A Novel Selective Ensemble Algorithm for Imbalanced Data Classification Based on Exploratory Undersampling

Algorithm 2

Rotation forest algorithm.

(i)   Input: 𝑋=𝑥_𝑖^𝑁_𝑖=1: the objects in the training data set (an 𝑁×𝑛 matrix).

    𝑌=𝑦_𝑖^𝑁_𝑖=1: the class labels of the training set (an 𝑁×1 matrix).

    𝑇: number of classifiers in the ensemble.

    𝐾: number of feature subsets.

    Φ={−1,+1}: the set of class labels.

(ii)  Training Phase:

(iii) For  𝑖=1  to  𝑇  do

   (1) Calculate the rotation matrix 𝑅^𝑎_𝑖:

    (a) Randomly split the feature set 𝐹 into 𝐾 subsets𝐹_𝑖,𝑗,𝑗=1,2,…,𝐾.

    (b) For 𝑗=1 to 𝐾 do

      Let 𝑋_𝑖,𝑗 be the data set 𝑋 for the features in 𝐹_𝑖,𝑗.

      Select a bootstrap sample 𝑋^†_𝑖,𝑗 of 75% number of objects in 𝑋_𝑖,𝑗.

      Apply PCA on 𝑋^†_𝑖,𝑗 and store the component coefficients in a matrix 𝐶_𝑖,𝑗.

    (c) Endfor

    (d) Arrange the 𝐶_𝑖,𝑗(𝑗=1,2,…,𝐾) into a block diagonal matrix 𝑅_𝑖.

    (e) Construct 𝑅^𝑎_𝑖 by rearranging columns of 𝑅_𝑖 to match the order of features in 𝐹.

  (2) Build the classifier ℎ_𝑖 using (𝑋𝑅^𝑎_𝑖,𝑌) as the training set.

(iv) Endfor

(v)  Output: For a given 𝑥, calculate its class label assigned by the ensemble classifier ℎ^∗:

       ℎ^∗(𝑥)=argmax^𝑇_𝑦∈Φ∑_𝑖=1𝐼ℎ_𝑖𝑥𝑅^𝑎_𝑖=𝑦,

  where 𝐼(⋅) is an indicator function.