implement semantics adequate to answer questions such as “Which were the moving objects that surely have been within a speci?ed region during a certain time” or “The list of all moving objects that could have been within a speci?ed region”, as proposed in [6]. The same does not hold for Brownian motion, as the parameters used for carrying out this method do not allow establishing an accurate bounding of lens areas. It is also possible to implement many other semantics. For instance, one may be interested in de?ning a probable operator and consider that probable means a probability of at least 80%, unlikely means at most 20%, or that a query yields true if the probability is at least 60%, among many others reasoning possibilities.

In this example, movingObjects(objId, objMotion) is a table holding the information about the spatiotemporal behavior of the moving objects3 , and probWithin is a function implemented using one of the methods proposed in this paper. Notice that, queries de?ning that a probability is equal to some precise value are not meaningful, as these values are merely estimations for what could have happen in a real situation. Although, the methods have been developed and tested for the validity of the bounding values, i.e., when P (t) = 0 and P (t) = 1. For instance, when one of the observations falls into a particular region during the speci?ed time interval, all methods assure that P (t) = 1. For methods grounding on a maximum velocity value, it is also possible to guaranty that when P (t) = 0, the moving object has not been within the speci?ed zone during that time.

5

Conclusion and future work

This paper deals with the computation of estimates about spatiotemporal events for moving objects systems. The focus is on the history of object’s movement and it is assumed that the objects move freely in space. We have followed two main approaches, a point-based and a trajectory-based approach, and proposed three methods, with di?erent features, that may be applied to a wide range of moving objects systems. We succeeded in obtaining non-uniform statistical distributions for estimating the location of moving objects, congruent with the features of real-world systems. These methods can be used to put into practice the operations proposed in di?erent query languages for dealing with uncertainty in the location of moving objects. We also expect that the issues presented in 3 For simplicity, we assume in this example that the this paper could give useful insights for develmovement of an object, denoted objMotion, is de?ned as an abstract data type, but other kinds of represen- oping new solutions in the future. tation might have been used. At a ?rst glance, it seamed that the point-

based approach was the most obvious solution for estimating the probability of presence of a moving object within a certain region. This also was the solution that was envisaged in the literature, but, as previous section shows, the method based on this approach is considerably more di?cult to implement than the others. Besides, this method can only be applied to answer probabilistic queries about the location of a moving object at a certain time instant, i.e., time intervals are not allowed. On the other hand, trajectory-based methods are simpler to implement and they are able to cope with a greater variety of applications: it is possible to answer probabilistic queries about the location of a moving object during any temporal domain speci?ed by users, and it is also possible to develop methods that do not require knowing the maximum velocity in advance. The purpose of this work was to look for statistical tools suitable for the implementation of probabilistic spatiotemporal queries. The emphasis was on feasibility – identi?cation of approaches and development of methods to put them into practice – and expressivity – which are the domains of application and the limitations, when applying these methods to the implementation of the di?erent semantics proposed in the literature for dealing with uncertainty –. Performance and reliability issues were left for future work. This task involves establishing policies for ?ne-tuning the controllable parameters de?ned for each method, in order to obtain the best response times without compromising the desired reliability of the results. To ful?l this task and to put in evidence the strengths and the weaknesses of each method, a benchmarking methodology, de?ning appropriate metrics for performance and reliability, is required. Other related tasks include looking for adequate ?lter steps for eliminating a large number of non-qualifying candidates,

using methods computationally non-exigent. This issue is also related with the development of data structures and e?cient access methods for movement data.

Acknowledgments
We would like to thank Cyril Branciard, Aur? elien Coquard, Yvain Thonnart, NghocMinh Vo, C? edric Lacage, Haykel Tagourti and Thomas Sirvent for their precious collaboration. They have worked on the development and implementation of the statistical tools presented in this paper, and their contribution was fundamental for the achievement of this work.

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