# Fuzzy MLKNN in Credit User Portrait +
## 3 Fuzzy MLKNN
### Fuzzy MLKNN  
The basic idea of the traditional MLKNN algorithm is to draw on the idea of the KNN algorithm to find K samples adjacent to the predicted sample, count the number of each label in the K samples, and then calculate the probability of the test sample containing each label through the maximum posterior probability [45]. The label whose predicted probability is greater than a certain threshold is the label of the predicted sample. The improved MLKNN algorithm using the intuitionistic fuzzy distance is mainly the process of finding K nearest neighbor samples in the traditional MLKNN algorithm. The measure of the distance between two sample points in high-dimensional space is improved from the traditional Euclidean measure to the fuzzy Euclidean measure to find more accurate nearest neighbor sample points with reference significance for label prediction. The specific mathematical form and symbolic expression of the algorithm follow, and the basic notation is shown in Table 2.  
Suppose the training set is \(X=\{x_{1},x_{2},x_{3}\ldots x_{n}\}\), indicating that there are \(n\) training samples. The feature data in each training sample is represented by \(\alpha=(\alpha_{1},\alpha_{2},\ldots\alpha_{l})\). There are \(t\) features in total, and each feature data is represented by an intuitionistic fuzzy number. The label set of the training sample is \(Y=\{y_{1},y_{2},y_{3}\ldots y_{n}\}\), It represents the label set corresponding to each sample. \(L=\{l_{1},l_{2},l_{3}\ldots l_{m}\}\) represents the label category vector, and \(m\) represents the number of label types. It is known that the training sample is \(x\in X\), and its corresponding label set is \(y_{x}\subseteq Y\), If \(y_{x}(l)=1\), it means that the label \(l\) is included in the label set of the sample \(x\), otherwise \(y_{x}(l)=0\), it means that the label set of sample \(x\) does not contain the label \(l\). Furthermore, let \(N(x)\) denote the set of K nearest neighbors of \(x\) identified in the training set, \(C_{x}(l)\) refers to the number of samples with label \(l\) in the K nearest neighbors of \(x\). \(H_{1}^{l}\) represents the event that sample \(x\) has the label \(l\), \(H_{0}^{l}\) represents the event that sample \(x\) has not the label \(l\). \(E_{j}^{l}\) represents the event that, among the K nearest neighbors of test sample \(x\), there are exactly \(j\) samples with label \(l\). At this time, the formula of MLKNN for multi-label classification obtained according to [PERSON] theorem is as follows:  
\[y_{x}(l)=\underset{b\in\{0,1\}}{argmax}\frac{P(H_{b}^{l})P(E_{q}^{l}|H_{b}^{ l})}{P(E_{q}^{l})}=\underset{b\in\{0,1\}}{argmax}P(H_{b}^{l})P(E_{q}^{l}|H_{b}^{ l}) \tag{2}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Variables** & **Description** \\ \hline \(X\) & Samples space \\ \(Y\) & Labels space \\ \(x_{i}\) & Arbitrary \(i\)-th sample \\ \(a_{i}\) & Feature vector of \(x_{i}\). The elements in \(a_{i}\) are composed of intuitionistic \\  & fuzzy numbers. \\ \(y_{i}\) & Label set of \(x_{i}\) \\ \(L\) & The label category vector \\ \(l\) & Arbitrary single category label \(l\in L\) \\ \(N(x)\) & The set of K nearest neighbors of \(x\) identified in the training set \\ \(C_{x}(l)\) & The number of sample with label \(l\) in neighbor set \(N(x)\) \\ \(H_{1}^{l}\) & the event that \(x\) has label \(l\) \\ \(H_{0}^{l}\) & The event that \(x\) has not label \(l\) \\ \(E_{j}^{l}\) & The event that, among the K nearest neighbors of \(x\), there are exactly \(j\) instances with label \(l\). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notation. Among them, \(b\) takes 0 or 1. If we want to get whether the label \(l\) belongs to sample \(x\), we only need to judge which value of \(b\) can maximize the value of the formula. If the formula value is the largest when \(b=1\), it is proved that \(y_{x}(l)=1\), that is, the sample \(x\) has the label \(l\). On the contrary, if the value of the formula is the largest when \(b=0\), it is proved that \(y_{x}(l)=0\), that is, the sample does not have the label \(l\).