Generalized dynamic attribute reduction based on similarity relation of intuitionistic fuzzy rough set

2.1 Presentation of dynamic reduction

As one of the important research contents of data mining and data reduction technology, attribute reduction refers to the deletion of redundant attributes and their attribute values in the decision table, but its prerequisite is to maintain the dependency relationship between condition attributes and decision attributes of the original decision table unchanged. At present, the reduction algorithm based on rough set theory can be divided into two kinds according to whether there is heuristic information or not. One is blind method, which does not use any heuristic information to obtain a reduction, but the result is unsatisfactory. The other is heuristic algorithm [13, 14], whose idea is to start with the core of the decision table and regard it as a reduction, then add attributes according to certain heuristic information, that is the importance of attributes, until the reduction of the decision table is obtained. For example, there are attribute reduction algorithms based on mutual information, based on discernibility matrix, and based on Pawlak attribute importance, etc. But the problem of these algorithms is that they can only reduce the small-scale compatible decision information system. When the decision information system has a large size of data, the generalization ability of the decision rules obtained by these algorithms is limited. At the same time, the decision information system contains a lot of noise data, so it is an urgent problem to get stability a reduction.

Aim to solve the above problems, Jan. G. Bazan has proposed dynamic reduction algorithms [15–17]. These algorithms are sampling large and complex decision information systems many times, and transform the reduction of complex systems into the intersection of several reductions of sub-decision information systems, and obtain more stable reduction. They have good adaptability to variable data sets, and provide an effective way to solve the attribute reduction for large-scale fuzzy information systems. The facts show that the reduction obtained by dynamic reduction has high stability. It has a good performance in big data processing, adaptability to variable data sets, stability of reductions, and anti-noise ability [18].

2.2 Determination of f-family range

–Determination of lower limit for f family

For attribute reduction of incremental data, each additional data are regarded as a sub-table, the original data are taken as a master table, and the parent table can also be regarded as a sub-table set too. Regardless of whether data increasing with time, attribute reduction can use sampling method to obtain sub-tables. If the original table has |U| data, and a new table should select about |U| /|F| data size each time. F is called the F family of decision information system DS. That is, for all decision subsystems P(DS) of DS, there are F ⊆ P (DS) and F ≠ Φ . |F| represents the capacity of the F family. According to the study of the F family capacity in document [19], it can get.

| F | ⩾ t α 2 · P G ( R ) · ( 1 - P G ( R ) ) / ( Δ MLE ( P G ( R ) ) ) 2(1)

Where, P _G  (R) is the probability of dynamic reduction R appearing in a sub-table of the original decision table. MLE (P _G  (R)) is the maximum likelihood estimation of P _G  (R). ΔMLE (P _G  (R)) is the maximum acceptable error of MLE (P _G  (R)). t _α represents the interval function related to the allowable maximum error, and it must satisfy, 1 - α = 2 / π ∫ - t α t α exp ( - t / 2 ) 2 dt(2)

When P _G  (R) =1/4, P _G  (R) • (1 - P _G  (R)) obtains the maximum value 1/4, and then the minimum value of |F| is obtained, | F | ⩾ t α 2 / - ( 4 ( Δ MLE ( P G ( R ) ) ) 2 )(3)

Where, the value of t _α can be querying through the normal distribution table.

–Determination of upper limit of f family

If too many sub-tables are extracted, the computational complexity will increase and the time complexity of the algorithm will be improved. From Bernonlli’s idea, when the decision sub-tables extracted from the F family is large enough, the continued extraction of sub-tables has little effect on the stability of reduction. Therefore, there must be an upper limit for the capacity of F family. From the stability of reduction, Bazan, et al. [19], had studied the upper limit of F family capacity.

Given the initial decision system S = (U, C ∪ D), where U is an universe, C is a non-null conditional attribute set, D is a non-null decision attribute set, and S′ = (U_S′, C ∪ D) is a full sample decision system. Where, there is U_S′ =∪ { U _B |B ∈ F  }, and U_S′ ⊆ U. The similarity index β₁ of decision classification ability between decision sub-system B and full sample decision system S′ is as follows:

β 1 = | Pos ( U B , C , D ) | / | Pos ( U S ′ , C , D ) | ⩾ | U B | / | U S ′ |(4)

Similarly, the decision-making ability index β₂ between full sample decision-making system S′ and initial decision-making system S,

β 2 = | Pos ( U S ′ , C , D ) | / | Pos ( U S , C , D ) | ⩾ | U S ′ | / | U S |(5)

The stability parameter of F family relative positive domain is,

SC S Pos ( F , B ) = | { B ∈ F , β 1 ⩾ | U B | / U S ′ , β 2 ⩾ | U S ′ | / U S } | / | F |(6)

Obviously, SC S Pos ( F , B ) ∈ [ 0 , 1 ] . However, when SC S Pos ( F , B ) ∈ [ 0 , 0.5 ] , it shows that the extracted subsystems can not meet the requirements of similarity and have no research value. SC S Pos ( F , B ) = 1 shows that all the decision-making sub-systems extracted have similar decision-making classification ability with the full sample decision-making system S′, and that all the decision-making sub-systems extracted have similar decision-making classification ability with the initial decision-making system S. SC S Pos ( F , B ) = 0 shows that all the decision-making sub-systems extracted do not have similar decision-making classification ability with the full sample decision-making system S′, and that the full sample decision-making system S′ does not have the same decision-making classification ability with the initial decision-making system S, and the similarity decision indexes of all decision sub-tables do not meet the requirements.

If SC S Pos ( F , B ) > λ (λ is a threshold), then there is

| F | = | { B ∈ F , β 1 ⩾ | U B | / U S ′ , β 2 ⩾ | U S ′ | / U S } | / λ(7)

When the range of F family capacity is determined, it can replace the capacity cardinality of F family specified by experts in probabilistic extraction method. When the number of F family reaches this range, there is no need to extract sub-tables. Therefore, a dynamic reduction method without prior knowledge is obtained. Document [20] demonstrates that the number of effective sub-tables extracted by this method is less.

–F Sub-table extraction strategy

What kind of sampling strategy adopted to extract the subfamily F is the premise of dynamic reduction. The quality of sampling strategy will also directly affect the accuracy of reduction results. When decision tables contain massive data or are constantly changing, the used sampling strategies are different. When a decision table contains a big data information or the decision table is changing, direct reduction of the initial decision table will produce a great reduction error, so the results of reduction can not be used as a guide for drawing up decision-making rules, otherwise the decision-making will fail because of the instability of data. Therefore, when decision table contains massive data information or the decision table is changing, the decision rules obtained by static reduction can not describe the characteristics of the object to be studied very well, but the initial decision table is processed for extracting the sub-table family, that is F family, as the object of reduction, and then the reductions of the sub-tables are intersected, the relatively stable reduction is finally obtained. The characteristics of the initial decision table can be well described by these reductions that exist in most sub-tables. At present, there are two methods for F family sub-tables extraction: trace extraction and probability sampling [20]. Trace extraction method does not take into account the principle of determining the stability coefficient of the decision sub-table, can not extract representative reductions, and the accuracy of dynamic reductions is low [20]. Based on this, this paper chooses the probability sampling method. After the total number of sub-tables and the number of sub-tables in F family are determined, the decision-making sub-tables are extracted according to the following strategies. In order to make the probability of a reduction of F family sub-tables more than 1/2 and exist simultaneously in reductions of the initial decision table, the number of objects in the decision sub-tables that needs to be randomly extracted is greater than 50% of the original decision table. The steps of the probabilistic sampling method are described as follows:

Algorithm 1 Probability Sampling Method

Step 1, N/5 decision sub-tables are randomly selected from initial decision table which size is 90%,80%,70%,60% and 50% of the size of initial decision table.

Step 2, to compute the reductions of these N decision sub-tables.

Step 3, to select reductions whose stability coefficient is not less than a fixed threshold from these reductions.

Step 4, to intersect these reductions obtained in step 3 and get the final dynamic reduction.

Before sampling, this sampling strategy calculates the minimum number of decision sub-tables that need to be extracted. It takes accuracy as the first consideration factor, and then determines the size of a single decision sub-table as effective as possible. They consider reducing the reduction error.

–F-dynamic reduct

Definition 1 (F-dynamic reduction) For a given decision-making information system DT = (U, C ∪ d), U is an universe, and C is the non-null conditional attribute set, and d is a non-null decision attribute. B = (U _B , C ∪ d) is an arbitrary decision information sub-system of decision information system DT, and where U _B  ⊆ U. F is a decision information sub-system family of DT. DR ( DT , F ) = RED ( DT , d ) ∩ ⋯ ∩ B ∈ F RED ( B , d )(8)

Where, any element in the system is called F-dynamic reduction of decision information system DT.

The final reduction of F-dynamic reduction is calculated by the intersection of all reductions of the initial decision information system and all reductions of decision information sub-system randomly extracted from it. Therefore, the reduction is the most stable and the generalization ability of the decision rules is the strongest. However, this method is too strict for the reduction of F family dynamic reductions, which is dependent on the initial decision information system. Document [21] uses dynamic reduction method to process UCI data sets. The results show that many data sets can not get F family dynamic reduction. Therefore, it need to generalize the dynamic reduction concept, and introduce (F–λ)-dynamic reduction, improve the ability of F family dynamic reduction.

Definition 2 ((F–λ) -dynamic reduce) For a given decision information system DT = (U, C ∪ d), where U is an universe, and C is the non-null conditional attribute set, and d is a non-null decision attribute. B = (U _B , C ∪ d) is an arbitrary decision information sub-system for decision information system DT, where U _B  ⊆ U. F is a decision information sub-system family of decision information system DT.

DR λ ( DT , F ) = { R ∈ RED ( DT , d ) : | { B ∈ F : R ∈ RED ( B , d ) } | | F | ⩾ λ }(9)

Where, any element is called the (F-λ)-dynamic reduction of decision information system DT, and λ ∈ [0, 1] is called the precision coefficient of (F–λ)-dynamic reduction. | { B ∈ F : R ∈ RED (B, d) } |/|F| is called the stability coefficient of (F–λ)-dynamic reduction R relative to F family.

(F–λ)-dynamic reduction has the properties as follows,

If F ={ DT }, then DR (DT, F) = RED (DT, d).

(F–λ)-dynamic reduction of decision information system is an extension of F family dynamic reduction, and reduces the dependence on initial decision information system in F family dynamic reduction. However, when the data increases, the decision rules generated by the new decision information system will be greatly different from those generated by the original decision information system, which requires further expansion of dynamic reduction to improve the adaptability and generalization ability of reduction.

Definition 3 (F-generalized dynamic reduction) For a given decision information system DT = (U, C ∪ d), U is an universe, and C is the non-null conditional attribute set, and d is a non-null decision attribute. B = (U _B , C ∪ d) is an arbitrary decision information sub-system for decision information system DT, where U _B  ⊆ U. F is a decision-making information sub-system family of decision information system DT. GDR ( DT , F ) = ∩ B ∈ F RED ( B , d )(10)

Where, any element is called F-generalized dynamic reduction of decision information system DT.

When the number of objects and attributes in the decision information system is very large, the complexity and difficulty computing the reduction of the decision system will become immeasurable. F-family generalized dynamic reduction is no longer to reduce the initial decision information system, but to intersect the reductions of all decision sub-systems randomly extracted. In a sense, generalized dynamic reduction is more extensive than dynamic reduction.

Definition 4 ((F–λ)-generalized dynamic reduction) For a given decision information system DT = (U, C ∪ d), U is an universe, and C is the non-null conditional attribute set, and d is a non-null decision attribute. B = (U _B , C ∪ d) is an arbitrary decision information sub-system for decision information system DT, where U _B  ⊆ U. F is a decision information sub-system family of decision information system DT.

GDR λ ( DT , F ) = { R ⊆ C : { B ∈ F : R ∈ RED ( B , d ) } | F | ⩾ λ }(11)

Where, any element is called (F–λ)-generalized dynamic reduction in the decision information system DT. Where λ ∈ [0, 1] is the precision coefficients of (F–λ)-generalized dynamic reduction, and | { B ∈ F : R ∈ RED (B, d) } |/|F| is called the stability coefficients of (F–λ)-generalized dynamic reduction R relative to F family.

(F–λ)-generalized dynamic reduction has the properties as follows,

DR (DT, F) ⊆ GRD (DT, F).

4.1 Algorithm process

The basic idea of dynamic attribute reduction algorithm for large data based on similarity relation of intuitionistic fuzzy attributes is described as follows. Firstly, the continuous attributes of decision tables are intuitionistic fuzzified according to the standardization method in document [23]. Then, the intuitionistic fuzzy decision information system is sampled by dynamic sampling technology, and the intuitionistic fuzzy decision information sub-system family {U₁, U₂, ⋯ , U _n  } is obtained. Secondly, for each decision information sub-system U _i , the cardinality of the relative positive domain for each level attribute combination in U _i is taken as the basis of heuristic search, and the attribute with the largest cardinality of the relative positive domain is selected and put into the candidate subset, ...  ...  . Until the combination of hierarchical attributes satisfies the criterion of the maximizing cardinality of the relative positive domain, and the reduction R _i of the subfamily U _i is obtained. This cycle lasts until all reductions {R₁, R₂, ⋯ R _n  } of the families of intuitionistic fuzzy decision information sub-system are computed. Finally, the relative reduction R of intuitionistic fuzzy decision information system is obtained by intersecting these reductions, that is R = R₁ ∩ R₂ ∩ ⋯ ∩ R _n .

Based on the above ideas, the process of generalized dynamic attribute reduction algorithm based on similarity relation of intuitionistic fuzzy rough set is as follow as Fig. 1.

OPEN IN VIEWER

Fig. 1

General dynamic attribute reduction based on fuzzy similarity relation of intuitionistic fuzzy rough set.

Generalized dynamic attribute reduction algorithm based on fuzzy similarity relation of intuitionistic fuzzy rough set and is described as follows,

Algorithm 2 Generalized dynamic attribute reduction algorithm based on fuzzy similarity relation of intuitionistic fuzzy rough set

Input: A hybrid information system S = (U, C, V _C , D, V _D , F), threshold k.

Output: An approximate minimum relative reduction R of S = (U, C, V _C , D, V _D , F).

Step 1, to preprocess data and construct fuzzy information system U.

Step 2, to carry out intuitionistic fuzzification for continuous attributes, to obtain the membership degree of each object to each intuitionistic fuzzy attribute, and then obtain the non-membership degree γ (x) using intuitionistic index π (x) and according to the formula γ (x) =1 - μ (x) - π (x). So that the value of each attribute can be expressed by pairs 〈μ (x) , γ (x)〉 and the intuitionistic fuzzy information system U ′ can be obtained after intuitionistic fuzzification.

Step 3, to determine F family capacity N, that is |F|, of intuitionistic fuzzy information system U ′ . Where, the determination of stability coefficient A _i of generalized dynamic reduction R relative to F family is very important. Document [21] considers that the system can obtain the most stable generalized reduction when Φ ∈[0.5,1], that is, the sampling coverage of the sub-family must be more than 50%. By using probability sampling method, N1 subtables with coverage 90%,80%,70%,60% and 50% in decision information system are randomly sampled to join F families, where N = 5•N1. Dynamic sampling generates F family of intuitionistic fuzzy decision system U ′ , i.e. subfamily A₁, A₂, ⋯ , A _c . For each X = {x₁, x₂, ⋯ , x _n }, the step 4 to step 7 of the loop is executed, and the resulting reductions are recorded as x _j  = (x_j1, x_j2, ⋯ , x _jm ) , j = 1, 2, ⋯ , n.

Step 4, to initialize, R _i  =∅, pos R i k ( D ) = 0 , C′ = C.

Step 5, according to the value of decision attribute D, the objects in the universe U′ are sorted and the set of equivalent classes {X|X ∈ U/D} is obtained.

Step 6, for all a _i  ∈ C′,

Step 6.1, to calculate the intuitionistic fuzzy similarity relation R_{iR i ∪{a i }} = R _{iR i}  ∩ R_{i{a i }} according to Equation (15).

Step 6.2, to calculate the cut set R iR i ∪ { a i } k of intuitionistic fuzzy similarity relation R_{iR i ∪{a i }} at the threshold level k.

Step 6.3, to calculate fuzzy similarity classes according to R iR i ∪ { a i } k .

Step 6.4, to calculate the relative positive domain pos R i ∪ { a i } k ( D ) according to Equation (17).

Step 6.5, to calculate the relative positive domain cardinality | pos R i ∪ { a i } k ( D ) | .

Step 7, for all a _i  ∈ C′, to find the maximum relative positive domain cardinality pos max k = max { | pos R i ∪ { a i } k ( D ) | | a i ∈ C ′ } .

Step 7.1, if there are more than one maximum value of | pos R i ∪ { a i } k ( D ) | , then any one conditional attribute a _k with the largest cardinality of relative positive domain is selected as the candidate conditional attribute.

Step 7.2, if pos max k > pos R i k ( D ) , then R _i  = R _i  ∪ {a _k }, pos _{R i}  (D) = pos_{R i ∪{a k }} (D), C′ = C - {a _k }, and return to step 6.

Step 7.3, if pos max k ⩽ pos R i k ( D ) , the algorithm terminates this loop, the reduction set R _q  (q = 1, ⋯ , n) is output.

Step 8, to intersect operation R = R 1 ∩ R 2 ∩ … ∩ R n , and the reduction R of intuitionistic fuzzy decision information system is obtained.

5.1 Experimental data and parameter setting

In order to verify the principle, the performance of the algorithm is verified by the fault diagnosis case of certain aircraft in ground state. The engine fault diagnosis model can be represented by the following nonlinear equations [24]: y = f ( T 1 * , N 1 , N 2 , Φ pc , P M , B )(19)

Where,

y——the output value. It indicates the engine state.

T₁^*——Total inlet temperature.

N₁——Low-pressure rotor speed.

N₂——High-pressure rotor speed.

Φ _pc ——Tail nozzle indication value.

P _M ——Lubricating oil pressure.

B——Vibration value of engine crankcast.

Let T =  (U, S, R, D) be an engine fault fuzzy information system. Where U = x₁, x₂, ...  , x _n is a set of engine fault data, which is abbreviated as universe. R = total inlet temperature, low-pressure rotor speed, high-pressure rotor speed, tail nozzle indication value, lubricating oil pressure, vibration value of engine crankcast is a continuous conditional attribute set, respectively noted as R ={ r₁, r₂, r₃, r₄, r₅, r₆ }. D = engine state y is a discrete decision attribute set. S = R ∪ D is an all attribute set. The intuitionistic fuzzy information system of engine fault is obtained from the data in document [24], as shown in Table 1.

The parameters are set as follows that fuzzy index π _ij  = 0.01, λ = 0 . 6, intuitionistic fuzzy partition threshold κ = 0 . 6. When the acceptable error ΔMLE(P_G(R)) is 11.11%, α = 9, t_α = 0.703, then |F| ⩽ 10. It is taken that |F| = 10. There are two subfamily tables for each sampling probability.

The subfamily is expressed as F - R_{(P G ,χ)}. Where, P _G represents the sampling probability, χ represents the sampling times. Two subfamilies which are respectively containing 90%,80%,70%,60%,50% samples were sampled respectively from original decision information system, and they are noted as F-R _(90 %,1) , F-R _(90 %,2) , F-R _(80 %,1) , F-R _(80 %,2) , F-R _(70 %,1) , F-R _(70 %,2) , F-R _(60 %,1) , F-R _(60 %,2) , F-R _(50 %,1) , F-R _(50 %,2) , and their reductions are noted as R_(90%,1), R_(90%,2), R_(80%,1), R_(80%,2), R_(70%,1), R_(70%,2), R_(60%,1), R_(60%,2), R_(50%,1), R_(50%,2).

According to the algorithm 2, the reduction of the intuitionistic fuzzy information system is the result of intersection operation of sub-family reductions after dynamic sampling. Under the condition that the threshold is (α,β)  =  (0.6, 0.4), the reduction results of all sub-families are shown in Table 2. The intuitionistic fuzzy rough information system after attribute reduction is shown in Table 3.

The generalized dynamic reduction of whole intuitionistic fuzzy information system is as GDR _IFRS  ={ r₁, r₆ }. While the reduction of generalized attribute reduction based on discernibility matrix of intuitionistic fuzzy rough set is GDR _IFRS  ={ r₁ }.

Rough set method is used to reduce the data in Table 1, and the attribute sets are reduced as {r₁, r₂, r₄, r₆}, {r₁, r₂, r₅, r₆}, {r₂, r₄, r₅, r₆} and {r₁, r₃, r₅, r₆}. Under the condition of 100% sampling probability, the generalized dynamic reduction of intuitionistic fuzzy information system is a subset of the static reduction of rough set. This shows that, compared with rough set reduction, generalized dynamic reduction can get more refined reduction.

The stability analysis of each sub-family reduction is shown in Table 2. The envelope of dynamic reduction is described by the number of attributes in dynamic reduction, as shown in Fig. 2. The envelope analysis shows that the generalized dynamic reduction based on the relative positive domain of intuitionistic fuzzy rough sets is relatively stable. When the sampling coverage is between 50% and 70%, the system reduction reaches the most stable state, as shown in Fig. 2(a). The generalized attribute reduction algorithm based on discernibility matrix of intuitionistic fuzzy rough set achieves the most stable state when the sampling coverage is more than 90%, as shown in Fig. 2(b). From Fig. 2, compared with the envelope range of the generalized dynamic reduction algorithm based on discernibility matrix of intuitionistic fuzzy rough set, the generalized dynamic reduction algorithm based on the relative positive domain of intuitionistic fuzzy rough sets is smaller than that of the generalized dynamic reduction algorithm based on discernibility matrix of intuitionistic fuzzy rough set. It shows that the stability of the generalized dynamic reduction algorithm in this paper is better than that of the algorithm based on discernibility matrix of the intuitionistic fuzzy rough set.

OPEN IN VIEWER

Fig. 2

Variation curves of samples and reduced envelopes.

In the process of dynamic reduction, the cardinality of the relative positive domain keeps increasing and then remains unchanged. There is no case that the number of attributes is increasing while the cardinality, dependence and importance of the relative positive domain is decreasing, to see Fig. 3.

OPEN IN VIEWER

Fig. 3

Variation curves of relative positive domain cardinality during reduction.

Rough set method is used to reduce the data of Table 4, the reductions are {r₁, r₂, r₄, r₆}, {r₁, r₂, r₅, r₆}, {r₂, r₄, r₅, r₆}, and {r₁, r₃, r₅, r₆}. Compared with rough set method, it is found that the algorithm proposed in this paper can get smaller reductions, and the number of reduction attribute is less, and the redundant information is less.

In order to further verify the validity of the extracted feature parameters, the 7∼9 samples are taken as test samples, the rest samples are taken as training samples, the attribute values in the reduction set are taken as BP neural network input, and the engine state is taken as output to compare and calculate. The results are shown in Table 4. In order to verify the scientificity of the algorithm and the validity of the feature parameters, the experiment adds the combination of some attributes excluded from the reduction to investigate the interference of redundant attributes.

The validation results show that, compared with rough set method, the proposed algorithm obtains fewer feature parameters, eliminates the interference of redundant attributes, and maintains the highest diagnostic accuracy of feature parameters obtained by rough set method. Redundant attributes r₂, r₄, r₅ have less interference to diagnostic accuracy, while redundant attribute r₃ has the greatest interference ability, and the combination precision of reduction containing attribute r₃ is very low. The diagnostic accuracies of reductions {r₂, r₃, r₆} and {r₃, r₆} are the same, it indicates that attribute r₂ is redundant. The diagnostic accuracies of reductions {r₂, r₃, r₅, r₆} and {r₂, r₃, r₆} is the same, it indicates that attribute r₅ is redundant.

The diagnostic accuracy of reduction {r₁, r₆} is higher than that of reduction {r₁, r₂, r₄, r₆}, and attribute r₂ is redundant, so the attribute r₄ is not only redundant, but also has greater interference ability. Compared with the diagnostic accuracy of attribute combination {r₂, r₃, r₄, r₆}, the diagnostic accuracy of attribute combination {r₂, r₃, r₅, r₆} is lower. Under the condition that r₂, r₃, r₄ are all redundant attributes, and r₅ is redundant attribute too, so they can be removed from reduction. The diagnostic accuracies of reduction sets {r₁} and {r₁, r₆} are the same, but it can not be explained whether attribute r₆ is redundant. Because all of r₂, r₃, r₄ are redundant attributes, the diagnostic accuracies of attribute sets {r₂, r₃, r₄, r₆} and {r₂, r₄, r₅, r₆} has reached or reached the same level of {r₁} and {r₁, r₆}, and they are the highest. This shows that when the contribution of attribute r₁ to correct diagnosis is 0, the attribute r₆ plays an important role in making correct diagnosis. They are also the key attributes in describing the fuzzy information system and can not be omitted.

In summary, there are still too many redundant attributes in the attribute reduction using rough set method. The generalized attribute reduction method based on discernibility matrix of intuitionistic fuzzy rough set has the problem of over-reduction, and the key attribute r₆ is omitted in the reduction set. However, the algorithm in this paper can get the relatively correct reduction set of intuitionistic fuzzy information system. In this way, fault diagnosis can be realized with fewer characteristic parameters, which greatly reduce the workload and improve the efficiency of fault diagnosis. At the same time, it can also solve the redundancy problem of diagnosis parameters.

Known from Table 1 and Table 3, the system needs 144 storage units to store before reduction. While after reduction, the system only needs 72 storage units, which reduces the storage space by 50%.

According to the classification and discrete results of document [28], the traditional rough set reduction algorithm is used to extract the diagnostic rules, if the total set {r₁, r₂, r₃, r₄, r₅, r₆} of conditional attributes as characteristic parameters, then there are 1024 diagnostic rules extracted from the decision table. Using the algorithm proposed in this paper, if the set {r₁, r₆} of reduction attribute is taken as characteristic parameters, then there are 32 diagnostic rules extracted from the decision table. The size of rule base is reduced by 98.88%.

This proves that the generalized dynamic attribute reduction algorithm based on relative positive domain of intuitionistic fuzzy rough set can get fewer rules after extracting the feature parameters. That is, 1024 diagnostic rules need to be stored before, while only 32 rules need to be stored, and the optimization effect is obvious. This greatly reduces the size of rule base, saves a lot of storage space, effectively solves the problem of big knowledge base, and improves the efficiency of diagnosis.

5.6 Time complexity analysis

Assumption S = (U, P ∪ Q, V, f) is an intuitionistic fuzzy knowledge representation system. Where, |U| = n, |P| = m, and the number of sub-families is |F|. In the dynamic sampling stage, each sampling almost needs to traverse the domain of the system. Therefore, the time complexity of the sub-family sampling process is O (mn|F|). In document [26], the maximum time complexity of the algorithm based on the intuitionistic fuzzy discernibility matrix of intuitionistic fuzzy rough set approximates to O (|F|m (m + 1) n²/2). The time complexity of the attribute reduction algorithm based on the dependency degree of intuitionistic fuzzy rough set is approximating to O (|F|m (m + 1) n²/2) [25]. The time complexity of the proposed algorithm in this paper is approximate to O (|F|m (m - 1) n²/2). Compared with other algorithms, the algorithm proposed in this paper has obvious advantages in time consumption and can save 27.69% time in the case in this paper.

5.7 Comprehensive analysis

A fuzzy rough set attribute reduction method based on fuzzy distance measure of similarity relation is proposed in document [27]. This method is similar to the method based on discernibility matrix and attribute dependency, but it is still powerless to solve the problem of large data sets. Therefore, it can only be compared with the proposed method for the small data set reduction results after generalized sampling.

The parameters are selected as follows p = 2, k = 3, δ = 0.001. According to the reduction results in Table 2, the algorithm in this paper obtains the same generalized dynamic reduction GDR _IFRS  ={ r₁, r₆ } as the method in document [27]. The difference is that the algorithm proposed in this paper has redundant attribute r₃ in the reduction of subsets, while the algorithm in document [27] have redundant attribute r₅ in the reductions of subsets, and the stability is also very good. This has been described above.

According to the fault diagnosis results in Table 4, the diagnostic accuracy of the algorithm in document [27] is the same as that of the algorithm in this paper. After removing redundant attribute {r₅}, the diagnostic accuracy is also improved.

The time complexity of the algorithm in document [27] is O (mn² +  (m² + m)/2). Combining with the sampling time complexity O (mn|F|) of the sub-family, the time complexity of the synthesis algorithm is O ( | F | m 2 n ( n 2 + 1 2 m + 1 2 ) ) , which is higher than that of the algorithm proposed in this paper.