## 2 Materials and Methods
### Random Projection Ensemble Classifier  
Machine learning has been extensively applied in many fields. In mathematical statistics, RP is a method for reducing the dimensionality of a series of points that lie in Euclidean space. Compared with other methods, the RP method is simpler and has less output error. It has been successfully used in the reconstruction of dispersion signals, facial recognition, and textual and visual information retrieval. Now we introduce the RP algorithm in detail.  
Let \(\Gamma=\left\{x_{i}\right\}_{i=1}^{N}\) be a series of column vectors in primitive data space with high dimension. \(x_{i}\in R^{n}\), \(n\) is the high dimension, and \(N\) denotes the number of columns. Dimensionality reduction embeds the vectors into a space \(R^{q}\), which has lower dimension than \(R^{n}\), where \(q<<n\). The output results are column vectors in the space with lower dimension.  
\[\widetilde{\Gamma}=\left\{\widetilde{x}_{i}\right\}_{i=1}^{N}\widetilde{x}_{ i}\in R^{q} \tag{9}\]  
where \(q\) is close to the intrinsic dimensionality of \(\Gamma\)[46; 47]. The embedded vectors refer to the vectors in the set \(\Gamma\).  
Here, if we want to employ the RP model to reduce the dimensionality of \(\Gamma\), a random vector set \(\gamma=\left\{r_{i}\right\}_{i=1}^{n}\) must be structured first, where \(r_{i}\in R^{q}\). There are two choices in structuring the random basis:  
1. The vectors \(\left\{r_{i}\right\}_{i=1}^{n}\) are spread on the \(q\)-dimensional unit spherical surface.
2. The components of \(\left\{r_{i}\right\}_{i=1}^{n}\) conform to Bernoulli +1/+2 distribution and the vectors are standardized such that \(\left\|r_{i}\right\|_{l_{2}}=1\) for \(i=1,\cdots,n\).  
We generated a \(q\times n\) matrix **R**, where \(q\) consists of the vectors in \(\gamma\). \(n\) was mentioned in the previous paragraph. The projecting result \(\widetilde{x_{i}}\) can be obtained by  
\[\widetilde{x_{i}}=R\cdot x_{i}. \tag{10}\]  
In our method, RP is used to construct a training set on which the classifiers will be trained. The use of RP lays the foundation for our ensemble model.  
Now, we illustrate the theory of our ensemble method. A training set \(\Gamma\) is given in Equation (9). We generate a \(G\) whose degree is \(n\times N\), and \(G\) comes from the column vectors in \(\Gamma\).  
\[G=\left(x_{1}|x_{2}\cdots|x_{N}\ \right). \tag{11}\]  
We then structure \(\left\{R_{i}\right\}_{i=1}^{k}\), whose size is \(q\times n\), where \(q\) and \(n\) are introduced in the preceding paragraph, \(k\) means the size of ensemble classifiers. The columns are standardized so as to the \(l_{2}\) norm is 1.  
In the ensemble classifiers, we constructed the training sets \(\{T_{i}\}_{i=1}^{k}\) by means of projecting \(G\) onto \(\{R_{i}\}_{i=1}^{k}\).  
\[T_{i}=R_{i}\cdot G\ \ \ \ \ i=1,\cdots,k. \tag{12}\]  
They are then fed into the base classifier, and the outcomes are a group of classifiers \(\left\{\ell_{i}\right\}_{i=1}^{k}\). For the sake of classifying a fresh set \(u\) with classifier \(\ell_{i}\), firstly, \(u\) will be inlaid into the target space \(R^{q}\). It can be obtained by embedding \(u\) into \(R_{i}\).  
\[\widetilde{u}=R_{i}\cdot u \tag{13}\]  
where \(\widetilde{u}\) is the result of embedding. The classification of \(\widetilde{u}\) can be garnered by \(\ell_{i}\). In this ensemble algorithm, the RP ensemble classifier will use the classification result of all classifiers \(\left\{\ell_{i}\right\}_{i=1}^{k}\) of \(\widetilde{u}\) to decide the final result with a majority voting scheme.  
In this study, the 1000 coefficients were divided into 100 non-overlapping blocks. We chose the projection from a block of size 10 that obtained the smallest test error with the leave-one-out test error estimate. We used the k-Nearest Neighbor (KNN) as a base classifier, where \(k\) = seq(1, 25, by = 3). Prior probability of interaction pairs in the calculated training samples dataset was considered as the voting parameter.