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I know that when two features are highly correlated with each other, one of them should be removed from the dataset so they don't add twice the weight. However, what if all my features share a correlation?

For example, when I calculate the Pearson coefficients I get:

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Does this mean that only one feature is actually relevant when I start clustering the dataset?

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Essentially, yes. One feature predicts to a reasonably high degree what the other features look like, so the additional features have limited discriminatory power. Obviously there is some effect, as they don't correlate perfectly, but it's minimal.

I would look for some other features if possible.

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