Double-Granule Conditional-Entropies Based on Three-Level Granular Structures

Rough set theory is an important approach for data mining, and it refers to Shannon’s information measures for uncertainty measurements. The existing local conditional-entropies have both the second-order feature and application limitation. By improvements of hierarchical granulation, this paper establishes double-granule conditional-entropies based on three-level granular structures (i.e., micro-bottom, meso-middle, macro-top), and then investigates the relevant properties. In terms of the decision table and its decision classification, double-granule conditional-entropies are proposed at micro-bottom by the dual condition-granule system. By virtue of successive granular summation integrations, they hierarchically evolve to meso-middle and macro-top, to respectively have part and complete condition-granulations. Then, the new measures acquire their number distribution, calculation algorithm, three bounds, and granulation non-monotonicity at three corresponding levels. Finally, the hierarchical constructions and achieved properties are effectively verified by decision table examples and data set experiments. Double-granule conditional-entropies carry the second-order characteristic and hierarchical granulation to deepen both the classical entropy system and local conditional-entropies, and thus they become novel uncertainty measures for information processing and knowledge reasoning.


Introduction
Rough set theory can effectively implement data mining for the imprecise, inconsistent, and incomplete information [1], and it has been extensively applied in artificial intelligence and machine learning [2][3][4][5][6][7][8]. In rough set theory, attribute reduction based on decision tables is a main topic for approximate reasoning and knowledge discovery, and there are three main construction strategies: from the positive region, information measure, and a discernibility matrix [9][10][11][12][13][14][15]. By virtue of the discernibility matrix, Wei et al. [16] proposed an incremental reduction algorithm for dynamic data; Ma et al. [17] utilized the compressed binary discernibility matrix to construct an incremental reduction algorithm for group dynamic data; moreover, Nie and Zhou [18] proposed a new discernibility matrix defined by local conditional-entropies to compute the reduction core.
Information theory originated from Shannon's entropy system [19], and it provides an effective method for uncertainty measurement, such as in attribute reduction. Currently, information theory has been introduced into rough set theory for uncertainty analyses and information processing [20][21][22][23][24][25]. As far as attribute reduction is concerned, Miao [26] offered the informational representation of knowledge reduction and decision reduction, where entropy and mutual-information are highlighted; Wang et al. [27] conducted a comparative study on attribute reduction from the algebra and The decision table (U, C ∪ D) and its granulation from A ⊆ C and D constitute the basic background for information measure construction. The probability space (U, 2 U , P) establishes the usual probability framework, where P : 2 U → Q, P(X) = |X| |U| , ∀X ⊆ U, and thus two usual probabilities are Definition 1 ([26,27,56]). The entropy on condition A, conditional-entropy on D given A, and mutual-information between A and D are respectively defined by In terms of the decision table (U, C ∪ D), the classical system of Shannon entropies has been introduced into rough set theory, as shown by Definition 1 and Theorem 1. As three basic information measures, the entropy, conditional-entropy, and mutual-information have uncertainty semantics and granulation monotonicity, so they are extensively used in attribute reduction and heuristic algorithms [26,27,42]. The granulation relation U/I ND(A) U/I ND(B) is equivalent to I ND(A) ⊇ I ND(B), that is, ∀B i * ∈ U/I ND(B), ∃A i ∈ U/I ND(A), s.t., B i * ⊆ A i , and it is usually induced by A ⊆ B ⊆ C; furthermore, relevant granulation monotonicity/ non-monotonicity becomes an important index to assess and apply uncertainty measures.
According to the decision table and its formal structure, Zhang and Miao [56] recently introduced three-level granular structures, i.e., and further investigated weighted-entropy constructions. As a result, the previous entropy system (Equation (3)) is actually located at macro-top and has an equivalent construction from the weighted-entropy system; at meso-middle, Zhang et al. [10] established three-way informational class-specific reducts to be compared with the algebraic class-specific reducts [9].
In particular, Nie and Zhou [18] proposed a new discernibility matrix for computing the reduction core, and they tactfully utilized a kind of novel information of so-called local conditional-entropy. As our preliminary, the relevant entropy and matrix are reviewed as follows, where let U/I ND(C) = {C k : k = 1, .., r} and the cardinality form is mainly adopted. Definition 2 ([18]). The local conditional-entropy on decision table (U, C ∪ D) is defined by: Definition 3 ([18]). The discernibility matrix DM = (r i j ) |U|×|U| on decision table (U, C ∪ D) is defined by: where dx = {D(y) : y ∈ [x] C } (∀x, y ∈ U) represents the set of decision values induced by conditional class [x] C while |dx| means the corresponding cardinality [63]. In Equation (6) is determined to represent the conditional-entropy of local decision table when accompanied by new universe C p ∪ C q after deleting attribute c; moreover,

Double-Granule Conditional-Entropies Based on Three-Level Granular Structures
The local conditional-entropy in Equation (5) implements effective uncertainty descriptions to guide the in-depth discernibility matrix and core calculation [18], thus exhibiting fundamental significance. However, this basic measure has three flawed aspects, and corresponding improvements for general applications.
(1) According to Equation (5), the locality mainly refers to less range C p ∪ C q in universe U.
More essentially, we can stand on the dual granules C p and C q to propose a novel notion of double-granule conditional-entropies, and it differs from the usual entropy system with only the single-granule representation which implies a kind of first-order style. Moreover, the measure properties are lacking in [18], and we will provide in-depth properties such as restriction bounds and granulation non-monotonicity. (2) Regarding granular structures, all decision classes D j (j = 1, · · · , m) (or decision classification U/I ND(D)) are considered, but condition granules involve only two factors C p and C q . A condition partition U/I ND(C)) needs considering in practice to provide a system description of knowledge granulation, so we also focus on granulation U/I ND(C) to introduce three-level granular structures for hierarchical constructions of double-granule conditional-entropies. (3) Finally, the initial concept is limited to only C for expressing the discernibility matrix and reduction core, and a general subset A ⊆ C has better theoretical and practical prospects, especially for the knowledge-based applications (such as attribute reduction or feature selection).
Along the above thoughts, this section mainly establishes double-granule conditional-entropies based on a universal attribute-subset A ⊆ C and investigates relevant algorithms and properties, and we particularly use a kind of three-level granular structures.
From a viewpoint of only condition granulation, basic descriptions of three-level granular structures are provided in Table 1, and relevant concepts are usually intuitionistic and descriptive according to a supporting figure with granular structures: Figure 1. Micro-bottom (A p , A q ) focuses on only two granules, meso-middle consists of one granule and a partition, while macro-top (U/IND(A) = {A p : p = 1, · · · , n}, U/I ND(A) = {A q : q = 1, · · · , n}) considers the same partition with different construction origins. The three-level granular structures carry a kind of hierarchical integration (or decomposition) relationship, and they provide n × n, n, and one parallel patterns, respectively; they will be presented in a table form with the n × n mainbody data as well as the edge statistics. Moreover, they differ from the existing three-level granular structures for decision tables, which consider not only the condition granulation (with A i and U/I ND(A)) but also decision granulation (with D j and U/I ND(D)) [56].

Double-Granule Conditional-Entropy at Micro-Bottom
The local conditional-entropies are actually at only micro-bottom, i.e., (C p , C q ) regarding C. As a basis of hierarchical development, this subsection improves local conditional-entropies to construct double-granule conditional-entropies at micro-bottom (A p , A q ) (p, q ∈ {1, · · · , n}), which comes from an arbitrary condition-attribute subset A ⊆ C. We first suppose weight coefficients where ω p + ω q = 1.

Definition 4.
At micro-bottom (A p , A q ), the double-granule conditional-entropy is defined by Proposition 1. The double-granule conditional-entropy based on A p becomes By using probabilistic and cardinal forms, Definition 4 proposes the double-granule conditional-entropy at micro-bottom. In contrast to the local conditional-entropy in [18], our measure generally adopts the same essence but a different viewpoint. In other words, Equation (9) with forms (A p , A p ) and |A q | + |A p | is equivalent to Equation (5) with styles A p ∪ A p and |A q ∪ A p | when but the former becomes different and coherent when moreover, it more tends to the double-granule description rather than the granule-union locality. In Equation (9), conditional-information measures represent the uncertainty of decision classification U/I ND(D) regarding condition granules A p and A q , respectively, and they are integrated into H (A p ,A q ) (D/A) by two complementary weight coefficients ω p and ω q . As a result, H (A p ,A q ) (D/A) embodies a kind of information fusion of double-granule A p , A q to describe decision classification U/I ND(D) and its uncertainty, from the perspective of conditional information. Therefore, H (A p ,A q ) (D/A) is naturally called the double-granule conditional-entropy, and it is actually located at micro-bottom (A p , A q ). In particular, the double-granule measures utilize the double-granule fusion to capture a new feature of second-order, because main entropy systems (such as those in Equation (3)) utilize only the single-granule description which correspondingly refers to the so-called first-order information. Proposition 1 focuses on a specific case of A q = A p , and the concrete result H (A p ,A p ) (D/A) degenerates into a one-order measure regarding conditional-entropy.

Proposition 2.
At micro-bottom, double-granule conditional-entropies offer n × n values, i.e., Since both A p and A q have n granules based on p = 1, · · · , n and q = 1, · · · , n, H (A p ,A q ) (D/A) offers number n × n (Proposition 2) to correspond to n × n micro-bottoms. The n × n kinds of double-granule conditional-entropies are arranged in Table 2, and the mainbody refers to an n × n square symmetric matrix where Based on Equation (9), Algorithm 1 resorts to a "for" loop to effectively offer a double-granule conditional-entropy H (A p ,A q ) (D/A) for two arbitrary granules A p , A q ∈ U/I ND(A). Furthermore, we can achieve all n × n entropies values by adding two "for" loops regarding p = 1, · · · , n and q = 1, · · · , n.
1: Compute U/I ND(A) to obtain two concrete granules A p , A q ∈ U/I ND(A), and determine ω p , ω q . 2: Compute U/I ND(D) to obtain all decision classes D j (j = 1, · · · , m).

Theorem 2.
At micro-bottom, the double-granule conditional-entropy has lower and upper bounds. Concretely, so In Theorem 2, the double bounds of H (A p ,A q ) (D/A) are acquired by the enlarging and reducing of weight coefficients. Regarding Equation (12), on the other hand, In other words, ω p and ω q have theoretical lower bounds |A p | 2|U| and |A q | 2|U| , respectively, but they usually have closer lower bounds usually, it may be practically restricted by a better measure: which offers We below provide another upper bound of H (A p ,A q ) (D/A), which may be better than Proof. As shown in Figure 2 and then the famous "Jensen's inequality" in mathematics could induce where In other words, we can get In Theorem 3, the convex property of information function When comparing Equations (7) and (17), we can surprisingly discover that H * (A p ,A q ) (D/A) highly adheres to which naturally comes from H C p ∪C q (D/(C − {c})) (Equation (7)). In fact, when A p = A q ; when where A p ∪ A q = A p , there is a difference between two measures, and we obtain Thus far, H (A p ,A q ) (D/A) has one lower bound H (A p ,A q ) (D/A) and two upper bounds ). An interesting question naturally emerges, i.e., can we necessarily determine the size relationship between H (A p ,A q ) (D/A) and H * (A p ,A q ) (D/A) to provide an exact bound? Unfortunately, the answer is negative, and the later example and experiment will reveal the size uncertainty. We simply provide a mechanism analysis. Let and its numerator/denominator be the corresponding sum of numerators/denominators of P p and P q . According to [64], we can obtain P pq ∈ [min(P p , P q ), max(P p , P q )] but P pq produces an uncertainty location between P p and P q . In view of the information function f (P) = −Plog 2 P and its maximum point ( 1 e , 1 eln2 ) (Figure 2), never having the necessary size relationships, so also never have the necessary size relationships. In summary, H (A p ,A q ) (D/A) and H * (A p ,A q ) (D/A) adopt different views to become irrelevant and interactive, and they together restrict H (A p ,A q ) (D/A). With the addition of lower bound of H (A p ,A q ) (D/A), there are in total three bounds to systematically emerge. Similar to H (A p ,A q ) (D/A) and its distributional Table 2, they can also be arranged in a table with an n × n square symmetric matrix, i.e., Table 3, and thus Table 3 correspondingly restricts Table 2.
Finally, consider relevant granulation monotonicity/non-monotonicity. In fact, micro-bottom and its double-granule conditional-entropies focus on only two condition granules and thus never consider the condition granulation and further monotonicity/non-monotonicity. Moreover, U/I ND(A) U/I ND(B) implies the granulation refining and granule decomposition from A to B; thus A p , A q ∈ U/I ND(A) and B p * , B q * ∈ U/I ND(B) exhibit complex correspondence and uncertainty change, so we cannot mine fine relationships between H (A p ,A q ) (D/A) and H (B p * ,B q * ) (D/B). Table 3. Three bounds of double-granule conditional-entropies at micro-bottom.

Double-Granule Conditional-Entropy at Meso-Middle
As analyzed above, double-granule conditional-entropies at micro-bottom never consider the condition granulation to lack robust functions of uncertainty descriptions. In terms of fixed decision granulation U/I ND(D), H (A p ,A q ) (D/A) at micro-bottom (A p , A q ) involves only two condition granules A p , A q and their interactive uncertainty information. For the function promotion, the condition granulation U/I ND(A) with systematic granules is worth introducing based on double-granule conditional-entropy H (A p ,A q ) (D/A). Thus, we will gradually strengthen the knowledge granulation U/I ND(A) to establish better double-granule conditional-entropies, by virtue of three-level granular structures (Table 1). This subsection discusses double-granule conditional-entropies at meso-middle Definition 5. At meso-middle (A p , U/I ND(A)), the double-granule conditional-entropy is defined by Corollary 1. At meso-middle, the double-granule conditional-entropy has an analytic expression:  In Proposition 3, double-granule conditional-entropies naturally exhibit number n to correspond to n meso-middles. The n values can be stored in an n-dimension vector to be related to the previous distributional Table 2. By enlarging Table 2, they are represented by the marginal vector of the bottom or right in Table 4, and they exactly correspond to the relevant row/column sum of micro-bottom's information values. According to Equations (21) and (23), Algorithm 2 resorts to two "for" loops to effectively offer a double-granule conditional-entropy H (A p ) (D/A) for an arbitrary granule A p ∈ U/I ND(A). In fact, the inner loop invokes Algorithm 1 to calculate an arbitrary double-granule conditional-entropy at micro-bottom, while the outer loop integrates n related bottomed measures to produce H (A p ) (D/A). Furthermore, we can achieve all n middle entropies values by adding a "for" loop regarding p = 1, · · · , n. Table 4. Marginal distribution of double-granule conditional-entropies at meso-middle and macro-top. 1: Compute U/I ND(A) to obtain all condition classes A i (i = 1, · · · , n) and a fixed granule A p ∈ U/I ND(A).

Theorem 5.
At meso-middle, the double-granule conditional-entropy has a lower bound and two upper bounds. Concretely, where Theorem 5 naturally comes from Theorems 2-4. The three bounds in Equation (25) hierarchically integrate previous three bounds at micro-bottom (Equations (11) and (17)) to correspondingly restrict H (A p ) (D/A). They can be supplemented into distributional Table 4, and following Table 5 provides the relevant part. Table 5. Three bounds of double-granule conditional-entropies at meso-middle and macro-top.

A n H (A n ) (D/A) H (A n ) (D/A) H (A n ) (D/A) H * (A n ) (D/A) Macro-Top H(D/A) H(D/A) H(D/A) H * (D/A)
At meso-middle, H (A p ) (D/A) introduces the condition granulation U/I ND(A), but it still needs condition granule A p . Thus, we cannot make a positive assertion regarding granulation monotonicity/non-monotonicity. In fact, U/I ND(A) U/I ND(B) also implies chaos between H (A p ) (D/A) and H (B p * ) (D/B).

Double-Granule Conditional-Entropy at Macro-Top
As analyzed above, double-granule conditional-entropies at meso-middle consider the condition granulation, but in an insufficient way, and H (A p ) (D/A) also depends on a single condition granule A p . For the thorough granulation and robust description, systematic measures H (A p ) (D/A) (p = 1, · · · , n) can be further integrated to generate double-granule conditional-entropies at macro-top. Based on the previous thought and result in Sections 3.1 and 3.2, this subsection further discusses double-granule conditional-entropies at macro-top (U/IND(A) = {A p : p = 1, · · · , n}, U/I ND(A) = {A q : q = 1, · · · , n}), which is given in Table 1.
We will directly provide the relevant integration definition, number distribution, calculation algorithm, three bounds, and we finally uncover an important conclusion of granulation non-monotonicity.

Definition 6. At macro-top (U/IND(A), U/I ND(A)), the double-granule conditional-entropy is defined by
Corollary 2. At macro-top, the double-granule conditional-entropy has an analytic expression: Theorem 6. Double-granule conditional-entropies have a hierarchical integration from micro-bottom and meso-middle to macro-top, i.e.,  In Proposition 4, the double-granule conditional-entropy naturally exhibits number 1 to correspond to the sole macro-top. In fact, the first top entropy comes from the fusion of either n middle entropies or n × n bottom entropies; thus, three-level entropies accord with three-level granular structures (Table 1) from the quantitative and structural perspective, and they embody the hierarchical integration. In particular, the sole conditional-entropy H(D/A) is put into the lower-right corner of Table 4, thus corresponding to the summations of central n × n micro values and marginal n meso values. According to Equations (26) and (28), Algorithm 3 resorts to three "for" loops to effectively offer the double-granule conditional-entropy H(D/A). The two inner loops invoke Algorithm 2 to calculate an arbitrary double-granule conditional-entropy at meso-middle (where the central loop invokes Algorithm 1 to construct micro-bottom's entropies), while the outer loop integrates n related meso-middle's information values to produce H(D/A). In other words, Algorithms 1-3 exhibit a kind of hierarchical evolution based on circulation development, and thus they constitute a novel kind of three-level algorithms. Compute ω p , ω q . 8: Let H p = 0, H q = 0. 9: for j ∈ {1, .., m} do 10:

11:
end for 12: Obtain H (A p ,A q ) (D/A) = ω p H p + ω q H q . 13: 14: end for 15: Theorem 7 naturally comes from Theorems 2-6. The three bounds in Equations (30)-(32) hierarchically integrate previous three bounds at meso-middle and micro-bottom, and thus they become three new uncertainty measures at macro-top (U/IND(A), U/I ND(A)) to correspondingly restrict H(D/A). They are supplemented into the bottom in the previous bound table: Table 5. At macro-top, the double-granule conditional-entropy completely breaks away from the condition granule dependence to establish the condition granulation description, so it becomes a powerful type of information measure for knowledge-based uncertainty representation. In terms of condition granulation, its non-monotonicity is finally revealed in Theorem 8, and the relevant evidence will be provided in the later example and experiment. Moreover, this fundamental non-monotonicity conclusion embodies information uncertainty, and it can be induced or explained by the previous complexity mechanism at micro-bottom and meso-middle. Based on macro-top and its granulation mechanism, the related three bounds (Equations (30)-(32)) and their monotonicity/non-monotonicity can be practically observed, and thus we also obtain the granulation non-monotonicity for H(D/A) and H(D/A); however, the case of upper bound H * (D/A) becomes a remaining problem.

Decision Table Example
In this section, the above theoretical constructions and properties are illustrated by a decision table example. By extracting a part of VOTING data set (which comes from UCI database [65]), we provide a practical decision table (U, C ∪ D) in Table 6 with According to this decision table, As an example, A = {c 1 , c 2 , c 3 , c 4 , c 5 } is chosen to generate condition granulation where n = |U/IND(A)| = 6. By virtue of three-level granular structures (Table 1), double-granule conditional-entropies and their three bounds are calculated by relevant algorithms and definitions, and they are compactly listed in Tables 7 and 8, respectively. The measures at micro-bottom, meso-middle, macro-top have numbers 36, 6, 1, respectively, and they correspond to the central 6 × 6 matrix, marginal 6-dimensional vector, lower-right-corner 1 digit, respectively. In part, we provide some processes of entropy calculation as follows.
By Tables 7 and 8, we can make relevant verification analyses. First, entropies and bounds naturally present hierarchical integration relationships from micro-bottom to meso-middle to macro-top. Indeed, conditional-entropies are correspondingly restricted by three bounds. Moreover, the two types of upper bounds exactly have no necessary size relationships, and a part but powerful proof is provided as follows: Table 7. Information values of double-granule conditional-entropies in the example.  Table 8. Three bounds of double-granule conditional-entropies in the example. Finally, the granulation non-monotonicity at macro-top (Theorem 8) is verified. For this, we chose a natural attribute-addition chain: CA k (k ∈ {1, 2, · · · , 11}) denotes the attribute subset in the chain, and its granulation is represented by U/I ND(CA k ) = {CA k,1 , · · · , CA k,p , · · · , CA k,|U/I ND(CA k )| }.
In other words, CA k,p corresponds to the kth chain element CA k to represent the pth condition granule in partition U/I ND(CA k ). According to the subset chain, Table 9 provides double-granule conditional-entropies, including both part values at micro-bottom (CA k,p , CA k,q ), meso-middle (CA k,p , U/I ND(CA k )) and all values (as well as the three bounds) at macro-top (U/IND(CA k ), U/I ND(CA k )). As a supporting detail, previous Tables 7 and 8 Table 10 for better observation and illustration.
(1) Since different chain subsets may have different equivalence partitions and granule numbers, the measures at micro-bottom and meso-middle consider condition granules to have a distinctive number and difficult correspondence. Table 9 focuses on the small and the same granule number, but relevant granules have different connotations. For example, the granules of the first one -CA k,1 (k = 1, 2, · · · , 11)-may be different. Thus, we cannot acquire the so-called granulation non-monotonicity assertion because of granulation incompletion, although the values at micro-bottom and meso-middle actually exhibit a kind of non-monotonic change in Table 9. (2) In contrast, macro-top offers the complete condition granulation, so we can effectively focus on value monotonicity/non-monotonicity for both double-granule conditional-entropies and their three bounds. Observing the bottom part of Table 9 in the enlargement chain direction, we can discover that the three types of information measures are all non-monotonic, i.e., H(D/CA k ), H(D/CA k ), H(D/CA k ) (except H * (D/CA k )).
More vividly, the entropy and its three bounds regarding the chain are depicted in Figure 3, so the related granulation non-monotonicity becomes clearer. For example, the macro entropy value H(D/CA k ) first increases and then decreases in the addition chain direction. Moreover, Table 9 and Figure 3 reflect the restriction properties of three bounds. Table 9. Double-granule conditional-entropies based on an attribute-enlargement chain in the example.

Data Experiments
In this section, the above theoretical results and their effectiveness are verified by data experiments. The new measures are mainly suitable for categorical (or nominal) data, which are usually used in the traditional rough set theory, and thus we adopt three relevant data sets from the UCI Machine Learning Repository [65], whose concrete descriptions on decision table (U, C ∪ D) are given in Table 11. Similar to the above example, we also adopt the attribute-addition chain and its relevant symbol such as U/I ND(CA k ) = {CA k,1 , · · · , CA k,p , · · · , CA k,|U/I ND(CA k )| }.
Note that this attribute-subset sequence (Equation (34)) can deeply and typically probe the hierarchical knowledge-granulation within a framework of the complete lattice (2 C , ⊆). As a representative manifestation, we provide two typical results regarding the first chain element CA 1 = {c 1 } and the last one CA |C| = C.
(1) Regarding VOTING, {c 1 } and C induce three and 342 granules, respectively, and relevant double-granule conditional-entropies and three bounds are provided in Tables 12 and 13, respectively. (2) Regarding SPECT, {c 1 } and C produce two and 169 granules, respectively, and relevant three-level measures and three bounds are provided in Tables 14 and 15, respectively. (3) Regarding Tic-Tac-Toe, {c 1 } and C determine three and 958 granules, respectively, and relevant entropies and bounds are provided in Tables 16 and 17, respectively. Meso-Middle · · · U/I ND(CA 9 ) CA 9,1 · · · CA 9,958 Meso-Middle From the perspective of macro-top, double-granule conditional-entropies and their three information bounds based on the attribute-enlargement chain are finally summarized in Figure 4. These tables and figures can be utilized to effectively verify all previous conclusions, including the hierarchy, algorithm, restriction, and non-monotonicity. In particular, double-granule conditional-entropies are confined by three bounds, thus supporting the boundedness (Theorems 2, 3, 5 and 7); moreover, the entropies and their matched double-bounds fluctuate up and down, thus proving relevant granulation non-monotonicity (Theorem 8).

Conclusions
The information measures implement fundamental uncertainty measurement in rough set theory and granular computing. The local conditional-entropies have the second-order feature, but they are limited to micro-bottom for describing discernibility matrix and reduction core [18]. In this paper, double-granule conditional-entropies achieve corresponding improvements of hierarchical/conditional granulation, and thus they become broader measures with uncertainty representation and information processing. They focus more on the double-granule interaction rather than granule-union locality, which is used in local conditional-entropies [18]. This strategy directly utilizes the second-order mechanism to implement more systematic and robust uncertainty measurements, especially when compared to the current mainstream of first-order information measures. In our studies, double-granule conditional-entropies and their hierarchies, granulation, algorithms, bounds, and non-monotonicity are acquired and verified at three-level granular structures (i.e., micro-bottom, meso-middle, macro-top), and these results underlie both the efficiency in information processing and effectiveness in knowledge-based data analyses. Furthermore, their future developments and in-depth applications can be explored as follows.
(1) In contrast to the relevant technology in [56], the hierarchical granulation of three-level granular structures focuses on the conditional granulation and relevant number, and it can be generalized for granular computing. (2) The double-granule conditional-entropies and their three bounds become new types of information measures with the second-order feature. In contrast to the traditional first-order entropy system, their description power and application advantage need further practical verification. (3) The double-granule conditional-entropies have three-restrictive bounds and granulation non-monotonicity, which have been experimentally verified by a granulation-hierarchical sequence (i.e., Equation (34)). These results are worth deeply utilizing in uncertainty measurement and data mining. (4) The double-granule conditional-entropies originate from the local conditional-entropies to carry a potential and distinctive advantage of discernibility matrix representation, and they also have the complete conditional granulation to have application prospects in knowledge reasoning or acquisition. Both their relationships with the discernibility matrix and their functions on attribute reduction need be deeply researched by promoting the previous studies in [18].