Classifying sleep stages from only EEG
The standard approach would be to


Choose your feature vectors.
Partition your data set into training and testing.
Calculate the feature vectors for all your data (both training and test).
Use the training data set to train your classification algorithm.
Apply the trained classification algorithm to the test data set and calculate your correct classifications and incorrect classifications to get your precision / recall.


The trick will be step 1: choosing your feature vectors, given that you've only got Fourier coefficients (and values derived from them).