有序信息系统中多粒度决策理论粗糙集模型及其应用

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本文主要探讨了有序信息系统中的多粒度决策理论粗糙集（Multigranulation Decision-theoretic Rough Set in Ordered Information System）。在当前的研究背景下，基于贝叶斯决策理论的决策粗糙集模型已经成为粗糙集理论发展的重要趋势。为了进一步扩展这一理论，作者Wentao Li 和 Weihua Xu 在《Fundamenta Informaticae》期刊的第139卷（2015年）67-89页上提出了一个创新的多粒度决策粗糙集模型。该模型的核心思想是将有序信息系统特有的结构引入到决策粗糙集的框架中。传统的粗糙集理论通常处理的是单一粒度的信息，而多粒度方法允许在不同层面捕捉复杂性，并考虑信息系统的内在顺序。在这个新模型中，关键的概念是引入了一个基本集分配函数，它在有序信息系统中起着桥梁作用，用于定义粗糙集的划分依据。文章详细阐述了如何在有序信息系统中构建这种多粒度粗糙集，包括如何根据系统的有序特性来定义不确定性、模糊性和依赖关系。此外，作者还探讨了如何通过这个函数来量化决策者对数据的理解和偏好，以及如何根据这些量化信息进行有效的决策分析。这不仅提升了粗糙集在决策支持系统中的实用性，而且有助于处理那些在单一粒度下难以解析的复杂决策问题。通过引入多粒度决策理论粗糙集，该研究为有序信息系统提供了更为精细和灵活的决策分析工具，为实际应用中的信息处理和决策优化提供了新的可能性。此外，文中可能还包含了计算方法、案例研究或与其他粗糙集理论比较的部分，以证明其有效性。这篇论文不仅扩展了粗糙集理论在决策领域的应用，而且还展示了如何利用有序信息系统的特性来提升决策的精度和效率。这对于推进粗糙集理论在信息技术、数据挖掘和人工智能等领域的发展具有重要意义。

W. Li and W. Xu / Multigranulation Decision-theoretic Rough Set in Ordered Information System 71

Let U be a non-empty and ﬁnite set of objects, one can deﬁne P as probability measure if the set-

valued function P maps from 2

to [0, 1]. P satisﬁes the two conditions: P (U ) = 1; if A ∩ B = ∅, then

P (A ∪ B) = P (A) + P (B). Then P is a probability measure of σ − algebra which is combined by the

family subset of U.

In an information system, given U as a non-empty and ﬁnite set of objects, where R is an equivalence

relation in U. Denote [x]

is the equivalence class with respect to x. And P is a probability measure

of σ − algebra which is combined by the family subset of U . The triple A

= (U, R, P ) is called

probability approximation space. And P = (X|Y ) is the conditional probability of whether concept X

happens or not depends on Y .

Deﬁnition 2.1 [36] Let 0 ≤ β < α ≤ 1, for any X ⊆ U, the lower and upper approximations based on

thresholds α, β with respect to A

= (U, R, P ) are deﬁned as follows

(α,β)

(X) = {x ∈ U|P (X|[x]

) ≥ α},

(α,β)

(X) = {x ∈ U|P (X|[x]

) > β}.

If pr

(α,β)

(X) = pr

(α,β)

(X), then X is a deﬁnable set, otherwise X is a rough set.

Accordingly, the probabilistic positive, negative and boundary region are

pos(X) = pr

(α,β)

(X) = {x ∈ U|P (X|[x]

) ≥ α};

neg(X) = U − pr

(α,β)

(X) = {x ∈ U|P (X|[x]

) ≤ β};

bn(X) = pr

(α,β)

(X) − pr

(α,β)

(X) = {x ∈ U|β < P (X|[x]

) < α}.

The parameters α, β in the probabilistic rough set theory above can be determined by special methods

according to some additional conditions. From Bayesian decision theory as one will introduce next, the

parameters α, β can be obtained.

In the Bayesian decision produce, a ﬁnite set of states can be written as Ω = {ω

, ω

, · · · , ω

}, and a

ﬁnite set of m possible actions can be denoted by A = {a

, a

, · · · , a

}. Let P(ω

|x) be the conditional

probability of an object x being in state ω

given that the object is described by x. Let λ(a

|ω

) denote

the loss, or cost for taking action a

when the state is ω

, the expected loss function associated with

taking action a

is given by

R(a

|x) =

j=1

λ(a

|ω

)P (ω

|x)

In Pawlak’s rough set theory, lower and upper approximation operators partition the universe U into

three disjoint sets. Using the conditional probability P (X|[x]

), the Bayesian decision procedure can

decide how to assign x into these disjoint three regions.

With respect to the membership of an object in X, we have a set of two states and a set of three

actions for each state. The set of states is given by Ω = {X, X

} indicating that an element is in X and

not in X, respectively. The set of actions with respect to a state is given by A = {a

, a

}, where

P , B and N represent the three actions in deciding x ∈ pos(X), deciding x ∈ bn(X), and deciding

x ∈ neg(X), respectively. The loss function regarding the risk or cost of actions in different states is

given by the 3 × 2 matrix:

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