between corresponding feature values for all feature vectors is created. The distance δ between two corresponding feature values is defined as follows.

$$\delta(V_{1},V_{2})=\sum_{i=1}^{n}|\frac{C_{1i}}{C_{1}}-\frac{C_{2i}}{C_{2}}|^{k}$$
$$\left(1\right)$$

In the above equation, V1 and V2 are the two corresponding feature values. C1 **is the total**
number of occurrences of feature value V1, and C1i**is the number of occurrences of feature**
value V1 for class i. A similar convention can also be applied to C2i and C2. k **is a constant,**
usually set to 1. This equation is used to compute the matrix of value differences for each nominal feature in the given set of feature vectors. Equation 1 gives a geometric distance on a fixed, finite set of values (Cost & Salzberg, 1993). Cost and Salzberg's modified VDM
omits the weight term w a f included in the δ **computation by Stanfill and Waltz, which has**
an effect of making δ **symmetric. The distance ∆ between two feature vectors is given by:**

$$\Delta(X,Y)=w_{x}w_{y}\sum_{i=1}^{N}\delta(x_{i},y_{i})^{r}$$
$$\left(2\right)$$
r(2)
r = 1 yields the Manhattan distance, and r **= 2 yields the Euclidean distance (Cost &**
Salzberg, 1993). wx and wy are the exemplar weights in the modified VDM. wy **= 1 for a**
new example (feature vector), and wx **is the bias towards more reliable examples (feature**
vectors) and is computed as the ratio of the number of uses of a **feature vector to the number**
of correct uses of the feature vector; thus, more accurate feature vectors will have wx ≈
1. For SMOTE-N we can ignore these weights in equation 2, as SMOTE-N is not used for classification purposes directly. However, we can redefine these weights to give more weight to the minority class feature vectors falling closer to the majority class feature vectors; thus, making those minority class features appear further away from the feature vector under consideration. Since, we are more interested in forming broader but accurate regions of the minority class, the weights might be used to avoid populating along neighbors which fall closer to the majority class. To generate new minority class **feature vectors, we can create** new set feature values by taking the majority vote of the feature vector in consideration and its k **nearest neighbors. Table 6.2 shows an example of creating a synthetic feature vector.**
Let F1 = A B C D E be the feature vector under consideration and let its 2 nearest neighbors be F2 = A F C G N
F3 = H B C D N
The application of SMOTE-N would create the following feature vector:
FS = A B C D N
Table 7: Example of SMOTE-N