A Novel Boundary Oversampling Algorithm Based on Neighborhood Rough Set Model: NRSBoundary-SMOTE

Algorithm 1

Neighborhood rough set boundary SMOTE algorithm for oversampling (NRSBoundary SMOTE).

Input: the training sample set: 𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡, the radius of neighborhood: w.

Output: new training sample set: 𝑁𝑒𝑤𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡.

Step  1: (Initialization)

 𝑆𝑎𝑚𝑝𝑙𝑒𝑆𝑒𝑡=𝜙; //𝑆𝑎𝑚𝑝𝑙𝑒𝑆𝑒𝑡 is the generated synthetic sample set.

 𝐵𝑜𝑢𝑛𝑑𝑆𝑒𝑡=𝜙; //𝐵𝑜𝑢𝑛𝑑𝑆𝑒𝑡 is the minority class sample set in boundary

       region which needs over-sampling.

 𝑃𝑜𝑠𝑆𝑒𝑡=𝜙;  //𝑃𝑜𝑠𝑆𝑒𝑡 is the majority class sample set in lower

       approximation of decision.

Step  2: (Compute the majority class sample set and minority class sample set)

 According to the decision values 1 to N, divide 𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡 into 𝑁 subsets: 𝑋₁,𝑋₂,…,𝑋_𝑁;

 Compute the minority class sample set 𝑋_min=min(|𝑋₁|,|𝑋₂|,…,|𝑋_𝑁|);

 Compute the majority class sample set 𝑋_max=𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡−𝑋_min;

Step  3: (Compute boundary region and lower approximation of decision)

 FOR each 𝑥_𝑖  in 𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡 DO

  According to formulas (2) and (3), compute the distance Δ₂(𝑥_𝑗,𝑥_𝑖) between

  𝑥_𝑖 and the other sample 𝑥_𝑗 in 𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡 (1≤𝑗≤|𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡|,𝑗≠𝑖);

  𝑑Max_𝑖=maxΔ₂𝑝_𝑗,𝑝_𝑖∣𝑗=1,2,…,|𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡|;

  𝑑Min_𝑖=minΔ₂𝑝_𝑗,𝑝_𝑖∣𝑗=1,2,…,|𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡|;

  According to formula (10), compute the threshold 𝛿_𝑖 of 𝑥_𝑖;

  Compute the neighborhood 𝛿𝑥_𝑖 of 𝑥_𝑖, 𝛿𝑥_𝑖={𝑥_𝑗∣𝑥_𝑗∈𝑈,Δ𝑥_𝑖,𝑥_𝑗≤𝛿_𝑖};

  IF (𝑥_𝑖∈𝑋_min)∧(𝛿𝑥_𝑖⊄𝑋_min) //minority class sample 𝑥_𝑖 which

                 belongs to boundary region.

   THEN 𝐵𝑜𝑢𝑛𝑑𝑆𝑒𝑡=𝐵𝑜𝑢𝑛𝑑𝑆𝑒𝑡∪𝑥_𝑖;

  ELSE IF (𝑥_𝑖∉𝑋_min)∧(𝛿𝑥_𝑖⊆𝑋_max)//majority class sample 𝑥_𝑖 which

                    belongs to lower approximation of decision.

   THEN 𝑃𝑜𝑠𝑆𝑒𝑡=𝑃𝑜𝑠𝑆𝑒𝑡∪{𝑥_𝑖};

  END IF

 END FOR

Step  4: (Generate synthetic samples from BoundSet)

 FOR each 𝑥_𝑖  in 𝐵𝑜𝑢𝑛𝑑𝑆𝑒𝑡 DO

  BOOL 𝐼𝑠𝐶𝑜𝑛𝑓𝑙𝑖𝑐𝑡=𝑓𝑎𝑙𝑠𝑒;

  Compute 𝑥_𝑖's 𝑘(𝑘=5) nearest neighborhoods with the same classification: 𝑦₁,𝑦₂,…,𝑦_𝑘;

  𝑇𝑒𝑚𝑝𝑂𝑏𝑗𝑒𝑐𝑡𝑆𝑒𝑡={𝑦₁,𝑦₂,…,𝑦_𝑘};

  WHILE 𝑇𝑒𝑚𝑝𝑂𝑏𝑗𝑒𝑐𝑡𝑆𝑒𝑡≠𝜙 DO

   Choose one sample denoted by 𝑦(𝑦∈𝑇𝑒𝑚𝑝𝑂𝑏𝑗𝑒𝑐𝑡𝑆𝑒𝑡) randomly;

   𝑇𝑒𝑚𝑝𝑂𝑏𝑗𝑒𝑐𝑡𝑆𝑒𝑡=𝑇𝑒𝑚𝑝𝑂𝑏𝑗𝑒𝑐𝑡𝑆𝑒𝑡−{𝑦};

   //Generate a synthetic sample.

   𝑥_new=𝑥_𝑖+rand(0,1)×𝑦−𝑥_𝑖;

   //Judge whether 𝑥_new affects the lower approximation of decision.

   FOR each 𝑥_𝑗  in 𝑃𝑜𝑠𝑆𝑒𝑡 DO

    IF 𝑥_new∈𝛿𝑥_𝑗 THEN

      𝐼𝑠𝐶𝑜𝑛𝑓𝑙𝑖𝑐𝑡=𝑡𝑟𝑢𝑒;

      BREAK;

    END IF

   END FOR

   //Add 𝑥_new to SampleSet

   IF 𝐼𝑠𝐶𝑜𝑛𝑓𝑙𝑖𝑐𝑡==𝑓𝑎𝑙𝑠𝑒 THEN

    𝑆𝑎𝑚𝑝𝑙𝑒𝑆𝑒𝑡=𝑆𝑎𝑚𝑝𝑙𝑒𝑆𝑒𝑡∪{𝑥_new};

   END IF

  END WHILE

 END FOR

Step  5: (Return)

 𝑁𝑒𝑤𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡=𝑆𝑎𝑚𝑝𝑙𝑒𝑆𝑒𝑡∪𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡;

 RETURN 𝑁𝑒𝑤𝑇𝑟𝑎𝑖𝑛𝑆𝑒𝑡.