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I'm rather new to PCA and was hoping to have some confusion cleared up. Lets say for example we have a feature matrix that's nx100 and I want to get it down to something a bit smaller, p-dimensions, without losing too much variance.

After applying PCA and receiving and new feature matrix nxp, I would use x_reduced to predict some target variable y.

My question is, after the transformation, the new reduced feature matrix has been rotated by the eigenvectors and is sitting on a new basis. Yet, our y has not changed relative to X_reduced.

I'm unsure about how y_original and x_reduced can be used for training since y has not changed with respect to x_reduced.

Is there a way to correct for this or am I not thinking about it correctly?

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The short answer is that the y_original and x_reduced are still connected to each other, so it is safe to train your data using y_original and x_reduced. While x_reduced is on a different scale, as you mentioned via eigenvectors, it still is representative of the data that was attached to that observation, just in a different format. You lose a lot of interpretability as far as what the actual numbers mean which is why it may seem confusing, but it's just a transformed representation of the x_original that (hopefully) contains enough of the x_original variability to make it useful.

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    $\begingroup$ No Problem! On a side note if you want to get a test error, you should create the PCA rotations on the x_original_training, then use those same rotations on x_original_test. If you split up your data into train/test after you transform everything, then you're using data from your training set to influence your test set before the prediction occurs, which will make your test error overly-optimistic. $\endgroup$ – TBSRounder Feb 19 '16 at 15:48
  • $\begingroup$ You can also check out supervised methods, such as LDA (Linear Discriminant Analysis) $\endgroup$ – Omri374 Feb 22 '16 at 18:50

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