4.4.3.1Background and status of concept lattice reduction

The DF concept lattice is a complex lattice structure, which places a serious obstacle in its specific application. Therefore, it is necessary to control the growth of the concept lattice nodes and to reduce the DF concept lattice. There are two common reasons for the expansion of the concept lattice. One is that there are too many attributes. In the worst case, the number of nodes in the DF concept lattice grows exponentially with the number of attributes. The second problem is the high degree of similarity between elements of the object; i.e. the difference between the different concepts is very small. If we allow nearly identical objects to be merged (similar to clustering problems), the size of the concept lattice will be reduced. This may also improve the application efficiency of the concept lattice and reduce the consumption of time and space. Therefore, reducing the DF concept lattice not only improves the application efficiency but also improves the application quality.

At this stage, there have been many studies on the reduction of the concept lattice. Oosthuizen proposed a method of pruning concept lattices [14], and Agrawal proposed a method of clustering association rules in concept lattices [15]. Ho et al. used fuzzy set theory [16] to reduce the fuzzy concept lattice. Concept lattice reduction techniques are widely used, especially in classification systems. These prevent the use of a large concept lattice, reducing the processing time and improving the precision of the concept lattice.

4.4.3.2Reduction based on clustering technique

In data analysis, conceptual objects with large similarities are often divided into different concepts. From a human point of view, however, this division is completely unnecessary. It is often considered that objects belong to the same concept. Based on this consideration, we can use some clustering techniques to classify similar concepts and form one concept class.

Definition 4.27 Consider a known dynamic fuzzy concept C = (O, A) in the dynamic fuzzy form background B:(X, Y, I) with node vector $E^{(\stackrel{\leftharpoonup }{c}.\stackrel{\rightharpoonup }{c})}=$$\left(E_{({\stackrel{\leftharpoonup }{y}}_1.{\stackrel{\rightharpoonup }{y}}_1)}^{(\stackrel{\leftharpoonup }{c}.\stackrel{\rightharpoonup }{c})},E_{({\stackrel{\leftharpoonup }{y}}_2.{\stackrel{\rightharpoonup }{y}}_2)}^{(\stackrel{\leftharpoonup }{c}.\stackrel{\rightharpoonup }{c})},\dots ,E_{({\stackrel{\leftharpoonup }{y}}_3.{\stackrel{\rightharpoonup }{y}}_3)}^{(\stackrel{\leftharpoonup }{c}.\stackrel{\rightharpoonup }{c})},\dots ,E_{({\stackrel{\leftharpoonup }{y}}_q.{\stackrel{\rightharpoonup }{y}}_q)}^{(\stackrel{\leftharpoonup }{c}.\stackrel{\rightharpoonup }{c})}\right).\text{For}\text{(}{\stackrel{\leftharpoonup }{y}}_i,{\stackrel{\rightharpoonup }{y}}_i)ϵY:$

$E_{\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_i\right)}^{\left(\overleftarrow{C},\stackrel{\rightharpoonup }{C}\right)}=\{\begin{array}{l}\frac{1}{n}{\displaystyle \sum _{j=1}^{\pi }I\left(\left({\overleftarrow{o}}_j,{\stackrel{\rightharpoonup }{o}}_j\right),\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_i\right)\right)\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_i\right)\in A,\left({\overleftarrow{o}}_j,{\stackrel{\rightharpoonup }{o}}_j\right)\in O}\\ \frac{1}{n}{\displaystyle \sum _{j=1}^{\pi }I\left(\left({\overleftarrow{x}}_j,{\stackrel{\rightharpoonup }{x}}_j\right),\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_i\right)\right)\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_i\right)\cancel{\in }A,\left({\overleftarrow{x}}_j,{\stackrel{\rightharpoonup }{x}}_j\right)\in X}\end{array},\left(4.23\right)$

where |X| = k, |Y| = q, |O| = n. The formula indicates that the value of $E_{({\stackrel{\leftharpoonup }{y}}_i.{\stackrel{\rightharpoonup }{y}}_i)}^c$ depends on whether attribute $({\stackrel{\leftharpoonup }{y}}_i,{\stackrel{\rightharpoonup }{y}}_i)$ is included in connotation $(\stackrel{\leftharpoonup }{A},\stackrel{\rightharpoonup }{A})$ of node $(\stackrel{\leftharpoonup }{C},\stackrel{\rightharpoonup }{C}).$ The dynamic fuzziness of the object is introduced into the node, which reflects the entire dynamic fuzzy characteristics of the object set on the basis of preserving the dynamic fuzzy features of a single object. Thus, the dynamic fuzzy lattice between the nodes has a theoretical basis for comparison. Each concept in DF is a node. Each node can be expressed in vector form, so that the similarity between nodes (or concepts) can be defined as a metric for concept clustering.

Definition 4.28 The distance between the dynamic fuzzy concept lattice nodes $({\stackrel{\leftharpoonup }{C}}_i,{\stackrel{\rightharpoonup }{C}}_i)\text{and}({\stackrel{\leftharpoonup }{C}}_j,{\stackrel{\rightharpoonup }{C}}_j)$ is defined as the Manhattan distance $d\left(({\stackrel{\leftharpoonup }{C}}_i,{\stackrel{\rightharpoonup }{C}}_i),({\stackrel{\leftharpoonup }{C}}_j,{\stackrel{\rightharpoonup }{C}}_j)\right)\text{=}$$d_1+d_2+\dots +d_{\text{i}}+\dots ++d_m$ between vectors $E^{({\stackrel{\leftharpoonup }{c}}_i.{\stackrel{\rightharpoonup }{c}}_{\text{i}})}\text{and}{E}^{({\stackrel{\leftharpoonup }{c}}_j.{\stackrel{\rightharpoonup }{c}}_j)},$ where di = $|E_{({\stackrel{\leftharpoonup }{y}}_i.{\stackrel{\rightharpoonup }{y}}_i)}^{({\stackrel{\leftharpoonup }{c}}_i.{\stackrel{\rightharpoonup }{c}}_i)}-E_{({\stackrel{\leftharpoonup }{y}}_i.{\stackrel{\rightharpoonup }{y}}_i)}^{({\stackrel{\leftharpoonup }{c}}_j.{\stackrel{\rightharpoonup }{c}}_j)}|,\text{and}{E}_{({\stackrel{\leftharpoonup }{y}}_i.{\stackrel{\rightharpoonup }{y}}_i)}^{({\stackrel{\leftharpoonup }{c}}_i.{\stackrel{\rightharpoonup }{c}}_i)}{E}_{({\stackrel{\leftharpoonup }{y}}_i.{\stackrel{\rightharpoonup }{y}}_i)}^{c_i},E_{({\stackrel{\leftharpoonup }{y}}_i.{\stackrel{\rightharpoonup }{y}}_i)}^{({\stackrel{\leftharpoonup }{c}}_j.{\stackrel{\rightharpoonup }{c}}_j)}{E}_{({\stackrel{\leftharpoonup }{y}}_i.{\stackrel{\rightharpoonup }{y}}_i)}^{c_j},$, are the i th components of the dynamic fuzzy parameters of concepts $({\stackrel{\leftharpoonup }{C}}_i,{\stackrel{\rightharpoonup }{C}}_i)=\left((O_i,O_i),({\stackrel{\leftharpoonup }{A}}_i,{\stackrel{\rightharpoonup }{A}}_i)\right)\text{and}\text{(}{\stackrel{\leftharpoonup }{C}}_j,{\stackrel{\rightharpoonup }{C}}_j)=$$\left(({\stackrel{\leftharpoonup }{O}}_j,{\stackrel{\rightharpoonup }{O}}_j),({\stackrel{\leftharpoonup }{A}}_j,{\stackrel{\rightharpoonup }{A}}_j)\right),$ respectively.

λs is the similarity threshold for clustering. It can be set manually or calculated by other methods. If the distance between the two concepts is less than λs, the two concepts are placed into the same concept category. If the distance between the two concepts is greater than λs, they belong to different concepts.

4.4.4.1Representation of dynamic fuzzy rules

Dynamic fuzzy association rules are the implication relations among dynamic rules: $(\stackrel{\leftharpoonup }{M},\stackrel{\rightharpoonup }{M})\Rightarrow (\stackrel{\leftharpoonup }{N},\stackrel{\rightharpoonup }{N}),\text{where}(\stackrel{\leftharpoonup }{M},\stackrel{\rightharpoonup }{M})$ is the premise of the dynamic association rules and $(\stackrel{\leftharpoonup }{N},\stackrel{\rightharpoonup }{N})$ is the conclusion. The meaning of the rule is that the DF membership of each attribute in $(\stackrel{\leftharpoonup }{M},\stackrel{\rightharpoonup }{M})$ is greater than the corresponding set of thresholds. The DF membership of each attribute in $(\stackrel{\leftharpoonup }{N},\stackrel{\rightharpoonup }{N})$ is also greater than the corresponding threshold. DF classification rules are special association rules. When the conclusion of a rule is a decision attribute (i.e. label attribute of dynamic fuzzy concept), a DF classification rule is obtained.

To extract concept rules effectively, the DF minimum support and DF minimum confidence must be specified by the user. The main problem of dynamic fuzzy concept rule extraction is that, given an object database, finding all conception rules may require a support degree that is not lower than the user specified DF minimum support and DF minimum credibility.

Definition 4.29 Given the dynamic fuzzy concepts $({\stackrel{\leftharpoonup }{c}}_i,{\stackrel{\leftharpoonup }{c}}_i)\text{and}({\stackrel{\leftharpoonup }{c}}_j,{\stackrel{\leftharpoonup }{c}}_j),$ and the dynamic fuzzy association rule $(\stackrel{\leftharpoonup }{M},\stackrel{\rightharpoonup }{M})\Rightarrow (\stackrel{\leftharpoonup }{N},\stackrel{\rightharpoonup }{N}),\text{where}(\stackrel{\leftharpoonup }{M},\stackrel{\rightharpoonup }{M})=Intent(({\stackrel{\leftharpoonup }{c}}_i,{\stackrel{\rightharpoonup }{c}}_i)),$$(\stackrel{\leftharpoonup }{N},\stackrel{\rightharpoonup }{N})=Intent(({\stackrel{\leftharpoonup }{c}}_j,{\stackrel{\rightharpoonup }{c}}_j))Intent(({\stackrel{\leftharpoonup }{c}}_i,{\stackrel{\rightharpoonup }{c}}_i)),$ the dynamic fuzzy association rules support and confidence are

$DFSUP\left(\left(\overleftarrow{M},\stackrel{\rightharpoonup }{M}\right)\Rightarrow \left(\overleftarrow{N},\stackrel{\rightharpoonup }{N}\right)\right)=\frac{\left|Extent\left(\left({\overleftarrow{c}}_j,{\stackrel{\rightharpoonup }{c}}_j\right)\right)\right|}{\left|\left(\overleftarrow{0},\stackrel{\rightharpoonup }{0}\right)\right|}\left(4.24\right)$

and

$DFCONF\left(\overleftarrow{M},\stackrel{\rightharpoonup }{M}\right)\Rightarrow \left(\overleftarrow{N},\stackrel{\rightharpoonup }{N}\right))=\frac{\left|Extent\left(\left({\overleftarrow{c}}_j,{\stackrel{\rightharpoonup }{c}}_j\right)\right)\right|}{\left|Extent\left(\left({\overleftarrow{c}}_j,{\stackrel{\rightharpoonup }{c}}_j\right)\right)\right|}.\left(4.25\right)$

The corresponding thresholds are dfsup and dfconf, respectively.

4.4.4.2Dynamic fuzzy rule extraction algorithm

To avoid generating redundant dynamic fuzzy association rules, two parameters are defined to control the generation of association rules. According to the definitions of mean and variance in statistics, the mean defines the fuzzy degree of the sample and the variance defines the degree of dispersion of the sample. These are introduced into dynamic fuzzy concept learning as follows.

Definition 4.30 Define two DF parameters: the dynamic fuzzy mean $(\overleftarrow{\alpha },\overrightarrow{\alpha })$$(({\overleftarrow{X}}_i,{\overrightarrow{X}}_i),({\overleftarrow{Y}}_i,{\overrightarrow{Y}}_i))$ and the dynamic fuzzy variance $(\stackrel{\leftharpoonup }{\delta },\stackrel{\rightharpoonup }{\delta })(({\stackrel{\leftharpoonup }{X}}_i,{\stackrel{\rightharpoonup }{X}}_i),({\overleftarrow{Y}}_i,{\overrightarrow{Y}}_i)):$

${\left(\overleftarrow{\alpha },\stackrel{\rightharpoonup }{\alpha }\right)}_{\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_i\right)}=\frac{1}{\left|\left({\overleftarrow{X}}_i,{\stackrel{\rightharpoonup }{X}}_i\right)\right|}{\displaystyle \sum _{\left({\overleftarrow{x}}_i,{\stackrel{\rightharpoonup }{x}}_i\right)\in \left({\overleftarrow{X}}_i,{\stackrel{\rightharpoonup }{X}}_l\right)}I\left(\left({\overleftarrow{x}}_l,{\stackrel{\rightharpoonup }{x}}_i\right),\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_i\right)\right),\left(4.26\right)}$

$\left(\overleftarrow{\alpha },\stackrel{\rightharpoonup }{\alpha }\right)=\frac{1}{\left|\left({\overleftarrow{Y}}_i,{\stackrel{\rightharpoonup }{Y}}_i\right)\right|}{\displaystyle \sum _{\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_l\right)\in \left({\overleftarrow{Y}}_t,{\stackrel{\rightharpoonup }{Y}}_l\right)}{\left(\overline{\alpha },\overline{\alpha }\right)}_{\left({\overleftarrow{y}}_t,{\stackrel{\rightharpoonup }{y}}_l\right)}},\left(4.27\right)$

$\begin{array}{l}{\left(\overleftarrow{\delta },\stackrel{\rightharpoonup }{\delta }\right)}_{\left({\overleftarrow{y}}_t,{\overleftarrow{\stackrel{\rightharpoonup }{y}}}_t\right)}=\left(\overleftarrow{\delta },\stackrel{\rightharpoonup }{\delta }\right)\left(\left\{I\left(\left({\overleftarrow{x}}_i,{\stackrel{\rightharpoonup }{x}}_i\right),\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_i\right)\right)|\left({\overleftarrow{x}}_i,{\stackrel{\rightharpoonup }{x}}_i\right)\in \left({\overleftarrow{X}}_i,{\stackrel{\rightharpoonup }{X}}_t\right)\right\}\right)\\ =\sqrt{\frac{{\displaystyle \sum _{\left({\overleftarrow{x}}_t,{\stackrel{\rightharpoonup }{x}}_t\right)\in \left({\overleftarrow{X}}_i,{\stackrel{\rightharpoonup }{X}}_t\right)}\left(I\left(\left({\overleftarrow{x}}_i,{\stackrel{\rightharpoonup }{x}}_i\right),\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_i\right)\right)-\left(\overleftarrow{\alpha },\stackrel{\rightharpoonup }{\alpha }\right)\right)}}{\left|\left({\overleftarrow{X}}_i,{\stackrel{\rightharpoonup }{X}}_i\right)\right|}},\left(4.28\right)\end{array}$

and

$\left(\overleftarrow{\delta },\stackrel{\rightharpoonup }{\delta }\right)\left(\left({\overleftarrow{x}}_i,{\stackrel{\rightharpoonup }{x}}_i\right),\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_i\right)\right)=\frac{1}{\left|\left({\overleftarrow{Y}}_i,{\stackrel{\rightharpoonup }{Y}}_i\right)\right|}{\displaystyle \sum _{\left({\overleftarrow{y}}_i,{\stackrel{\rightharpoonup }{y}}_l\right)}{\left(\overleftarrow{\delta },\stackrel{\rightharpoonup }{\delta }\right)}_{\left({\overleftarrow{y}}_t,{\stackrel{\rightharpoonup }{y}}_t\right)},\left(4.29\right)}$

where $|({\stackrel{\leftharpoonup }{X}}_i,{\stackrel{\rightharpoonup }{X}}_i)|$ is the length of element $({\stackrel{\leftharpoonup }{X}}_i,{\stackrel{\rightharpoonup }{X}}_i)\text{and}\text{|}({\stackrel{\leftharpoonup }{Y}}_i,{\stackrel{\rightharpoonup }{Y}}_{\text{i}})|$ is the length of element $({\stackrel{\leftharpoonup }{Y}}_i,{\stackrel{\rightharpoonup }{Y}}_i).(\stackrel{\leftharpoonup }{\alpha },\stackrel{\rightharpoonup }{\alpha })$ denotes the dynamic average fuzzy degree of concept $({\stackrel{\leftharpoonup }{c}}_i,{\stackrel{\rightharpoonup }{c}}_{\text{i}}),\text{and}\text{(}\overline{\delta }\text{,}\overline{\delta })$ illustrates the discrete degree of dynamic fuzzy membership of concept $({\stackrel{\leftharpoonup }{c}}_i,{\stackrel{\rightharpoonup }{c}}_i)$

Definition 4.31 $\forall ({\stackrel{\leftharpoonup }{c}}_i,{\stackrel{\rightharpoonup }{c}}_i)=(({\stackrel{\leftharpoonup }{X}}_i,{\stackrel{\rightharpoonup }{X}}_i),({\stackrel{\leftharpoonup }{Y}}_i,{\stackrel{\rightharpoonup }{Y}}_i)),$, if the parameter $(\stackrel{\leftharpoonup }{\alpha },\stackrel{\rightharpoonup }{\alpha })(({\stackrel{\leftharpoonup }{X}}_i,{\stackrel{\rightharpoonup }{X}}_i),$$({\stackrel{\leftharpoonup }{Y}}_i,{\stackrel{\rightharpoonup }{Y}}_{\text{i}}))$ is greater than or equal to the given threshold $(\stackrel{\leftharpoonup }{\theta },\stackrel{\rightharpoonup }{\theta })$ and the parameter $(\stackrel{\leftharpoonup }{\delta },\stackrel{\rightharpoonup }{\delta })$ is smaller than the threshold $(\stackrel{\leftharpoonup }{\psi },\stackrel{\rightharpoonup }{\psi }),$ then this is a candidate node.