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I am reading the original paper by Chawla and others for SMOTE. I am trying to understand how to generate these synthetic examples for over-sampling the minority class. The paper says:

"Synthetic samples are generated in the following way: Take the difference between the feature vector (sample) under consideration and its nearest neighbor. Multiply this difference by a random number between 0 and 1, and add it to the feature vector under consideration. This causes the selection of a random point along the line segment between two specific features".

I understand the idea, take your sample, the nearest neighbor, pick a random point in between, what I don't understand is how these nearest neighbors are defined.

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You need to compute the Euclidean distance between each point and sort theses distances.

That said, you can find an implementation in the imbalanced-learn toolbox. More precisely, you can see the different steps of the implementation that you mentioned: (i) fit a KNN, (ii) find the NN of each sample, (iii) generate a new samples.

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The nearest neighbor will be the sample with the smaller Euclidean distance.

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