Subsethood Measures of Spatial Granules

Given a universe of discourse $X=\{x_{1},\cdots,x_{n}\}$, the granules and binary relations on $X$ are one-to-one corresponding, where the granules corresponding to fuzzy equivalence relations are called fuzzy equivalence granules and the granules corresponding to equivalence relations are called equivalence granules. Each equivalence granule is a partition of a subset of $X$, and, in particular, a partitions of $X$ is also called a quotient granule on $X$. For the sake of simplicity, we only discuss equivalence granules in this paper, that is, the atomic granules of a granule are its equivalence classes.

Assume $A$ and $B$ are two subsets of $X,R_{A}$ and $R_{B}$ are equivalence relations on $A$ and $B$ respectively, and the equivalence granules corresponding to $R_{A}$ and $R_{B}$ are $A_{R}=\{a_{1},\cdots,a_{k}\}$ and $B_{R}=\{b_{1},\cdots,b_{l}\}$ respectively. For convenience, $A_{R}$ can also be denoted by $A$, and use granule $A$ or set $A$ to to distinguish them so as not to cause ambiguity, that is, the granule $A$ is a partition of the set $A$. The operations of meet, join, quotient meet and quotient join are respectively defined as follow:

Where the quotient meet and quotient join operations are for (fuzzy) equivalence granules while the meet and join operations are for other granules. Obviously, for equivalence granules, the quotient meet is the same with meet but the join and quotient join are different.

Given an information system $I=(X,\textit{{R}})$, where $\textit{{R}}=\{R_{1},\cdots,R_{m}\}$ is a family of equivalence relations on subsets of $X=\{x_{1},\cdots,x_{n}\}$. This information system can be viewed from the micro and the macro perspectives respectively. From the micro perspective, we think about all the subsets of $X$, denoted as $\sigma(X)$. $(\sigma(X),\supseteq)$ is a complete lattice, and all the elements in $\sigma(X)$ can be hierarchized under set inclusion relation.

Assume the equivalence granules corresponding to $R_{i}$ are $P_{i}(i=1,\cdots,m)$, respectively, $R$ is the intersection of all $R_{i}(i=1,\cdots,m)$, and $P$ is the quotient meet of all $P_{i}(i=1,\cdots,m)$. For any $A\in\sigma(X),A$ is called $R$-definable if it is one of the equivalence classes in $P$ or a union of two or more equivalence classes in $P$. Assume $d(\sigma(X))$ is a family of all definable sets in $\sigma(X)$ and $d_{0}(\sigma(X))$ is a family of the empty set and all definable sets. Then $(d_{0}(\sigma(X)),\supseteq)$ is a complete bounded sublattice of $(\sigma(X),\supseteq)$, and $d_{0}(\sigma(X))$, which is closed under under union and intersection operations, is called the micro knowledge space generated from $I=(X,\textit{{R}})$. Therefore, $\sigma(X)$ can be divided into two categories: $d(\sigma(X))$ and $\widetilde{d}(\sigma(X))$, i.e., the family of all undefinable sets. By rough set theory, $\widetilde{d}(\sigma(X))$ can be further divided into $d_{r}(\sigma(X))$, i.e., the set of roughly definable sets, and $\widetilde{d}_{r}(\sigma(X))$, i.e., the set of roughly or totally undefinable sets.

Obviously, for any $A\in\sigma(X)$, its upper approximation is to find its least upper bound in $d_{0}(\sigma(X))$, and its lower approximation is to find the greatest lower bound in $d_{0}(\sigma(X))$. $(\underline{R}(A),\overline{R}(A))$ is called an approximation space of $A$.

When $I$ is complete, i.e., all $R_{i}(i=1,\cdots,n)$ are equivalence relations on $X$, every atomic granule in $d(\sigma(X))$ can be obtained from the atomic granules of $P$, and then we can replace $d(\sigma(X))$ with $P$. However, we should examine two extreme cases: $\forall B\in P,A\supset B$ and $\forall B\in P,B\supset A$. We can define its upper approximation as $A$ in the first case and define its lower approximation as $A$ in the second case. When $I$ is incomplete, not all of the atomic granules in $d(\sigma(X))$ can be obtained from the atomic granules of $P$. Therefore, we cannot replace $d(\sigma(X))$ with $P$. It can be seen that the classical rough set model is only for complete information systems while the above model is not only for complete information systems but also for incomplete information systems. When $X$ a domain, we can divide it into $n$ subdomains, which can be regarded as $n$ objects, and the above model is also applicable. All the extended models developed from the classical rough set model can be accordingly defined by $d(\sigma(X))$ so as to be applicable to any information system, which will be discussed in another paper.

Assume $\Pi(\sigma(X))$ is the family of all equivalence granules on $X$ and $\Pi_{0}(\sigma(X))$ is the family of the empty granule and all equivalence granules on $X$. Then viewing $I$ from the macro perspective, the whole space is $\Pi_{0}(\sigma(X))$. There is no set inclusion relation between two granules, and we must define new relation.

$(\Pi_{0}(\sigma(X)),\succeq)$ is a complete bounded lattice [45], all the elements in $\Pi_{0}(\sigma(X))$ and the vertices of the unit $n$-dimensional hypercube are one-to-one corresponding, and $\Pi_{0}(\sigma(X))$ can be hierarchized by coarse-fine relation. For any granule $A\in\Pi(\sigma(X))$, it is called $R$-definable under this information system if $A\succ P$. Assume $d(\Pi(\sigma(X)))$ is a family of all definable granules in $\Pi(\sigma(X))$ and $d_{0}(\Pi(\sigma(X)))$ is a family of $P$ and all definable granules. Then $(d_{0}(\Pi(\sigma(X))),\succeq)$ is a complete bounded sublattice of $(\Pi_{0}(\sigma(X)),\succeq)$, and $d_{0}(\Pi(\sigma(X)))$, which is closed under under quotient meet and quotient join operations, is called the macro knowledge space generated from $I$. Therefore, $\Pi_{0}(\sigma(X))$ can be divided into two categories: $d(\Pi(\sigma(X)))$ and $\widetilde{d}(\Pi_{0}(\sigma(X)))$, i.e., the family of all undefinable granules. While $\widetilde{d}(\Pi_{0}(\sigma(X)))$ can be further divided into $d_{r}(\Pi(\sigma(X)))$, i.e., the set of roughly definable granules, and $\widetilde{d}_{r}(\Pi_{0}(\sigma(X)))$, i.e., the set of roughly or totally undefinable granules.

For any granule $A$ in $\Pi(\sigma(X))$, its upper approximation is to find its lowest upper bound in $d_{0}(\Pi(\sigma(X)))$, and its lower approximation is to find the greatest lower bound in it.

The upper and lower approximations in the above model are not obtained from one of its tangent planes but from the $n$-dimensional space. Therefore, the model is also called the spatial rough granule model which can be applied to any structural information system and non-structural information system as well. In particular, we have $\underline{R}(A)=A\wedge_{t}P$ and $\overline{R}(A)=A\vee_{t}P$ when $I$ is complete.

Subsethood measures should satisfy some axioms to make them to be meaningful. Sinha and Dougherty [48] presented nine axioms for subsethood and the last five ones further restrict subsethood measures, and Young [12] mainly discussed the first four. Different scholars may define different axioms in different fields [9, 15, 49, 11, 24]. However, we can divided these axioms into two classes: basic axioms and extended axioms. Basic axioms are similar, and extended axioms may be different by the properties of empirical objects.

In many situations, it is more convenient to consider a normalized measure for which the maximum value is $1$ and the minimum is $0$. For any two atomic granules $a,b\in\sigma(X)$, the basic axioms of a subsethood measure should satisfy: a subsethood measure must reach the maximum value if and only if $a\subseteq b$, it reaches the minimum value if and only if $a\cap b=\emptyset$, and it belongs to $[0,1]$; it should show the monotonicity because the set inclusion is a partial order relation.

For the classical set inclusion, a set $a$ is either a subset of another set $b$ or not, i.e., $sh(b,a)$ is either 1 or 0, and the conditions (A1) and (A2) are dual each other. Some authors [50, 51] used a single implication:

That is, $sh(b,a)$ reaches the maximum value if $a\subseteq b$. However we may still have $sh(b,a)=1$ even though $\neg(a\subseteq b)$. Gomolińska [52, 53] considered the other single implication:

In this case, we can get $a\subseteq b$ from $sh(b,a)=1$, and the other way around is not true. None of the two single implications can faithfully reflect whether a set is a subset of another besides the double implication.

For the general set inclusion, one set can be a subset of another one to some degree, that is, $sh(b,a)$, the degree of the inclusion, can be any value between 0 and 1. When researching on subsethood measure, (A1) is the only condition for normalized measure, which is to extend subsethood function. If our purpose is to measure the degree of coarse-fine relation of two granules and the boundary conditions defined in Definition III.1 are the minimum requirements that subsethood measures can truthfully reflect the basic properties of inclusion degree or coarse-fine degree unless we do not consider the special case $a\cap b=\emptyset$. If our purpose is to judge whether a granule is coarser than or finer than another granule, then the axiom (A1) is enough for normalized measure, i.e. boundary condition, and the focus is on monotonicity.

In [24], Yao and Deng discussed four monotone properties of subsethood measures among three sets $a,b,c\in\sigma(X)$ as follows.

(M1)

$b\subseteq c\Rightarrow sh(b,a)\leq sh(c,a);$

(M2)

$b\subseteq c\wedge(b\cap a=c\cap a)\Rightarrow sh(a,c)\leq sh(a,b);$

(M3)

$b\subseteq c\Rightarrow sh(a,c)\leq sh(a,b);$

(M4)

$a\subseteq b\subseteq c\Rightarrow sh(a,c)\leq sh(a,b);$

Comparing with the conditions (M1) and (M3), we know the monotonicity of function $sh(a,b)$ is reversed with that of function $sh(b,a)$, and we have (A3) $\Rightarrow$ (A4) and (A4) $\Rightarrow$ (A3). Therefore, (A3) or (A4) alone can be thought as the monotonically increasing condition of subsethood. In condition (M2), $b\cap a=c\cap a$ is the greatest lower bound of $a,b$ and $c$, which reminds us to think about its dual question, that is, their corresponding least upper bound $b\cup a=c\cup a$. Therefore, we have the following monotone properties.

(A5) 

$b\subseteq c\wedge(b\cap a=c\cap a)\Rightarrow sh(b,a)\leq sh(c,a)$;

(A6) 

$b\subseteq c\wedge(b\cap a=c\cap a)\Rightarrow sh(a,c)\leq sh(a,b)$;

(A7) 

$b\subseteq c\wedge(b\cup a=c\cup a)\Rightarrow sh(b,a)\leq sh(c,a)$;

(A8) 

$b\subseteq c\wedge(b\cup a=c\cup a)\Rightarrow sh(a,c)\leq sh(a,b)$;

(A9) 

$a\subseteq b\subseteq c\Rightarrow sh(b,a)\leq sh(c,a)$;

(A10)

$a\subseteq b\subseteq c\Rightarrow sh(a,c)\leq sh(a,b)$;

(A11)

$b\subseteq c\subseteq a\Rightarrow sh(b,a)\leq sh(c,a)$;

(A12)

$b\subseteq c\subseteq a\Rightarrow sh(a,c)\leq sh(a,b)$.

The axioms (A5), (A7), (A9) and (A11) are weaker versions of (A3), i.e., (A3) $\Rightarrow$ (A5), (A7), (A9) and (A11); the axioms (A6), (A8), (A10) and (A12) are weaker versions of (A4), i.e., (A4)$\Rightarrow$ (A6), (A8), (A10) and (A12). Therefore, we can only discuss the axioms (A1), (A2), (A3) and (A4). The axioms (A5) and (A6) are the dual questions of (A7) and (A8) respectively, and the axioms (A9) and (A10) are the dual questions of (A11) and (A12) respectively.

Yao and Deng [24] reviewed existing subsethood measures including $sh_{l}$[18, 19, 20, 9, 1, 15, 2, 52, 5, 6, 54, 3, 12, 55], $sh_{\cap}$ [52, 56], $sh_{\cup}$ [15, 5, 57, 4], $sh_{\cap}^{c}$ [15], and $sh_{\cup}^{c}$  [15, 58, 51] that have been considered in many studies. Most of them focus on fuzzy sets, but not on crisp sets. Yao and Deng gives the five subsethood measures of two crisp sets and have the corresponding probabilistic interpretations as follows.

If any of the value of subsethood measures is equal to 1, and we can judge the atomic $a$ is a subset of $b$. It can be seen that only $sh_{\cap}$ satisfies both (A1) and (A2).

Then these $sh_{i}^{\prime}(\cdot,\cdot)=1-sh_{i}(\cdot,\cdot)(i=1,\cdots,5)$, which can be called supsethood, are the monotonically decreasing measures corresponding to $sh_{i}(b,a)(i=1,\cdots,5)$, respectively, and every $sh_{i}^{\prime}(b,a)(i=1,\cdots,5)$ can be used to define conditional fineness. For these $sh_{i}^{\prime}(i=1,\cdots,5)$, we have

(A1^′)

$sh_{i}^{\prime}(b,a)=0\Longleftrightarrow a\subseteq b.$

For $sh_{2}^{\prime}$, we also have

(A2^′)

$sh_{2}^{\prime}(b,a)=1\Longleftrightarrow a\cap b=\emptyset.$

A subsethood measure of two sets is a quantitative generalization of the set inclusion relation, and a subsethood measure of two granules should be a quantitative generalization of the coarse-fine relation.

The monotonically increasing and monotonically decreasing measures corresponding to conditional granularity and conditional fineness respectively can be defined as follows.

We also have (A3) $\Rightarrow$ (A4) and (A4) $\Rightarrow$ (A3), and (A3^′) $\Rightarrow$ (A4^′) and (A4^′) $\Rightarrow$ (A3^′). Therefore, (A3) or (A4) alone can be the monotonically increasing condition, and (A3^′) or (A4^′) alone can be the monotonically decreasing condition.

For any equivalence granules $A,B,C$ on $X$, the conditions (A5), $\cdots$, (A12) and the conditions (A5^′), $\cdots$,(A12^′) are as follows.

(A5)

$C\succeq B\wedge(B\wedge A=C\wedge A)\Rightarrow sh(B,A)\leq sh(C,A)$;

(A6)

$C\succeq B\wedge(B\wedge A=C\wedge A)\Rightarrow sh(A,C)\leq sh(A,B)$;

(A7)

$C\succeq B\wedge(B\vee A=C\vee A)\Rightarrow sh(B,A)\leq sh(C,A)$;

(A8)

$C\succeq B\wedge(B\vee A=C\vee A)\Rightarrow sh(A,C)\leq sh(A,B)$;

(A9)

$C\succeq B\succeq A\Rightarrow sh(B,A)\leq sh(C,A)$;

(A10)

$C\succeq B\succeq A\Rightarrow sh(A,C)\leq sh(A,B)$;

(A11)

$A\succeq C\succeq B\Rightarrow sh(B,A)\leq sh(C,A)$;

(A12)

$A\succeq C\succeq B\Rightarrow sh(A,C)\leq sh(A,B)$;

(A5^′)

$C\succeq B\wedge(B\wedge A=C\wedge A)\Rightarrow sh(C,A)\leq sh(B,A)$;

(A6^′)

$C\succeq B\wedge(B\wedge A=C\wedge A)\Rightarrow sh(A,B)\leq sh(A,C)$;

(A7^′)

$C\succeq B\wedge(B\vee A=C\vee A)\Rightarrow sh(C,A)\leq sh(B,A)$;

(A8^′)

$C\succeq B\wedge(B\vee A=C\vee A)\Rightarrow sh(A,B)\leq sh(A,C)$;

(A9^′)

$C\succeq B\succeq A\Rightarrow sh(C,A)\leq sh(B,A)$;

(A10^′)

$C\succeq B\succeq A\Rightarrow sh(A,B)\leq sh(A,C)$;

(A11^′)

$A\succeq C\succeq B\Rightarrow sh(C,A)\leq sh(B,A)$;

(A12^′)

$A\succeq C\succeq B\Rightarrow sh(A,B)\leq sh(A,C)$.

The conditions (A5), (A7), (A9) and (A11) are weaker versions of (A3), i.e., (A3) $\Rightarrow$ (A5), (A7), (A9) and (A11); the axioms (A6), (A8), (A10) and (A12) are weaker versions of (A4), i.e., (A4) $\Rightarrow$ (A6), (A8), (A10) and (A12). The conditions (A5), (A6), (A7) and (A8) are a special case of (A9), (A10), (A11) and (A12), respectively. The conditions (A5) and (A6) are the dual questions of (A7) and (A8) respectively, and the conditions (A9) and (A10) are the dual questions of (A11) and (A12) respectively. While the axiom (Ai) is reversed with (Ai^′) ($i=1,\cdots,12$). The first four are their basic properties.

Given two equivalence granules $A=\{a_{1},\cdots,a_{k}\}$ and $B=\{b_{1},\cdots,b_{l}\}$ on $X$. Then there are $|a_{i}\cap b_{j}|(a_{i}\cap b_{j})(i=1,\cdots,k,j=1,\cdots,l)$ in $A\wedge B$, where $|a_{i}\cap b_{j}|$ is the cardinality of $a_{i}\cap b_{j}$. We can normalize these $|a_{i}\cap b_{j}|(i=1,\cdots,k,j=1,\cdots,l)$ and get a probability distribution which is called a probability distribution of the granule $A\wedge B$ denoted as $P_{A\wedge B}$.

where $p(a_{i}\cap b_{j})$ indicates the probability of the intersection of $a_{i}$ and $b_{j}$ contained in $X$. We have the following result.

Given two equivalence granules $A=\{a_{1},\cdots,a_{k}\}$ and $B=\{b_{1},\cdots,b_{l}\}$ on $X$. Then, for each $sh_{m}(m=1,\cdots,5)$, the conditional granularity of $B$ with respect to $A$ is defined by the expectations of $sh_{m}(m=1,\cdots,5)$ with respect to the probability distribution of $A\wedge B$.

In general, we can take $sh_{m}^{\prime}(\cdot,\cdot)=1-sh_{m}(\cdot,\cdot)(m=1,\cdots,5)$. Then, the expectations of $sh_{m}^{\prime}(\cdot,\cdot)=1-sh_{m}(\cdot,\cdot)(m=1,\cdots,5)$ with respect to the probability distribution of $A\wedge B$ is $E_{P_{A\wedge B}}({sh_{m}^{\prime}}(\cdot,\cdot))$

Given two equivalence granules $A=\{a_{1},\cdots,a_{k}\}$ and $B=\{b_{1},\cdots,b_{l}\}$ on $X$. The conditional fineness of $B$ with respect to $A$ can be defined by

By the above definition, we can easily get the following theorems.

Now we start to prove $G_{i}(B|A)(i=1,\cdots,5)$ satisfies the axiom (A1), (A3) and (A4), and $F_{i}(B|A)(i=1,\cdots,5)$ satisfies the axiom (A1^′), (A3^′) and (A4^′).

The proofs are seen in Appendix Proof of the Theorem III.6. By the above theorem, we can get the following corollary.