# MIPART: A Partial Decision Tree-Based Method for Multiple-Instance Classification
## 3 Materials and Methods  
### Multiple-Instance Classification  
Traditional supervised learning methods rely on training sets composed of feature vectors linked to specific class labels. In contrast, multiple-instance classification (MIC) uses training sets composed of bags of feature vectors, with each bag assigned a single class label. Unlike conventional single-instance classification, MIC provides no information on the individual feature vectors' class labels within each bag. A MIC algorithm is designed to learn from datasets structured into these labeled bags, each containing multiple instances [36]. Figure 1 illustrates the difference between single-instance and multiple-instance classification, where only the bags hold labels, with each potentially containing various instances. The number of instances within each bag may change autonomously, and a single instance can appear in multiple bags. To formalize the distinction illustrated in Figure 1, we can express single and multiple-instance classification using mathematical notation. In traditional classification, each individual data point is denoted as \(z\), representing a point in the feature space \(Z\), where \(Z=\left\{z_{1},z_{2},\,\ldots,\,z_{p}\,\right\}\) and \(z_{j}\in Z\). The training dataset \(T\) consists of instance-label pairs, expressed as \(T=\left\{(z_{1},\,c_{1}),(z_{2},c_{2}),\ldots,(z_{p},c_{p})\,\right\}\), where the set of class labels is \(C=\left\{c_{1},c_{2},\ldots,\,c_{k}\right\}\) for \(j=1,2,3,\ldots,\,k\). In contrast, multiple-instance classification \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Author(s)** & **Year** & **Method** & **Application of MIL** & **Evaluation Metric** \\ \hline \multirow{4}{*}{[PERSON] et al. [35]} & \multirow{4}{*}{2023} & MIL & & Accuracy \\  & & CLAM\_SB & & \\  & & CLAM\_MB & Medical image & precision \\  & & DSMIL & classification & F1-score \\  & & TransMIL & & AUC \\  & & TGMIL & & \\ \hline Bagci Das and & \multirow{2}{*}{2023} & SimpleMI & Human activity & \\  & Birant [28] & & MIWrapper & recognition & Accuracy \\ \hline \multirow{4}{*}{[PERSON] et al. [30]} & \multirow{4}{*}{2011} & PART (MISimple) & \\  & & AdaBoostM1 \& PART (MISimple) & \\  & & Bagging and PART (MIWrapper) & \\  & & AdaBoostM1 and PART (MIWrapper) & \\  & & PART (MIWrapper) & \\  & & & PART (MIWrapper) & \\  & & SMO & \\  & & NaiveBayes & performance & Specificity \\  & & MILR & \\  & & MIDD & \\  & & MIEMD & \\  & & MDD & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 1: Single- and multiple-instance classification. organizes the data into bags rather than individual instances. Each bag is represented as \(B\), which is a collection of \(m\) instances, such that \(B=\{z_{1},z_{2},\dots,\ z_{m}\}\) with \(z_{i}\in B\) for \(i=1,\ 2,\ 3,\dots,\ m\). Here, the training dataset \(S\) consists of labeled bags, denoted as \(S=\big{\{}\langle B_{j},c_{j}\rangle\big{\}}_{j=1}^{n}\), where \(c_{j}\) represents the class label assigned to the bag, \(B_{j}\). For example, in a binary classification scenario, the labels \(c_{j}\) can be expressed as either 0 or 1, such as \(c_{1}=0\) and \(c_{2}=1\). A variety of multi-instance learning algorithms have been proposed in the literature, including Citation-KNN [5], MIBoost [6], MIEMDD [7], MILR [8], MISVM [9], MITI [10], and SimpleMI [37], among others. In this research, we focused on applying these classifiers to 12 well-known multi-instance datasets. Among these, SimpleMI stands out due to its efficiency and straightforward approach, which summarizes multiple instances in a bag into a single representation using the arithmetic mean. The mean, \(M\), for each bag, \(B\), is calculated in Equation (1) as follows:  
\[M=\frac{1}{m}\sum_{i=1}^{m}z_{i} \tag{1}\]  
where \(z_{i}\in B\) represents the individual instances in the bag, and \(m\) is the total number of instances. This summarization allows SimpleMI to effectively retain critical information while facilitating the application of conventional classifiers.