In a two-dimensional geographic space, a fuzzy set is a subset of the two-dimensional Euclidean space. Each point in the space has a membership value that represents the degree to which this location point belongs to a given object. The membership values of fuzzy objects with indeterminate boundaries are always larger than 0 and smaller than 1. Hence, for a given fuzzy object, the corresponding membership values can be represented as a field. Because the fuzzy object itself is conceptualized based on an original field (e.g., the temperature field), the membership field can be viewed as a mapping from the original field to the range

$[0,1]$. Let the original field be

$f:x\to v$, then the membership field can be formalized as:

According to map algebra, operations on raster data can be categorized into the following four groups: local, focal, zonal, and global [

40]. Because raster is a type of field representation of the geographic space, we can generalize the same categories of operations to be applied to field data. Additionally, the conceptualization can either be a one-step or multi-step process. In the former case, objects are identified directly based on the field values, whereas for the latter, lower-order objects need to be identified first and higher-order objects are conceptualized based on the extracted lower-order objects (e.g., trees versus a woodland). Taking into account different field operations and conceptualization processes, we identified five categories of fuzzy regions: direct field-cutting objects, focal operation based field-cutting objects, element-clustering objects, object-referenced objects, and dynamic boundary objects. We also constructed a conceptual membership function for each category of fuzzy regions.

#### 3.1.1. Direct Field-Cutting Objects

A direct field-cutting object (DFCO) is the simplest type of fuzzy object. DFCOs can be identified directly based on the attribute values of a field. The membership function for a direct field-cutting object can be defined as:

where

g is the membership value at location

$(x,y)$, obtained by applying an object identification function

m on the original field

f. Obviously,

m is a local operation, meaning that the membership value of

$(x,y)$ only depends on the attribute value of this particular location in the original field. The attribute values of neighboring locations have no impact on the membership value of

$(x,y)$. In the geographic space, a plateau (

Figure 3a) is a typical direct field-cutting object. The degree to which a certain location belongs to a plateau is only determined by its elevation. For example, the membership value of a location belonging to the Tibetan Plateau (

$\mu T$) is a function of the elevation of this location, as shown in

Figure 3b. Hence, in order to delineate the boundary of the Tibetan Plateau, we need to obtain a cut set according to a pre-defined threshold (e.g., 0.5) of the membership degrees. The result is equivalent to cutting the original field directly based on the local operation

m. This is why we named this category of fuzzy regions “direct field-cutting objects”. Another DFCO example is climatic zones, such as the semi-tropical zone. A climatic zone is determined on the basis of the average temperature over time. In practice, many non-spatial concepts can be characterized using membership functions similar to DFCOs, such as “tall”, “large”, and so on. These concepts only depend on the attribute values of the original field.

#### 3.1.2. Focal Operation-Based Field-Cutting Objects

A focal operation-based field-cutting object (FoFCO) is slightly more complicated than DFCOs because the identification of such objects requires focal operations. When conducting a focal operation, the membership degree of a location not only depends on the attribute value of this specific location, but also relies on the values in a regularly-shaped neighborhood in the original field. Focal operations are very common in raster data processing. A typical focal operation is calculating slopes from a grid DEM dataset. There are two crucial parameters in a focal operation: the definition of the neighborhood and the operator to be executed on the original field. The neighborhood type can be a rook, bishop, or queen, whereas the operator can be a sum, maximum, median, and so on (Shekhar and Chawla 2003). The membership function for this type of fuzzy region is defined as follows:

where the neighborhood and the operator are denoted by

N and

$Oper$, respectively.

In ecological science, topographic aspect is an important factor that influences a species’ spatial distribution. Aspect values can be calculated by applying a focal operation to a DEM field. Furthermore, we can identify and extract vague regions such as “south slope area” (

Figure 4) and “east slope area” based on the aspect values. Their spatial extents are indeterminate due to the inherent vagueness of the concept “south” or “east”. These vague regions are typical examples of FoFCOs. Similarly, a “steep slope area” is also a type of fuzzy object in this category.

Here, we use “south slope” as an example to demonstrate how Equation (4) can be instantiated and customized in practice. Assume that a grid DEM is represented by a function

$f(u,v),u,v\in \mathbf{N}$, with the top-left point as the origin,

$(0,0)$. For a given location

$(x,y)$ in the study area, its queen neighbourhood can be defined as:

and the aspect represented by its compass direction (i.e., the value of north is 0) can be computed using:

Note that there are other algorithms which can be used to calculate aspect values. Here, we adopt the method from [

41] as an example. We can then define a trapezoid membership function (

Figure 4b) for the direction “south” as follows:

where the aspect values can be calculated using the

$Oper$ function (i.e., Equations (6)–(9)).

#### 3.1.3. Element-Clustering Objects

An element-clustering object (ECO) is defined as an object consisting of several smaller objects. An ECO and its smaller components thus form a part-whole relationship, similar to the ontological relation between stars and galaxies in astronomy. Generally, the “part” objects should satisfy the following conditions: (1) They should be easily identifiable in human cognition or in machine pattern recognition; (2) They belong to the same category (e.g., all trees) so all component objects play an equal role when forming the “whole” object; and (3) Two neighboring objects should be close enough to be considered in the same cluster.

According to Gestalt psychology, human beings tend to group similar items together. An ECO can therefore be viewed as an instance of a Gestalt, as these component objects are usually similar to each other (

Figure 5a). For an ECO, the spatial distribution of the component objects determines its boundary. Even though the boundaries of the component objects are often determinate, the boundary of the entire ECO is indeterminate and vague. This vagueness mainly comes from the “gaps” between the component objects inside the ECO (e.g., the grassland between trees in a forest). Although many analytical tools can be applied to approximate an ECO’s boundary, it still remains an open question whether or not these gaps should be included when defining an ECO.

As mentioned earlier, a piece of woodland (or forest) is a typical ECO (

Figure 5b), in which the trees in the woodland are the element (component) objects. The membership function of an ECO can be defined as:

where

S stands for a set of the component objects, and

m determines the membership value at location

$(x,y)$ based on the spatial distribution of

S (e.g., the density of trees). Note that the choice of the clustering algorithm will inevitably influence the resulting membership function. In most cases, the membership degree is positively correlated with the density of the element objects, which can be calculated by different density measurements, such as the kernel density estimation (KDE). In addition, parameters of the applied density measurement (e.g., the bandwidth of KDE) can also impact the extracted ECOs. Given that this paper focuses on proposing a conceptual framework of membership functions, the choice of the clustering algorithm can be determined by researchers based on their practical needs.

Because the element objects are identified from the original field, we have:

where

$f(u,v)$ is the original field, and function

c identifies the component objects from

f at location

$(x,y)$. A typical example of

c is the extraction of buildings from remote sensing imagery. As can be seen, the membership values in Equation (11) eventually depend on the attribute values of the original field. Compared to FoFCOs, ECOs are based on zonal operations because identifying a component object requires a search in an irregular neighborhood in the original field. Note that the component objects of an ECO can include not only physical objects such as buildings and trees, but also human activities such as “drinking” or “shopping”. A typical example is to extract “nightlife districts” from social media check-in data [

42].

Previous research has demonstrated that many geographic phenomena are scale-dependent [

43]. For ECOs, another factor that should be considered is the size of the measurement unit (to avoid confusion, we do not use the term “scale” here). If the measurement unit is too small (e.g., smaller than the size of gaps between element objects), it is difficult to extract interesting distribution patterns since the granularity is too fine. Similarly, we may lose too much detail if the size is excessively large. Therefore, we need to apply an appropriate measurement size when extracting ECOs, which depends on the sizes of the component objects and the sizes of the gaps between them.

Additionally, a structurally more complex field will naturally lead to more complex membership functions, so that an extracted ECO from this field may have holes and inner boundaries [

44]. The boundary of the holes can be either crisp or vague. Assume that there is an ECO,

${O}_{1}$, consisting of a number of type

I component objects,

${O}_{2}$. Meanwhile, there is another type

II object,

${O}_{3}$, inside

${O}_{1}$, and thus causes gaps between element objects

${O}_{2}$. In general, whether

${O}_{3}$ can be considered an inner boundary of

${O}_{1}$ depends on the size of

${O}_{3}$. If

${O}_{3}$ is big enough and breaks the “continuity” of

${O}_{1}$, we should consider

${O}_{3}$ as a hole that creates an inner boundary for

${O}_{1}$. For example, assume that

${O}_{1}$ is a piece of woodland and

${O}_{3}$ is a large lake. From the point of view of the lake, the inner boundary is crisp; however, from the point of view of the woodland, the boundary is vague. We suggest considering the vagueness of the inner boundaries in the conceptualization process so that the membership function can be constructed in a consistent way for both inner and outer boundaries.

#### 3.1.4. Object-Referenced Objects

An object-referenced object (ORO) belongs to another category of vague objects that require a multi-step conceptualization. First, we need to identify a reference object from the original field. Second, we extract the target object based on its (qualitative) spatial relation to the reference object. There are three common types of qualitative spatial relations: topological, cardinal direction, and qualitative distance. They have been studied extensively in GIScience and qualitative spatial reasoning (QSR) [

45,

46]. For each type of qualitative spatial relation, researchers defined a set of jointly exhaustive and pairwise disjoint basic relations to support the algebraic operations, such as overlap (topological relation), north-east (cardinal direction relation), and close (qualitative distance relation). Many vague regions are identified based on a reference object and a qualitative spatial relation, such as the Bay Area (based on the topological and distance relations to the San Francisco Bay), the Far East (based on the qualitative distance relation to Europe), and northern and southern California.

Figure 6a shows the synthetic spatial view of membership changes for southern California and northern California purely based on the internal cardinal direction and distance to the borders [

47], while

Figure 6b shows the corresponding vague cognitive regions using social media data [

14]. Darker colors represent a higher degree of membership, and vice versa. The cognitive vagueness of the border between northern and southern California comes from multiple factors, including not only the different interpretations of the internal cardinal directions within California [

47], but also socioeconomic and cultural factors.

The vagueness of an ORO comes from two aspects—the vagueness of the reference objects and the vagueness of the spatial relations. Firstly, if the reference object is vague, the target object is inevitably vague. Secondly, spatial relations, except for topological relations, are inevitably vague. For example, we cannot precisely delineate the boundary between “southern California” and “northern California”, or differentiate between “close to home” and “far away from home” in the metric space. Previous literature addressing this type of fuzziness tries to integrate the fuzzy set theory and QSR in a semi-quantitative way [

48,

49,

50]. In other words, the membership degree of a location being included in an ORO is equivalent to the degree of the relation between this location and the reference object being categorized as a certain type of spatial relation. The membership function for an ORO can be written as:

where

O stands for the reference object and

R is the spatial relation between the reference object and the target object in the ORO. Because

O is usually identified from a field, it can be defined as:

where

C is a function representing the conceptualization of

O from the original field. Although ECOs and OROs are both vague regions that require a multi-step conceptualization, they are fundamentally different.

Figure 7 shows a class diagram demonstrating the relationships between first-order objects and second-order objects for ECOs and OROs, respectively. The similarity between the two is that an ECO is identified based on the spatial relation “closeness” (i.e., an element object should be merged with other element objects if they are close enough), whereas an ORO can involve other types of spatial relations, such as topological, directional, and qualitative distance relations.

#### 3.1.5. Dynamic Boundary Objects

We did not consider the time dimension in the above four categories of fuzzy regions; however, an object’s boundary can be indeterminate because it changes over time. We defined such objects as dynamic boundary objects (DBOs) and identified four types of changes for an areal object based on its location and shape: (1) discrete change (e.g., the merge or split of parcels); (2) simple movement without a change in shape; (3) movement with a change in shape; and (4) expansion or shrinkage without a significant location shift. A similar categorization can be found in [

27]. The latter three are all continuous changes. In these three cases, the spatial extent of a dynamic region is determinate at each time point. However, its position and boundary are indeterminate during a long time period. This type of temporal vagueness is also described in [

51]. A typical example of dynamic changing objects is a lake (

Figure 8). Since a lake expands and shrinks periodically, it is difficult to determine its exact boundary during a relatively long time period.

Therefore, a dynamic two-dimensional field can be modeled by a three-dimensional field (two spatial dimensions plus one temporal dimension) [

52]. The corresponding membership function is defined as:

where

$f(u,v,t)$ is a dynamic field and

C represents a procedure to extract objects from

f. To compute the membership degree of a location, a simple way is to calculate the proportion of the entire time period, during which this location is covered by a given dynamic object. For instance, if a location is covered by a seasonal lake for 100 days in one year, then we can assume that the membership degree associated with this location is

$100/365.25\approx 0.274$.