This section remembers basic concepts that go beyond primarily FCA, FCM, and LSTM. This includes fuzzy formal contexts, concept clustering, and gating mechanisms.
Fuzzy formal contexts and fuzzy formal decision contexts
Definition 1
36 Domain was given \(u \),fuzzy set \({\ tilde {x}} \) Above \(u \) is defined as
$$\begin{aligned}{\tilde{x}}=\{\langle x,\mu_{{\tilde{x}}}(x)\rangle\midx\inu\},\end{aligned}$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
(1)
where \(\mu _{{\tilde {x}}}:u \rightArrow [0,1]\)and \(\mu _{{\tilde {x}}}(x)\) Represents the degree of membership of the object \(x \) In \({\ tilde {x}} \).
Definition 2
5 A fuzzy formal context can be defined as a triple \(u,a,{\tilde {i}})\)where \(u = \{x_1, x_2, \dots, x_n \} \) A set of objects, \(a = \{a_1, a_2, \dots, a_m \} \) It is a set of attributes \({\ tilde {i}} \) It is a fuzzy relationship between \(u \) and \(a \),representative \({\tilde {i}}(x,a)= \mu _{{\tilde {i}}}(x,a)\) For each \(x \ in u \) and \(a \ in \). here, \(\mu _{{\tilde {i}}}(x,a)\in [0, 1]\) Represents a membership degree and indicates the strength of relationships between objects \(x \) and attributes \(a \).
A fuzzy formal context is given \(u,a,{\tilde {i}})\)if \({\ tilde {i}} \) Take the value \(\{0, 1 \} \)it is equivalent to the formal context, \((u,a,i)\). In other words, fuzzy formal contexts generalize the concept of formal contexts. Let me \(2^u \) Represents the power set of \(u \)and \(f^a \) Shows all fuzzy sets \(a \). You can define a fuzzy concept using two fuzzy operators.
Definition 3
5 Let me \(u,a,{\tilde {i}})\) It will be a fuzzy formal context. for \(x \subseteq u \), \({\ tilde {b}} \in f^a \), \(x, {\tilde {b}})\) It is called the fuzzy concept of \(u,a,{\tilde {i}})\) In both cases \({\mathcal {l}}(x)= {\tilde {b}}\) and \({\mathcal {h}}({\tilde {b}} = x \) Owned. here, \({\mathcal {l}} \), \({\mathcal {h}} \) The fuzzy operator defined between \(2^u \) and \(f^a \),satisfaction
$$\begin{aligned}\begin{aligned}{\mathcal{l}}(x,a) &=\bigwedge_{x\in x}{\tilde{i}}(x,a),\quad\{h}}({\tilde{b}}) &=\{x\inu\mid\forall a\in{\tilde{b}}(a)\le {\tilde{i}}(x,a)\}. \end {aligned} \end {aligned} $$
(2)
For the fuzzy concept \(x, {\tilde {b}})\)if present \(x \ in u \) Such a thing \(x = {\mathcal {h}}({\mathcal {l}}(x)\)after that \(x, {\tilde {b}})\) It is called the granular concept of an object. On the other hand, if it exists \(a \ in \) Such a thing \(x = {\mathcal {h}}(a)\)after that \(x, {\tilde {b}})\) It is called the granular concept of attributes. More attention is paid to the granularity concept of an object, as the granularity concept can be derived from specific operators of the granularity concept of an object. The set of concepts formed by the granular concept of an object is called the granular concept space, presented as:
$$\begin{aligned}{\mathcal{g}}_{\mathcal{l}\mathcal{h}} =\{({\mathcal{h}}}({\mathcal{l}}}), {x\}.\end{aligned}$$
(3)
Proposal 1
37 Let me \({\mathcal {l}} \) and \({\mathcal {h}} \) It will be two fuzzy operators. for \(x_1, x_2 \subseteq x \), \(b\subseteq a\),there is:
$$\begin{aligned}\begin{array}{l}(1)\quad x_1\subseteq x_2\rightArrow{\mathcal{l}}(x_2)\subseteq{\mathcal{l}}}(x_1){\tilde{b}}_1\subseteq{\tilde{b}}_2\rightarrow{\mathcal{h}}({\tilde{b}}_2)\subseteq{\mathcal{h}}({\tidde}}\\(3)\quad(x_1\cup x_2)\supseteq{\mathcal{l}}(x_1)\ cap{\mathcal {l}}(x_2), \\(4)\quad{\mathcal {\mathcal {\tilde {b}}_2) = {\mathcal {h}}({\tilde {b}}_1)\cap{\mathcal {h}}({\tilde {b}}_2). \end {array} \end {aligned} $$
To apply conceptual knowledge to a particular task, it is common to associate one formal context with another. For example, in decision-making tasks where decisions are made based on conditional attributes, conditional data is treated as a fuzzy formal context and is associated with the formal context of decision information. In a real application, different types of context associations can form different formal decision contexts. For example, the association between fuzzy formal contexts and formal contexts can define formal decision contexts of fuzzy cres, whereas the association between formal contexts and fuzzy formal contexts can define a clear, fuzzy formal decision context. Given that the relationship between sample and decision attributes is usually a deterministic binary relationship in decision-making tasks, this paper focuses primarily on the formal context of fuzzy cull decisions. This is strictly defined as follows:
Definition 4
twenty four The formal decision context for the Fuzzy Crisk is Quintuple \((u,a,{\tilde {i}},d,j)\)where \(a \) Conditional attribute set \(d \) It's a set of decision-making attributes and is satisfied \(a \cap d = \emptyset \).
Gives the formal decision context for Fuzzy-Crisp \((u,a,{\tilde {i}},d,j)\)To distinguish between concept generation operators and decision attribute sets for conditional attribute sets, we show operators at \(u,a,{\tilde {i}})\) As \({\mathcal {l}}^c \) and \({\mathcal {h}}^c \)The granular concept space for that object is shown as follows: \({\mathcal {g}}_{\mathcal {l}\mathcal {h}}^c \). Similarly, the operator is on \((u, d, j)\) It is shown as \({\mathcal {l}}^d \) and \({\mathcal {h}}^d \)The granular concept space for that object is shown as follows: \({\mathcal {g}}_{\mathcal {l}\mathcal {h}}^d \).
Example 1
Table 1 shows the formal decision context for the phasic criss. \((u,a,{\tilde {i}},d,j)\)where \(a = \{b_1, b_2, b_3, b_4 \} \), \(d = \{d_1, d_2 \} \). Conditional granular concept space \({\mathcal {g}}_{\mathcal {l}\mathcal {h}}^c \) And the detailed concept space of decisions \({\mathcal {g}}_{\mathcal {l}\mathcal {h}}^d \) of \((u,a,{\tilde {i}},d,j)\) It is as follows:
$$\begin{aligned}\begin{aligned}{\mathcal{g}}_{\mathcal{l}\mathcal{h}}^{\textrm{c}}}}&=\left\{\{l}{l}\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{\{
Fuzzy C-means clustering
Fuzzy c-means is a widely used fuzzy clustering method aimed at splitting samples \(c \) Clusters with soft clustering. Unlike traditional hard clustering methods, in which each sample is strictly assigned to a cluster, FCM belongs to multiple clusters with different membership degrees with sample points. Therefore, the main step in FCM is to continually iterate through the membership matrix. Dataset given \({\textbf {x}} = \{x_1 = ({\textbf {x}}_1, y_1), x_2 = ({\textbf {x}}_2, y_2), \dots, x_i = ({\textbf {x}}_i, , x_n = ({\textbf {x}}_n, y_n)\} \)where \({\textbf {x}}}_i =(x_{i1}, x_{i2}, \dots, x_{ij}, \dots, x_{im})\) Sample feature vector \(x_i \) and \(y_i \) Corresponding labels. The main processes of FCM are explained as follows:
At the start of the algorithm, FCM initializes the membership matrix \({\textbf {u}} = [{\textbf{u}}_1, {\textbf{u}}_2, \dots , {\textbf{u}}_n]^t \)where \({\textbf {u}}_i =(u_{x_i}^1, u_{x_i}^2, \dots, u_{x_i}^c)\) Represents the degree of membership of the sample \(x_i \) In \({\textrm {c}} \) cluster. Cluster Center \({\textbf {v}} = [ {\textbf{v}}_1, {\textbf{v}}_2, \dots , {\textbf{v}}_k, \dots , {\textbf{v}}_{\textrm{c}}]^t \)where \({\textbf {v}}_k =(v_{k1}, v_{k2}, \dots, v_{kj}, \dots, v_{km})\) is initialized using a membership matrix with the following expression at the center of each cluster \({\textbf {v}} _k \):
$$\begin{aligned}{\textbf{v}}_k=\frac{\sum_{i=1}^n u_{x_i}^{k,p}{\textbf{x}}}_i}{\sum_{i=1}^n u_{x_i}^{k,p}},\end {aligned}$$
(4)
where \(p \) It is a fuzzification parameter, and the higher the value of \(p \)the greater the ambiguity of the classification.
To obtain reasonable clustering results, FCM repeatedly optimizes the membership matrix and clustering center location. Membership degree \(u_ {x_i}^{k, t+1} \) Sample \(x_i \) In \(k \)-th cluster in \({t+1} \) The iteration is updated using the formula.
$$\begin{aligned} u_{x_i}^{k, t+1} =\frac{1}{\left(\sum_{j=1}^{\textrm{c}}\frac{d_{ik}}}{d_{ij}}^{\frac{2}{p-1}}},\end{aligned} $$
(5)
where
$$\begin{aligned} d_{ik} = \left\|\mathbf{x_i} – \mathbf{v_k} \right\|_2 = \left(\sum_{j=1}^m(x_{ij} -v_{kj})^2 \right)^{1/2} \end {aligned} $$
(6)
Represents the Euclidean distance between \(x_i \) and Cluster Center \(\mathbf {v_k} \).
The iterative process is achieved by optimizing the following objective function:
$$\begin{aligned} j({\textbf{u}}, {\textbf{v}})=&\sum_{i=1}^n\sum_{k=1}^{\textrm{c}}}} u_{x_i}^{k} d_ &\text{st}\end{aligned} $$
$$ \ begin {aligned} u_ {x_i}^{k} \in [0, 1],\end {aligned} $$
(7)
$$ \begin {aligned} 0 <&\sum _ {i = 1}^{n} u_ {x_i}^{k}
By minimizing the objective function \(j({\textbf {u}}, {\textbf {v}})\),Algorithms iteratively adjust the clustering center and membership matrix, so that membership for each sample point is dynamically updated based on the distance to the cluster center, and the clustering process is completed.
Two-way long-term memory network
Long-term long-term memory (LSTM) networks are a type of recurrent neural networks (RNNs) designed to handle long-term dependencies of continuous data. An important component of LSTM is the LSTM cells. The LSTM cells include input gates, forgetting gates, and output gates that coordinate information flows and update network states. Input gate \(that\) Controls the flow of new information into cell states and determines how much the current input affects the cell. It is calculated as follows:
$$\begin{aligned} i_t =\sigma\left(w_i\left[ h_{t-1}, {\textbf{x}}_t \right] + b_i \right), \end {aligned} $$
(8)
where \(w_i \) and \(b_i \) Input gate weight matrix and bias terms, and \(\sigma\) An activation function.
Forgotten gate \(f_t \) Determine how much you should forget about your previous cellular state, allowing the model to retain relevant long-term information. It is calculated as follows:
$$\begin{aligned} f_t = \sigma\left(w_f\left[ h_{t-1}, {\textbf{x}}_t \right] + b_f \right). \end {aligned} $$
(9)
Output gate \(o_t \) Controls the hidden output of the LSTM cell and determines what information should be passed to the next step. It is calculated as follows:
$$\begin{aligned} o_t = \sigma\left(w_o\left[ h_{t-1}, {\textbf{x}}_t \right] + b_o \right). \end {aligned} $$
(10)
Furthermore, short-term memory \(h_t \) and long-term memory \(c_t \) It will be updated as follows:
$$\begin{aligned} h_t =&o_t\odot\tanh(c_t),\end{aligned}$$
(11)
$$\begin{aligned} c_t =&f_t\odot c_{t-1} + i_t\odot{\tilde{c}}_t,\end{aligned} $$
(12)
where \(\ odot \) Indicates element-by-element multiplication. \(\ tanh \) tangent activation function of hyperbola of candidate cell states; \({\ tilde {c}} _t \) This is the candidate value for the cell state.
bilstm extends standard LSTM by capturing both forward and backward time dependencies in sequence series data. This allows BILSTM to better understand sequences and improve its accuracy and ability to capture long-term dependencies by integrating context information from both directions.
