Modeling Words for Qualitative Distance Based on Interval Type-2 Fuzzy Sets

Abstract

Modeling qualitative distance words is important for natural language understanding, scene reconstruction and many decision support systems (DSSs) based on a geographic information system (GIS). However, it is difficult to establish the relationship between qualitative distance words and quantitative distance for special applications since the meanings of these words are influenced by both subjective and objective factors. Some existing methods are reviewed, and the Hao–Mendel approach (HMA) is improved to model qualitative distance words for four travel modes by using interval type-2 fuzzy sets (IT2 FSs), aiming at addressing the individual and interpersonal uncertainty among qualitative distance words. The area of the footprint of uncertainty (FOU), fuzziness (entropy), and variance are adopted to measure the uncertainties of qualitative distance words. The experimental results show that the improved HMA algorithm is better than the original HMA algorithm and can be used in spatial information retrieval and GIS-based DSSs.

1. Introduction

Spatial relations are important for geographical information systems (GISs), artificial intelligence (AI), decision support systems (DSSs) and image analysis [1,2,3,4,5]; and they are usually classified as topological relations, directional relations, and distance relations which are further divided into quantitative and qualitative distance relations. The qualitative distance relations describe the proximity between spatial objects using natural language [6], such as “The Tianjin Binhai International Airport is far away from the Tianjin Railway Station” and “Many hotels are near the Tianjin Railway Station”. In these statements, “far” and “near” are vague words that describe qualitative distance relations. From a linguistic point of view, a large number of linguistic distance concepts are available to qualitatively describe distance [7,8]. In some knowledge-based geospatial analysis and mining systems, the number of the required qualitative distance words depends on the level of granularity; e.g., using “far” and “near” at a rough level and “very close”, “close”, “medium”, “far”, and “very far” at a relatively fine level. In addition, another key issue is to determine the semantics of each qualitative distance variable, which is important for natural language understanding, scene reconstruction and many GIS-based DSSs [2,3]. Although natural language is gradually becoming an important data source for GISs [9] and some studies interpreted fuzzy semantics of natural language spatial relation (NLSR) terms using the fuzzy random forest (FRF) algorithm and fuzzy sets [5,10], most GISs currently cannot handle natural language effectively, and most linguistic descriptions about spatial relations cannot be used in current GISs. This issue has not yet been solved; therefore, how to integrate the spatial relationships described in natural language into a GIS is a challenge in the field of GIS [11]. The relationship between qualitative distance words in natural languages and quantitative distance must be established [12].

In the past 20 years, many scholars have performed much work to study the connotation of qualitative distance words, including “near”, “close to”, “adjacent”, and “surrounding” [13,14,15]. Different methods are used to establish the relationships between qualitative distance words and quantitative distance, and to build context-contingent translation mechanisms and computational models between linguistic proximity descriptors and metric distance measures [16]. However, establishing such a relationship is a very difficult task because it is influenced by many factors, such as the application scene (environment), context, psychological and physiological characteristics of the subject, culture, travel destination and travel mode [6,11,13,15,16,17]. From the psychometric testing of perceived proximity, Gahegan noted that the definition of proximity is related to five aspects [13]. Clementini et al. emphasized that the concept of proximity is context dependent [6]. Yao and Thill classified the context factors of proximity perception into objective and subjective factors [11]. Overall, the factors that affect the connotation of qualitative distance are very complex.

Due to the ambiguity of qualitative distance words, many people adopt fuzzy sets to establish the relationships between qualitative distance words and quantitative distance, and these methods generally fall into three types. The first type establishes the distance membership functions directly for the decision system based on the experience of experts [18]. In actual operation, the parameters of the distance membership functions should be adjusted according to the effect of the decision systems; thus, this type of method is very subjective. The second type establishes the connotation expression of the qualitative distances by mining Web text resources related to the qualitative distance descriptions [14,19]. The advantage of this approach is that the resources of the sampling are very rich, but it is difficult to define valid membership functions because different application scenarios may require different definitions. Another issue is that guaranteeing the authenticity and objectivity of Web text is difficult. For example, when real estate agents are advertising houses, the location of the house relative to other locations of importance can be falsified to promote sales. Therefore, even if a supermarket or school is four kilometers away from a house, the agents may say that the supermarket or the school is near the house. The third type investigates a specific population using questionnaires and obtains knowledge regarding their qualitative distances; a mapping relationship between qualitative distance words and quantitative distance is then established by means of statistics and regression [11,16,20,21,22]. Compared to the methods mentioned earlier, the methods of this type allow for the dynamic construction of context-contingent proximity models based on sample data. These methods are more suitable for GIS-based DSSs, as these systems are often oriented toward a particular population and a specific task. One problem that exists in the method of Yao and Thill [11,16] is that it is difficult to quantify these factors because many factors themselves are vague. At the same time, how these factors affect the meaning of qualitative distance words, which may involve the mechanism of cognition, is difficult to determine.

From the perspective of the semantic meaning of spatial vocabulary, the semantics of words that describe the spatial locations and spatial relationships of crisp or fuzzy objects are typically uncertain. Usually, a linguistic distance measure implies a different spatial range for different people. Figure 1 shows three classical fuzzy sets (i.e., type-1 fuzzy sets, T1 FSs) for the distance word “near” perceived by three people, indicating that the same words may have different meanings for different people in the real world. Moreover, there is some semantic uncertainty between these T1 FS models, i.e., interpersonal uncertainty of words [23,24]. Mendel [25] introduced six principles associated with evaluation methods used to obtain FS models for a word in computing with words (CWW). The T1 FS models of words violate principle three: “Words contain linguistic uncertainties, i.e., words mean different things to different people, and so the FS model that is used to represent a word must be able to incorporate both of these uncertainties”. The T1 FS has no ability to handle this kind of semantic uncertainty because the membership degree of a T1 FS is a single value.

The main goal of this study is oriented towards CWW; that is, the direct use of natural language words as objects of computation, based on interval type-2 fuzzy set (IT2 FS), is adopted to model qualitative distance words, and then the semantic uncertainty of distance words can be addressed. The Hao–Mendel approach (HMA) [25,26] is an effective means to estimate the fuzzy set after collecting data related to the words because it is the only method to-date that leads to normal interval type-2 fuzzy sets. However, HMA considers only the uncertainty of the upper membership function of an IT2 FS, not the entire uncertainty of the IT2 FS. To address this, the HMA algorithm is first improved by the entire uncertainty of IT2 FS, called the improved HMA, and then the IT2 FSs for distance words are established by the improved HMA in this study. The remainder of this article is organized as follows. In the second section, some basic concepts related to this article are introduced, such as IT2 FS and CWW. In the third section, the HMA algorithm is improved for distance words. In the fourth section, an experiment is conducted to verify the improved HMA algorithm and then the improved HMA algorithm is compared with the original HMA algorithm using three uncertainty indices. The results show that the improved HMA algorithm is better than the original algorithm. The characteristics of the qualitative distance models established using the improved HMA algorithm are then analyzed based on these three indices. Conclusions are presented in the fifth section. Appendix A is the questionnaire and all Abbreviations and their definitions used in this paper are listed in Appendix B.

2. Background

In this study, IT2 FSs are used to model distance words. This section introduces concepts such as CWW and IT2 FS. In addition, several uncertain indices related to qualitative distance words are introduced.

2.1. CWW

CWW is a methodology in which the objects of computation are words and prepositions drawn from a natural language. CWW was first introduced by Zadeh [27,28] and has been widely used in market investments, decision-making analyses, automatic control designs, fuzzy queries, data mining, and other fields. According to the existing literature, CWW techniques fall into three categories: membership-function-based models, symbolic linguistic computing models and 2-tuple linguistic models. Each category of models is unique in its advantages and limitations [29,30,31]. Li et al. proposed a new CWW framework based on the 2-tuple linguistic model [32]. Such a CWW framework allows people to address personalized individual semantics (PIS) to allow CWW to maintain the idea that words mean different things to different people.

2.2. IT2 FS

The T1 FS theories have been applied to many domains because of their ability to model fuzziness. However, the membership degree of a T1 FS is a single value, so T1 FS cannot manage the errors associated with the membership values of fuzzy objects, as shown in Figure 2a. Zadeh introduced the type-2 fuzzy sets and type-N fuzzy sets to overcome this flaw of T1 FS [28]. The primary membership function of a common type-2 fuzzy set (T2 FS) is a type-1 fuzzy set, which makes computation extremely difficult. Thus, the application of the T2 FS to a real task is comparatively difficult.

The IT2 FS is a special case of T2 FS and it uses an interval to express the primary membership value (Figure 2b). The IT2 FS is better than the T2 FS in complexity and computation, so it is adopted in this study. Many authors refer to interval type-2 fuzzy sets as type-2 fuzzy sets and add the qualifying term ‘generalized’ only when discussing non-interval type-2 fuzzy sets [33,34]. Fuzzifying the spatial data that contain certain types of errors is necessary and can achieve error containment. For example, geological, topographical and environmental parameters are required to assess regional debris flow hazards, and some parameters have vague boundaries. Therefore, the IT2 FS is a more reasonable choice to solve fuzzy geographical problems. The interval-valued fuzzy set (IVFS) is a special case of the IT2 FSs [35]. In this paper, the definition of IT2 FS introduced by Mendel is adopted.

Definition 1.

(Interval type-2 fuzzy set) [26]: an IT2-FS$\tilde{A}$in the universe$X\ne \varnothing$is given by:

$$\tilde{A}=\left\{((x,u),{\mu }_{\tilde{A}}(x,u)=1)|x\in X,u\in U\equiv [0,1]\right\}$$

(1)

where$U$is the universe of discourse for the secondary variable$u$. Note that$U$is a subset of [0, 1].

For the sake of convenience, the IT2 FS is represented as $\tilde{A}(x)=[A^{-}(x),A^{+}(x)](0\le A^{-}(x)\le A^{+}(x)\le 1)$, and $A^{-}(x)$ and $A^{+}(x)$ are the lower membership function (LMF) and upper membership function (UMF), respectively. The footprint of uncertainty (FOU) of an IT2 FS $\tilde{A}$ is the uncertainty in the primary memberships $\tilde{A}$, and it has always had the connotation of a geometric object bounded by a UMF and an LMF, as shown in Figure 2b.

The interval type-2 fuzzy set is adopted to model qualitative distance words as shown in Figure 2c. It is a normal, trapezoid interval type-2 fuzzy number that can be expressed as:

$$\tilde{A}=[A^{-}(x),A^{+}(x)]=[(a_1^{-},a_2^{-},a_3^{-},a_4^{-}),(a_1^{+},a_2^{+},a_3^{+},a_4^{+})]$$

(2)

where $A^{-}(x)$ and $A^{+}(x)$ are the LMF and UMF, respectively. They can be expressed as:

$$A^{-}(x)=\{\begin{array}{cc}\frac{x-a_1^{-}}{a_2^{-}-a_1^{-}}& a_1^{-}<x<a_2^{-}\\ 1& a_2^{-}\le x\le a_3^{-}\\ \frac{a_4^{-}-x}{a_4^{-}-a_3^{-}}& a_3^{-}<x<a_4^{-}\\ 0& otherwise;\end{array}$$

(3)

and

$$A^{+}(x)=\{\begin{array}{cc}\frac{x-a_1^{+}}{a_2^{+}-a_1^{+}}& a_1^{+}<x<a_2^{+}\\ 1& a_2^{+}\le x\le a_3^{+}\\ \frac{a_4^{+}-x}{a_4^{+}-a_3^{+}}& a_3^{+}<x<a_4^{+}\\ 0& otherwise;\end{array}$$

(4)

2.3. IT2 FS Uncertainty Measures

Three uncertainty indices of IT2 FSs are used in this study to measure the uncertainties of IT2 FS models corresponding to qualitative distance words: area of the FOU, fuzziness (entropy) and variance [36]. The fuzziness (entropy) is an uncertainty measure of the membership domain, and variance is an uncertainty measure of the Euclidean space domain.

2.3.1. Fuzziness (Entropy) of an IT2 FS

The fuzziness (entropy) of an FS can quantify the amount of ambiguity of a fuzzy set. Many definitions of fuzziness for IT2 FSs have been proposed [36,37,38].

Definition 2.

The fuzziness of an IT2 FS$\tilde{A}$proposed by Burillo and Bustince, denoted by$F_{BB}(\tilde{A})$, is defined as [37]:

$$F_{BB}(\tilde{A})={\displaystyle \sum _{i=1}^N}[A^{+}(x_i)-A^{-}(x_i)]$$

(5)

Definition 3.

The fuzziness of an IT2 FS$\tilde{A}$proposed by Wu and Mendel [36], denoted by$F_{WM}(\tilde{A})$, is the union of all cardinalities of its embedded T1 FSs$A_e$, i.e.,:

$$F_{WM}(\tilde{A})\equiv {\displaystyle \bigcup _{\forall A_e}}f(A_e)=[f_l(\tilde{A}),f_r(\tilde{A})]$$

(6)

where$f_l(\tilde{A})=\underset{\forall A_e}{min}f(A_e)$,$f_r(\tilde{A})=\underset{\forall A_e}{max}f(A_e)$, and$f(A_e)=1-\frac{1}{N}{\sum }_{i=1}^N|2A_e(x_i)-1|$. To compute$f_l(\tilde{A})$and$f_r(\tilde{A})$, let$A_{e1}$be defined as:

$$A_{e1}(x)=\{\begin{array}{cc}{A}^{+}(x)& A^{+}(x)isfurtherawayfrom0.5than{A}^{-}(x)\\ A^{-}(x)& otherwise\end{array}$$

(7)

and let$A_{e2}$be defined as:

$$A_{e2}(x)=\{\begin{array}{cc}{A}^{+}(x)& both{A}^{+}(x)and{A}^{-}(x)arebelow0.5\\ A^{-}(x)& both{A}^{+}(x)and{A}^{-}(x)areabove0.5\\ 0.5& other\end{array}$$

(8)

Then,$f_l(\tilde{A})=f(A_{e1})$, and$f_r(\tilde{A})=f(A_{e2})$.

2.3.2. Variance of an IT2 FS

The variance in a T1 FS $A$ measures its compactness; i.e., a smaller (larger) variance means that $A$ is more (less) compact.

Definition 4.

The variance$V(\tilde{A})$of an IT2 FS$\tilde{A}$is the union of relative variance of all its embedded T1 FSs$A_e$; i.e.,:

$$V(\tilde{A})\equiv {\displaystyle \bigcup _{\forall A_e}}{V}_{\tilde{A}}(A_e)=[v_l(\tilde{A}),v_r(\tilde{A})]$$

(9)

where$V_{\tilde{A}}(A_e)=\frac{{\sum }_{i=1}^N{[x_i-c(\tilde{A})]}^2{A}_e(x_i)}{{\sum }_{i=1}^N{A}_e(x_i)}$is the relative variance of an embedded T1 FS$A_e$relative to the IT2 FS$\tilde{A}$, and$c(\tilde{A})$is the average centroid of$\tilde{A}$, which can be calculated using the KM algorithms [39,40].$v_l(\tilde{A})=\underset{\forall A_e}{min}{v}_{\tilde{A}}(A_e),$$v_r(\tilde{A})=\underset{\forall A_e}{max}{v}_{\tilde{A}}(A_e)$are the minimum and maximum relative variances of all$A_e$, respectively.

3. Modeling Qualitative Distance Words

The HMA algorithm is used to model adverbs as IT2 FSs in the interval [0, 10], such as “very little” and “moderately”. It is composed of two parts: data processing and fuzzy sets. As discussed before, the semantic connotation of distance words may be affected by the travel mode; that is, different travel modes may have different spatial ranges. For example, generally, we estimate that the spatial range of the walking distance of ordinary human subjects is [0, 3] km, and the spatial range of the cycling distance is [0, 10] km. A spatial distance larger than 20 km may be very far for the two travel modes. The upper limit of “very far” is uncertain and can even be infinitely large. Therefore, the HMA algorithm is not suitable for modeling qualitative distance. In addition, the HMA algorithm calculates the parameters of an IT2 FS by using only the standard deviation intervals and the standard deviation of the UMF of the IT2 FS, but ignores the standard deviation of the LMF of the IT2 FS. In other words, this algorithm does not consider the entire uncertainty of the IT2 FS. As a result, the HMA algorithm must be improved to model qualitative distance. For the sake of discussion, the qualitative distance linguistic variable consists of five words in this study, i.e., “very near (Vnear)”, “near”, “medium”, “far”, and “very far (Vfar)”. The membership functions of each word can be determined using the improved HMA algorithm.

3.1. Collecting Data

Four travel modes are considered in this study: walk, bicycle, public transportation and train. The first three are interurban travel modes, whereas the fourth is a travel mode between cities. Via questionnaires, the distance interval sets corresponding to the five qualitative distance words for each travel mode are obtained. As mentioned in Section 1, the factors affecting the qualitative distance relationships are very sophisticated, and the impact of each factor is impossible to quantify. Therefore, the data uncertainty should be reduced by the following assumptions: (1) The scope of the survey is limited to grade 3 and 4 university students, as most of them are aged 21 to 23 years old and are familiar with the city environment; (2) the geographical environment is normal; i.e., mountains, deserts or other special environments are not considered; (3) the physical condition of each respondent is normal, and the weather is also normal. Some special physical conditions are not considered, including illnesses and hunger or special weather conditions such as snow, low temperatures and strong winds; (4) For the four travel modes, the lower limit is 0, and the upper limits are 5 km, 10 km, 30 km and 2000 km, respectively; (5) the intervals obtained from one person for the five words cannot overlap, and there is no gap between them; (6) a normal goal is set for each travel mode, as urgent tasks affect our perception of distance. The questionnaire used in this study is attached in Appendix A.

3.2. The Improved HMA Algorithm

As discussed previously, the original HMA algorithm does not consider the entire uncertainty of the IT2 FS; in this section it is improved by the entire uncertainty of the IT2 FS, and called the improved HMA. This improved HMA is still based on the questionnaire, and five interval sets corresponding to the five distance words for each travel mode are obtained. For the j-th word, the i-th subject provides interval endpoints $a_{i,j}$ and $b_{i,j}$, and the group of n subjects provides ${[a,b]}_{i,j}=[a_{ij},b_{ij}]$. In this study, the variables are normalized by the upper limit value to l = 0 and r = 10.

3.2.1. Data Processing

During data processing, statistics and probability are used to handle these interval sets and to reduce model uncertainty corresponding to distance words. This part involves five steps:

(1)

Correcting and removing bad data. As not everyone takes a survey seriously, some bad data may seriously pollute the interval sets and affect the statistics. To ensure the correctness of the statistical results, the intervals of the i-th subject should be removed if one of the following four conditions is true: (1) The data intervals of the i-th subject overlap with each other; (2) there is some gap between these intervals; (3) the value $a$ of the interval that corresponds to “very far” is greater than the threshold; (4) all intervals of the i-th subject are much larger or smaller than others. If the value $b$ of the interval corresponding to “very far” is greater than the threshold (different thresholds for four travel modes are set in Section 3.1), then $b$ should be set as the threshold. If the value $a$ of the interval corresponding to “very near” is greater than 0, then $a$ should be set to 0. $n^{\prime }$ subject intervals will remain.

(2)

Normalizing the data. As mentioned previously, the thresholds for the four travel modes are different. For the sake of convenience, these data should be normalized into a unified range. All $n^{\prime }$ subject intervals are normalized into the same range $[0,10]$ using the threshold value, and l = 0 and r = 10 are set.

(3)

Performing outlier processing. To remove some of the noise in the data set, outlier processing is necessary. In this section, the box and whisker tests [41] are performed for the remaining $n^{\prime }$ intervals, i.e., only the intervals satisfying Equation (10) are kept

$$\{\begin{array}{c}{a}_i\in [0.25Q_a-1.5IQ{R}_a,0.75Q_a+1.5IQ{R}_a]\\ b_i\in [0.25Q_b-1.5IQ{R}_b,0.75Q_b+1.5IQ{R}_b]\\ L_i\in [0.25Q_L-1.5IQ{R}_L,0.75Q_L+1.5IQ{R}_L]\end{array}$$

(10)

where $Q_a$ ($Q_b$, $Q_L$) and $IQ{R}_a$ ($IQ{R}_b$, $IQ{R}_L$) are the quartile and interquartile ranges for the left (right) endpoints and the interval length, with $L_i=b_i-a_i$.

After outlier processing, $m^{\prime }$ data intervals will remain for which the following data statistics will then be computed: $m_a$, ${\sigma }_a$, $m_b$, ${\sigma }_b$, $m_L$, ${\sigma }_L$ (mean and standard deviation of the $m^{\prime }$ left endpoints, right endpoints and the lengths of these intervals).

(4)

Performing tolerance limit processing on the remaining${\mathit{m}}^{\mathbf{\prime }}$intervals simultaneously. The goal of this stage is to guarantee that the data intervals fall within an acceptable two-sided tolerance limit. Only intervals satisfying Equation (11) are kept.

$$\{\begin{array}{c}{a}_i\in [m_a-k{\sigma }_a,m_a+k{\sigma }_a]\\ b_i\in [m_b-k{\sigma }_b,m_b+k{\sigma }_b]\\ L_i\in [m_L-k{\sigma }_L,m_L+k{\sigma }_L]\end{array}$$

(11)

where $k$ is the tolerance factor (refer to Reference [32] for detailed information). After tolerance limit processing, $m^{″}$ data intervals will remain, and the data statistics $m_a$, ${\sigma }_a$, $m_b$, ${\sigma }_b$, $m_L$, ${\sigma }_L$ will then be recomputed.

(5)

Performing reasonable interval processing. The goal of this stage is to remove data intervals that do not overlap or overlap only slightly with other data intervals. First, the point that best separates the left and the right endpoints set must be found.

$${\xi }^{*}=\raisebox{1ex}{$\left\{(m_b{\sigma }_a^2-m_a{\sigma }_b^2)\pm {\sigma }_a{\sigma }_b{[{(m_a-m_b)}^2+2({\sigma }_a^2-{\sigma }_b^2)log({\sigma }_a/{\sigma }_b)]}^{1/2}\right\}$}\!\left/ \!\raisebox{-1ex}{$({\sigma }_a^2-{\sigma }_b^2)$}\right.$$

(12)

where $m_a\le {\xi }^{*}\le m_b$. Then, only intervals satisfying Equation (13) are kept. This step reduces the $m^{″}$ interval endpoints to $m$ interval endpoints. The data statistics $m_a$, ${\sigma }_a$, $m_b$, ${\sigma }_b$ are then recomputed.

$$a_i\le {\xi }^{*}\le b_i$$

(13)

Therefore, the original n data intervals are reduced to a set of m data intervals via this data pre-processing.

3.2.2. Establishing IT2 FSs for Distance Words

As mentioned previously, the original HMA uses the upper membership function to estimate the IT2 FS parameters of words; that is, the standard deviation of a sample is equal to the standard deviation of the UMF in the original HMA, but it does not consider the LMF uncertainty. Consequently, this measurement is unilateral, and the uncertainties of these IT2 FSs may be exaggerated. As we know, the FOU of an IT2 FS is constructed by the LMF and UMF, and it is a description of the overall uncertainty of an IT2 FS. The uncertainties of an IT2 FSs may not be exaggerated if the FOU is considered for estimating its parameters. Therefore, in this study, the entire uncertainty (the FOU area) is adopted to estimate the parameters of IT2 FSs of words. First, the type of FOU of each word must be determined. Next, the overlap interval in ${[a,b]}_i$ ($0<i\le m$) must be found. Then, the parameters of the IT2 FS should be determined. The improved HMA consists of the following four steps:

(1)

Determining the type of FOU of each word. Three kinds of FOU exist: left-shoulder, right-shoulder and interior FOU, as shown in Figure 3. For an interval set ${[a,b]}_i(0<i\le m)$, the one-sided tolerance intervals are computed first, namely, $\underset{\_}{a}=m_a-k(m){\sigma }_a$ and $\overline{b}=m_b+k(m){\sigma }_b$. $m_a$, ${\sigma }_a$, $m_b$ and ${\sigma }_b$ are calculated in the last step of the data processing step, and $k(m)$ is a one-sided tolerance. Detailed information can be found in Reference [42]. Next, the types of FOU that correspond to each word can be classified as follows:

$$FOU(w)=\{\begin{array}{cc}Left-shoulderFOU& \underset{\_}{a}\le 0\\ Right-shoulderFOU& \overline{b}\ge 10\\ InteriorFOU& otherwise\end{array}$$

(14)

(2)

Determining the overlap interval $[o_l,o_r]$ of the interval set $\left\{{[a,b]}_i\right\},0<i\le m$.

$$[o_l,o_r]=\{\begin{array}{cc}\left[0,\underset{0\le i\le 1}{min}{b}_i\right]& Left-shoulderFOU\\ \left[\underset{0\le i\le 1}{max}{a}_i,\underset{0\le i\le 1}{min}{b}_i\right]& InteriorFOU\\ \left[\underset{0\le i\le 1}{max}{a}_i,10\right]& Right-shoulderFOU\end{array}$$

(15)

(3)

Removing the overlap interval $[o_l,o_r]$ from each of the $m$ intervals. For a left-shoulder FOU (Figure 3a), the interval set becomes a smaller interval set $\left\{{\left[o_r=\underset{0<i\le m}{min}{b}_i,b\right]}_i\right\}(0<i\le m)$; for a right-shoulder FOU (Figure 3b), the interval set becomes $\left\{{\left[a,o_l=\underset{0<i\le m}{max}{a}_i\right]}_i\right\}(0<i\le m)$; for an interior FOU (Figure 3c), the set becomes two new sets of smaller intervals $\left\{{\left[a,o_l=\underset{0<i\le m}{max}{a}_i\right]}_i\right\}(0<i\le m)$ and $\left\{{\left[o_r=\underset{0<i\le m}{min}{b}_i,b\right]}_i\right\}(0<i\le m)$.

(4)

Determining the parameters of the IT2 FS. As shown in Figure 4, the unknown parameters for the left-shoulder FOU (or right-shoulder FOU) are $b_l$ and $b_r$ ($a_l$ and $a_r$). For the interior FOU, four unknown parameters exist: $a_l$, $a_r$, $b_l$ and $b_r$. The set $\left\{{\left[a,o_l=\underset{0<i\le m}{max}{a}_i\right]}_i\right\}(0<i\le m)$ is used to determine $a_l$ and $a_r$ and $\left\{{\left[o_r=\underset{0<i\le m}{min}{b}_i,b\right]}_i\right\}(0<i\le m)$ is used to determine $b_l$ and $b_r$.

Let $\tilde{L}$ be a right-shoulder FOU or the left-hand portion of an interior FOU. The corresponding small interval set is $\left\{{\left[a,o_l=\underset{0<i\le m}{max}{a}_i\right]}_i\right\}(0<i\le m)$. The mean length of the interval set is $m_L=({\sum }_{i=1}^m|a_i-o_l|)/m$. The mean and the sample standard deviation of the set are $m_{LH}=o_l-m_L$ and $S_{LH}=\sqrt{\frac{1}{m}{\sum }_{i=1}^m{(a_i-o_l-m_L)}^2}$, respectively.

The centroid of $\tilde{L}$, $C_{\tilde{L}}$ and the average centroid, $\mathrm{c}(\tilde{L})$, are induced by the mean of T1 embedded set of $\tilde{L}$ (more information refers to the “Appendix A” of literature [14]) and expressed as:

$$C_{\tilde{L}}=\left[c(UMF(\tilde{L})),c(LMF(\tilde{L}))\right]=[(2o_l+a_l)/3,(2o_l+a_r)/3]$$

(16)

$$\mathrm{c}(\tilde{L})=\frac{1}{2}\left[c(UMF(\tilde{L}))+c(LMF(\tilde{L}))\right]=\frac{2}{3}{o}_l+\frac{1}{6}(a_l+a_r)$$

(17)

In Reference [14], the standard deviation of the UMF is used to calculate unknown parameters. Obviously, this value cannot represent the uncertainty of the FOU, while the area of the FOU can reflect the global uncertainty of the membership degree of this FOU. Thus, the area of the FOU is adopted here and is computed as:

$$A(\tilde{L})=(a_r-a_l)/2$$

(18)

Then, $a_l$ and $a_r$ can be calculated as:

$$\{\begin{array}{c}{m}_{LH}\equiv c(\tilde{L})\\ S_{LH}\equiv A(\tilde{L})\end{array}$$

(19)

such that

$$a_l=max(0,3m_{LH}-2o_l-S_{LH})$$

(20)

$$a_r=max(0,3m_{LH}-2o_l+S_{LH})$$

(21)

The $max$ operation prevents $a_l$ and $a_r$ from being negative. Similarly, $b_l$ and $b_r$ can be calculated as:

$$b_l=min(10,3m_{RH}-2o_r-S_{RH})$$

(22)

$$b_r=min(10,3m_{RH}-2o_r+S_{RH})$$

(23)

where $m_{RH}=o_r-m_R$ and $S_{RH}=\sqrt{\frac{1}{m}{\sum }_{i=1}^m{(b_i-o_r-m_R)}^2}$ are the mean and sample standard deviation of a left-shoulder FOU or the right-hand portion of an interior FOU, respectively, and $m_R=({\sum }_{i=1}^m|b_i-o_r|)/m$.

Thus, the IT2 FS that corresponds to the qualitative distance word can be expressed as:

$$\tilde{A}=\{\begin{array}{ccc}[(0,0,o_r,b_l,),(0,0,o_r,b_r,)]& & Left-shoulderFOU\\ [(a_r,o_l,o_r,b_l,),(a_l,o_l,o_r,b_r,)]& & InteriorFOU\\ [(a_r,o_l,r,r,),(a_l,o_l,r,r,)]& & Right-shoulderFOU\end{array}$$

(24)

4. Experimental Results and Analysis

A questionnaire survey was conducted at the School of Geographic and Environmental Sciences, Tianjin Normal University in October 2017. The subjects of this questionnaire survey are grade 3 and 4 university students, and the hometown of most of these subjects is Tianjin City, so they are familiar with the city. The language of this questionnaire is Chinese, and the language used in the questionnaire listed in the Appendix A is the second language of these subjects. During the survey, six assumptions or conditions regarding the respondents were stressed and 86 valid questionnaires were collected and analyzed using four Excel forms. Then, based on the improved HMA algorithm, the parameters of IT2 FSs corresponding to the five qualitative distance words for the four travel modes were calculated (

Table 1. Parameters of IT2 FSs corresponding to five qualitative distance words for four travel modes.

Travel Mode	Distance Word	Parameters of UMF (km)				Parameters of LMF (km)				${\mathit{L}}_{\mathit{C}}$ (km)	${\mathit{L}}_{\mathit{S}}$ (km)
		${\mathit{a}}_{\mathbf{1}}^{\mathbf{+}}$	${\mathit{a}}_{\mathbf{2}}^{\mathbf{+}}$	${\mathit{a}}_{\mathbf{3}}^{\mathbf{+}}$	${\mathit{a}}_{\mathbf{4}}^{\mathbf{+}}$	${\mathit{a}}_{\mathbf{1}}^{\mathbf{+}}$	${\mathit{a}}_{\mathbf{2}}^{\mathbf{-}}$	${\mathit{a}}_{\mathbf{3}}^{\mathbf{-}}$	${\mathit{a}}_{\mathbf{4}}^{\mathbf{-}}$
Walk	Vnear	0.00	0.00	0.01	0.61	0.00	0.00	0.01	0.40	0.01	0.61
	Near	0.00	0.40	0.50	0.63	0.00	0.40	0.50	0.54	0.10	0.63
	Medium	0.15	0.70	0.80	1.26	0.27	0.70	0.80	1.06	0.10	1.11
	Far	0.01	1.50	1.60	2.83	0.45	1.50	1.60	2.63	0.10	2.82
	Vfar	0.71	3.00	5.00	5.00	1.90	3.00	5.00	5.00	2.00	4.29
Cycle	Vnear	0.00	0.00	0.10	2.09	0.00	0.00	0.10	1.51	0.10	2.06
	Near	0.00	0.80	1.00	1.16	0.20	0.80	1.00	1.00	0.20	1.16
	Medium	0.74	2.00	2.20	3.07	1.26	2.00	2.20	2.83	0.20	2.33
	Far	0.20	3.50	4.00	6.74	1.22	3.50	4.00	5.91	0.50	6.53
	Vfar	0.00	8.00	10.00	10.00	0.94	8.00	10.00	10.00	4.00	9.26
Public transportation	Vnear	0.00	0.00	1.00	6.02	0.00	0.00	1.00	4.39	1.00	6.08
	Near	1.78	3.00	4.00	6.85	2.66	3.00	4.00	6.10	1.00	5.06
	Medium	1.20	6.00	7.00	13.73	3.39	6.00	7.00	12.12	1.00	12.53
	Far	6.59	10.0	12.00	20.12	7.41	10.0	12.00	16.34	2.00	12.22
	Vfar	6.27	25.0	30.0	30.00	11.79	25.0	30.00	30.00	5.00	23.73
Train	Vnear	0	0	100	129	0	0	100	110	100.0	130.10
	Near	68.6	120	150	476.1	87.4	120	150	354.9	30.00	415.20
	Medium	44.1	500	600	1039	251.9	500	600	841	100.0	1009.4
	Far	28.2	800	1000	1313	251.8	800	1000	1023	200.0	1438.8
	Vfar	0	1800	2000	2000	536	1800	2000	2000	300.0	1913.6

). These IT2 FSs are presented in Figure 4, Figure 5, Figure 6 and Figure 7.

4.1. General Description of the Qualitative Distance Model

The results were independently reviewed by an entirely separate group of 10 adults, and they all agreed that the results were consistent with actual situations. Regarding the IT2 FSs for each travel mode, the membership value is equal to 1 in the interval [$a_2^{-}$, $a_3^{-}$] ([$a_2^{+}$, $a_3^{+}$]), and the length of these intervals is $L_C=a_3^{+}-a_2^{+}$. The interval [$a_1^{+}$, $a_4^{+}$] is the supporting interval, with the length being $L_S=a_4^{+}-a_1^{+}$. These two intervals have direct directives. Using walking distance as an example, the membership function of the IT2 FS that corresponds to “medium” takes the value of 1 for the distance interval [0.70, 0.80] km, indicating that when most people consider a distance within this interval, they estimate that it is suitable for walking. The interval [1.50, 1.60] km can be estimated to belong to the range of “far”, for which people are likely to choose other transportation modes. When the distance is longer than 3 km, the corresponding range is “very far”; at these distances, people generally forego walking and choose other travel modes. For cycling, the distance interval [2.00, 2.20] km is certainly suitable, but when the distance is approximately [3.50, 4.00] km, people may hesitate to ride a bicycle. In general, when the distance is greater than 6 km, people choose public transportation, such as buses, subways and taxis.

For travel between cities, the distance interval [120, 150] km is estimated to correspond to “near”, as more than 20 thousand km of high-speed railways have been built in China, covering almost all the major cities in China. The speed of these railway trains is approximately 300 km/h; thus, people often say that Beijing is close to Tianjin. When the distance is greater than 1000 km, people may hesitate to travel along the high-speed railway, as the time spent and cost of traveling by train may be greater than those when traveling by airplane.

Figure 4, Figure 5, Figure 6 and Figure 7 and Table 1 show that the IT2 FSs are asymmetrical. For the four travel modes, $L_C$ and $L_S$ gradually increase as the distance increases from very near to very far, with a more obvious increase in $L_S$ than that in $L_C$. This result intuitively illustrates that the words “far” and “very far” are more ambiguous than “near” and “very near” for all four travel modes.

4.2. Comparison with HMA

In Euclidean space, the domain of quantitative distance is a semi-closed set, i.e., $[0,\infty )$. Thus, it is difficult to determine the upper limit of “Vfar” in real applications, and different upper limits of “Vfar” can yield different IT2 FSs, which may lead to different measures of uncertainty. Consider the train travel mode as an example. If its upper limit is less than 1800 km, the FOU, cardinality, fuzziness and variance of the IT2 FSs may be different for different upper limits. If its upper limit is larger than 1800 km, the FOUs of different IT2 FSs, which are modeled using different upper limits, are equal; in contrast, their cardinality and variance are not equal. Notably, the values of the LMF and UMF of the IT2 FSs corresponding to “Vfar” for the four travel models are equal to 1 when the quantitative distance is larger than $a_2^{-}(a_2^{+})$. Thus, the upper limits of “Vfar”, $a_2^{-}(a_2^{+})$, should be set for convenience of analysis. Then, the area of the FOU, cardinality, fuzziness, and variance of these IT2 FSs, modeled using the improved HMA and the original HMA, are calculated, as shown in

Table 2. Uncertainty indices of IT2 FSs corresponding to five distance words for four travel models modeled by the HMA and improved HMA.

Travel Mode	Uncertainty Index	HMA						Improved HMA
		Vnear	Near	Medium	Far	Vfar	Total	Vnear	Near	Medium	Far	Vfar	Total
Walk	Area of FOU	0.05	0.10	0.28	1.00	0.83	2.26	0.10	0.05	0.16	0.32	0.60	1.22
	Fuzziness 1	0.34	0.31	0.32	0.35	0.36	1.70	0.35	0.22	0.33	0.35	0.36	1.61
	Fuzziness 2	1.78	3.27	9.43	33.44	27.86	75.77	3.59	1.56	5.19	10.56	19.99	40.88
	Variance	0.01	0.02	0.05	0.35	0.36	0.79	0.02	0.02	0.04	0.27	0.25	0.59
Cycle	Area of FOU	0.47	0.34	0.33	1.21	0.15	2.50	0.29	0.18	0.38	0.92	0.47	2.24
	Fuzziness 1	0.34	0.30	0.33	0.34	0.37	1.68	0.34	0.36	0.32	0.34	0.37	1.73
	Fuzziness 2	5.98	4.50	4.14	15.27	1.86	31.76	3.75	3.61	4.89	11.54	5.87	29.65
	Variance	0.22	0.05	0.17	1.42	3.39	5.26	0.19	0.05	0.17	1.38	3.17	4.96
Public transportation	Area of FOU	0.74	1.80	3.48	3.06	4.26	13.35	0.82	0.81	1.90	2.30	2.76	8.58
	Fuzziness 1	0.30	0.29	0.34	0.31	0.36	1.61	0.30	0.29	0.34	0.31	0.37	1.61
	Fuzziness 2	3.00	7.30	14.04	12.41	17.05	53.82	3.38	3.29	7.72	9.33	11.05	34.78
	Variance	1.52	1.07	5.96	6.16	18.37	33.09	1.55	0.88	5.04	6.18	15.73	29.38
Train	Area of FOU	20.13	10.35	168.96	469.50	280.13	949.08	9.49	70.04	202.93	256.37	267.98	806.81
	Fuzziness 1	0.10	0.31	0.33	0.32	0.37	1.43	0.06	0.33	0.33	0.31	0.37	1.39
	Fuzziness 2	1.31	0.56	9.50	26.47	15.56	53.40	0.56	3.94	11.42	14.39	14.89	45.20
	Variance	884	5557	28156	74643	145232	254472	867	6240	28931	53368	145663	235068

Note: Fuzziness 1 and Fuzziness 2 are computed based on the definitions of Wu and Mendel [24] and Burillo and Bustince [2], respectively.

. The two rows, Fuzziness 1 and Fuzziness 2, are computed based on the definitions in References [36,38] respectively, and the value in the Total column is the sum of the columns Vnear, Near, Medium, Far and Vfar.

As mentioned previously, the FOU is the uncertainty measure of the primary memberships of a T2 FS. Thus, a small FOU area indicates low uncertainty regarding the IT2 FS; in geometric terms, an IT2 FS with a smaller FOU area is thinner than that with a larger FOU area. Table 2 shows that the total area of the FOU of these IT2 FSs modeled by the improved HMA is smaller than that obtained by the original HMA. Moreover, the fuzziness (entropy) and variance of these IT2 FSs modeled by the improved HMA are smaller than those by the original HMA, having a strong correlation with the area of the FOU. Therefore, in terms of uncertainty, the improved HMA is better than the original HMA.

4.3. Compactness and Fuzziness Analysis

As introduced earlier, the quantity variance is an uncertainty measure in the spatial domain. For the four travel modes, from “Vnear” to “Vfar”, the variance increases; for example, the variance of the IT2 FS of “Vnear” is 867, whereas that of “Vfar” is 145,663. The variances of IT2 FSs for traveling by train and walking experience the most significant increases, followed by bicycle travel. Particularly, for bicycle and public transportation, the variance value of the IT2 FS of “Vnear” is larger than that of “near”. This shows that for these four travel modes, people’s cognitive ambiguity regarding “very far” or “far” is larger than that of “near”, whereas for bicycle and train modes, the cognitive ambiguity of “Vnear” is larger than that of “near”. The variation tendency of fuzziness is similar to that of variance, especially for Fuzziness 2. The values of the IT2 FS of “Vfar” in the row of Fuzziness 2 are significantly larger than those of “Vnear”.

In summary, no matter what the travel mode is, the semantic uncertainty of “Vfar” is the greatest, whereas the uncertainty of “near” (“Vnear”) is the smallest. In other words, the spatial scope of “near” is easy for people to agree on, but there is a large difference in the spatial ranges of “far” and “Vfar”.

4.4. Comparison with Other Methods

In Section 4.3 and Section 4.4, we present a full comparison of the improved and original HMAs and show that the improved HMA is better than the original. As discussed in the Introduction, there are three types of methods typically used to establish the relationships between qualitative distance words and quantitative distances. In this section we present a comparison with these methods.

The parameters of the distance membership functions in the first type directly depend on the experience of the experts and are extremely subjective. In addition, a complex decision support application often involves multiple experts, and these experts have different understandings of qualitative distance words. As Yao and Thill [11] noted, a linguistic distance measure, e.g., near, implies different trip distances for different people. Therefore, the fuzzy membership functions of distance words based on their personal cognitive habits are likely to be inconsistent, as shown in Figure 8. In other words, membership functions established by this method do not reflect the semantic uncertainty among experts. The data used in the proposed method are from multiple subjects, and the semantic uncertainty of subjects can be expressed through the upper and lower membership functions, thereby yielding a more objective method. The second type method discussed in section one involves establishing membership functions by collecting data from Web text. The context of a distance word in Web text may be different, and the semantics of qualitative distance words depend on context [11,16,21]. Therefore, it is difficult to establish membership functions of distance words for a specific context or specific task based on this method. Additionally, the membership functions established in this manner are type 1; therefore, they cannot express the semantic uncertainty of words among different people.

The third type methods discussed in section one are used to establish fuzzy sets corresponding to qualitative distance words through questionnaires. In the spatial relation acquisition station (SRAS) [9,28], subjects answer questions as YES or NO based on a map, such as “Is the city of Blackshear close to Douglas?”, then the fuzzy distance relation between geographical features can be derived by fuzzy logic method. While Robinson’s effort is worthwhile in many respects, it does not incorporate the influence of the context of the query on the mapping of proximity spatial relations [11]. But beyond that, it has no ability to establish fuzzy sets for distance words. Yao and Thill [16] highlighted the importance of context and noted that considering the relationship between the context and the metric properties of data is necessary. Additionally, the Adaptive Neurofuzzy Inference System (ANFIS), a Takagi-Sugeno fuzzy inference system, was adopted, and the linguistic proximity measure was predicted based on the metric distance and contextual factors and expressed as $Linguisti\mathrm{c}\_Distance=\mathrm{f}(Metric\_Distance,Context\_Factors)$ based on fuzzy rules. The prediction is affected by the number of factors, types of factors, fuzzification method, and the rules and reasoning of the model. For example, as previously noted, it is difficult to fuzzify these factors, and unsuitable fuzzification methods may introduce new uncertainty. However, the translation from metric distance to linguistic distance in Reference [16] is inexplicit. Grütter et al. [43] claimed that this process does not support the translation of local prepositions, such as “near” or “far”, into distance measures processed in metric systems and that the information required to link the numerous contextual variables may not be available in practical applications. Hence, it is often difficult to implement these processes on large scales. In Yao and Thill’s other method [11], namely the Ordered Logit Regression approach, probability functions of distance words are induced by Ordered Logit Regression; however, the maximum value of the first four functions is less than 1, which is contrary to human cognition, because in general, we can always determine a spatial range and the distance between this range to the reference object is near or far away.

Although many methods abovementioned could be used to establish FS models for distance words, to the best of our knowledge, they are related to T1 FS models and violate principle three introduced in Section 1. Because the membership function of a type-1 FS does not provide flexibility for simultaneously incorporating both types of linguistic uncertainty, it is therefore scientifically incorrect to model a word using a T1 FS [23]. Our method uses an IT2 FS to model distance words, can handle both types of linguistic uncertainty, and is context dependent. Overall, the proposed method is better than the other methods discussed.

4.5. Practical Application

In this section, a case study on natural language spatial queries was used to illustrate the usefulness and effectiveness of our presented methods. The Haihe River is the main river in Tianjin City, and the surrounding area of the upstream part of the Haihe River is the city center. The infrastructure in this region is very mature. There are many famous and desirable primary schools in this region. If a school is too close to the river, children will sneak off to the riverside to play. This is very dangerous for children, so parents tend not to favor such schools. As a result, it is useful to determine which primary school is a medium distance from (not too near) the upstream section of the Haihe River, and the results can provide advice for parents’ guidance to choose primary schools. In Reference [33], we divided the river into upstream, midstream and downstream sections by using the fuzzy partition method as shown in Figure 9.

In Section 4.1, the IT2 FS corresponding to “medium” distance by cycle is $\tilde{A}=[(0.74,2.0,2.2,3.07),(1.26,2.0,2.2,2.83)]$. The region with medium distance to the Haihe River by bicycle can be determined by the fuzzy logic method [33], shown in Figure 10a,b. Schools located in the medium distance, which is modeled as IT2 FS to the upstream of the Haihe River, are listed in

Table 3. The primary schools located in the “medium” distance to the upstream of the Haihe River which is modeled as IT2 FS.

ID	Name	Lower Membership Grade	Upper Membership Grade	Error of Membership
8511245	YONG JI XIAO XUE	1.0000	1.0000	0.0000
8515596	YUE YANG DAO XIAO XUE	1.0000	1.0000	0.0000
8530346	YI YANG XIAO XUE	1.0000	1.0000	0.0000
8406934	TAO HUA YUAN XIAO XUE	1.0000	1.0000	0.0000
8540833	KUN MING LU XIAO XUE	0.9849	0.9899	0.0050
8510894	YUAN CHENG XIAO XUE	0.9803	0.9816	0.0013
8512237	HUA XIA WEI LAI YI SHU XIAO XUE	0.9270	0.9791	0.0521
8542812	SHANG HAI DAO XIAO XUE	0.9661	0.9774	0.0113
8514663	KUN MING LU XIAO XUE	0.9568	0.9767	0.0199
8520570	YUE YANG DAO XIAO XUE	0.9166	0.9551	0.0385
8514586	XIN XING XIAO XUE	0.9213	0.9475	0.0262
8538733	EN DE LI XIAO XUE	0.8736	0.9326	0.0590
8513199	YI YANG XIAO XUE	0.8721	0.9312	0.0590
8514613	DONG FANG XIAO XUE	0.8954	0.9303	0.0349
8519552	SI HAO LU XIAO XUE	0.8939	0.9288	0.0349
8537185	SHANG HAI DAO XIAO XUE FEN XIAO	0.8939	0.9288	0.0349
8426650	HU ZHU DAO XIAO XUE	0.8643	0.9259	0.0616
8538022	TIAN JIN SHI SHI YAN XIAO XUE	0.8609	0.9251	0.0642
8426553	HU ZHU DAO XIAO XUE	0.9239	0.9239	0.0000
8509788	HE DONG SHI YAN XIAO XUE	0.8534	0.9211	0.0677
8513580	KUN WEI LU YI XIAO	0.8534	0.9211	0.0677
8508222	HAI YANG YI XIAO JIAO XUE QU	0.8661	0.9107	0.0446
8539007	WU MA LU XIAO XUE	0.8661	0.9107	0.0446
8533082	QIU ZHEN XIAO XUE	0.8661	0.9107	0.0446
8418068	ZHONG XIN DONG DAO XIAO XUE	0.8986	0.8986	0.0000
8469819	HE MU DAO XIAO XUE	0.8986	0.8986	0.0000
8512972	LI SHUI DAO XIAO XUE	0.8811	0.8811	0.0000
8408161	YOU AI DAO XIAO XUE	0.8691	0.8691	0.0000
8509640	KUN WEI LU YI XIAO	0.7183	0.8483	0.1300
8536258	NAN KAI ZHONG XIN XIAO XUE	0.6984	0.8376	0.1392
8515544	JIAN SHAN XIAO XUE	0.6167	0.8285	0.2118
8510652	HE BEI QU DI ER SHI YAN XIAO XUE	0.6703	0.8224	0.1522
8538777	XIN CUN XIAO XUE	0.6703	0.8224	0.1522
8522712	TIAN JIN SHI DA DI ER FU XIAO	0.5965	0.8114	0.2148
8512402	HONG HU LI XIAO XUE	0.5714	0.7692	0.1978
8512553	SHAO GONG ZHUANG XIAO XUE	0.6390	0.7593	0.1203
8515227	WU MA LU XIAO XUE	0.6355	0.7570	0.1215
8424178	YU YING LI XIAO XUE	0.6355	0.7570	0.1215
8531501	YA XI YA XIAO XUE	0.5376	0.7510	0.2134
8538975	WU MA LU XIAO XUE	0.6081	0.7387	0.1306
8411260	LING SHUI DAO XIAO XUE	0.6865	0.7369	0.0504
8514552	NAN KAI ZHONG XIN XIAO XUE	0.5076	0.7348	0.2273
8411750	ZHU JIANG DAO XIAO XUE	0.7249	0.7249	0.0000
8513795	WEN CHANG GONG MIN ZU XIAO XUE	0.4684	0.7138	0.2454
8511466	NAN KAI ZHONG XIN XIAO XUE	0.5402	0.6935	0.1533
8514988	WAI YU XUE YUAN FU ER XIAO	0.5402	0.6935	0.1533
8538703	MA CHANG DAO XIAO XUE	0.5402	0.6935	0.1533
8411270	DONG HAI LI XIAO XUE	0.3633	0.6930	0.3297
8514270	WEN SHAN LI XIAO XUE	0.4016	0.6778	0.2762
8511329	XI KANG LU XIAO XUE	0.5000	0.6667	0.1667
8517795	QIAN CHENG XIAO XUE	0.3789	0.6656	0.2866
8510697	HU ZHU DAO XIAO XUE	0.2631	0.6493	0.3862
8510891	DA QIAO DAO XIAO XUE	0.2631	0.6493	0.3862
8408068	ER HAO QIAO XIAO XUE	0.5244	0.6473	0.1230
8538694	KUN MING LU XIAO XUE	0.4602	0.6401	0.1799
8511172	TIAN TAI XIAO XUE	0.3282	0.6382	0.3101
8509898	HE PING QU XIN ZHONG XIAO XUE	0.2857	0.6154	0.3297
8510408	KUN MING LU XIAO XUE	0.4175	0.6117	0.1942
8511274	NING YUAN XIAO XUE	0.3979	0.5986	0.2007
8509892	HE PING QU ZHONG XIN XIAO XUE	0.2032	0.5709	0.3678
8540181	ZHONG YING XIAO XUE	0.2032	0.5709	0.3678
8408448	WU SHI XIAO XUE	0.5642	0.5655	0.0013
8511404	DA QIAO DAO XIAO XUE	0.1169	0.5485	0.4316
8531345	HONG XING XIAO XUE	0.3205	0.5470	0.2265
8408042	HE DONG WU SHI XIAO XUE	0.5440	0.5440	0.0000
8514559	HONG YUAN LI XIAO XUE	0.1326	0.5330	0.4003
8426222	JIN MEN XIAO XUE	0.0824	0.5170	0.4345
8519570	TONG WANG XIAO XUE	0.2665	0.5110	0.2445
8522698	XIN SHI JI XIAO XUE	0.1178	0.5088	0.3910
8522746	TIAN JIN SHI DA DI ER FU XIAO	0.1178	0.5088	0.3910
8510681	QI ZHI XUE XIAO	0.0859	0.5078	0.4219

, we can see that the membership degree of each school belonging to the “medium” distance to the upstream of Haihe River is an interval value, i.e., the membership grade of “KUN MING LU XIAO XUE” is [0.9568, 0.9767], and error of membership grade is 0.0199. When the upper membership grade decrease from 1 to 0.5, the error of membership grade increases from 0 to 0.4219. It means that when the upper membership grade is near to 0.5, the semantic uncertainty is great. On the contrary, when the distance words are modeled as T1 FS, the membership grade is a singleton and the error of membership grade cannot be measured. So, modeling distance words as IT2 FS is more reasonable and flexible than modeling as T1 FS.

On the other hand, we obtain 71 primary schools in this region by optimistic query when the upper membership grade is greater than 0.5, shown in Figure 10c. and we can obtain 50 primary schools by pessimistic query when the lower membership grade is greater than 0.5, shown in Figure 10d. So, there are 21 difference between the optimistic and pessimistic query. However, when the distance word is modeled as T1 fs, due to it not being able to express the semantic uncertainty, users can only query in one way, and the result of the query is more arbitrary and monotonous than the method proposed in this paper.

5. Conclusions

People use qualitative distance words to describe the approximate positions of objects relative to one another in natural languages. The meanings of qualitative distance words depend on objective factors. If these words are used by people in different contexts, their connotations will be different. Therefore, a unified fuzzy set representation of qualitative distance words is impossible to establish for all GIS-based decision support systems. In other words, the fuzzy set representation of distance words can be obtained for specific tasks. First, aiming at the shortcomings of the original HMA algorithm, the HMA method is improved by introducing the area of the FOU. Then, the improved HMA algorithm is used to establish the IT2 FS representation of qualitative distance words. Three uncertainty indices, i.e., the area of the FOU, fuzziness (entropy), and variance, are used to measure the uncertainties of these IT2 FSs. Experimental results show that the values of these indices for the IT2 FSs obtained by the improved HMA algorithm are smaller than those obtained by the original HMA algorithm. This indicates that the improved HMA algorithm is better than the original algorithm. In addition, it shows that the semantic uncertainty of “Vfar” is the greatest, while that of “very near” (“Vnear”) is the smallest, no matter which travel mode is considered. Hence, the cognitive fuzziness of “far” is larger than that of “near”. In other words, the spatial scope of “near” is easy for people to agree on, but there is a large difference in the spatial range of “far”.

The qualitative distance, topological relationship and direction relationship are often used together to describe a spatial position in natural languages. Therefore, in a follow-up study, the IT2 FS representation of the qualitative distance established in this study will be used to study the semantic spatial relationship by being combined with topological and direction relationships.