Combining hierarchies for spatial reasoning of hierarchical relations among places in different contexts

Abstract

Spatial reasoning of hierarchical relations is an important research topic in geographic information science. Existing literature focuses mainly on hierarchical relations according to ordered containment of point sets based on the 9-Intersection Model (9IM). However, in natural language descriptions, hierarchical relations expressed by prepositions such as in can have spatial configurations conflicting with those hierarchical relations based on 9IM. This paper proposes an algorithm employing combined hierarchies to infer hierarchical relations among places in different contexts. Two graph databases containing combined hierarchies conforming to administrative reach and geometric enclosure are established with places represented by polygons with misaligned boundaries on prototypical and real data. The 13170 pairs of conflicting hierarchical relations with the same start nodes but different end nodes detected in the combined hierarchies on real data validate the proposed algorithm for spatial reasoning of hierarchical relations in different contexts. This research facilitates consistent spatial reasoning of hierarchical relations in different contexts for spatial information retrieval and question answering.

Keywords:

• Place hierarchy
• spatial reasoning
• context

1. Introduction

Hierarchical relations among places may conform to different rules, such as geometric enclosure, or administrative reach. For example, a hierarchical relation can be established if a city is administrated by a country. Or, a hierarchical relation can be created if a city is enclosed but not administrated by a country. In the existing literature, while hierarchies of places conforming to administrative reach have been defined and established (principle of subsidiarity) (Ma 2005, Li et al. 2015), no hierarchies of places conforming to geometric enclosure have been defined or established. Specifically, hierarchical relations of places conforming to geometric enclosure include but are not limited to the topological relations between enclaves and their surrounding places. As far as an enclave and its surrounding place are concerned, the two places are mutually exclusive.

Hierarchical relations among places are often expressed by spatial prepositions in natural language descriptions (NLD). From these spatial prepositions, it is often not clear to which hierarchy the description is referring. For example, in the NLD ‘there are 22 Belgian exclaves in the Netherlands’ (Wikipedia contributors 2021) (italicizing of the preposition added by the author), there is a hierarchical relation between the 22 Belgian exclaves and Belgium according to administrative reach, and there is a hierarchical relation between the 22 Belgian exclaves and the Netherlands according to geometric enclosure. Both hierarchical relations are correct in their own contexts but conflict with each other if contexts are neglected. Furthermore, the differentiation of contexts of hierarchical relations among places in NLD, which is required for reasoning with the corresponding relational constraints of these hierarchies, cannot be achieved by algorithms yet. If spatial reasoning of hierarchical relations in all contexts is carried out, there may be some results by humans or algorithms that are correct in specific contexts but contradict corresponding reasoning results in other contexts.

This research hypothesizes that it is possible to infer different hierarchical relations among places expressed by prepositions such as in in NLD. Two research questions are explored. First, what are the rules to form different hierarchies of places that are expressed as in in NLD? Second, what is the algorithm of disambiguating contexts of the use of in in NLD?

In order to address the two questions, this paper proposes to establish hierarchies of places represented by polygons possibly with misaligned boundaries according to two different contexts, namely, geometric enclosure and administrative reach. This research combines these hierarchies, which will allow to disambiguate both contexts in qualitative spatial reasoning.

Meeting the demands of establishing hierarchies of places conforming to geometric enclosure and differentiating both contexts of hierarchical relations according to administrative reach and geometric enclosure is important since the spatial reasoning of hierarchical relations will be improved in consistency and completeness for conflicting results due to contexts. Further, the solution to the spatial reasoning of hierarchical relations among places in different contexts with combined hierarchies is not limited to the differentiation of contexts of geometric enclosure or administrative reach of administrative units, but also can be generalized to other places such as banks and restaurants, or other contexts such as mountain classifications.

The main contribution of this research is a method that establishes combined hierarchies conforming to both geometric enclosure and administrative reach, disambiguates different contexts, and infers hierarchical relations with combined hierarchies. The paper also presents two graph databases containing combined hierarchies conforming to administrative reach and geometric enclosure. The proposed method to infer hierarchical relations in different contexts is important because it supports the resolution of multiple contexts and contradicting hierarchical relations if different contexts are ignored. Hence, this research can facilitate consistent inference of hierarchical relations in different contexts for spatial information retrieval and question answering.

For example, people may search for special places to travel. A special place A can be located in another place B of which A is not part in NLD. There is a hierarchical relation expressed as in between A and B conforming to geometric enclosure. Meanwhile, there is a hierarchical relation between A and place C of which A is part conforming to administrative reach. As shown in Figure 1, the land parcel A administrated by Baarle-Nassau C in orange color is enclosed by part of Baarle-Hertog B in blue color. According to existing formal topological models, since B and C do not overlap because there is no administrative relation between the two, there should be no part of C overlapping with B. Then there should be no hierarchical relation between A and B. So the hierarchical relation between A and B is conflicting with the hierarchical relation between A and C. According to existing formal topological models, only the hierarchical relation between A and C is consistent with 9IM, which can be correctly represented and searched. Therefore, the hierarchical relation between A and B cannot be represented or searched. People’s search for special places to travel such as A will receive no suitable answer based on existing literature. This also applies to recent generative artificial intelligence (AI) models, such as ChatGPT (OpenAI 2022). However, the method proposed in this research can maintain such conflicting hierarchical relations due to different contexts and infer hierarchical relations in specific contexts. By analyzing administrative units around the world, the proposed method is able to find a set of administrative units such as A for people to choose to travel.

Figure 1. The enclave in Baarle-Hertog.

In the following sections, firstly, a literature review of conceptualization and formalization of hierarchies, inconsistency in reasoning with hierarchies, definitions of places, and formal topological models is presented in Section 2. Then, the definition and establishment of the hierarchy conforming to geometric enclosure of polygons, and the algorithm to infer hierarchical relations in both contexts of geometric enclosure and administrative reach are introduced in Section 3. Next, in Section 4, the algorithm is tested by inferring with combined hierarchies of places represented as polygons in both contexts of geometric enclosure and administrative reach on both prototypical and real data. Finally, the novel hierarchy and the inferring algorithm are discussed in Section 5, and conclusions of this research are given in Section 6.

2. Literature review

This section will review existing literature on the conceptualization and formalization of hierarchies, on the inconsistency in reasoning hierarchical relations in different contexts, on the definitions of places, and on formal topological models.

2.1. Conceptualization of hierarchies

In general, a hierarchy is a structure of objects where each object has a rank at some ordered levels. In existing literature, hierarchies have been conceptualized differently in various fields including philosophy (Kolnai 1971), architecture (Ching 2014), graphic design (Lupton and Phillips 2008), mathematics (Cucker 1992), social science (Satoshi Miura 2013), biological systems (Deichmann 2017), computer science (Bartels et al. 2004, Ivy Wigmore 2012), organizational science (Kanter 1991), and systems theory (Smith and Sage 1973). In the various fields where hierarchies are conceptualized, the ordered importance, power or control of items is key to forming the hierarchical structures.

Hierarchies are also essential in cognitive science as the significant structure of organizing information of reality in mental maps (Stevens and Coupe 1978, Hirtle and Jonides 1985). In Geographical Information Science (GIScience), hierarchies are employed to facilitate cartography in various granularities (Cohn 1995, Butzler et al. 2011), spatial reasoning (Timpf and Frank 1997, Winter 1999, Winter and Bittner 2002, Yuan et al. 2014), route description (Tomko et al. 2008, Winter et al. 2008) and understanding functionality of urban places (Kusenbach 2008, Shimizu et al. 2020).

There are various notions of hierarchies in GIScience. One of the notions of hierarchies is based on the ordered levels of abstraction of geographic objects for multiscale cartography. Van Oosterom (2005) proposes a generalized partition tree algorithm to construct a multi-scale data structure for area partitioning. The scale to which each geometry belongs is determined by an important function which is dependent on the area and the length of boundaries of the geometry. Butzler et al. (2011) develop a hierarchy of labels based on population and economic importance of place points to determine the presentation of a map in different scales. Such hierarchies of abstracted representations for multiscale cartography are employed for the visualization of spatial information instead of reasoning with hierarchies. Therefore, they are out of scope of this research.

A second notion of hierarchies is based on the ordered relations of containment. For geometries with crisp boundaries, Stevens and Coupe (1978) propose a hierarchical representation of locations of administrative units where spatial relations between units administrated by the same superordinate units are stored explicitly. Meanwhile, spatial relations not stored explicitly are inferred with reference to the relations among the units and their superordinate units. This hierarchy and the inference process are employed to account for some distortions in humans’ judgments of spatial relations. Hamzei et al. (2018) generate a hierarchy of general places such as buildings and parks which are represented by geometries. Qualitative spatial relations are derived for the geometries of the hierarchy to conform to human perception and description of places. Both kinds of hierarchies by Stevens and Coupe (1978) and by Hamzei et al. (2018) are based on the rigid geometric computation of containment relations conforming to the 9-intersection model (9IM) (Egenhofer and Herring 1992). However, such hierarchies cannot represent some hierarchical relations expressed in NLD such as the ones between enclaves and their surrounding polygons according to geometric enclosure.

A third notion of hierarchies is based on the ordered part-whole relations in the composition of objects. While ordered relations of containment emphasize spatial attributes, the ordered part-whole relations rely more on non-spatial attributes. The part-whole relations are rooted in mereology, and there are different kinds of part-of relations, that is, component–integral relation, member–collection relation and portion–mass relation (Gerstl and Pribbenow 1995). The component–integral relation is the most widely adopted one to build hierarchies of geographical objects. For places with vague boundaries, Wu et al. (2019) use check-in footprints of Sina Weibo and fuzzy formal concept analysis to generate hierarchies of place names, which adopts part-whole relations and allows for overlap among bordered areas of places. In the research, the nonspatial attribute, namely, the check-in frequency of Sina Weibo users at a place is employed. But this algorithm also conforms to 9IM. It is unable to represent some hierarchical relations conforming to geometric enclosure.

A fourth notion of hierarchy is based on the ordered salience of objects such as landmarks or streets (Tomko et al. 2008, Winter et al. 2008, Tomko and Winter 2009). This kind of hierarchy is similar to the ones applied in the field of architecture or graphic design. In comparison with the hierarchy of cartography, this kind of hierarchy better conforms to human cognition of the salience of places. For places without hierarchical relations, the ranks of places in hierarchies according to salience may differ greatly from the ones according to other attributes such as age of construction. But for places with hierarchical relations, the containment relation referred to in the hierarchical relations of places only conforms to 9IM as well.

The last notion of hierarchy is emerging from software design. While the hierarchical trusted execution environments called nested enclaves (Park et al. 2020) are in a three-level structure according to geometric enclosure, no mathematical definition is proposed for the hierarchy. No spatial reasoning of hierarchical relations is mentioned in that paper, either.

2.2. Formalization of hierarchies

In GIScience, the formation of hierarchies can be classified into three categories, namely, bottom-up, top-down, and intrinsic. To establish hierarchies of map series, Timpf (1999) proposes three types of hierarchies of geometries formed in a bottom-up manner, that is, aggregation hierarchy, generalization hierarchy and filter hierarchy. Meanwhile, there are some hierarchies formed in a top-down manner to partition problems into a hierarchy of subproblems to simplify computation tasks (Ogniewicz and Kübler 1995, Rossi and Torsello 2012, Zhou et al. 2013). Intrinsic hierarchies are hierarchies formed without human assignment of ranks and have invariant rules under natural evolution, such as the rank-size rule in cities’ hierarchies (Batty 2006) and the 80/20 principle in street hierarchies (Jiang 2009). Except that hierarchies based on the ordered levels of abstraction are established in a bottom-up manner and that intrinsic hierarchies are formed naturally, hierarchies with other notions in the above section usually can be established by humans in both bottom-up and top-down manner. But the established hierarchies may be different due to the difference between bottom-up and top-down manners.

In this research, the hierarchy conforming to geometric enclosure is established in a bottom-up manner, and the hierarchy conforming to administrative reach is established in a top-down manner.

2.3. Inconsistency in reasoning by hierarchical relations

In existing literature, different perspectives of inconsistency have been researched (Lenski 1954, Costa and C. N. 1974, Kydland and Prescott 1977, Higgins and Thompson 2002, Higgins et al. 2003, Grant and Hunter 2006). In general, definitions of inconsistency are based on differences or conflicts between at least two objects or results.

In GIScience, the study of Moar and Bower (1983) refers to inconsistency as the fact that spatial knowledge derived from human memory contradicts mathematical laws in geometry. This kind of inconsistency is similar to the inconsistency of hierarchical relations derived from existing formal models of topological relations and NLD. However, the quantitative contradictions studied by Moar and Bower (1983) may not be engaged in the qualitative hierarchies of places. Chen et al. (2015) judge a spatial relation to be consistent if no contradicting spatial relations can be inferred from the database where the spatial relation is stored. They deal with consistent binary spatial relations including cardinal and relative direction, qualitative distance and topological relations. There is another research on reasoning over different directional frames of reference in which contradicting descriptions are dealt with such as A is to the left of B and B is to the left of A (Hua et al. 2018). Both qualitative reasoning problems (Chen et al. 2015, Hua et al. 2018) are due to the indeterminacy caused by incomplete information of the location of objects or observers, but not the different semantics of spatial relation terms. As pointed out by Ragni and Knauff (2013), humans may have multiple mental models in spatial reasoning and the inference of the indeterminate spatial configuration may engage in testing the validity of multiple mental models. In each mental model or frame of reference, the semantics of spatial relation terms such as in, north, and left, are the same. However, in this research, there are different semantics of hierarchical relations represented by hierarchies, i.e., hierarchical relations between places due to different definitions of hierarchies. In existing literature (Chen et al. 2015, Ragni and Knauff 2013, Hua et al. 2018), only hierarchical relations conforming to containment relations based on 9IM are considered in the place graph and no hierarchies are engaged in the resolution of the inconsistency of the hierarchical relations between the one conforming to containment relations based on 9IM with the one conforming to geometric enclosure such as enclaves and their surrounding places. Therefore, a new definition of inconsistency in reasoning hierarchical relations is proposed in Section 3.3.

2.4. Definitions of places

Places play an essential conceptual role in multiple disciplines such as geography, social science, environmental psychology, and recently information science (Hamzei et al. 2019). With emphasis on different aspects of places, definitions and formal models of places vary in different fields. In existing literature in GIScience, places are mostly identified by names and represented by geometries with names. Besides places with crisp boundaries such as the Commonwealth of Australia, places without crisp boundaries such as Mount Alps are also modeled with egg-yolk theory (Cohn and Gotts 1996), supervaluation (Bennett 2001, Varzi 2001), fuzzy logic (Montello et al. 2003, Wu et al. 2019), and rough location (Bittner and Stell 2002).

However, there are still challenges of capturing information related to places, including ambiguity, indeterminacy, preferences, and salience of places (Winter and Freksa 2012). Goodchild (2015) points out that named places, instead of coordinates, are more important when it comes to human perception, reasoning and behavior in the world. Winter and Freksa (2012) propose to formulate characteristics of places by contrast so that places can be localized without specific geometries. To capture information on places, triplets or n-plets are applied to establish graph databases (Kim et al. 2015, Chen et al. 2018). In these place graphs, places are represented by nodes with attributes such as names and edges with attributes such as spatial relations with other places, more similar to human perception and communication of places.

Recently, Purves et al. (2019) give a definition of places that places are shared unique geographic objects with location and identifiable properties and/or relationships but not necessarily with boundaries. Under such a definition, representations of places in traditional GIS and graph databases can be unified. Therefore, such a definition of places is employed in this research. The data analyzed in this research consists of places represented as polygons in traditional GIS and the hierarchies established in this research consist of places in graph databases.

2.5. Formal topological models

In existing literature, there are general topological models (Egenhofer and Herring 1992, Randell et al. 1992, Kainz et al. 1993) to categorize different topological relations between points, lines, and regions, while some specific topological models represent detailed topological relations between lines or regions in particular (Worboys 2012, 2013, Lewis et al. 2013, Dube et al. 2015, Sindoni and Stell 2017, Lewis 2019). Since the topological relations between enclaves and surrounding places focused on this research are special in that the exterior boundaries of enclaves are a subset of the boundaries of surrounding places, these topological models are not directly applicable to the differentiation of topological relations between enclaves and surrounding places from ones between non-enclaves without further development of definitions or rules about enclaves.

Although there are a few formal definitions of surrounds relations (Worboys and Duckham 2021) or enclaves (Wang 2023) which are related to the topological relations between enclaves and surrounding places, they have limitations in terms of identifying the topological relations between enclaves and surrounding places generally in non-ideal data. Worboys and Duckham (2021) developed the region maptree to represent the qualitative surrounds relations between disjoint regions. The method is applicable to six different definitions of surrounds in literature, namely, completely enclosed, partially enclosed, visually surrounded, contained in convex hull, contained in convex hull of union, and region-Voronoi-diagram-surrounded. Although the topological relations between enclaves and surrounding places are completely enclosed, enclaves and surrounding places intersect at the exterior boundaries of enclaves instead of being disjoint. Wang (2023) proposed the Boundary Extended 16-Intersection Model (BE-16IM) to distinguish 74 different spatial configurations of enclaves and surrounding places. But the method cannot be applied to non-ideal data where boundaries of regions are misaligned such as Figure 2(b). Not to mention that none of the research has discussed the definitions of surrounds relations or enclaves with regard to the spatial reasoning of hierarchical relations in different contexts.

Figure 2. Examples of geometries with aligned or misaligned boundaries. The area in light grey is polygon A which administrates polygons A₁, A₂, A₃ and A₄. The area in dark grey is polygon B. The area in black is polygon C.

3. Methodology

The hypothesis of this research is tested by assessing the performance of the algorithm of disambiguating contexts of the use of in in NLD. If the algorithm of disambiguating contexts of the use of in in NLD is applicable, then the hypothesis that it is possible to infer different hierarchical relations among places expressed by prepositions such as in in NLD is valid. Otherwise, the hypothesis is invalid.

This research defines a hierarchy conforming to administrative reach as a graph of nodes connected with directed relations where a node representing polygon A_i ($i=1,2,\dots ,n$) is a child of another node representing polygon A if A administrates A_i. The hierarchical relations conforming to administrative reach can be extracted from administrative attributes without geometric computations.

This research defines a hierarchy conforming to geometric enclosure as a graph of nodes connected with directed relations where a node representing polygon A_i ($i=1,2,\dots ,n$) is a child of another node representing polygon A if the exterior boundary of A_i is completely enclosed by the exterior boundary of A. Specifically, the intersection of A_i and the closed component of A’s exterior boundary is A_i. It should be noted that different versions of surrounds relations between polygons summarized by Worboys and Duckham (2021) are not included in this hierarchy except for the completely enclosed spatial configurations.

The combined hierarchies conforming to both administrative reach and geometric enclosure are formed by merging the above two graphs. Each pair of nodes representing the same polygon in the two graphs is merged into one node according to the unique identifier of the polygon. The hierarchical relations conforming to administrative reach and geometric enclosure are both kept in the combined graph.

Here is an example of the differences between the hierarchy conforming to administrative reach and the hierarchy conforming to geometric enclosure. As shown in Figure 2(a), polygon A in light grey administrates four polygons, A₁, A₂, A₃, and A₄. Polygon B in dark grey and polygon C in black are sitting in the hole of A, and C is sitting in the hole of B. A, B and C are administrative units at the same level. On the one hand, according to administrative reach, a hierarchy can be derived as shown in Figure 3(a). On the other hand, since the exterior boundaries of A₁, A₂, A₃, A₄, B and C are all enclosed by the exterior boundary of A, a hierarchy conforming to geometric enclosure can be derived as shown in Figure 3(b). Both hierarchies are different in terms of the relation between A, B, and C. In the hierarchy conforming to administrative reach, there is no hierarchical relation among A, B, or C. But in the hierarchy conforming to geometric enclosure, there are hierarchical relations among A, B, and C.

Figure 3. Examples of hierarchies conforming to geometric enclosure or administrative reach.

In the following parts of this section, firstly, the processes to establish the hierarchy conforming to geometric enclosure are introduced. Then, the algorithm of inferring hierarchical relations in both contexts of geometric enclosure and administrative reach is presented. Lastly, a definition of inconsistency in reasoning hierarchical relations in different contexts is provided.

3.1. Building the hierarchy conforming to geometric enclosure

In theory, polygons representing administrative units at different levels have aligned boundaries. But in reality, polygons representing administrative units at different levels usually have slightly misaligned boundaries, as shown in Figure 2(b). Although the boundary of A is the same in Figure 2(a,b), the boundary of A₄ in Figure 2(b) overlaps with the exterior of A₄ in Figure 2(a). As a result, the exterior boundary of A₄ is not enclosed by A totally in Figure 2(b). Accordingly, the hierarchy conforming to geometric enclosure derived from Figure 2(b), as shown in Figure 4, is different from the one derived from Figure 2(a), as shown in Figure 3(b).

Figure 4. The hierarchy conforming to geometric enclosure with misaligned boundaries.

In the existing literature, no formal model can derive the hierarchical relations in the hierarchy conforming to geometric enclosure, neither for polygons with aligned boundaries nor misaligned boundaries. Although there are methods to align misaligned boundaries automatically or manually in literature (Wadembere and Ogao 2014, Irene et al. 2017), the hierarchical relation between a polygon B enclosed by another polygon A without the boundaries of B being a subset of the boundaries of A (for example, Figure 5(b,c)) remains to be derived by automatic computation. Therefore, this research proposes the algorithm shown in Figure 7 to establish the hierarchy conforming to geometric enclosure (for example, Figure 5). Besides generating hierarchical relations conforming to geometric enclosure for polygons in partition, this algorithm is also able to derive hierarchical relations conforming to geometric enclosure where polygons overlap, an example demonstrated in prototypical data in Section 4.

Figure 5. Examples of hierarchical relations conforming to geometric enclosure. The area in light grey is polygon A which administrates polygons A₁, A₂, A₃, and A₄. The area in dark grey is polygon B or polygon D. The area in black is polygon C.

According to the algorithm, firstly, polygons representing administrative units at different levels are congregated into set U. Secondly, for each polygon A_i ($i=1,2,\dots ,n$) in U, a corresponding polygon will be derived by filling the closed component of the exterior boundary of A_i, and be put in set $U\prime .$ Thirdly, a spatial index will be built on $U\prime$ to accelerate spatial computation on the data set afterwards. Fourthly, selections of the direct parents of polygons in U will be carried out in iteration. At the beginning of an iteration, for polygon A_i in U, a set of candidate polygons that intersect with but not equal A_i will be selected from $U\prime .$ Next, to accelerate the computation, candidate polygons that intersect with A_i but have areas smaller than a threshold α of the area of A_i will be removed. Then, only one polygon A with the smallest area among the candidate polygons that overlap A_i with more than a threshold β of the area of A_i will be selected as the direct parent of A_i. Finally, the hierarchical relation where A is the direct parent of A_i is derived in the hierarchy conforming to geometric enclosure. This process will run iteratively for every polygon in U.

In the algorithm, there are two thresholds α and β to be determined. Since the exterior boundary of a polygon is enclosed by the exterior boundary of its direct parent polygon according to the definition of the hierarchy conforming to geometric enclosure, there should not be more than one direct parent for a polygon in U. Therefore, the minimum overlapping area between a polygon A_i and its parent A should be larger than 50 percent of the area of A_i. But to determine a specific value larger than 50 percent and no larger than 100 percent, this research defines two metrics, overlapping ratio (OR) and area ratio (AR), for reference. (1) $$\mathit{\text{OR}}(U)=\mathrm{\text{min}}(\mathit{\text{OR}}(A_i)),(i=0,1,\dots ,n)$$(1) (2) $$\mathit{\text{OR}}(A_i)=\mathit{\text{area}}(\mathit{\text{intersect}}(A_i,A\prime ))/\mathit{\text{area}}(A_i)$$(2) where $A\prime$ is the polygon which is in $U\prime$ and corresponds to A which administrates A_i.

The OR of data set U is the minimum OR for each polygon in U. The OR for a polygon A_i is the value of the area ratio between the intersection of A_i and $A\prime ,$ and A_i. $A\prime$ has the same exterior boundary as A but without any holes inside its exterior boundary. (3) $$\mathit{\text{AR}}(U)=\mathrm{\text{min}}(\mathit{\text{AR}}(A_i)),(i=0,1,\dots ,n)$$(3) (4) $$\mathit{\text{AR}}(A_i)=\mathit{\text{area}}(A\prime )/\mathit{\text{area}}(A_i)$$(4) where $A\prime$ is in the same meaning as above.

The AR of data set U is the minimum AR for each polygon in U. The AR for a polygon A_i is the value of the area ratio between $A\prime$ and A_i.

According to Equations (1(1) $$\mathit{\text{OR}}(U)=\mathrm{\text{min}}(\mathit{\text{OR}}(A_i)),(i=0,1,\dots ,n)$$(1) , 2(2) $$\mathit{\text{OR}}(A_i)=\mathit{\text{area}}(\mathit{\text{intersect}}(A_i,A\prime ))/\mathit{\text{area}}(A_i)$$(2) , 3(3) $$\mathit{\text{AR}}(U)=\mathrm{\text{min}}(\mathit{\text{AR}}(A_i)),(i=0,1,\dots ,n)$$(3) , and 4)(4) $$\mathit{\text{AR}}(A_i)=\mathit{\text{area}}(A\prime )/\mathit{\text{area}}(A_i)$$(4) , OR and AR for U can be calculated. The value of α had better be set smaller than AR(U). The value of β had better be set smaller than OR(U). Otherwise, some relevant polygons will be incorrectly removed in the computation. But just slightly smaller values are helpful for removing plenty of irrelevant polygons in the computation.

Table 1 presents the process of deriving the corresponding polygon $A\prime$ with the filled closed component of the exterior boundary of a polygon A. There are operations for different spatial configurations where A is a simple polygon, a polygon with holes such as A in Figure 5(a), or a complex polygon such as A in Figure 5(d). In a spatial configuration where a closed component of A’s exterior is bounded by A’s exterior boundary such as B enclosed by A in Figure 5(d), the BE-16IM by Wang (2023) is employed to identify the closed component to further derive $A\prime .$

Table 1. The process of deriving the corresponding polygon $A\prime$ with filled closed component of exterior boundary of a polygon A.

1. If A is a simple polygon without interior boundary, let $A\prime$ be A, return $A\prime .$

2. If A is a polygon with holes, that is, with interior boundaries, create a simple

polygon $A\prime$ with the exterior boundary of A. Return $A\prime .$

3. If A is a complex polygon, for each component Ai of A, run step 1 and 2,

to get Ci. Then, union all Ci to produce C.

Since it is possible that a closed component of the exterior of C is bounded by

the exterior boundary of C, the following steps are carried out.

3.1 Create a simple polygon with the bounding box of C. Let it be D.

3.2 Get the difference between D and C to be E.

3.3 For each component Ei of E, check with the BE-16IM by Wang (2023)

whether Ei is an enclave surrounded by C. If yes, keep Ei.

Union all Ei where Ei is an enclave surrounded by C to produce F.

3.4 Union F and C to produce $A\prime .$ Return $A\prime .$

The BE-16IM is key to derive $A\prime$ in a spatial configuration such as Figure 5(d) because it is able to identify enclaves that do not sit in the holes of other polygons. As shown in Figure 5(d), A has no interior boundary because a valid polygon cannot have an interior boundary that touches the exterior boundary at more than one point (Gillies 2024). Therefore, there is no hole in A. Since the four components A₁, A₂, A₃ and A₄ connect at a finite number of points and their exterior boundaries form a closed point set where B sits, B is an enclave surrounded by A. Before the hierarchical relation conforming to geometric enclosure between B and A can be generated, $A\prime$ shown in Figure 6(d) should be derived according to A. However, with the exterior boundary of A, the polygon bounded by the exterior boundary of A will be generated as C in Figure 6(a), which is the same as A. Therefore, extra steps employing BE-16IM are important to derive $A\prime .$ According to the steps shown in Table 1, a polygon D of which the exterior boundary is the bounding box of C is generated, as shown in Figure 6(b). Then, the difference between C and D is generated as E which consists of E₁, E₂, E₃, and E₄, shown in Figure 6(c). Next, according to the definition of enclaves based on BE-16IM, each component of E is tested whether it is an enclave surrounded by C. As a result, only E₄ is an enclave surrounded by C. Therefore, E₄ and C are unioned to derive $A\prime$ which is shown in Figure 6(d).

Figure 6. Polygons generated to derive $A\prime$ for Figure 5d.

3.2. Inferring hierarchical relations in both contexts

Using the algorithm in Section 3.1, a hierarchy conforming to geometric enclosure can be established. Then, the hierarchy conforming to geometric enclosure and the hierarchy conforming to administrative reach can be combined to infer hierarchical relations in both contexts. In either the hierarchy conforming to administrative reach or the hierarchy conforming to geometric enclosure, hierarchical relations can be inferred with transitivity, that is, if B is in A, and C is in B, then C is in A. However, in the combined hierarchies, when contexts are not resolved, whether the hierarchical relations conform to administrative reach or geometric enclosure is undetermined. Therefore, the resolution of contexts needs to be carried out before hierarchical relations can be inferred with direct match or with transitivity.

As shown in Figure 8, the hierarchy conforming to administrative reach and the hierarchy conforming to geometric enclosure are differentiated with inAdmin and inBond relations. Both hierarchies are combined to represent both contexts. The disambiguation of both contexts is facilitated by hierarchical relations in specific contexts. For example, combine a and b in Figure 3 into Figure 9. The solid edges and dashed edges represent inBond and inAdmin relations, respectively. Because there are no hierarchical relations among A, B, and C according to administrative reach, no dashed edges are linking them.

Given a condition that B is in A, infer whether C is in A? According to the given condition, the context of it can be disambiguated with the combined hierarchies, that is, the hierarchy with inBond relation. Then, according to the hierarchy with inBond relation, inBond relation between C and A can be reasoned. Therefore, the answer to the inferring question is Yes, C is in A.

For the disambiguation of both contexts, polygons with different parents in both hierarchies are important. If using polygons with the same parents in both hierarchies, the disambiguation of both contexts will be invalid. Similarly, this inferring algorithm is important for polygons with different parents in both hierarchies. For polygons with the same parents in both hierarchies, there is no need to disambiguate contexts. In the following section, a definition of inconsistency will be introduced to capture conflicting results of inferring hierarchical relations in both contexts.

For a formal proof of the correctness of the algorithm, let us consider all possible edges to which a node links in the combined hierarchies. A node in the combined hierarchies must be in one of the following four linking states, that is, linking to none edges, linking to a single edge of inAdmin relation, linking to a single edge of inBond relation, and linking to both edges of inAdmin and inBond relations. Suppose a given condition referring to a polygon with the same parents of both inAdmin and inBond relations or without any parents of inAdmin and inBond relations, then contexts cannot be disambiguated according to the given condition. Otherwise, if the given condition refers to either inAdmin or inBond relation of a polygon with different parents, contexts can be determined as inAdmin or inBond relation correspondingly. Based on the result of context disambiguation, the hierarchical relation in question can be inferred by searching the corresponding inAdmin or inBond relation between the polygons in question. If the polygons in question do not link according to the disambiguated context of inAdmin or inBond relation, then the answer to the inferences of inAdmin or inBond relation about the polygons in question should be No. Otherwise, the answer should be Yes.

3.3. A Definition of inconsistency

This research defines inconsistency as different end nodes in both hierarchies for the same start nodes by the same depth. The depth equals the number of edges between two nodes. For example, according to geometric enclosure, the depth from C to A in Figure 9 is two. However, according to administrative reach, the node with depth two from C is not A because there are no edges or paths of inAdmin relations that are linking C and A. Therefore, there is inconsistency in inferring the relation between A and C.

4. Testing on prototypical and real data

This research applies the proposed definitions and algorithms to both self-created prototypical data and real data to test their performance.

4.1. Testing on prototypical data

The section establishes a hierarchy conforming to administrative reach and a hierarchy conforming to geometric enclosure on a self-created prototypical data set.

4.1.1. Data

The self-created data is shown in Figure 10. Polygons A in dark gray, E in dim gray, and D in black are not part of each other. E consists of B and C. While C’s exterior boundary should equal the interior boundary of B if its boundary is aligned, the misaligned boundary of C causes a tiny blank area and a tiny overlapping area between B and C. According to administrative reach in politics, there should not be overlapping between units. Suppose there is a second corporational administrative reach that causes overlaps between D and B, and D and A. But among A, B, C, D, and E, there are only two hierarchical relations conforming to administrative reach, that is, B administrated by E and C administrated by E.

4.1.2. Results

According to the algorithm in Figure 7, in step 1, polygons A, B, C, D, and E are all congregated into set U. Then, in step 2, polygons with filled closed components of exterior boundaries corresponding to A, B, C, D, and E are derived in set $U\prime$ which includes $A\prime ,B\prime ,C\prime ,D\prime ,$ and $E\prime$ shown in Figure 11. Next, to determine the values of α and β, AR(U) which equals min(AR(B), AR(C)) and OR(U) which equals min(OR(B), OR(C)) are calculated because there are only two hierarchical relations conforming to administrative reach, B administrated by E and C administrated by E. As a result, both α and β are set as 1.0, no larger than AR(U) and OR(U) which are 1.14285714 and 1.0, respectively. After the spatial index on set $U\prime$ is built in step 3, the possible parent node of each polygon in set U is searched iteratively according to steps 4.1 to 4.4. For example, for B in U, the set of candidate polygons in step 4.1 includes A because $A\prime$ intersects with B, C because $C\prime$ intersects with B, D because $D\prime$ intersects with B, and E because $E\prime$ intersects with B. In step 4.2, C and D are removed from the set of candidate polygons, because the area of $C\prime$ is smaller than α of the area of B, and the area of $D\prime$ is smaller than α of the area of B. According to step 4.3, E is selected, because the area of $E\prime$ is smaller than the area of $A\prime ,E\prime$ overlaps B with more than β of the area of B, and E is not administrated by B. So in step 4.4, a hierarchical relation conforming to geometric enclosure is derived between E and B. E is the direct parent of B.

Figure 7. The process of building hierarchy conforming to geometric enclosure.

After running the algorithm in Figure 7, hierarchical relations conforming to geometric enclosure are derived on the prototypical data, which are C in B, B in E, E in A, and D in A. The combined hierarchies conforming to both administrative reach and geometric enclosure are stored in a graph database in Neo4j Desktop 1.4.5, a graph database platform (Neo4j, Inc 2020), and shown in Figure 12. Each direct hierarchical relation in the hierarchy conforming to administrative reach is an In_admin edge connecting from a child node to its parent node. Similarly, each direct hierarchical relation in the hierarchy conforming to geometric enclosure is an In_bound edge connecting from a child node to its parent node.

According to the results, only B has the same parent node E conforming to both administrative reach and geometric enclosure. Therefore, it is important to determine the context of hierarchical relations so that hierarchical relations of A, C, D, and E can be inferred consistently. For example, given a condition that E is in A, infer whether C is in B? Since E is in A according to the hierarchical relation conforming to geometric enclosure, then the context of the inference is geometric enclosure. Therefore, C is in B. Otherwise, given C is in E, infer whether E is in A? Since C is in E according to the hierarchical relation conforming to administrative reach, then the context of the inference here is administrative reach. Therefore, E is not in A.

4.2. Testing on real data

This section establishes a hierarchy conforming to administrative reach in politics and a hierarchy conforming to geometric enclosure on a real data set. Both hierarchies are combined to infer hierarchical relations and quantify the inconsistency in inferring hierarchical relations with negligence of different contexts.

4.2.1. Data

The data used in this section is the database of global administrative areas (GADM) version 3.6 (GADM 2018). Six shapefiles for the world were downloaded, each for one level of subdivision. In total, there are 386735 administrative units. The granularity of the six shapefiles named gadm36_0, gadm36_1, gadm36_2, gadm36_3, gadm36_4, gadm36_5 is from low to high in order. While gadm36_0 includes 256 administrative units at the country or independent administrative unit level, gadm36_5 only includes administrative units administrated by two countries, that is, France and Rwanda, at the commune or village level. Therefore, for some administrative units at a level with lower granularity than gadm36_5, there is no further subdivision at a level with higher granularity.

For each administrative unit in the data set, there is a unique identifier in their attributes, which can be used to identify the administrative unit. The attribute names of unique identifiers for administrative units in gadm36_0, gadm36_1, gadm36_2, gadm36_3, gadm36_4, gadm36_5 are GID_0, GID_1, GID_2, GID_3, GID_4, GID_5, respectively. Also, there are names of the administrative units. Besides, for administrative units A_i ($i=1,2,\dots ,n$) at levels with higher granularity than gadm36_0, the non-spatial attributes of the administrative units administrating A_i are included in the attributes of A_i. Therefore, the hierarchical relations conforming to administrative reach can be extracted according to the non-spatial attributes of the administrative units in the data set.

4.2.2. Results

With GADM, a hierarchy conforming to administrative reach is established, which consists of 386735 administrative units and 386479 direct hierarchical relations, according to the non-spatial attributes of administrative units. Meanwhile, with GADM, a hierarchy conforming to geometric enclosure is established, which consists of 386735 administrative units and 386482 direct hierarchical relations, according to the algorithm introduced in Section 3.1. An exception occurs to the geometry with GID_3 CAN.8.1.2_1. No parent node is found for it with the algorithm of building hierarchy conforming to geometric enclosure during a long period of time because of its complex geometry. As a result, ArcGIS 10.8.1 is employed to implement some geometric computation including projection, intersection, and area calculation to determine the parent node. With regard to thresholds α and β in the algorithm, α is set as 0.9 and β as 0.95, slightly smaller than AR(U) and OR(U) which are 0.99999347 and 0.99996538, respectively.

Both hierarchies are stored in a graph database in Neo4j Desktop 1.4.5. In the graph database, each administrative unit is a node with attributes gid, name, and type, and a label representing their administrative level. Each direct hierarchical relation conforming to administrative reach is an In_admin edge, and each direct hierarchical relation conforming to geometric enclosure is an In_bound edge connecting from a child node to its parent node in the hierarchy. Since the gid of each administrative unit is unique, to combine both hierarchies conforming to administrative reach and geometric enclosure, nodes with the same gid values in both hierarchies are merged into one node. Therefore, in the combined hierarchies, there are 386735 nodes and 772961 edges.

As shown in Figure 13, a part of the combined hierarchies in France are visualized. There are edges in yellow and grey representing In_bond and In_admin relations between nodes representing administrative units in ordered administrative levels. Each edge starts from a node A to a node B which administrates or has an exterior boundary enclosing the exterior boundary of A. While plenty of the edges in yellow and grey have the same start nodes and end nodes, a few edges do not. Among these edges, inconsistency due to different contexts occurs.

As shown in Figure 14, the node representing San Marino has no In_admin relation to end to another node because San Marino is a country, not administrated by any other country or independent administrative unit. However, the node representing San Marino has an In_bond relation to end to the node representing Italy because the exterior boundary of the polygon representing Italy encloses that of San Marino. As a result, in the context of geometric enclosure, the inferring result that San Marino is in Italy is correct, which conflicts with the inferring result in the context of administrative reach. Similarly, Waskatenau, a village surrounded by Smoky Lake County which does not administrate Waskatenau in Canada, has no In_admin relation but an In_bond relation to Smoky Lake County. In the context of geometric enclosure, the inferring result that Waskatenau is in Smoky Lake County is correct. But in the context of administrative reach, Waskatenau is in Smoky Lake County is incorrect. In terms of applying the algorithm of inferring hierarchical relations with combined hierarchies in Figure 8, for instance, given a condition that San Marino is in Italy, infer whether Waskatenau is in Smoky Lake County? Since San Marino is in Italy according to the hierarchical relation conforming to geometric enclosure, then the context of the inference is geometric enclosure. Therefore, Waskatenau is in Smoky Lake County. Otherwise, given San Marino is not in Italy, infer whether Waskatenau is in Smoky Lake County? Since San Marino is not in Italy according to the hierarchical relation conforming to administrative reach, then the context of the inference here is administrative reach. Therefore, Waskatenau is not in Smoky Lake County.

Figure 8. The process of inferring hierarchical relations with combined hierarchies.

Figure 9. The combined hierarchies. The solid edges represent inBond relations. The dashed edges represent inAdmin relations.

Figure 10. The prototypical data. A complex polygon A in dark gray consisting of four triangles surrounds a polygon with a hole B in dim gray which surrounds a polygon C in dim gray. B and C comprises E. A polygon D in black overlaps B and A.

Figure 11. Polygons in $U\prime .$

Figure 12. The combined hierarchies on prototypical data.

Figure 13. A part of the combined hierarchies in visualization. The nodes in pink, yellow, green, blue, purple, and grey represent administrative units in levels one, two, three, four, five, and six, respectively, in granularity ordered from high to low. The edges in yellow and grey represent In_bond and In_admin relations, respectively.

Figure 14. An example of inconsistency in the hierarchical relations in different contexts. The nodes in pink, yellow, green, and blue represent administrative units in levels one, two, three, and four, respectively. The edges in yellow and grey represent In_bond and In_admin relations, respectively.

The hierarchical relation is transitive, that is, a start node and an end node on a directed path of a hierarchy also have hierarchical relation. The number of edges on the path is called the depth. It is possible that hierarchical relations starting from a node can go along several edges and end at different nodes in different contexts. Therefore, this research computed inferring results for each node in the combined hierarchies in all possible depths from zero to five. The number of inconsistency in depths from zero to five is 3, 3148, 4143, 4357, 1486, and 33. Therefore, there will be 13170 pairs of conflicting results inferred in different contexts. But if any hierarchical relation from the 13170 pairs of conflicting results is given as a condition, another hierarchical relation will be consistently and correctly inferred with the combined hierarchies because a resolution of contexts can be carried out to infer the subsequent hierarchical relation.

5. Discussion

This research tackles the inference of hierarchical relations in different contexts with combined hierarchies conforming to both administrative reach and geometric enclosure. Although there is existing literature on hierarchies in different fields such as architecture and graphic design, spatial reasoning of hierarchical relations is mainly studied in GIScience. However, the spatial reasoning of hierarchical relations in GIScience based on containment relation according to 9IM only conforms to administrative reach, which is unable to represent and infer some hierarchical relations in NLD, such as ones between enclaves and their surrounding places. Therefore, the novel hierarchy conforming to geometric enclosure defined in this research helps to incorporate hierarchical relations with the enclosure of exterior boundaries. Moreover, by employing hierarchical relations with the same start nodes but different end nodes in both hierarchies conforming to administrative reach and geometric enclosure, the resolution of contexts can be carried out before spatial reasoning of hierarchical relations in either hierarchy, which eliminates inconsistent reasoning results if different contexts are neglected. Although this research is inspired by the NLD of hierarchical relations in different contexts, this research concentrates on the modeling, resolution, and consistent inference of hierarchical relations in different contexts. The transitivity and consistent expressions of hierarchical relations in NLD in different contexts are newly validated in the research by Wang and Zhang (2023). While further research can be carried out to combine the proposed algorithms with NLD samples, algorithms proposed in this research should be able to infer hierarchical relations in different contexts more consistently than existing algorithms based on hierarchical relations in a single context.

Although spatial reasoning with inconsistency has been studied in GIScience (Moar and Bower 1983, Chen et al. 2015, Hua et al. 2018), this research originally explores the spatial reasoning with inconsistency caused by different semantics of hierarchical relations, instead of the inconsistency caused by indeterminacy on information based on single semantics of hierarchical relations. Therefore, the accuracy of question answering of hierarchical relations can be improved based on this research in cases of different contexts of hierarchical relations. In addition, in terms of the applicability of the algorithms proposed in this research, they can process general polygons consisting of single or multiple components, with or without interior boundaries. The algorithms are also directly applicable to data with both aligned and misaligned boundaries.

Although this research defines the hierarchy conforming to administrative reach to represent the hierarchical relations according to ordered containment of point sets based on the 9IM, the proposed method can be generalized to other hierarchies and applications. For example, assume that some investors want to build places in the countrysides which are adjacent to lakes and in or enclosed by forests so that people can go on a retreat there. The hierarchical relations according to ordered containment of point sets based on the 9IM can be defined for land parcels in the same land cover types in ordered granularities. But the hierarchical relations conforming to geometric enclosure can still be defined according to the enclosure of exterior boundaries for land parcels in different land cover types in ordered granularities. Without the proposed method, hierarchical relations conforming to geometric enclosure between the countrysides and forests and between the lakes and forests cannot be found in spatial queries. But with the proposed method, such hierarchical relations can be found in spatial queries and further analyzed to meet the investors’ needs.

Another example, among the conflicts occurring in multiple cadastral systems, some may be due to the neglection of different contexts instead of wrong information. For instance, hierarchical relations between places in the context of environment protection may be different from those in the context of economic development. To resolve such conflicts, the resolution of different contexts, that is, multiple cadastral systems, is important. The proposed method sheds light on resolving such conflicts. Each cadastral system can be established as a hierarchy according to ordered containment of point sets based on the 9IM in terms of different socio-economic factors or time frames. The multiple cadastral systems can be merged into combined hierarchies. Then, with a search of the conflicting hierarchical relations in the combined hierarchies, the contexts can be identified. If in all contexts, only one of the conflicting hierarchical relations exists, then the conflict is caused by the wrong information. Even in a single cadastral system, the combined hierarchies conforming to ordered containment of point sets based on the 9IM and geometric enclosure can be helpful. For example, land parcels enclosed by but not part of other land parcels can be identified with the proposed method for urban planning. Such land parcels have hierarchical relations with other land parcels enclosing them according to geometric enclosure. Meanwhile, such land parcels have hierarchical relations with other land parcels of which they are part according to ordered containment of point sets based on the 9IM.

Although this research uses polygons to establish the hierarchies, it bridges the gap between different hierarchical relations expressed in NLD and the modeling of hierarchical relations in different contexts in computers. Moreover, the combined hierarchies can be enriched with information from NLD in the future. Hierarchical relations expressed in NLD can be differentiated according to different contexts by humans and added to the combined hierarchies to provide better query results. For example, in history, West Berlin was an enclave within East Germany. But now West Berlin has disappeared and there is no polygon representing it in the GADM dataset. The hierarchical relation conforming to geometric enclosure between West Berlin and East Germany can be added to the combined hierarchies manually. With enriched information, the algorithm of resolving contexts proposed in this research can have more applications as well. For example, given Hong Kong was inside Britain, infer whether Macao was inside Portugal. A resolution of chronological contexts is needed before a correct answer can be produced for the question. While Hong Kong is part of China now, Hong Kong was administrated by Britain from 1959 to 1997. At that time, Macao was administrated by Portugal, while Macao has been returned to China now. According to the conflicts between the hierarchical relation between Hong Kong with China and Hong Kong with Britain, the chronological context of the question can be determined as the one of Hong Kong with Britain. Then, the inference result can be generated as Yes according to the determined chronological context. In addition, the algorithm of resolving contexts proposed in this research may be employed to develop algorithms of retrieving hierarchical relations in NLD in different contexts automatically. The automatically retrieved hierarchical relations can be added to the combined hierarchies as well.

However, there are limitations of this research which call for future work. Firstly, the algorithms of establishing the hierarchy conforming to geometric enclosure and of inferring hierarchical relations in different contexts are applied to only places that are administrative units and represented as polygons in this research. However, places that are not administrative units or not represented as polygons may be dealt with in the future. For example, to establish a hierarchy conforming to geometric enclosure with places represented as lines, the hierarchy can be defined as a graph of nodes connected with directed relations where a node representing line A_i ($i=1,2,\dots ,n$) is a child of another node representing line A if the interior of A_i is completely enclosed by the interior of A. Secondly, in the algorithm of establishing the hierarchy conforming to geometric enclosure, this research defined OR and AR to help derive the direct hierarchical relations conforming to geometric enclosure. If the misaligned boundaries of polygons in the data set are aligned before deriving the hierarchical relations, the values of OR and AR can be set to one directly without extra computation. Thirdly, this research defined a novel hierarchy conforming to geometric enclosure to resolve the contexts of administrative reach and geometric enclosure. Other hierarchies such as classification systems are not incorporated but may be defined and resolved in future research. Lastly, this research focuses on the spatial reasoning of hierarchical relations in different contexts. Future work can be done to tackle the inference of more spatial relations such as directional relations with topological relations in different contexts.

6. Conclusion

Hierarchy is an important concept in different fields. In GIScience, spatial reasoning of hierarchical relations has been researched, but the issue of conflicting hierarchical relations due to different contexts remains. Existing literature focuses on a single kind of hierarchical relations based on 9IM, and falls short of incorporating a second kind of hierarchical relations conforming to geometric enclosure and consistently inferring hierarchical relations in different contexts. This research defined a novel hierarchy conforming to geometric enclosure and proposed an algorithm to establish combined hierarchies conforming to administrative reach and geometric enclosure. Moreover, this paper proposed an algorithm of inferring hierarchical relations in different contexts where contexts are differentiated by using hierarchical relations with the same start nodes but different end nodes in the combined hierarchies. With a data set of 386735 polygons representing administrative units, this paper established corresponding combined hierarchies and identified 13170 pairs of conflicting hierarchical relations which can be employed to resolve different contexts for consistent inference of hierarchical relations. In the future, this method may be generalized to other places such as banks represented as points in data sets, or to other contexts to represent hierarchical relations that are conflicting if contexts are neglected, and to infer hierarchical relations consistently when a hierarchical relation marking a specific context is provided. In addition, further work may be carried out to establish combined hierarchies for places without crisp boundaries or in rasters to reduce the inconsistency in spatial reasoning of hierarchical relations due to different contexts.

Acknowledgement

The author acknowledges the comments from anonymous reviewers and editors for helping improve the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data and codes availability statement

The codes developed in this research are made available on the Figshare (https://doi.org/10.26188/20448231)

Additional information

Funding

This work is funded by the Melbourne Research Scholarship and PhD Write Up Award.

Notes on contributors

Xiaonan Wang

Xiaonan Wang is a PhD student at the University of Melbourne. Her research interests are spatial relations and semantics. Her contribution to this paper includes research design, implementation, draft writing, and revision.