## 2 Background Fundamentals
### Randomized Low-Rank Approximations
The reduced dataset from the randomized low-rank approximation is partitioned into training data of 7200(60%) and testing data of 4800(40%) and fed into the deep ensemble model. The overall learning architecture of the proposed ensemble is depicted in Figure 13. The architecture makes use of two levels of classifiers (\(1\)). Each training fold is applied into the random forest (RF) as a base learner (\(2\)). The output of the base learner in each fold is combined with the DEBi-LSTM using a meta classifier and produces the output. The proposed architecture is similar to a multi-layer perceptron, with Level 0 representing the input layer, Level 1 representing the hidden layer, and the output layer. The meta-classifier (DEBi-LSTM) uses the 'Relu' activation function by taking the input from the base learner and producing the output in the final layer. Let us discuss the design of the proposed double-edged bi-directional LSTM before the training process. #### 3.4.2 Double-Edged Bi-Directional LSTM

The first hidden layer of the bi-directional connection is configured with 128 LSTM memory units through the 'Relu' activation function, whereas the return sequence must be true to allow passage into the output from the prior layer to the next subsequent layer. The first layer is narrowed using the dropout layer as 0.2. The second hidden layer is stacked with 64 LSTM memory units by the 'Relu' activation function, and the dropout layer is configured as 0.1. Hereafter, two dense layers (32 and 3, respectively) are separated by a dropout layer equal to 0.2. A fully connected layer followed by this above architecture for obtaining the prediction output is called a dense layer. The design of the proposed DEBi-LSTM is represented in Figure 14.

Figure 13: Architecture of proposed ensemble model. #### 3.4.3 Proposed Training Process

The training phase took the input of both the approximated (AP) data from RLRA and the non-approximated reduced (NAP) data to be fed into Level 1 of the base learner. Given the dataset \(Input_{i}\)\({}^{0}\), the procedure selected random n samples of equal size for each dataset. Each sample was partitioned into training (60%) and testing (40%) data. \(Input_{i}\)\({}^{0}\) = \((Train_{i}\)\({}^{0},Test_{i}\)\({}^{0})\). The Level 0 classifier was generated using RF to each \(Train_{i}\)\({}^{0}\) with a \(\lambda\) learning rate. Hence, the first fold with the \(\lambda\) learning rate was called a \(\lambda\) team. A collection of \(\lambda\) teams was \(GRP_{i}\), \(1\leq i\leq n\), represented as \(GRP_{i}\) = \(RF_{i1}\), \(RF_{i2}\), \(.\)\(RF_{ik}\), where \(i=1\) to 5, \(k\) : \(\lambda\) team. Each \(Test_{i}\)\({}^{0}\) produced the prediction as \(F_{out}\)\({}^{0}\) of a sample \(n\) to be fed as input to the next level, Level 1. Once the Level 1 fed with the previous layer prediction was finished with the meta-classifier, double-edge bi-directional LSTM (DEBi-LSTM) trained the \(F_{out}\)\({}^{1}\) as \(Train_{i}\)\({}^{1}\). Subsequently, the \(Test_{i}\)\({}^{1}\) generated the data based on the meta-classifier to produce the final output, \(F_{out}\)\({}^{2}\). The overall training process is illustrated in Algorithm 3. Figure 8.0 Design of proposed DEBi-LSTM. ```
1:Parameter:
2:\(Input_{data}\ Ap_{i}{}^{0}:\) Input matrix of \([\hat{A}_{Ap}]_{p\times q}{}^{Ap}\) of approximated data. 3:\(Input_{data}\ NAP_{i}{}^{0}:\) Input matrix of \([\hat{A}_{Ap}]_{p\times q}{}^{NAp}\) of non-approximated data. 4:\([Train_{Ap}]_{i}{}^{0}:\) Training set for approximated data for Level 0. 5:\([Test_{Ap}]_{i}{}^{0}:\) Testing set for approximated data for Level 0. 6:\(C1:\) Base learner classifiers. 7:\(RF_{ij}:\) Random forest classifier for each fold of the training set. 8:\((RF_{i}:\) Team of the base learner classifier. 9:\([F_{out}]_{ij}{}^{1}]^{Ap}:\) Prediction of Level 0 base learner classifier. 10:\(Input_{data}\ Ap_{i}{}^{1}\): The output matrix produced from Level 0 to be input to the Level 1 model.