Modeling and querying fuzzy spatiotemporal objects

Abstract

Spatiotemporal object management has been a topic of increasing interest in many application areas such as meteorological information systems, geographic information systems, and dynamic social networks. Because of increasing requirement for these applications, modeling and querying spaitotemporal objects are two core issues of spatio-temporal object management. However, spatiotemporal objects are not always crisp, but fuzzy and uncertain. This brings about the need for modeling and querying fuzzy spatiotemporal objects. In this paper, we propose a novel model for representing fuzzy spatiotemporal objects and their topological relations. Based on this model, we investigate how to design basic and complex fuzzy query operators so that it is possible to describe the evolution of fuzzy spatiotemporal objects over time. We also show how to integrate these query operators into the existing query language SQL to enable fuzzy spatiotemporal queries.

Keywords

Spatiotemporal management fuzzy spatiotemporal object topological relations modeling querying

1 Introduction

Spatiotemporal objects are commonly found in many spatiotemporal application domains such as meteorological information systems, geographic information systems, and dynamic social networks [19, 20]. Because of the increasing requirements of these applications, the need for management of spatiotemporal objects becomes important and has received much attention [13].

The management of spatiotemporal objects focuses on modeling and querying the evolution of spatial objects over time [21]. In modeling spatiotemporal objects, two common approaches have been widely used: discrete models [13, 14] and continuous models [12, 17]. The former type of model considers time as the third geometric dimension and represents discretely changing spatial positions and extents over time. On the other hand, the latter type of model represents continuous changes of spatial objects over time. At the same time, one important property of spatiotemporal objects is their topological relationships [16]. Currently, several approaches to representing topological relations between spatiotemporal objects have been proposed, such as the algebraic-based model [12] and the space-time combination method [4]. All of the approaches are based on the assumption that spatiotemporal objects have crisp boundaries.

However, spatiotemporal objects are not always crisp, but fuzzy and uncertain [5, 8]. Take as an example windy regions (and wavy regions) whose boundaries are a fuzzy concept. In meteorological systems, the boundaries of a hurricane object cannot be precisely determined and its location can change over time [3]. This fuzzy characteristic brings about the need for modeling fuzzy spatiotemporal objects in many applications.

Handling fuzzy spatiotemporal data effectively involves modeling of fuzzy spatiotemporal objects and their topological relations. It also involves designing a complete set of fuzzy query operators that represent the temporal evolution of fuzzy spatial objects [9]. Currently, several efforts have been made to represent fuzzy spatiotemporal objects and their topological relations, such as fuzzy set theory-based methods [22, 23], point-set topology [1], and motion-based approaches [9, 18].

The motivation for this work is the development of a fuzzy spatiotemporal object model that allows the identification of topological relationships between objects and the querying of fuzzy spatiotemporal objects through topological relationships. In this paper, we present a generic model for representing fuzzy spatiotemporal objects and their topological relations. Based on this model, we investigate a querying mechanism of fuzzy spatiotemporal objects containing the design of fuzzy query operators and the implementation of query language.

The rest of the paper is organized as follows. Section 2 gives a generic model for representing fuzzy spatiotemporal objects and their topological relations. In Section 3, we present a querying mechanism of fuzzy spatiotemporal objects containing the design of fuzzy query operators and the implementation of query language. Section 4 concludes the paper.

2 Modeling fuzzy spatiotemporal objects

In this section, we first review a basic model of fuzzy spatial objects (see Section 2.1), and then take it as a basis to propose formal definitions of fuzzy spatiotemporal objects, including moving fuzzy points and moving fuzzy regions, in Section 2.2. Since the topological relation is one of the most fundamental properties of fuzzy spatiotemporal objects, we identify topological relations between fuzzy spatiotemporal objects in Section 2.3.

2.1 Fuzzy spatial objects

We summarize the definitions of fuzzy spatial objects based on fuzzy topological space [2 , 24]. Please note that we only focus on fuzzy lines and fuzzy regions.

Definition 1. ( Fuzzy line ). A fuzzy line L can be represented by a boundary ∂L, an exterior L^- and an interior L⁰. For any ∂L, there is a membership function μ_L(x, y) such that 0 < μ_L ⩽ 1, where x and y denote the x-coordinate and y-coordinate of fuzzy line L, respectively. The graphical representation of fuzzy line L is shown in Fig. 1a.

Definition 2. ( Fuzzy region ). A fuzzy region R can be represented by a determinate part R⁰, a fuzzy part ∂R, and an exterior R^-, where the fuzzy part ∂R consists of the inner boundaries ∂R_i and outer boundaries ∂R_o. ∂R_i and ∂R_o correspond to the fuzzy degrees of μ_Ri = 1 and μ_Ro = 0, respectively. Figure 1b shows the graphical representation of a fuzzy region.

2.2 Modeling of fuzzy spatiotemporal objects

As mentioned above, fuzzy spatial objects (also called fuzzy spatial data types) are considered as static objects, which include fuzzy lines and fuzzy regions. Compared to them, fuzzy spatiotemporal objects are viewed as moving objects, representing the change of fuzzy spatial objects over time.

Definition 3. ( Fuzzy Spatiotemporal Objects ). A fuzzy spatiotemporal object is a function of fuzzy spatiotemporal data types F from fuzzy time t to fuzzy spatial objects S: $F : t \to S,$

where S denotes fuzzy lines and fuzzy regions and F represents all total functions from t to S. The symbol of t represents fuzzy time.

Fuzzy spatiotemporal objects describe continuous movement of fuzzy spatial objects. Thus, we define the fuzzy spatiotemporal data types as moving fuzzy lines and moving fuzzy regions. Those data types have the following meanings:

Moving fuzzy line (mfline): the positions and routes of a fuzzy line change over time, which forms a fuzzy line motion trajectory.

Moving fuzzy region (mfregion): the positions of a fuzzy region change over time, which forms a fuzzy region motion trajectory. For instance, a hurricane can be viewed as a moving fuzzy region since its boundaries can be fuzzy.

In order to clearly represent the continuity of fuzzy spatiotemporal objects, we utilize a discrete representation called sliced presentation. Here, we take an example to illustrate the fuzzy spatiotemporal objects.

As shown in Fig. 2, for an mfregionO, time is sliced into three sections, denoted by t₁, t₂ and t₃ (t₁ < t₂ < t₃). The time axis is regarded as the third geometric dimension. When time =t₁ and t₂, the position of O is shown in the left graphic and the middle graphic, respectively. The right graphic illustrates the position of O at time =t₃. Fig. 2 represents the trajectory of a fuzzy region O in a periodof time.

2.3 Modeling of topological relations between fuzzy spatiotemporal objects

The topological relation is one of the fundamental properties between fuzzy spatiotemporal objects [10] [11]. In this section, we propose a new 9-intersection model for identifying topological relations between fuzzy spatiotemporal objects.

Definition 4. (Fuzzy Spatiotemporal Relations). Let A, B denote two fuzzy spatiotemporal objects represented in Definition 3, and ∩_t denote the intersection during time interval t. Let ∂A, A^-, and A⁰ (∂B, B^- and B⁰) be a fuzzy part, exterior, and determinate part of A (B). Fuzzy spatiotemporal relation R between A and B during time interval t is defined as: $R (A, B, t) = [\begin{matrix} A^{0} \cap_{t} B^{0} A^{0} \cap_{t} \partial B A^{0} \cap_{t} B^{-} \\ \partial A \cap_{t} B^{0} \partial A \cap_{t} \partial B \partial A \cap_{t} B^{-} \\ A^{-} \cap_{t} B^{0} A^{-} \cap_{t} \partial B A^{-} \cap_{t} B^{-} \end{matrix}]$ (1)

Every element of the 3*3-intersection matrix contains either a 0 or a 1, which denote whether that intersection (∩_t) is empty or non-empty. ∩_t indicates that the topological relationship holds during time interval t.

Based on the new 9-intersection model, a total of 2⁹ = 512 possible topological relations can be obtained. However, not all of the topological relations can be represented in reality. Thus, we need to identify the possible topological relationships. In the following, we first give the conditions which the above 3*3-intersection matrix must satisfy.

Each part of A (∂A, A^- and A⁰) must intersect with at least one part of B (∂ B, B^- and B⁰), and vice versa. If A⁰, ∩_tB⁰ and A^- ∩_tB⁰ are 1, then ∂ A ∩_tB must be 1, and vice versa.

If A⁰ ∩_t ∂B and ∂A ∩_tB⁰ are 1, then ∂A ∩_t ∂B must be 1. If A⁰ ∩_tB⁰ is 0 and A⁰ ∩_t ∂B is 1, then ∂A ∩_t ∂B must be 1.

If A⁰ ∩_tB⁰ and ∂A ∩_tB⁰ are 1, A^- ∩_tB⁰ must be 0, and vice versa. If A⁰ ∩_t ∂B and A^- ∩_t ∂B are 1, then ∂A ∩_t ∂B must be 1, and vice versa.

Now, we discuss the identification of topological relationships between two fuzzy spatiotemporal objects. Due to the limitations on the length of this paper, we only investigate the topological relationships between moving fuzzy lines and moving fuzzy regions.

Based on above conditions, by eliminating those relationships that are not realistic, we can identify 18 different topological relations between mfline and mfregion. These relationships between a moving fuzzy line segment and a moving fuzzy region are shown in Fig. 3.

3 Querying fuzzy spatiotemporal objects

In order to retrieve fuzzy spatiotemporal objects through topological relations, it is necessary to design appropriate query operators based on topological relations. When fuzzy spatio-temporal objects are static or their topological relations do not change during a time interval, basic query operators should be used (see Section 3.1). When fuzzy spatiotemporal objects are dynamic or their topological relations change over time, complex query operators should be used to describe the evolution of topological relationships (see Section 3.2). Through fuzzy query operators, the querying of fuzzy spatiotemporal objects can be implemented (see Section 3.3).

3.1 Designing of basic fuzzy query operators

As mentioned in section 2.3, we have identified 18 topological relations between mfline and mfregion. In fact, designing every query operator for each topological relation between fuzzy spatiotemporal objects is not practical because there are too many query operators. It is necessary to design some appropriate query operators to meet most requirements of spatiotemporal queries. Thus, we define basic fuzzy query operators according to eight spatial topological predicates {disjoint, meet, overlap, covers, coveredby, contains, inside, equal} [10]. The eight topological predicates are selected since they have mathematical definitions and have been widely used in spatial databases and GIS. The basic fuzzy query operators can describe fuzzy spatiotemporal topological relations that do not change during a time interval.

Now, we discuss the designing of basic fuzzy query operators between mfline and mfregion. The method includes two steps: (i) defining basic fuzzy query operators and (ii) grouping topological relationships identified into basic fuzzy query operators.

Definition 5. Let L and R denote a moving fuzzy line and a moving fuzzy region, respectively, and let t be a temporal point or temporal interval. The time domains of L and R are denoted by domain (L) and domain (R). The basic fuzzy query operators from the moving fuzzy line to the moving fuzzy region are defined as follows.

$\begin{matrix} Disjoint (L, R, t) = \forall t \in domain (L) ⊓ \\ domain (R) : L^{0} \cap_{t} R^{0} = Ø \land L^{0} \cap_{t} \partial R = Ø \land \\ \partial L \cap_{t} R^{0} = Ø \land \partial L \cap_{t}, \partial R = Ø . \end{matrix}$ $\begin{matrix} F_D isjoint (L, R, t) = \exists t \in domain (L) ⊓ \\ domain (R) : (L^{0} \cap_{t} R^{0} = Ø \land \partial L \cap_{t} R^{0} = Ø) \lor \end{matrix}$

$\begin{matrix} (L^{0} \cap_{t} R^{0} = Ø \land \partial L \cap_{t} R^{0} = Ø \land L^{0} \cap_{t} \\ R^{-} = Ø \land \partial L \cap_{t} R^{-} = Ø) . \end{matrix}$ $\begin{matrix} Inside (L, R, t) = \forall t \in domain (L) ⊓ domain (R) : \\ L^{0} \cap_{t} R^{-} = Ø \land L^{0} \cap_{t} \partial R = Ø \land \partial L \cap_{t} \\ R^{-} = Ø \land \partial L \cap_{t} \partial R = Ø . \end{matrix}$ $\begin{matrix} F_I nside (L, R, t) = \exists t \in domain (L) ⊓ \\ domain (R) : (L^{0} \cap_{t} R^{0} = Ø \land \partial L \cap_{t} R^{0} = Ø \land \\ L^{0} \cap_{t} R^{-} = Ø \land \partial L \cap_{t} R^{-} = Ø) \lor \\ (L^{0} \cap_{t} R^{-} = Ø \land \partial L \cap_{t} R^{-} = Ø) . \end{matrix}$ $\begin{matrix} Meet (L, R, t) = \forall t \in domain (L) ⊓ domain (R) : \\ \partial L = Ø \land \partial R = Ø \land L^{0} \cap_{t} R^{0} = Ø . \end{matrix}$ $F_M eet (L, R, t) = \exists t \in domain (L) ⊓ domain (R) :$ $FLR_D isjoint (L, R, t) .$ $\begin{matrix} Intersect (L, R, t) = \forall t \in domain (L) ⊓ \\ domain (R) : L^{0} \cap_{t} R^{0} \neq Ø \land L^{0} \cap_{t} R^{-} \neq Ø \land \\ L^{0} \cap_{t} \partial R \neq Ø . \end{matrix}$ $\begin{matrix} F_I ntersect (L, R, t) = \exists t \in domain (L) ⊓ \\ domain (R) : \end{matrix}$ $\begin{matrix} \neg Disjoint (L, R, t) \land \neg Meet (L, R, t) \land \neg \\ F_Meet (L, R, t) \land \neg Inside (L, R, t) \land \neg \\ Intersect (L, R, t) . \end{matrix}$

The above definition formally gives 8 different basic fuzzy query operators between the moving fuzzy line and the moving fuzzy region. Here, we only account for the last three operators.

F _ Meet holds if a moving fuzzy line is possibly disjoint with a moving fuzzy region during a period of time t. Please note that we express the case of an intersection between mfline and mfregion by means of the operator Intersect instead of Overlap because the former can express more complete meaning than the latter. The Intersect operator holds if two objects satisfy the following condition: the determinate part of mfline intersects with the determinate part, fuzzy part and outer part of mfregion. The operator of F _ Intersect describes a case which excludes 4 definite operators and an F _ Meet operator mentioned before.

Based on the definition above, the 18 topological relations shown in Fig. 3 can be grouped into 8 basic operators. The result of grouping these topological relations into the 8 basic operators is shown in Table 1.

3.2 Designing of complex fuzzy query operators

In this section, we will investigate how these basic fuzzy query operators can be combined into complex ones. In fact, complex fuzzy query operators are the sequences of basic operators that hold during a time interval. These query operators can be successfully used to characterize developments of fuzzy spatiotemporal objects. In the following, we give a formal definition of complex fuzzy spatiotemporal operators between fuzzy spatiotemporalobjects.

Definition 6. ( Complex fuzzy spatiotemporal operators ).

Let α, β∈ {mfline, mfregion} be two fuzzy spatiotemporal objects, and let I = [I_S, I_E] be a whole time interval containing the start time I_S and the end time I_E such that I is the intersection of the time domain over α and β, I = timeDomain (α) ⊓ timeDomain (β). Let P be a finite set of the basic fuzzy spatiotemporal operator defined in Section 3.1, P = {p₁, p₂, …, p_n} , n ≥ 2. A complex fuzzy spatiotemporal operator C between α and β is defined as:

$\begin{matrix} C \overset{def}{=} \exists t_{1}, t_{2}, \dots, t_{n} \in I \land \\ t_{1} < t_{2} < \dots < t_{n} \land p_{1} (α, β, t_{1}) \land \\ p_{2} (α, β, t_{2}) \land \dots \land p_{n} (α, β, t_{n}) . \end{matrix}$

In fact, a complex operator is a motion verb that represents the change of fuzzy spatiotemporal topological relations over a period of time. One motion verb can represent at least two basic fuzzy spatiotemporal operators at one specified time. This is the reason why we set n ≥ 2 in the above definition. In addition, we can know that sequential composition satisfies associativity. As a result, we can easily define complex fuzzy spatiotemporal operators by successively composing the simple topological operators we have obtained. Hence, the above definition can also be written in the following form: $\begin{matrix} C \overset{def}{=} \exists t_{1}, t_{2}, \dots, t_{n} \in I \land \\ t_{1} < t_{2} < \dots < t_{n} \land C_{1} (α, β, t_{1}) \land \\ C_{2} (α, β, t_{2}) \land \dots \land C_{n} (α, β, t_{n}), \end{matrix}$

where C₁, C₂, …, C_n denote the simple topological operators we have obtained.

As is well known, the temporal interval comprises many basic temporal intervals of basic fuzzy spatiotemporal operators. The start is the beginning of the first basic fuzzy spatiotemporal operator, and the end is the end of the last operator. Hence, we can define a temporal composition to combine basic or simple fuzzy spatiotemporal operators.

Definition 7. ( Temporal Composition ). Let α, β be two fuzzy spatiotemporal objects. Let P and Q be basic or simple fuzzy spatiotemporal operators, and let the symbol ⊳ be the temporal composition operator. Then, $\begin{matrix} P ⊳ Q \overset{def}{=} \exists t_{1}, t_{2} \in I \land t_{1} < t_{2} : P (α, β, t_{1}) \land \\ Q (α, β, t_{2}) . \end{matrix}$

The definition describes the temporal compositions of basic fuzzy spatiotemporal operators, which are denoted by the symbol ⊳. The aim of the temporal composition is to concisely represent the evolution of fuzzy spatiotemporal objects. The symbol can make full use of the definitions of basic operators to denote the evolution. On the other hand, the definition is conducive to establishing alternating sequences of fuzzy spatiotemporal operators.

Now, we will define complex fuzzy spatiotemporal operators in detail. For lack of space, we will only give limited geometric representations of fuzzy spatiotemporal topological relationships. In the case of mfline/mfregion, there are 16 new complex operators to be defined. The formal definition is asfollows:

Touch (α, β, I) = Disjoint ⊳ F_Disjoint ⊳ Meet ⊳ F_Disjoint ⊳ Disjoint,

F _ Touch (α, β, I) = Disjoint ⊳ F_Meet ⊳ Disjoint,

Snap (α, β, I) = Disjoint ⊳ F_Disjoint ⊳ Meet,

F _ Snap (α, β, I) = (Disjoint ⊳ F_Meet) ∨ (F_Disjoint ⊳ F_Meet),

Release (α, β, I) = Meet ⊳ F_Disjoint ⊳ Disjoint,

F _ Release (α, β, I) = (Meet ⊳ F_Disjoint) ∨ (F_Meet ⊳ F_Disjoint),

Excurse (α, β, I) = Meet ⊳ F_Meet ⊳ Disjoint ⊳ FPL_Meet ⊳ Meet

F _ Excurse (α, β, I) = Meet ⊳ F_Disjoint ⊳ Meet,

IsIntersecting (α, β, I) = Disjoint ⊳ F_Intersect ⊳ Intersect,

F _ IsIntersecting (α, β, I) = Disjoint ⊳ F_Disjoint ⊳ F_Intersect,

Enter (α, β, I) = Disjoint ⊳ F_Disjoint ⊳ Meet ⊳ F_Inside ⊳ Inside,

F _ Enter (α, β, I) = Disjoint ⊳ F_Disjoint ⊳ Meet ⊳ F_Inside,

Leave (α, β, I) = Inside ⊳ F_Inside ⊳ Meet ⊳ F_Disjoint ⊳ Disjoint,

F _ Leave (α, β, I) = Inside ⊳ FLR_Inside ⊳ Meet ⊳ F_Disjoint,

Cross (α, β, I) = Enter ⊳ Leave,

F _ Cross (α, β, I) = Enter ⊳ FLR_Leave,

For example, we give the graphical representations of F_Leave evolution between a moving fuzzy line and a moving fuzzy region, as shown in Fig. 4. From the figure, we can observe that the development of a moving fuzzy line segment and a moving fuzzy region includes four states: Inside, F_Inside, Meet, and F_Disjoint. As a result, the whole evolution of them remains F_Leave during the time interval[t₁, t₄].

3.3 Querying of fuzzy spatiotemporal objects

Querying fuzzy spatiotemporal objects through fuzzy query operators (topological relationships) is one of the basic tasks in spatiotemporal databases or GIS. Currently, the existing database query languages (e.g., SQL) have some difficulties in answering temporal queries because they cannot support fuzzy spatiotemporal query operators. In this section, we can solve these difficulties by integrating basic and complex fuzzy query operators into the existing query language SQL. More precisely, we extend SQL by the set of basic and complex fuzzy query operators mentioned in Section 3.1 and 3.2. The aim of the extended SQL is to enable the querying of the evolutions of fuzzy spatiotemporal objects.

In the following, we consider two application scenarios and give two query examples. Our basic objective is to illustrate the use of fuzzy query operators in spatiotemporal databases.

The first scenario is described as follows: There is a meteorological event in which a wave region (No. 01559) occurs in the Mediterranean Sea at 30.10.2015 08 : 00. In this Sea, there exist three important ferry routes, denoted L₁, L₂ and L₃. Here, the wave region can be represented as a moving fuzzy region. We need to retrieve all the ferry routes that are restricted possibly for transportation due to wave. Clearly, basic fuzzy query operator F_Disjoint can be used to achieve this query. At first, we assume that the information stored has the following database schema:

Meteorological-event (id: string, trajectory: mfregion)

Ferry-route (id: string, line: line)

Next, we give an SQL sentence.

SELECT Ferry-route. id

FROM Ferry-route, Meteorological-event

WHEREF_Disjoint (Meteorological-event. trajectory, Ferry-route. line, 30.10.2015 08 : 00 : 00) AND Meteorological-event. id = “01559”.

The SQL sentence demonstrates that the fuzzy query operator F_Disjoint will return all ferry routes that possibly disjoint with a wave region (No. 01559) at 30.10.2015 08 : 00am. It should be noted that experts can give a degree of membership to represent the operator F_Disjoint, such as [0.6–0.7].

The second scenario is described as follows: the Seattle port is in a hurricane-prone. We need to retrieve all hurricanes that have possible crossed the Seattle ports from 2015-05-12 08 : 00 am to 2015-08-12 08 : 00 am. Assume that we have the following schemas:

Ports (id: string, name: string, line: line)

Hurricanes (id: string, name: string, trajectory: mfregion)

This query can be expressed as follows:

SELECT Hurricanes. id

FROM Ports, Hurricanes

WHEREF_Cross (Ports. line, Hurricanes. trajectory, 2015-05-12 08 : 00 : 00, 2015-08-12 08 : 00 : 00) AND Ports. Name = “Seattle port”.

The SQL sentence demonstrates the use of fuzzy query operator F_Cross and gets all Hurricanes that possibly cross the Seattle port from 2015-05-12 08 : 00 : 00 to 2015-08-12 08 : 00 : 00. Please note that experts can give a degree of membership (e.g., 0.6) to represent F_Cross.

In this section, we only give some query examples and provide some ideas about query processing issues. The detailed study of query processing, query implementation and query evaluation will be undertaken in future work.

4 Conclusion

In this paper, we propose a generic model of fuzzy spatiotemporal objects and a querying mechanism of fuzzy spatiotemporal objects. The generic model defines two kinds of fuzzy spatiotemporal objects, moving fuzzy lines and moving fuzzy regions, on the basis of well-defined fuzzy spatial data types. On this basis, we construct a new 9-intersection model of topological relations between fuzzy spatiotemporal objects. More precisely, we identify 18 basic relations between a moving fuzzy line segment and a moving fuzzy region. The querying mechanism described includes the design of fuzzy query operators and the implementation of query language. From a semantic point of view, we design basic fuzzy query operators based on spatial topological predicates. Then, based on basic operators, we design complex fuzzy query operators, which are the sequences of basic operators holding at the beginning and end of a temporal interval. We also show how to integrate basic and complex operators into the existing query language SQL.

Our approach is only applicable to topological relations between evolving fuzzy lines and fuzzy regions in spatiotemporal databases. In our future work, we will study fuzzy query operators between evolving fuzzy regions and give algorithms to determine basic and complex fuzzy query operators. Also, we will design new fuzzy query operators that are applicable to the application fields about image/video systems and wireless networks.

Footnotes

Acknowledgments

The work is supported by the National Natural Science Foundation of China (61672139).

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