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My understanding of principal component analysis is that it is a feature extraction technique in which new features are created by linearly combining the original features. I am working with a dataset in which there are 20,000 genes (predictors) and about 137 tissue samples (observations) which can be 1 of 8 cell types.

It seems to me that PCA is not influenced by the classes of the observations, but only by the predictor values. Is there a way to determine if a PC is strongly associated in classifying a class (such as PC1, PC6 are highly representative of cell type 1, and PC2 and PC8 are highly representative of cell type 2)?

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Instead of going from Principal Component Analysis you can go for Linear Discriminant Analysis to reduce the dimensionality of data. Linear Discriminant Analysis or LDA as it is popularly known takes the classes into consideration and tries to give feature components(linear combination of original vectors) that maximizes the inter-class distance and reduces the intra-class distance. You can take the first few LDA components and remove the other components as irrelevant and noise. Then these LDA features(only the first few) can be used as features to any other classification algorithm such as logistic regression,SVM,etc

Note: LDA works best when the observations more or less follow Guassian distributioon within a class.

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