Compromised multi-granulation rough sets based on an attribute-extension chain

Abstract

The multi-granulation rough sets serve as important hierarchical models for intelligent systems. However, their mainstream optimistic and pessimistic models are respectively too loose and strict, and this defect becomes especially serious in hierarchical processing on an attribute-expansion sequence. Aiming at the attribute-addition chain, compromised multi-granulation rough set models are proposed to systematically complement and balance the optimistic and pessimistic models. According to the knowledge refinement and measure order induced by the attribute-enlargement sequence, the basic measurement positioning and corresponding pointer labeling based on equilibrium statistics are used, and thus we construct four types of compromised models at three levels of knowledge, approximation, and accuracy. At the knowledge level, the median positioning of ordered granulations derives Compromised-Model 1; at the approximation level, the average positioning of approximation cardinalities is performed, and thus the separation and integration of dual approximations respectively generate Compromised-Models 2 and 3; at the accuracy level, the average positioning of applied accuracies yields Compromised-Model 4. Compromised-Models 1–4 adopt distinctive cognitive levels and statistical perspectives to improve and perfect the multi-granulation rough sets, and their properties and effectiveness are finally verified by information systems and data experiments.

Keywords

Multi-granulation rough set statistical compromised modeling attribute-addition chain granular computing tri-level analysis

1 Introduction

The rough set theory is an important theory for intelligent system processing [13], and its basis is the knowledge granulation induced by a specific equivalence relation. In granular computing, multiple equivalence relations and their systematic constructions are practically required, and thus multiple granulations emerge. In particular, multi-granulation rough sets were proposed by Qian et al. [15], and their basic models –the optimistic and pessimistic multi-granulation rough sets –adopt different logical fusions of multiple equivalence relations to establish dual approximations [16, 17]. Nowadays, the multi-granulation rough sets have been deeply studied and applied in the extended modeling [4 , 28], uncertainty measurement [6 , 27], three-way analysis [14 , 32], attribute reduction [7 , 23], decision making [20 , 30], and classification learning [5 , 19], etc, where multiple information factors (such as the fuzziness, neighborhood, and covering) were introduced as well. For example, Yao and She [28] offered four rough set models in multi-granulation spaces by using both the combination orders of relation and approximation and the set operations of intersection and union; Huang et al. [6] introduced the intuitionistic fuzzy measures to discuss multi-granulation intuitionistic fuzzy decision-theoretic rough sets; Sun et al. [22] proposed a neighborhood multi-granulation rough sets-based attribute reduction method by using Lebesgue and entropy measures in incomplete neighborhood decision systems; Senthil Kumar and Hannah Inbarani [19] explored the cardiac arrhythmia classification by applying the optimistic and pessimistic multi-granulation rough sets.

Regarding the multi-granulation rough sets, the classical optimistic and pessimistic models respectively use logical fusions of disjunction and conjunction to determine extreme values of dual approximations [15 –17], so they have outstanding advantages of construction simpleness and attitude clearness. On the other side, the optimistic and pessimistic multi-granulation models cause some shortcomings including the attitude extreme, value broadness, and feature deviation; unfortunately, these drawbacks may be magnified especially when the multiple granulations prominently exhibit a strength deviation or a biased distribution. Thus, novel compromised multi-granulation models can characterize systematic equilibriums to provide the intermediate objectivity and attitude improvement, i.e., they can balance and improve the existing optimistic and pessimistic models, so they are worth deeply constructing to underlie wide applications of intelligent systems. However, there hardly have relevant studies, because how to reasonably make the compromised modeling in complex environments becomes a difficulty. This paper aims to make a preliminary attempt to construct compromised multi-granulation rough set models by adopting a neutral and objective attitude based on the statistics principle, and it mainly resorts to a specific but valuable scenario regarding an attribute-enlargement sequence; thus, we can provide more robust application models for intelligent and uncertainty processing.

A distinctive kind of multiple granulations is formed by an attribute-extension chain, and the relevant granulation monotonicity underlies optimization constructions [1 , 24]. For example, Wang [24] studied monotonic uncertainty measures and their attribute reduction in probabilistic rough sets, and thus the attribute-extension chain essentially functions in both the measure construction and reduct search. In fact, the attribute-extension chain contains a refinement sequence of multiple granulations to transcend the simplicity of parallel or dispersive attributes, and it can deeply and effectively probe the knowledge hierarchy of the power set space with overall attributes and all attribute subsets; therefore, underlying total-order features are beneficial to granular computing of complex non-linear interactions, and the attribute-extension chain becomes a usual and important tool for the relevant granulation construction, hierarchy processing, and knowledge discovery, etc. In terms of the attribute-enlargement chain, the optimistic and pessimistic models respectively become too loose and strict, so compromised models are urgently required to objectively reflect the data draw and system characteristic. This paper mainly focuses on the attribute-addition chain to establish compromised multi-granulation rough sets. By statistical characteristics of equilibriums, the basic measurement positioning and matching pointer labeling are used to construct four types of compromised models at three levels of the knowledge, approximation, and accuracy; furthermore, the relevant modeling properties and improvement effectiveness are verified by information systems and data experiments. Regarding contributions, compromised multi-granulation rough sets are proposed to improve both the optimistic and pessimistic multi-granulation rough sets and thus they enrich multi-granulation rough sets, mainly by virtue of an attribute-enlargement sequence; hence, four types of compromised models emerge to offer more application spaces for granular computing and cognitive analysis.

The remainder of this paper is organized as follows. Section 2 reviews the multi-granulation rough sets, mainly the optimistic and pessimistic models. Section 3 constructs four types of compromised multi-granulation rough sets based on an attribute-addition chain, including three parts of the basic modeling, comprehensive summary, and example illustration. Section 4 implements the data experiment verification by using seven UCI datasets. Finally, Section 5 concludes this paper.

2 Multi-granulation rough sets

In this section, multi-granulation rough sets are reviewed by Refs. [15 –17].

Let S = (U, AT, f) be a complete information system. Herein, U is a finite non-empty universe of objects, AT is a finite non-empty set of attributes, and f is an information function. Each attribute subset Q ⊆ AT determines a binary indiscernibility relation IND (Q) = {(x, y) ∈ U² : ∀ a ∈ Q, f (x, a) = f (y, a)}. This equivalence relation induces the equivalence class [x] _Q and equivalence partition U/IND (Q); the former means a granule containing sample x, while the latter implies a granulation called knowledge. Thus, the rough set modeling mainly resorts to the granular block and granulation structure to approximate a target concept X ⊆ U, and the approximation and accuracy serve as core notions. Let ||, ∼, ⌊⌋ denote the cardinality, complement, rounding functions, respectively.

Definition 1. ([13]) Regarding an attribute subset Q, the lower and upper approximations of X are defined by $\begin{matrix} \underline{Q} (X) & = {x \in U : [x]_{Q} \subseteq X}, \\ \bar{Q} (X) & = {x \in U : [x]_{Q} \cap X \neq \emptyset}, \end{matrix}$ (1) the corresponding accuracy of X is $α (Q, X) = \frac{| \underline{Q} (X) |}{| \bar{Q} (X) |} .$ (2)

Proposition 1. ([13]) The dual approximations constitute dual bounds of the central concept, i.e., $\underline{Q} (X) \subseteq X \subseteq \bar{Q} (X)$ .

Definition 1 and Proposition 1 reflect classical rough sets. By the knowledge granulation, the lower and upper approximations bidirectionally approach the basic set, thus determining the rough set modeling. The accuracy becomes the approximation cardinality ratio to represent a sort of knowledge-based completeness degrees for a concept, thus becoming an applied measure for the exactness and certainty characterizations.

The above traditional modeling concerns only a granulation induced by an attribute subset. In practice, multiple granulations induced by multiple attribute subsets are required. Accordingly, the multi-granulation rough sets are generated by aggregating an attribute subset sequence Q₁, Q₂, ⋯ , Q_m ⊆ AT and its multiple granulations U/IND (Q₁), U/IND (Q₂), ⋯, U/IND (Q_m), and they utilize logical fusions of concept approximations to establish the mainstream optimistic and pessimistic models.

Definition 2. ([16]) Regarding the family of attribute subsets Q₁, Q₂, ⋯ , Q_m, the optimistic lower and upper approximations of X are defined by

$\begin{matrix} {\underline{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) & = {x : \lor [x]_{Q_{i}} \subseteq X, 1 \leq i \leq m}, \\ {\bar{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) & = \sim ({\underline{\sum_{i = 1}^{m} Q_{i}}}^{O} (\sim X)), \end{matrix}$ (3) and the corresponding accuracy of X becomes

$\begin{matrix} α {(\sum_{i = 1}^{m} Q_{i}, X)}^{O} = \frac{| {\underline{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) |}{| {\bar{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) |}; \end{matrix}$ (4) in contrast, the pessimistic lower and upper approximations and the related accuracy of X are defined by

$\begin{matrix} {\underline{\sum_{i = 1}^{m} Q_{i}}}^{P} (X) & = {x : \land [x]_{Q_{i}} \subseteq X, 1 \leq i \leq m}, \\ {\bar{\sum_{i = 1}^{m} Q_{i}}}^{P} (X) & = \sim ({\underline{\sum_{i = 1}^{m} Q_{i}}}^{P} (\sim X)), \\ α {(\sum_{i = 1}^{m} Q_{i}, X)}^{P} & = \frac{| {\underline{\sum_{i = 1}^{m} Q_{i}}}^{P} (X) |}{| {\bar{\sum_{i = 1}^{m} Q_{i}}}^{P} (X) |} . \end{matrix}$ (5)

Proposition 2. ([15 –17]) The optimistic and pessimistic multi-granulation approximations can be integrated by single-granulation approximations, i.e., $\begin{matrix} {\underline{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) = ⋃_{i}^{m} \underline{Q_{i}} (X), & {\bar{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) = ⋂_{i}^{m} \bar{Q_{i}} (X), \\ {\underline{\sum_{i = 1}^{m} Q_{i}}}^{P} (X) = ⋂_{i}^{m} \underline{Q_{i}} (X), & {\bar{\sum_{i = 1}^{m} Q_{i}}}^{P} (X) = ⋃_{i}^{m} \bar{Q_{i}} (X) . \end{matrix}$ They and their accuracies offer strength relationships: ${\underline{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) \supseteq {\underline{\sum_{i = 1}^{m} Q_{i}}}^{P} (X)$ , ${\bar{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) \subseteq {\bar{\sum_{i = 1}^{m} Q_{i}}}^{P} (X)$ , $α {(\sum_{i = 1}^{m} Q_{i}, X)}^{O} \geq α {(\sum_{i = 1}^{m} Q_{i}, X)}^{P}$ .

As a supplement, the strength relation of knowledge granulation is recalled [33]. In S = (U, AT, f), let R₁, R₂ ⊆ AT be two attribute subsets. Knowledge granulation U/R₂ is finer than granulation U/R₁ (noted as U/R₁ ≽ U/R₂), if ∀ [x] _{R
₂} ∈ U/IND (R₂), ∃ [x′] _{R
₁} ∈ U/IND (R₁), s.t., [x] _{R
₂} ⊆ [x′] _{R
₁}. The knowledge refinement U/R₁ ≽ U/R₂ can be naturally achieved by attribute enlargement R₁ ⊆ R₂, and it includes a strict case U/R₁ ≻ U/R₂ when U/IND (R₁) ≠ U/IND (R₂). The symmetrical knowledge coarsening is similarly and oppositely determined, and it uses symbols ⪯, ≺.

3 Four types of compromised multi-granulation rough sets based on an attribute-addition chain

The multi-granulation rough sets have two main models. By Definition 2, the optimistic and pessimistic models respectively adopt logical disjunctions and conjunctions to determine dual approximations. This concise modeling adheres to extreme attitudes and maximum/minimum characteristics, so it has some limitations of broadness and bias. In-depth models are worth constructing to comprehensively characterize the multi-granulation system.

Aiming at the hierarchical application of an attribute-addition chain, we next propose compromised multi-granulation models to well balance the classical optimistic and pessimistic models. Concretely, statistical characteristics of equilibriums are utilized for the approximation positioning, and thus four types of compromised models are established at three levels to achieve their compromised properties and mutual relationships. In S = (U, AT, f), the attribute-enlargement sequence is supposed to be

$Q_{1} \subseteq Q_{2} \subseteq \dots \subseteq Q_{m} \subseteq AT .$ (6)

3.1 Basic modeling

Herein, the basic compromised modeling is implemented. For this purpose, basic properties of the attribute-addition chain are first provided to clarify the relevant reasonability and implementation.

Lemma 1. Regarding Q₁ ⊆ Q₂ ⊆ ⋯ ⊆ Q_m, we have

IND (Q₁) ⊇ IND (Q₂) ⊇ ⋯ ⊇ IND (Q_m), U/IND (Q₁) ≽ ⋯ ≽ U/IND (Q_m), $\underline{Q_{1}} (X) \subseteq \underline{Q_{2}} (X) \subseteq \dots \subseteq \underline{Q_{m}} (X)$ , $\bar{Q_{1}} (X) \supseteq \bar{Q_{2}} (X) \supseteq \dots \supseteq \bar{Q_{m}} (X)$ , α (Q₁, X) ≤ α (Q₂, X) ≤ ⋯ ≤ α (Q_m, X).

Lemma 2. Regarding Q₁ ⊆ Q₂ ⊆ ⋯ ⊆ Q_m, the optimistic multi-granulation model exhibits

${\underline{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) = \underline{Q_{m}} (X)$ , ${\bar{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) = \bar{Q_{m}} (X)$ , $α {(\sum_{i = 1}^{m} Q_{i}, X)}^{O} = α (Q_{m}, X)$ ,

while the pessimistic model offers

${\underline{\sum_{i = 1}^{m} Q_{i}}}^{P} (X) = \underline{Q_{1}} (X)$ , ${\bar{\sum_{i = 1}^{m} Q_{i}}}^{P} (X) = \bar{Q_{1}} (X)$ , $α {(\sum_{i = 1}^{m} Q_{i}, X)}^{P} = α (Q_{1}, X)$ .

Corollary 1. Regarding Q₁ ⊆ Q₂ ⊆ ⋯ ⊆ Q_m, the optimistic and pessimistic models are respectively determined by carrier subsets Q_m and Q₁ or corresponding positioning pointers $i_{O}^{*} = m, i_{P}^{*} = 1 .$ (7)

By Lemma 1, the attribute-enlargement chain causes some total-order changes of feature strengths, such as the weakening of equivalence relations, refinement of knowledge granulations, expansion of lower approximations, contraction of upper approximations, and increase of accuracy measures. Thus by Lemma 2 and Corollary 1, the optimistic and pessimistic models naturally intercept the extreme values at two endpoints; that is, the optimistic/pessimistic model is completely embodied by the largest/least subset at the right/left endpoint. Regarding the two models, their strength characteristic of interactions still exists on the attribute-addition chain, but it becomes particularly extreme; thus, their maximum/minimum statistics with the great deviation cannot well evaluate the system characteristics.

To effectively characterize the integrality of the attribute-enlargement chain, we next make the compromised modeling by the equilibrium statistic and balance positioning, and we mainly resort to an explicit identification, i.e., a positioning pointer of natural number indexes (as well as its corresponding chain-element subset). Accordingly, suitable chain points (rather than endpoints) are optimally selected to determine compromised models and their connotations. Then, four compromised models are positioned by pointers $i_{k}^{*} \in {1, 2, \dots, m}$ ( $k = 1, \underline{2}, \bar{2}, 3, 4$ ) and labeled by symbols C_k (k = 1, 2, 3, 4), where letter C implies word “compromise”.

The attribute-expansion chain contains several total-order strengths (such as the knowledge refinement), and the optimistic and pessimistic models rely on the chain endpoints to embody the degeneration and extreme. In contrast, a simple compromised model is to locate the chain midpoint, and this median statistic can directly indicate some systematicness and representativeness. Accordingly, the first model is proposed as follows.

Definition 3. (Compromised-Model 1) Let $i_{1}^{*} = ⌊ \frac{m + 1}{2} ⌋$ be the median of index set {1, 2, ⋯ , m}. Regarding Q₁ ⊆ Q₂ ⊆ ⋯ ⊆ Q_m, pointer $i_{1}^{*}$ (as well as its subset $Q_{i_{1}^{*}}$ ) determines Compromised-Model 1. That is, the relevant approximation and accuracy are $\begin{matrix} {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) = \underline{Q_{i_{k}^{*}}} (X), & {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) = \bar{Q_{i_{k}^{*}}} (X), \\ α {(\sum_{i = 1}^{m} Q_{i}, X)}^{C_{k}} = α & (Q_{i_{k}^{*}}, X) = \frac{| \underline{Q_{i_{k}^{*}}} (X) |}{| \bar{Q_{i_{k}^{*}}} (X) |}, \end{matrix}$ (8) where k = 1.

Compromised-Model 1 uses the chain median to directly establish the approximation and accuracy, so it acquires an intuitionistic equilibrium between the optimistic and pessimistic models. The modeling simply considers the ordinal ranking statistic of knowledge granulations to offer advantages of the direct mechanism and easy operation, but it never involves the concept description. Improved models need adding relevant concept information, and two-level compromised models are next constructed by introducing the approximation cardinalities and accuracy values.

Rough set models mainly embrace dual approximations. For the compromised modeling, the approximation level can be utilized, and its relevant cardinalities underlie statistical evaluations. By Lemmas 3.1 and 3.2, Corollary 3.1, the approximation cardinalities have total orders: $| \underline{Q_{1}} (X) | \leq | \underline{Q_{2}} (X) | \leq \dots \leq | \underline{Q_{m}} (X) |$ , $| \bar{Q_{1}} (X) | \geq | \bar{Q_{2}} (X) | \geq \dots \geq | \bar{Q_{m}} (X) |$ ,

while the optimistic and pessimistic models are positioned at the extreme bounds. To acquire the balance and representativeness, two means of approximation cardinality sequences can provide two systematic locations for compromised lower and upper approximations. Next, two types of compromised models are constructed by two strategies, i.e., the respective and uniform positioning pointers for dual approximations.

Definition 4. (Compromised-Models 2 and 3) For Q₁ ⊆ Q₂ ⊆ ⋯ ⊆ Q_m, suppose that the average cardinality $| \underline{\sum_{i = 1}^{m} Q_{i}} (X) | = \frac{1}{m} \sum_{i = 1}^{m} | \underline{Q_{i}} (X) |$

of lower approximations is closely realized by s (s ∈ [1, m]) indexes: $i_{\underline{2} 1}^{*}, \dots, i_{\underline{2} s}^{*} \in {1, 2, \dots, m}$ , which simultaneously yield the shortest distance: $min_{i = 1, 2, \dots, m} | | \underline{Q_{i}} (X) | - | \underline{\sum_{i = 1}^{m} Q_{i}} (X) | |$ ;

thus, a locating tag is set up by the middle index, i.e., $\begin{matrix} i_{\underline{2}}^{*} = i_{\underline{2} ⌊ \frac{1 + s}{2} ⌋}^{*} . \end{matrix}$ (9) Similarly, the case of upper approximation generates the cardinality mean, carrier indexes, and locating pointer as follows: $| \bar{\sum_{i = 1}^{m} Q_{i}} (X) | = \frac{1}{m} \sum_{i = 1}^{m} | \bar{Q_{i}} (X) |$ , $i_{\bar{2} 1}^{*}, \dots, i_{\bar{2} t}^{*}$ , $i_{\bar{2}}^{*} = i_{\bar{2} ⌊ \frac{1 + t}{2} ⌋}^{*}$ .

(1) The pointer pair $(i_{\underline{2}}^{*}, i_{\bar{2}}^{*})$ (as well as matching subsets $Q_{i_{\underline{2}}^{*}}, Q_{i_{\bar{2}}^{*}}$ ) yields Compromised-Model 2, and the corresponding approximation and accuracy become $\begin{matrix} {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{2}} (X) = \underline{Q_{i_{\underline{2}}^{*}}} (X), & {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{2}} (X) = \bar{Q_{i_{\bar{2}}^{*}}} (X), \\ α {(\sum_{i = 1}^{m} Q_{i}, X)}^{C_{2}} & = \frac{| \underline{Q_{i_{\underline{2}}^{*}}} (X) |}{| \bar{Q_{i_{\bar{2}}^{*}}} (X) |}, \end{matrix}$ (10) which accords with Equation (8) by revising k to 2 or $\underline{2}$ , $\bar{2}$ . (2) Double pointers $i_{\underline{2}}^{*}, i_{\bar{2}}^{*}$ derive a median pointer: $\begin{matrix} i_{3}^{*} = ⌊ \frac{i_{\underline{2}}^{*} + i_{\bar{2}}^{*}}{2} ⌋, \end{matrix}$ (11) and this label (as well as its carrier subset $Q_{i_{3}^{*}}$ ) defines Compromised-Model 3, whose approximation and accuracy are determined by Equation (8) by setting up k = 3.

In Definition 4, the average statistics of dual approximation cardinalities are utilized to construct two compromised models, where pointers $i_{\underline{2}}^{*}$ and $i_{\bar{2}}^{*}$ may be unequal. Compromised-Model 2 considers the respective realizations of dual approximations, by two-dimensional pairs $(i_{\underline{2}}^{*}, i_{\bar{2}}^{*})$ and $(Q_{i_{\underline{2}}^{*}}, Q_{i_{\bar{2}}^{*}})$ . Compromised-Model 3 captures a common implementation of dual approximations, by one-dimensional indexes $i_{3}^{*}$ and $Q_{i_{3}^{*}}$ which respectively adhere to a direct median of $(i_{\underline{2}}^{*}, i_{\bar{2}}^{*})$ and a concise compromise of $(Q_{i_{\underline{2}}^{*}}, Q_{i_{\bar{2}}^{*}})$ .

Furthermore, the accuracy fuses dual approximation cardinalities to carry the basic certainty measurement and valuable application message, so it is worth introducing for the compromised modeling. By Lemmas 3.1 and 3.1, Corollary 3.1, the accuracy exhibits the monotonic increase on the attribute-addition chain, while the accuracies in optimistic and pessimistic models also reach the maximum and minimum with extreme statistics. Accordingly, the statistical average accuracy on the attribute chain can inspire a positioning pointer to implement the compromised modeling, and this process is similar to the above ones regarding Compromised-Models 2 and 3.

Definition 5. (Compromised-Model 4) For Q₁ ⊆ Q₂ ⊆ ⋯ ⊆ Q_m, suppose that the average accuracy $α (\sum_{i = 1}^{m} Q_{i}, X) = \frac{1}{m} \sum_{i = 1}^{m} α (Q_{i}, X)$

is closely realized by l (l ∈ [1, m]) indexes: $i_{41}^{*}, \dots, i_{4 l}^{*} \in {1, 2, \dots, m}$ , which yield the optimal distance: $min_{i = 1, 2, \dots, m} | α (Q_{i}, X) - α (\sum_{i = 1}^{m} Q_{i}, X) |$ ,

and a locating pointer is set up by the middle index: $\begin{matrix} i_{4}^{*} = i_{4 ⌊ \frac{1 + l}{2} ⌋}^{*} . \end{matrix}$ (12) Thus, pointer $i_{4}^{*}$ (as well as its carrier subset $Q_{i_{4}^{*}}$ ) defines Compromised-Model 4, whose approximation and accuracy are determined by Equation (8) by setting up k = 4.

Compromised-Model 4 considers the accuracy to well characterize the systematic completeness or the average certainty on the overall attribute-expansion chain, and thus it balances and improves the optimistic and pessimistic models. This new model has the fundamental significance for the systematic measurement and related application embracing concepts.

3.2 Comprehensive summary

Thus far, four types of compromised multi-granulation rough set models have been established, mainly on an attribute-extension chain. Herein, they are summarized to offer their comparative analysis, integral property, and systematic algorithm.

At first, the compromised modeling mechanisms are clarified in Fig. 1. Based on multiple granulations from the attribute-increase line, the optimistic and pessimistic models involve only two chain endpoints, i.e., they are respectively realized by ordinal locations of (m, Q_m) and (1, Q₁), and thus their relevant extreme and span go against the integral representation of the multi-granulation system. In contrast, our compromised models rationally pursue the multi-granulation systematicness and representativeness, and the relevant modeling mainly depends on the vivid moderation positioning. According to the total-order strength contained in the extension chain, several equilibrium statistics are reasonably used to construct four compromised models, and we refer to the tri-level analysis [29, 34].

Fig. 1

Four compromised multi-granulation models at three levels and their statistical positioning mechanisms.

(1) At the knowledge level, the granulation becomes finer on the attribute-increase chain. Thus, ordinal median $i_{1}^{*}$ and carrier subset $Q_{i_{1}^{*}}$ derive Compromised-Model 1. This model adheres to the medium granulation strength to become concise but superficial. (2) At the approximation level, the dual approximations and their cardinalities exhibit the monotonic change on the attribute chain. Thus, pointer pair $(i_{\underline{2}}^{*}, i_{\bar{2}}^{*})$ and subset group $(Q_{i_{\underline{2}}^{*}}, Q_{i_{\bar{2}}^{*}})$ represent cardinality averages of dual approximations, and they offer Compromised-Model 2. This model thoroughly characterizes the systematic multi-granulation average of approximation cardinalities, so it becomes exact and reasonable. Furthermore by medially unifying $i_{\underline{2}}^{*}$ and $i_{\bar{2}}^{*}$ ( $Q_{i_{\underline{2}}^{*}}$ and $Q_{i_{\bar{2}}^{*}}$ ), median $i_{3}^{*}$ and carrier $Q_{i_{3}^{*}}$ determine Compromised-Model 3. This model deeply considers the strength balance of dual approximations to highlight the single positioning and carrier, so it becomes uniform in terms of the compromise of dual approximations. (3) At the accuracy level, the accuracy information increases on the attribute-extension chain. Thus, pointer $i_{4}^{*}$ and subset $Q_{i_{4}^{*}}$ optimally represent the accuracy mean to offer Compromised-Model 4. This model well characterizes the systematic accuracy average of multiple granulations, so it underlies the uncertainty description and optimization application for concepts.

The four compromised models adopt different cognition levels and compromise viewpoints to provide multiple use choices. By comparisons, Compromised-Models 2 and 4 have more advantages. By adhering to the central approximation cardinalities, Compromised-Model 2 separately deals with dual approximations, and this independent compromise brings more balances. By integrating the dual approximation information, Compromised-Model 4 highlights the accuracy measurement, and its compromise pointer and subset are systematically extracted to efficiently underlie the metric processing and practical application.

Proposition 3. Compromised-Models 1–4 all have the compromise function for the optimistic and pessimistic models. That is, when k = 1, 2, 3, 4 we have ${\underline{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) \supseteq {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) \supseteq {\underline{\sum_{i = 1}^{m} Q_{i}}}^{P} (X)$ , ${\bar{\sum_{i = 1}^{m} Q_{i}}}^{O} (X) \subseteq {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) \subseteq {\bar{\sum_{i = 1}^{m} Q_{i}}}^{P} (X)$ , $α {(\sum_{i = 1}^{m} Q_{i}, X)}^{O} \geq α {(\sum_{i = 1}^{m} Q_{i}, X)}^{C_{k}} \geq α {(\sum_{i = 1}^{m} Q_{i}, X)}^{P}$ .

By Proposition 3, our compromised models all satisfy the perfect equilibrium for the optimistic and pessimistic models, which are actually extreme. Therefore, Compromised-Models 1–4 are theoretically effective, and they practically have different comprise focuses and application emphases. In essence, they resort to the suitable knowledge positioning to pursue the systematic characterization, so they adhere to the balanced and representative single-granulation. Concretely, Compromised-Model k depends on only single granulation $U / IND (Q_{i_{k}^{*}})$ (where k = 1, 3, 4), while Compromised-Model 2 utilizes two sorts of single granulation: $U / IND (Q_{i_{\underline{2}}^{*}})$ and $U / IND (Q_{i_{\bar{2}}^{*}})$ . By virtue of these single-granulation transformations, the compromised models can achieve the next basic properties regarding approximation operations. Note that single-granulation approximations usually have multiple properties regarding set operations [13].

Proposition 4. Let X, Y ⊆ U. When k = 1, 2, 3, 4 we have

(1) ${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) \subseteq X \subseteq {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X)$ ,

(2) ${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (\emptyset) = {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (\emptyset) = \emptyset$ ,

${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (U) = {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (U) = U$ ,

(3) ${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (\sim X) = \sim {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X)$ ,

${\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (\sim X) = \sim {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X)$ ,

(4) $\underline{{\sum_{i = 1}^{m} Q_{i}}^{C_{k}}} \underline{{\sum_{i = 1}^{m} Q_{i}}^{C_{k}}} (X)$

$= {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) = {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X)$ ,

$\bar{{\sum_{i = 1}^{m} Q_{i}}^{C_{k}}} \bar{{\sum_{i = 1}^{m} Q_{i}}^{C_{k}}} (X)$

$= {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) = {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X)$ ;

when k = 1, we further have

(5) $X \subseteq Y \Rightarrow {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) \subseteq {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (Y)$ ,

$X \subseteq Y \Rightarrow {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) \subseteq {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (Y)$ ,

(6) ${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X \cap Y)$

$= {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) \cap {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (Y)$ ,

${\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X \cap Y)$

$\subseteq {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) \cap {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (Y)$ ,

(7) ${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X \cup Y)$

$\supseteq {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) \cup {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (Y)$ ,

${\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X \cup Y)$

$= {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X) \cup {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (Y)$ .

In Proposition 4, relevant operation properties are revealed for the compromised models, where Compromised-Model 2 separately considers lower and upper approximations by double pointers $i_{\underline{2}}^{*}$ and $i_{\bar{2}}^{*}$ . Compromised-Model 1 never considers concepts, so it has all basic properties. The surplus three models necessarily concern concepts, so they cannot involve the extended parts with two concepts X, Y, i.e., Items (5) (6) (7). As far as Item (5) is concerned, $X \subseteq Y \Rightarrow \underline{Q_{i}} (X) \subseteq \underline{Q_{i}} (Y), \forall i \in {1, 2, \dots, m},$

and this implies a sort of increase movability between two sequences of lower approximation cardinalities: $| \underline{Q_{1}} (X) | \leq | \underline{Q_{2}} (X) | \leq \dots \leq | \underline{Q_{m}} (X) |$ ,

$| \underline{Q_{1}} (Y) | \leq | \underline{Q_{2}} (Y) | \leq \dots \leq | \underline{Q_{m}} (Y) |$ ;

thus, pointer $i_{\underline{2}}^{*}$ adheres to the above cardinality means, but it cannot acquire a necessary size relationship between cases X and Y, i.e., between ${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{2}} (X)$ and ${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{2}} (Y)$ ; furthermore, the upper-approximation pointer $i_{\bar{2}}^{*}$ , integrated-approximation pointer $i_{3}^{*}$ , and accuracy pointer $i_{4}^{*}$ can be similarly analyzed, so Item (5) does not hold for Compromised-Models 2–4. In general, in-depth properties on multiple concepts need practical verifications and future explorations.

Compromised-Models 1–4 are completely defined by their positioning pointers, so pointers’ size relationships determine model relationships. In terms of complete interactions, pointer $i_{1}^{*}$ considers only the ordinal number to have no connections with other concept-based pointers; pointer $i_{4}^{*}$ focuses on the concentrated accuracy information to have no necessary size relationships with cardinality pointers $i_{\underline{2}}^{*}$ , $i_{\bar{2}}^{*}$ , $i_{3}^{*}$ , although the accuracy comes from the cardinality ratio of dual approximations; only the approximation level implies inherent interactions. Pointers $i_{\underline{2}}^{*}$ and $i_{\bar{2}}^{*}$ respectively refer to means of smaller and greater cardinality sequences, and they offer uncertainty relationships; however, both must bound their middle pointer $i_{3}^{*}$ , i.e., $\min (i_{\underline{2}}^{*}, i_{\bar{2}}^{*}) \leq i_{3}^{*} = ⌊ \frac{i_{\underline{2}}^{*} + i_{\bar{2}}^{*}}{2} ⌋ \leq \max (i_{\underline{2}}^{*}, i_{\bar{2}}^{*});$ (13) thus, Compromised-Models 2 and 3 have certain relationships as follows.

Proposition 5. For Compromised-Models 2 and 3,

If $i_{\underline{2}}^{*} \leq i_{\bar{2}}^{*}$ then $i_{\underline{2}}^{*} \leq i_{3}^{*} \leq i_{\bar{2}}^{*}$ , and thus

${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{2}} (X) \subseteq {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{3}} (X)$ ,

${\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{2}} (X) \subseteq {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{3}} (X)$ .

If $i_{\underline{2}}^{*} \geq i_{\bar{2}}^{*}$ then $i_{\underline{2}}^{*} \geq i_{3}^{*} \geq i_{\bar{2}}^{*}$ , and thus

${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{2}} (X) \supseteq {\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{3}} (X)$ ,

${\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{2}} (X) \supseteq {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{3}} (X)$ .

The compromised models can be calculated by a comprehensive algorithm. Thus, Algorithm 1 follows relevant definitions and formulas, and the locating pointers act as basic tools to produce the compromised models (mainly the approximation and accuracy). Step 1 directly provides the first pointer $i_{1}^{*}$ ; Step 2 pursues the measure sums, where the knowledge partition and circulation summation are needed; Step 3 establishes the mean statistics; Step 4 utilizes the loop to probe the minimum distances and their exact realizations regarding the approximation and accuracy; Step 5 accordingly obtains median pointers $i_{\underline{2}}^{*}$ , $i_{\bar{2}}^{*}$ , $i_{4}^{*}$ ; Step 6 offers the last pointer $i_{3}^{*}$ ; finally, Step 7 utilizes all pointers to offer the compromised approximation and accuracy, and Step 8 outputs the results. Algorithm 1 is effective according to the relevant mechanism and process, and it can yield Compromised-Models 1–4. This algorithm parallelly computes all the four models; by the part intercept, it can also derive only a single model. Thus, Compromised-Model 1 mainly requires Step 1, Compromised-Models 2 and 4 parallelly concern Steps 2–5, 7, 8, while Compromised-Model 3 needs basic treatments for $(i_{\underline{2}}^{*}, i_{\bar{2}}^{*})$ to induce pointer $i_{3}^{*}$ in Step 6.

Algorithm 1 Calculation of the four types of compromised
multi-granulation rough set models
Input: A complete information system S = (U, AT, f) with
specific concept X ⊆ U and attribute-addition chain Q₁ ⊆ Q₂
⊆ ⋯ ⊆ Q_m ⊆ AT.
Output: Pointer-based Compromised-Models 1–4.
1: Set up $i_{1}^{*} = ⌊ \frac{m + 1}{2} ⌋$ .
2: Circularly compute the approximations $\underline{Q_{i}} (X)$ , $\bar{Q_{i}} (X)$ ,
cardinalities $\| \underline{Q_{i}} (X) \|$ , $\| \bar{Q_{i}} (X) \|$ , accuracy α (Q_i, X), and their
individual metric sums on the attribute-addition chain.
3: Generate three types of means regarding the cardinality and
accuracy, by Definitions 4 and 5.
4: By the loop contrast, achieve the shortest distances
regarding the above averages and their realization indexes,
according to Definitions 4 and 5. For example, obtain the
minimum distance $min_{i = 1, 2, \dots, m} \| \| \underline{Q_{i}} (X) \| - \| \underline{\sum_{i = 1}^{m} Q_{i}} (X) \| \|$ and
its carrier indexes $i_{\underline{2} 1}^{}, \dots, i_{\underline{2} s}^{}$ by Definition 4.
5: Medially extract pointer $i_{\underline{2}}^{} = i_{\underline{2} ⌊ \frac{1 + s}{2} ⌋}^{}$ by Equation (8), and
similarly exact $i_{\bar{2}}^{}$ and $i_{4}^{}$ .
6: Set up $i_{3}^{} = ⌊ \frac{i_{\underline{2}}^{} + i_{\bar{2}}^{*}}{2} ⌋$ by Equation (11).
7: By pointers $i_{1}^{}$ , $(i_{\underline{2}}^{}, i_{\bar{2}}^{})$ , $i_{3}^{}$ , $i_{4}^{*}$ , obtain the approximation and
accuracy in Equation (8), where k concerns 1, 2, 3, 4 as well
as $\underline{2}, \bar{2}$ .
8: return ${\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X)$ , ${\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{k}} (X)$ , and
$α {(\sum_{i = 1}^{m} Q_{i}, X)}^{C_{k}}$ , where k = 1, 2, 3, 4.

3.3 Example illustration

Herein, Compromised-Models 1–4 are illustrated by a table example. The complete information system S = (U, AT, f) is given in Table 1, and it is a decision table to have universe U = {x₁, x₂, ⋯ , x₁₅} (i.e., U = {1, 2, ⋯ , 15}), condition attribute set C = {c₁, c₂, ⋯ , c₉} ⊆ AT, and decision attribute set D = {d} ⊆ AT. (1) The special multiple granulations are formed by a natural attribute-addition chain: $\begin{matrix} Q : Q_{1} = {c_{1}} \subseteq \dots \subseteq Q_{| C |} = {c_{1}, \dots, c_{| C |}} . \end{matrix}$ (14) (2) Target concepts consist of three decision classes: X₁ = {x₁, x₃, x₄, x₇, x₈, x₁₃} = {1, 3, 4, 7, 8, 13},

X₂ = {x₂, x₆, x₉, x₁₀, x₁₁} = {2, 6, 9, 10, 11},

X₃ = {x₅, x₁₂, x₁₄, x₁₅} = {5, 12, 14, 15}.

Regarding the attribute chain and decision class, the initial approximation sets are given in Table 2, the statistical cardinality and precision are shown in Table 3, while the final model results are presented in Table 4.

Table 1

An information system of a decision table

U	c ₁	c ₂	c ₃	c ₄	c ₅	c ₆	c ₇	c ₈	c ₉	d
x ₁	1	0	0	0	0	1	0	0	1	1
x ₂	0	1	0	0	1	0	1	0	0	2
x ₃	0	0	1	1	0	0	0	1	0	1
x ₄	0	0	1	1	0	0	0	1	0	1
x ₅	1	0	0	0	0	1	1	0	0	3
x ₆	0	1	0	0	1	0	0	0	1	2
x ₇	0	1	0	0	1	0	0	0	0	1
x ₈	1	0	0	1	0	1	1	0	1	1
x ₉	0	0	1	0	0	0	0	1	0	2
x ₁₀	0	0	1	0	0	0	1	0	0	2
x ₁₁	1	1	0	1	0	1	0	0	1	2
x ₁₂	0	0	0	0	1	0	0	1	0	3
x ₁₃	0	0	0	0	1	0	0	1	0	1
x ₁₄	0	0	1	0	0	1	1	0	0	3
x ₁₅	1	1	0	1	0	0	0	0	1	3

Table 2

Approximations of decision classes based on the attribute chain

X _j	Q _i	$\underline{Q_{i}} (X_{j})$	$\bar{Q_{i}} (X_{j})$
X ₁	Q ₁	∅	U
	Q ₂	∅	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14}
	Q ₃	∅	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14}
	Q ₄	{3, 4, 8}	{1, 2, 3, 4, 5, 6, 7, 8, 12, 13}
	Q ₅	{3, 4, 8}	{1, 2, 3, 4, 5, 6, 7, 8, 12, 13}
	Q ₆	{3, 4, 8}	{1, 2, 3, 4, 5, 6, 7, 8, 12, 13}
	Q ₇	{1, 3, 4, 8}	{1, 3, 4, 6, 7, 8, 12, 13}
	Q ₈	{1, 3, 4, 8}	{1, 3, 4, 6, 7, 8, 12, 13}
	Q ₉	{1, 3, 4, 7, 8}	{1, 3, 4, 7, 8, 12, 13}
X ₂	Q ₁	∅	U
	Q ₂	∅	{2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15}
	Q ₃	∅	{2, 3, 4, 6, 7, 9, 10, 11, 14, 15}
	Q ₄	∅	{2, 6, 7, 9, 10, 11, 14, 15}
	Q ₅	∅	{2, 6, 7, 9, 10, 11, 14, 15}
	Q ₆	{9, 10, 11}	{2, 6, 7, 9, 10, 11}
	Q ₇	{2, 9, 10, 11}	{2, 6, 7, 9, 10, 11}
	Q ₈	{2, 9, 10, 11}	{2, 6, 7, 9, 10, 11}
	Q ₉	{2, 6, 9, 10, 11}	{2, 6, 9, 10, 11}
X ₃	Q ₁	∅	U
	Q ₂	∅	{1, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15}
	Q ₃	∅	{1, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15}
	Q ₄	∅	{1, 5, 9, 10, 11, 12, 13, 14, 15}
	Q ₅	∅	{1, 5, 9, 10, 11, 12, 13, 14, 15}
	Q ₆	{14, 15}	{1, 5, 12, 13, 14, 15}
	Q ₇	{5, 14, 15}	{5, 12, 13, 14, 15}
	Q ₈	{5, 14, 15}	{5, 12, 13, 14, 15}
	Q ₉	{5, 14, 15}	{5, 12, 13, 14, 15}

Table 3

Information statistics of approximation cardinalities and accuracy values on the attribute-addition chain

X _j	Measure	Q ₁	Q ₂	Q ₃	Q ₄	Q ₅	Q ₆	Q ₇	Q ₈	Q ₉	Mean	Pointers
												$i_{\underline{2}}^{}$ , $i_{\bar{2}}^{}$ , $i_{4}^{*}$
X ₁	$\| \underline{Q_{i}} (X_{1}) \|$	0	0	0	3	3	3	4	4	5	2.44	5
	$\| \bar{Q_{i}} (X_{1}) \|$	15	13	13	10	10	10	8	8	7	10.44	5
	α (Q_i, X₁)	0	0	0	0.3	0.3	0.3	0.5	0.5	0.71	0.29	5
X ₂	$\| \underline{Q_{i}} (X_{2}) \|$	0	0	0	0	0	3	4	4	5	1.78	6
	$\| \bar{Q_{i}} (X_{2}) \|$	15	12	10	8	8	6	6	6	5	8.44	4
	α (Q_i, X₂)	0	0	0	0	0	0.5	0.67	0.67	1	0.31	6
X ₃	$\| \underline{Q_{i}} (X_{3}) \|$	0	0	0	0	0	2	3	3	3	1.22	6
	$\| \bar{Q_{i}} (X_{3}) \|$	15	12	12	9	9	6	5	5	5	8.67	4
	α (Q_i, X₃)	0	0	0	0	0	0.33	0.6	0.6	0.6	0.24	6

Table 4

Four compromised models based on positioning pointers

X _j	Model	Pointer	Lower approximation	Upper approximation	Accuracy
X ₁	1	$i_{1}^{*} = 5$	{3, 4, 8}	{1, 2, ⋯ , 8, 12, 13}	0.3
	2	$i_{\underline{2}}^{*} = 5$	{3, 4, 8}	{1, 2, ⋯ , 8, 12, 13}	0.3
		$i_{\bar{2}}^{*} = 5$
	3	$i_{3}^{*} = 5$	{3, 4, 8}	{1, 2, ⋯ , 8, 12, 13}	0.3
	4	$i_{4}^{*} = 5$	{3, 4, 8}	{1, 2, ⋯ , 8, 12, 13}	0.3
X ₂	1	$i_{1}^{*} = 5$	∅	{2, 6, 7, 9, 10, 11, 14, 15}	0
	2	$i_{\underline{2}}^{*} = 6$	{9, 10, 11}	{2, 6, 7, 9, 10, 11, 14, 15}	0.375
		$i_{\bar{2}}^{*} = 4$
	3	$i_{3}^{*} = 5$	∅	{2, 6, 7, 9, 10, 11, 14, 15}	0
	4	$i_{4}^{*} = 6$	{9, 10, 11}	{2, 6, 7, 9, 10, 11}	0.5
X ₃	1	$i_{1}^{*} = 5$	∅	{1, 5, ⋯ , 13, 14, 15}	0
	2	$i_{\underline{2}}^{*} = 6$	{14, 15}	{1, 5, ⋯ , 13, 14, 15}	0.222
		$i_{\bar{2}}^{*} = 4$
	3	$i_{3}^{*} = 5$	∅	{1, 5, ⋯ , 13, 14, 15}	0
	4	$i_{4}^{*} = 6$	{14, 15}	{2, 6, 7, 9, 10, 11}	0.333

Tables 2, 3, 4 can illustrate some details, and the Algorithm 1’s process and modeling pointers are focused on for analyses. (1) First consider the knowledge level. The index set {1, 2, ⋯ , 9} on the attribute chain exhibits median 5, and thus pointer $i_{1}^{*} = 5$ and representativeness Q₅ determine Compromised-Model 1 (Equation (8)): $({\underline{\sum_{i = 1}^{m} Q_{i}}}^{C_{1}} (X_{j}), {\bar{\sum_{i = 1}^{m} Q_{i}}}^{C_{1}} (X_{j}))$ $= (\underline{Q_{5}} (X_{j}), \bar{Q_{5}} (X_{j}))$ ,

$α {(\sum_{i = 1}^{m} Q_{i}, X_{j})}^{C_{1}} = α (Q_{5}, X_{j})$ , where j = 1, 2, 3. All the three classes X₁, X₂, X₃ adhere to the same case, but they get different approximations and accuracies. (2) Then consider the two levels of approximation and accuracy. Regarding decision concept X₁, we obtain two sequences of approximation enlargement and cardinality increase: $\underline{Q_{1}} (X_{1}) = \emptyset \subseteq \dots \subseteq \underline{Q_{9}} (X_{1}) = {1, 3, 4, 7, 8}$ ,

$| \underline{Q_{1}} (X_{1}) | = 0 \leq \dots \leq | \underline{Q_{9}} (X_{1}) | = 5$ . By Definition 3.1, the cardinality mean is 2.44, the minimum distance 3 - 2.44 = 0.56 is realized by value 3 and its three carrier subsets Q₄, Q₅, Q₆, i.e., ${i_{\underline{2} 1}^{*}, \dots, i_{\underline{2} s}^{*}} = {4, 5, 6}$ , and the median determines pointer $i_{\underline{2}}^{*} = 5$ (Equation (8)). Similarly, we can get the upper pointer $i_{\bar{2}}^{*} = 5$ and accuracy pointer $i_{4}^{*} = 5$ , and pair $(i_{\underline{2}}^{*} = 5, i_{\bar{2}}^{*} = 5)$ yields uniform pointer $i_{3}^{*} = 5$ . Therefore, index 5 and subset Q₅ are applied to Compromised-Models 2–4 to get the same approximation and accuracy, which are similar to those of Compromised-Model 1. Furthermore regarding concept X₂, we can similarly achieve pointers $(i_{\bar{2}}^{*}, i_{\underline{2}}^{*}) = (6, 4)$ , $i_{3}^{*} = 5$ , $i_{4}^{*} = 6$ . These results imply the difference of Compromised-Models 2–4, and Equations (8) and (10) are concretized to $({\underline{\sum_{i = 1}^{9} Q_{i}}}^{C_{2}} (X_{2}), {\bar{\sum_{i = 1}^{9} Q_{i}}}^{C_{2}} (X_{2}))$

$= (\underline{Q_{6}} (X_{2}), \bar{Q_{4}} (X_{2}))$ ,

$α {(\sum_{i = 1}^{9} Q_{i}, X)}^{C_{2}} = \frac{| \underline{Q_{6}} (X_{2}) |}{| \bar{Q_{4}} (X_{2}) |} = 0.375$ ,

$({\underline{\sum_{i = 1}^{9} Q_{i}}}^{C_{3}} (X_{2}), {\bar{\sum_{i = 1}^{9} Q_{i}}}^{C_{3}} (X_{2}))$

$= (\underline{Q_{5}} (X_{2}), \bar{Q_{5}} (X_{2}))$ ,

$α {(\sum_{i = 1}^{9} Q_{i}, X_{2})}^{C_{3}} = α (Q_{5}, X_{2}) = 0$ ,

$({\underline{\sum_{i = 1}^{9} Q_{i}}}^{C_{4}} (X_{2}), {\bar{\sum_{i = 1}^{9} Q_{i}}}^{C_{4}} (X_{2}))$

$= (\underline{Q_{6}} (X_{2}), \bar{Q_{6}} (X_{2}))$ ,

$α {(\sum_{i = 1}^{9} Q_{i}, X_{2})}^{C_{4}} = α (Q_{6}, X_{2}) = 0.5$ .

Finally regarding X₃, its pointers also become $(i_{\bar{2}}^{*}, i_{\underline{2}}^{*}) = (6, 4)$ , $i_{3}^{*} = 5$ , $i_{4}^{*} = 6$ , i.e., they are the same as those regarding X₂. Thus, Compromised-Models 2–4 have different outputs, and their connotations regarding X₃ are different from those regarding X₂.

For the properties in Propositions 3–5, the compromise feature and interaction relationship are clear, while the operation rule needs further verifications.

4 Data experiment verification

In this section, Compromised-Models 1–4 are effectively verified by data experiments. Seven datasets are chosen from UCI Machine Learning Repository [http://archive.ics.uci.edu/ml], and their information is shown in Table 5. The relevant setting and processing are similar to those in the above example illustration, but the optimistic and pessimistic models are properly highlighted. We still adopt the natural attribute-addition chain in Equation (14) and all decision classes X_j.

Table 5
Basic information of seven UCI datasets

No. Name Sample number Condition attribute number Decision class number

(a) Glass 214 9 6

(b) Zoo 101 16 7

(c) Lymphography 148 18 8

(d) Dermatology 366 34 6

(e) Wdbc 569 31 2

(f) SolarFlare 1066 9 6

(g) Cmc 1473 9 3

No.	Name	Sample number	Condition attribute number	Decision class number
(a)	Glass	214	9	6
(b)	Zoo	101	16	7
(c)	Lymphography	148	18	8
(d)	Dermatology	366	34	6
(e)	Wdbc	569	31	2
(f)	SolarFlare	1066	9	6
(g)	Cmc	1473	9	3

Regarding the attribute-addition chain, the approximation cardinalities and accuracy values, as well as their monotonicity changes, all become measurement bases for the compromised modeling, especially for main Compromised-Models 2–4 with the pointer positing. These basic measures have been thoroughly computed, and relevant results and further statistical information are given in Tables 6 –12, where only eight samples on attribute c₃₄ in (d) Dermatology exhibit missing values and thus are rationally filled. Herein, a decision class is chosen as an example for vivid demonstrations; in usual, the first class X₁ is chosen, while only (f) SolarFlare uses X₅ to pursue a better manifestation. (1) Three types of measure chains are depicted in Fig. 2, where La and Ua respectively denote lower and upper approximation cardinalities. (2) All chain-based three-dimensional points with form (La, Ua, Accuracy) constitute a line chart on surface function $Accuracy = \frac{La}{Ua}$ , and their corresponding extended ranges are depicted in Fig. 3.

Table 6

Statistics of approximations and accuracies on (a) Glass

X _j	Measure	Q ₁	Q ₂	⋯	Q ₇	Q ₈	Q ₉	Mean	Pointers
									$i_{\underline{2}}^{}$ , $i_{\bar{2}}^{}$ , $i_{4}^{*}$
X ₁	$\| \underline{Q_{i}} (X_{1}) \|$	64	70	⋯	70	70	70	69.333	5
	$\| \bar{Q_{i}} (X_{1}) \|$	76	70	⋯	70	70	70	70.667	5
	α (Q_i, X₁)	0.842	1	⋯	1	1	1	0.982	5
X ₂	$\| \underline{Q_{i}} (X_{2}) \|$	65	76	⋯	76	76	76	74.778	5
	$\| \bar{Q_{i}} (X_{2}) \|$	86	76	⋯	76	76	76	77.111	5
	α (Q_i, X₂)	0.756	1	⋯	1	1	1	0.973	5
X ₃	$\| \underline{Q_{i}} (X_{3}) \|$	13	17	⋯	17	17	17	16.556	5
	$\| \bar{Q_{i}} (X_{3}) \|$	21	17	⋯	17	17	17	17.444	5
	α (Q_i, X₃)	0.619	1	⋯	1	1	1	0.958	5
X ₄	$\| \underline{Q_{i}} (X_{4}) \|$	11	13	⋯	13	13	13	12.778	5
	$\| \bar{Q_{i}} (X_{4}) \|$	15	13	⋯	13	13	13	13.222	5
	α (Q_i, X₄)	0.733	1	⋯	1	1	1	0.970	5
X ₅	$\| \underline{Q_{i}} (X_{5}) \|$	6	9	⋯	9	9	9	8.667	5
	$\| \bar{Q_{i}} (X_{5}) \|$	12	9	⋯	9	9	9	9.333	5
	α (Q_i, X₅)	0.5	1	⋯	1	1	1	0.944	5
X ₆	$\| \underline{Q_{i}} (X_{6}) \|$	24	29	⋯	29	29	29	28.444	5
	$\| \bar{Q_{i}} (X_{6}) \|$	35	29	⋯	29	29	29	29.667	5
	α (Q_i, X₆)	0.686	1	⋯	1	1	1	0.965	5

Table 7

Statistics of approximations and accuracies on (b) Zoo

X _j	Measure	Q ₁	Q ₂	⋯	Q ₁₄	Q ₁₅	Q ₁₆	Mean	Pointers
									$i_{\underline{2}}^{}$ , $i_{\bar{2}}^{}$ , $i_{4}^{*}$
X ₁	$\| \underline{Q_{i}} (X_{1}) \|$	0	0	⋯	41	41	41	35.688	3
	$\| \bar{Q_{i}} (X_{1}) \|$	101	81	⋯	41	41	41	47.625	3
	α (Q_i, X₁)	0	0	⋯	1	1	1	0.863	3
X ₂	$\| \underline{Q_{i}} (X_{2}) \|$	0	20	⋯	20	20	20	18.750	9
	$\| \bar{Q_{i}} (X_{2}) \|$	58	20	⋯	20	20	20	22.375	9
	α (Q_i, X₂)	0	1	⋯	1	1	1	0.938	9
X ₃	$\| \underline{Q_{i}} (X_{3}) \|$	0	0	⋯	5	5	5	2.875	8
	$\| \bar{Q_{i}} (X_{3}) \|$	58	38	⋯	5	5	5	17.063	6
	α (Q_i, X₃)	0	0	⋯	1	1	1	0.540	8
X ₄	$\| \underline{Q_{i}} (X_{4}) \|$	0	0	⋯	13	13	13	5.688	5
	$\| \bar{Q_{i}} (X_{4}) \|$	58	38	⋯	13	13	13	22.938	6
	α (Q_i, X₄)	0	0	⋯	1	1	1	0.438	5
X ₅	$\| \underline{Q_{i}} (X_{5}) \|$	0	0	⋯	4	4	4	1.750	5
	$\| \bar{Q_{i}} (X_{5}) \|$	58	38	⋯	4	4	4	19.000	8
	α (Q_i, X₅)	0	0	⋯	1	1	1	0.438	5
X ₆	$\| \underline{Q_{i}} (X_{6}) \|$	0	0	⋯	8	8	8	5.250	8
	$\| \bar{Q_{i}} (X_{6}) \|$	101	81	⋯	8	8	8	25.375	6
	α (Q_i, X₆)	0	0	⋯	1	1	1	0.510	7
X ₇	$\| \underline{Q_{i}} (X_{7}) \|$	0	0	⋯	10	10	10	5.125	10
	$\| \bar{Q_{i}} (X_{7}) \|$	58	38	⋯	10	10	10	22.875	7
	α (Q_i, X₇)	0	0	⋯	1	1	1	0.459	8

Table 8

Statistics of approximations and accuracies on (c) Lymphography

X _j	Measure	Q ₁	Q ₂	⋯	Q ₁₆	Q ₁₇	Q ₁₈	Mean	Pointers
									$i_{\underline{2}}^{}$ , $i_{\bar{2}}^{}$ , $i_{4}^{*}$
X ₁	$\| \underline{Q_{i}} (X_{1}) \|$	0	0	⋯	56	56	56	27.167	9
	$\| \bar{Q_{i}} (X_{1}) \|$	144	144	⋯	58	58	58	92.5	10
	α (Q_i, X₁)	0	0	⋯	0.966	0.966	0.966	0.420	11
X ₂	$\| \underline{Q_{i}} (X_{2}) \|$	0	0	⋯	36	36	36	16.167	11
	$\| \bar{Q_{i}} (X_{2}) \|$	144	144	⋯	38	38	38	86.444	9
	α (Q_i, X₂)	0	0	⋯	0.947	0.947	0.947	0.363	12
X ₃	$\| \underline{Q_{i}} (X_{3}) \|$	0	0	⋯	18	18	18	9.444	9
	$\| \bar{Q_{i}} (X_{3}) \|$	142	142	⋯	18	18	18	55.278	7
	α (Q_i, X₃)	0	0	⋯	1	1	1	0.407	12
X ₄	$\| \underline{Q_{i}} (X_{4}) \|$	0	0	⋯	10	10	10	4.611	9
	$\| \bar{Q_{i}} (X_{4}) \|$	146	120	⋯	10	10	10	49.833	7
	α (Q_i, X₄)	0	0	⋯	1	1	1	0.373	11
X ₅	$\| \underline{Q_{i}} (X_{5}) \|$	0	0	⋯	8	8	8	3.389	11
	$\| \bar{Q_{i}} (X_{5}) \|$	61	45	⋯	8	8	8	24.889	9
	α (Q_i, X₅)	0	0	⋯	1	1	1	0.344	12
X ₆	$\| \underline{Q_{i}} (X_{6}) \|$	0	0	⋯	8	8	8	3.111	11
	$\| \bar{Q_{i}} (X_{6}) \|$	142	100	⋯	8	8	8	40.167	7
	α (Q_i, X₆)	0	0	⋯	1	1	1	0.335	12
X ₇	$\| \underline{Q_{i}} (X_{7}) \|$	0	0	⋯	8	8	8	4.278	9
	$\| \bar{Q_{i}} (X_{7}) \|$	146	91	⋯	8	8	8	31.222	5
	α (Q_i, X₇)	0	0	⋯	1	1	1	0.427	11
X ₈	$\| \underline{Q_{i}} (X_{8}) \|$	0	0	⋯	2	2	2	1.111	8
	$\| \bar{Q_{i}} (X_{8}) \|$	61	33	⋯	2	2	2	10.722	7
	α (Q_i, X₈)	0	0	⋯	1	1	1	0.519	9

Table 9

Statistics of approximations and accuracies on (d) Dermatology

X _j	Measure	Q ₁	Q ₂	⋯	Q ₃₂	Q ₃₃	Q ₃₄	Mean	Pointers
									$i_{\underline{2}}^{}$ , $i_{\bar{2}}^{}$ , $i_{4}^{*}$
X ₁	$\| \underline{Q_{i}} (X_{1}) \|$	0	1	⋯	112	112	112	93.088	9
	$\| \bar{Q_{i}} (X_{1}) \|$	366	358	⋯	112	112	112	142	8
	α (Q_i, X₁)	0	0.002	⋯	1	1	1	0.799	9
X ₂	$\| \underline{Q_{i}} (X_{2}) \|$	0	0	⋯	61	61	61	47.706	13
	$\| \bar{Q_{i}} (X_{2}) \|$	362	346	⋯	61	61	61	96.676	7
	α (Q_i, X₂)	0	0	⋯	1	1	1	0.724	13
X ₃	$\| \underline{Q_{i}} (X_{3}) \|$	0	1	⋯	72	72	72	63.618	6
	$\| \bar{Q_{i}} (X_{3}) \|$	366	360	⋯	72	72	72	97.147	5
	α (Q_i, X₃)	0	0.003	⋯	1	1	1	0.870	6
X ₄	$\| \underline{Q_{i}} (X_{4}) \|$	0	0	⋯	49	49	49	39.971	9
	$\| \bar{Q_{i}} (X_{4}) \|$	362	303	⋯	49	49	49	81.824	8
	α (Q_i, X₄)	0	0	⋯	1	1	1	0.744	13
X ₅	$\| \underline{Q_{i}} (X_{5}) \|$	0	2	⋯	52	52	52	42.794	11
	$\| \bar{Q_{i}} (X_{5}) \|$	366	334	⋯	52	52	52	86.471	7
	α (Q_i, X₅)	0	0.006	⋯	1	1	1	0.749	13
X ₆	$\| \underline{Q_{i}} (X_{6}) \|$	0	0	⋯	20	20	20	16.588	7
	$\| \bar{Q_{i}} (X_{6}) \|$	362	284	⋯	20	20	20	49.353	6
	α (Q_i, X₆)	0	0	⋯	1	1	1	0.820	7

Table 10

Statistics of approximations and accuracies on (e) Wdbc

X _j	Measure	Q ₁	Q ₂	⋯	Q ₂₉	Q ₃₀	Q ₃₁	Mean	Pointers
									$i_{\underline{2}}^{}$ , $i_{\bar{2}}^{}$ , $i_{4}^{*}$
X ₁	$\| \underline{Q_{i}} (X_{1}) \|$	187	212	⋯	212	212	212	211.167	16
	$\| \bar{Q_{i}} (X_{1}) \|$	241	212	⋯	212	212	212	212.967	16
	α (Q_i, X₁)	0.776	1	⋯	1	1	1	0.993	16
X ₂	$\| \underline{Q_{i}} (X_{2}) \|$	328	357	⋯	357	357	357	356.033	16
	$\| \bar{Q_{i}} (X_{2}) \|$	382	357	⋯	357	357	357	357.833	16
	α (Q_i, X₂)	0.859	1	⋯	1	1	1	0.995	16

Table 11

Statistics of approximations and accuracies on (f) SolarFlare

X _j	Measure	Q ₁	Q ₂	⋯	Q ₇	Q ₈	Q ₉	Mean	Pointers
									$i_{\underline{2}}^{}$ , $i_{\bar{2}}^{}$ , $i_{4}^{*}$
X ₁	$\| \underline{Q_{i}} (X_{1}) \|$	0	0	⋯	11	11	26	8.111	5
	$\| \bar{Q_{i}} (X_{1}) \|$	1066	1066	⋯	992	992	949	1017.667	5
	α (Q_i, X₁)	0	0	⋯	0.011	0.011	0.027	0.008	5
X ₂	$\| \underline{Q_{i}} (X_{2}) \|$	0	0	⋯	20	20	40	13.667	5
	$\| \bar{Q_{i}} (X_{2}) \|$	1031	1031	⋯	1005	1005	960	1011.889	5
	α (Q_i, X₂)	0	0	⋯	0.020	0.020	0.042	0.014	5
X ₃	$\| \underline{Q_{i}} (X_{3}) \|$	0	0	⋯	3	3	13	3.000	7
	$\| \bar{Q_{i}} (X_{3}) \|$	1066	1066	⋯	967	967	876	991.778	4
	α (Q_i, X₃)	0	0	⋯	0.003	0.003	0.015	0.003	7
X ₄	$\| \underline{Q_{i}} (X_{4}) \|$	0	0	⋯	1	1	2	0.667	6
	$\| \bar{Q_{i}} (X_{4}) \|$	700	700	⋯	600	600	530	627.889	4
	α (Q_i, X₄)	0	0	⋯	0.002	0.002	0.004	0.001	5
X ₅	$\| \underline{Q_{i}} (X_{5}) \|$	0	0	⋯	15	15	24	8.000	5
	$\| \bar{Q_{i}} (X_{5}) \|$	1066	1003	⋯	304	304	166	503.222	4
	α (Q_i, X₅)	0	0	⋯	0.049	0.049	0.145	0.032	6
X ₆	$\| \underline{Q_{i}} (X_{6}) \|$	0	0	⋯	0	0	1	0.111	4
	$\| \bar{Q_{i}} (X_{6}) \|$	1066	1045	⋯	679	679	595	793.333	4
	α (Q_i, X₆)	0	0	⋯	0	0	0.002	0.000	4

Table 12

Statistics of approximations and accuracies on (g) Cmc

X _j	Measure	Q ₁	Q ₂	⋯	Q ₇	Q ₈	Q ₉	Mean	Pointers
									$i_{\underline{2}}^{}$ , $i_{\bar{2}}^{}$ , $i_{4}^{*}$
X ₁	$\| \underline{Q_{i}} (X_{1}) \|$	0	50	⋯	554	595	596	371.778	4
	$\| \bar{Q_{i}} (X_{1}) \|$	1472	1465	⋯	715	671	670	1000.778	4
	α (Q_i, X₁)	0	0.034	⋯	0.775	0.887	0.890	0.477	4
X ₂	$\| \underline{Q_{i}} (X_{2}) \|$	0	1	⋯	248	280	281	144.778	5
	$\| \bar{Q_{i}} (X_{2}) \|$	1469	1253	⋯	428	386	385	738.000	4
	α (Q_i, X₂)	0	0.001	⋯	0.579	0.725	0.730	0.322	6
X ₃	$\| \underline{Q_{i}} (X_{3}) \|$	0	2	⋯	417	455	455	250.111	4
	$\| \bar{Q_{i}} (X_{3}) \|$	1472	1370	⋯	618	573	573	910.667	4
	α (Q_i, X₃)	0	0.001	⋯	0.675	0.794	0.794	0.379	5

Fig. 2

Approximation cardinalities and accuracy values based on the attribute-addition chain and decision class X₁ or X₅.

Fig. 3

Three-dimensional surfaces of approximation cardinalities and accuracy values regarding decision class X₁ or X₅.

For each dataset, the locating pointers become key for the multiple multi-granulation models. At first, the optimistic and pessimistic models respectively adhere to pointers $i_{O}^{*} = | C |$ and $i_{P}^{*} = 1$ (Equation (8)), while $i_{1}^{*}$ is simple for constructing Compromised-Model 1. Then, pointers $i_{\underline{2}}^{*}$ , $i_{\bar{2}}^{*}$ , $i_{4}^{*}$ become focuses, especially for Compromised-Models 2–4. By the statistical positioning, relevant pointers $i_{\underline{2}}^{*}$ , $i_{\bar{2}}^{*}$ , $i_{4}^{*}$ of seven datasets are respectively provided in final parts of Tables 6 –12. Furthermore, pointers $i_{O}^{*}$ , $i_{P}^{*}$ , $i_{1}^{*}$ , $(i_{\underline{2}}^{*}, i_{\bar{2}}^{*})$ , $i_{3}^{*}$ , $i_{4}^{*}$ all are summarized in Table 13, and they are also exhibited in Fig. 4.

Table 13

Statistical results of six multi-granulation rough set models based on location pointers

Datasets	X _j	Model-O	Model-P	Model-1	Model-2	Model-3	Model-4
		$i_{O}^{*}$ (La, Ua ; α)	$i_{P}^{*}$ (La, Ua ; α)	$i_{1}^{*}$ (La, Ua ; α)	$(i_{\underline{2}}^{}, i_{\bar{2}}^{})$ (La, Ua ; α)	$i_{3}^{*}$ (La, Ua ; α)	$i_{4}^{*}$ (La, Ua ; α)
(a) Glass	X ₁	9 (70,70; 1)	1 (64,76; 0.842)	5 (70,70; 1)	(5,5) (70,70; 1)	5 (70,70; 1)	5 (70,70; 1)
	X ₂	9 (76,76; 1)	1 (65,86; 0.756)	5 (76,76; 1)	(5,5) (76,76; 1)	5 (76,76; 1)	5 (76,76; 1)
	X ₃	9 (17,17; 1)	1 (13,21; 0.619)	5 (17,17; 1)	(5,5) (17,17; 1)	5 (17,17; 1)	5 (17,17; 1)
	X ₄	9 (13,13; 1)	1 (11,15; 0.733)	5 (13,13; 1)	(5,5) (13,13; 1)	5 (13,13; 1)	5 (13,13; 1)
	X ₅	9 (9,9; 1)	1 (6,12; 0.5)	5 (9,9; 1)	(5,5) (9,9; 1)	5 (9,9; 1)	5 (9,9; 1)
	X ₆	9 (29,29; 1)	1 (24,35; 0.686)	5 (29,29; 1)	(5,5) (29,29; 1)	5 (29,29; 1)	5 (29,29; 1)
(b) Zoo	X ₁	16 (41,41; 1)	1 (0,101; 0)	8 (41,41; 1)	(3,3) (38,47; 0.809)	3 (38,47; 0.809)	3 (38,47; 0.809)
	X ₂	16 (20,20; 1)	1 (0,58; 0)	8 (20,20; 1)	(9,9) (20,20; 1)	9 (20,20; 1)	9 (20,20; 1)
	X ₃	16 (5,5; 1)	1 (0,58; 0)	8 (4,9; 0.444)	(8,6) (4,10; 0.400)	7 (1,10; 0.100)	8 (4,9; 0.444)
	X ₄	16 (13,13; 1)	1 (0,58; 0)	8 (0,17; 0)	(5,6) (0,23; 0)	5 (0,32; 0)	5 (0,32; 0)
	X ₅	16 (4,4; 1)	1 (0,58; 0)	8 (0,17; 0)	(5,8) (0,17; 0)	6 (0,23; 0)	5 (0,32; 0)
	X ₆	16 (8,8; 1)	1 (0,101; 0)	8 (6,11; 0.545)	(8,6) (6,15; 0.400)	7 (6,11; 0.545)	7 (6,11; 0.545)
	X ₇	16 (10,10; 1)	1 (0,58; 0)	8 (8,13; 0.615)	(10,7) (8,28; 0.286)	8 (8,13; 0.615)	8 (8,13; 0.615)
(c) Lymphography	X ₁	18 (56,58; 0.966)	1 (0,144; 0)	9 (25,96; 0.260)	(9,10) (25,95; 0.263)	9 (25,96; 0.260)	11 (37,75; 0.493)
	X ₂	18 (36,38; 0.947)	1 (0,144; 0)	9 (12,84; 0.143)	(11,9) (15,84; 0.179)	10 (12,84; 0.143)	12 (20,49; 0.408)
	X ₃	18 (18,18; 1)	1 (0,142; 0)	9 (10,45; 0.222)	(9,7) (10,59; 0.169)	8 (7,59; 0.119)	12 (13,29; 0.448)
	X ₄	18 (10,10; 1)	1 (0,146; 0)	9 (3,39; 0.077)	(9,7) (3,54; 0.056)	8 (2,54; 0.037)	11 (7,18; 0.389)
	X ₅	18 (8,8; 1)	1 (0,61; 0)	9 (2,26; 0.077)	(11,9) (4,26; 0.154)	10 (2,26; 0.077)	12 (5,12; 0.417)
	X ₆	18 (8,8; 1)	1 (0,142; 0)	9 (1,35; 0.029)	(11,7) (3,40; 0.075)	9 (1,35; 0.029)	12 (4,16; 0.250)
	X ₇	18 (8,8; 1)	1 (0,146; 0)	9 (4,20; 0.200)	(9,5) (4,35; 0.114)	7 (3,22; 0.136)	11 (5,12; 0.417)
	X ₈	18 (2,2; 1)	1 (0,61; 0)	9 (1,5; 0.200)	(8,7) (1,8; 0.125)	7 (0,8; 0)	9 (1,5; 0.200)
(d) Dermatology	X ₁	34 (112,112; 1)	1 (0,366; 0)	17 (112,112; 1)	(9,8) (103,148; 0.696)	8 (67,148; 0.453)	9 (103,131; 0.786)
	X ₂	34 (61,61; 1)	1 (0,362; 0)	17 (59,62; 0.952)	(13,7) (49,109; 0.450)	10 (43,76; 0.566)	13 (49,68; 0.721)
	X ₃	34 (72,72; 1)	1 (0,366; 0)	17 (72,72; 1)	(6,5) (70,100; 0.700)	5 (52,100; 0.520)	6 (70,76; 0.921)
	X ₄	34 (49,49; 1)	1 (0,362; 0)	17 (48,51; 0.941)	(9,8) (39,90; 0.433)	8 (32,90; 0.356)	13 (43,60; 0.717)
	X ₅	34 (52,52; 1)	1 (0,366; 0)	17 (52,52; 1)	(11,7) (41,84; 0.488)	9 (40,73; 0.548)	13 (46,65; 0.708)
	X ₆	34 (20,20; 1)	1 (0,362; 0)	17 (20,20; 1)	(7,6) (19,76; 0.250)	6 (2,76; 0.026)	7 (19,21; 0.905)
(e) Wdbc	X ₁	30 (212,212;1)	1 (187,241;0.776)	15 (212,212;1)	(16,16) (212,212;1)	16 (212,212;1)	16 (212,212;1)
	X ₂	30 (357,357;1)	1 (328,382;0.859)	15 (357,357;1)	(16,16) (357,357;1)	16 (357,357;1)	16 (357,357;1)
(f) SolarFlare	X ₁	9 (26,949;0.027)	1 (0,1066;0)	5 (9,1002;0.009)	(5,5) (9,1002;0.009)	5 (9,1002;0.009)	5 (9,1002;0.009)
	X ₂	9 (40,960;0.042)	1 (0,1031;0)	5 (17,1014;0.017)	(5,5) (17,1014;0.017)	5 (17,1014;0.017)	5 (17,1014;0.0168)
	X ₃	9 (13,876;0.015)	1 (0,1066;0)	5 (2,980;0.002)	(7,4) (3,1002;0.003)	7 (3,967;0.003)	5 (2,980;0.002)
	X ₄	9 (2,530;0.004)	1 (0,700;0)	5 (1,605;0.002)	(6,4) (1,647;0.002)	5 (1,605;0.002)	5 (1,605;0.002)
	X ₅	9 (24,166;0.145)	1 (0,1066;0)	5 (6,348;0.017)	(5,4) (6,513;0.012)	6 (6,308;0.0195)	4 (6,513;0.012)
	X ₆	9 (1,595;0.002)	1 (0,1066;0)	5 (0,706;0)	(4,4) (0,831;0)	4 (0,831;0)	4(0,831;0)
(g) Cmc	X ₁	9 (596,670;0.890)	1 (0,1472;0)	5 (459,891;0.515)	(4,4) (436,946;0.461)	4 (436,946;0.461)	4 (436,946;0.461)
	X ₂	9 (281,385;0.730)	1 (0,1469;0)	5 (156,570;0.274)	(5,4) (156,630;0.248)	6 (187,514;0.364)	4 (123,630;0.196)
	X ₃	9 (455,573;0.794)	1 (0,1472;0)	5 (281,796;0.353)	(4,4) (248,853;0.291)	5 (281,796;0.353)	4 (248,853;0.291)

Fig. 4

Pointer curves of six types of multi-granulation models regarding all decision classes.

According to the above pointer results, the approximation and accuracy of all the multi-granulation rough set models (including Compromised-Models 1–4 as well as the optimistic and pessimistic models) can be eventually achieved. The relevant statistical results of approximation cardinality and accuracy value are offered in Table 13, where Model-k (k = O, P, 1, 2, 3, 4) label related models. In fact, we have obtained all modeling results, including the approximation sets. As an example or supplement, we provide the case of (c) Lymphography regarding decision class X₁. (1) For the optimistic model, the lower and upper approximations respectively contain 56 and 58 samples: {5, 7, 8, ⋯ , 145, 147}, {5, 7, 8, ⋯ , 145, 147}.

(2) The pessimistic lower and upper approximation respectively become ∅ and U. (3) For Compromised-Model 1, the lower and upper approximations respectively yield 25 and 96 elements: {13, 19, 24, ⋯ , 139, 141}, {4, 5, 6, ⋯ , 145, 147}.

(4) For Compromised-Model 2, the lower and upper approximations respectively have 25 and 95 elements: {13, 19, 24, ⋯ , 139, 141}, {4, 5, 6, ⋯ , 145, 147}.

(5) For Compromised-Model 3, the lower and upper approximations respectively carry 25 and 96 samples: {13, 19, 24, ⋯ , 139, 141}, {4, 5, 6, ⋯ , 145, 147}.

(6) For Compromised-Model 4, the lower and upper approximations respectively offer 37 and 75 ones: {13, 19, 20, ⋯ , 139, 141}, {5, 7, 8, ⋯ , 145, 147}.

Finally, the main characteristics of proposed methods are highlighted to offer some comparisons. The attribute-addition chain brings multiple total-orders while the optimistic and pessimistic models correspond to the extreme bounds, so the four types of compromised models adopt different order indexes and their own compromised location pointers to pursue the systematic balance. Concretely, Compromised-Models 1–4 respectively concern the ordinal numbers and their averages of knowledge, approximation, approximation fusion, and accuracy, and they can be completely determined and effectively exhibited by relevant label pointers. The relevant modeling mechanism, especially Algorithm 1, naturally induces the experiment results. According to the ultima Table 14 and Fig. 4, we can make the following comparative analyses to verify both the effectiveness and property of our compromised modeling. (1) First compare the compromised models with the optimistic and pessimistic models, and we achieve $i_{1}^{*}, i_{\underline{2}}^{*}, i_{\bar{2}}^{*}, i_{3}^{*}, i_{4}^{*} \in (i_{P}^{*}, i_{O}^{*})$ . This result shows that all the compromised models indeed improve the optimistic and pessimistic models in different degrees, and Proposition 3 holds. (2) Then internally compare the four compromised models. Proposition 4 is verified to be true, and thus Compromised-Model 1 relies on a fixed pointer to acquire more properties of approximation operators. Regarding pointers, there are no necessary size relationships between two pair-elements, and there are multiple pairs including $(i_{1}^{*}, i_{\underline{2}}^{*})$ , $(i_{1}^{*}, i_{\bar{2}}^{*})$ , $(i_{1}^{*}, i_{3}^{*})$ , $(i_{1}^{*}, i_{4}^{*})$ , $(i_{\underline{2}}^{*}, i_{\bar{2}}^{*})$ , $(i_{\underline{2}}^{*}, i_{3}^{*})$ , $(i_{\underline{2}}^{*}, i_{4}^{*})$ , $(i_{\bar{2}}^{*}, i_{3}^{*})$ , $(i_{\bar{2}}^{*}, i_{4}^{*})$ , $(i_{3}^{*}, i_{4}^{*})$ ; in contrast, the related results also justify the sole certain relationship between $i_{3}^{*}$ and $i_{\underline{2}}^{*}, i_{\bar{2}}^{*}$ , i.e., Equation (13) and Proposition 5 hold. According to the compromised comparison, Compromised-Models 1, 2, 4 exhibit the difference and independence, while Compromised-Model 3 can be viewed as a further approximation-balance model of Compromised-Model 2; in essence, these phenomena are determined and explained by the relevant modeling principles.

5 Conclusions

In terms of the attribute-addition chain, four types of compromised multi-granulation rough sets have been established at three levels of knowledge, approximation, and accuracy by four kinds of statistical strategies. These new models characterize both the systematicness of knowledge bases and the generality of data laws, so they balance and improve the optimistic and pessimistic models, which are respectively too loose and strict. Regarding the proposed compromised methods, their total-order mechanism and pointer-orientation characteristic can be systematically clarified by Fig. 1 and Algorithm 1, and the relevant essence induces the main definitions, properties, examples, and experiments.

This study is worth future studying from five aspects. (1) Compromised-Models 1–4 adopt the three levels of knowledge, approximation, and accuracy, and thus their relevant tri-level analysis can be deeply implemented, especially by referring to relevant methods in [29, 34]. (2) For clear manifestations, the compromised modeling mainly embraces the average locating pointers and corresponding carrier subsets, and thus it can further consider the essential granular division and other statistical strategies. (3) The compromised modeling is closely related to the multi-granular modeling which offers some recent proposals [2, 12], so the former can further utilizes the latter to make more discussions and developments on granular computing. (4) Our preliminary works are restricted to a basic precondition of an attribute-extension sequence, and thus the general case without the hierarchical hypothesis is needed to be extensively explored. (5) The compromised models, in both the special and general cases, are worth promoting and utilizing to the dependency-based machine learning (such as the feature selection), and thus the approximation cardinality and accuracy at the concept/class level can be integrated into systematic measures at the classification level (such as the dependency degree and information entropy).

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Footnotes

Acknowledgments

The authors thank both the editors and reviewers for their valuable suggestions, which substantially improve this paper.

This work was supported by National Natural Science Foundation of China (61673285 and 11671284), Sichuan Science and Technology Program of China (20YFG0290 and 19YYJC2845), and a joint research project of Laurent Mathematics Center of Sichuan Normal University and National-Local Joint Engineering Laboratory of System Credibility Automatic Verification.

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Compromised multi-granulation rough sets based on an attribute-extension chain

Abstract

Keywords

1 Introduction

2 Multi-granulation rough sets

Table 5 Basic information of seven UCI datasets No. Name Sample number Condition attribute number Decision class number (a) Glass 214 9 6 (b) Zoo 101 16 7 (c) Lymphography 148 18 8 (d) Dermatology 366 34 6 (e) Wdbc 569 31 2 (f) SolarFlare 1066 9 6 (g) Cmc 1473 9 3

Conflict of interest

Footnotes

Acknowledgments

References

Table 5
Basic information of seven UCI datasets

No. Name Sample number Condition attribute number Decision class number

(a) Glass 214 9 6

(b) Zoo 101 16 7

(c) Lymphography 148 18 8

(d) Dermatology 366 34 6

(e) Wdbc 569 31 2

(f) SolarFlare 1066 9 6

(g) Cmc 1473 9 3