Principal Components Analysis
نویسنده
چکیده
Let X be an n× p matrix whose rows are iid random vectors Xi· with mean μ ∈ R and covariance Σ ∈ Sp+— for example, they might be (Xi·) iid ∼ No(μ,Σ). For many problems (such as multivariate regression of some Y on X) we might wish to reduce the dimension p of these rows. For example, if we have a vector of p = 1000 possible explanatory variables about each individual, we may hope that a small subset of these or perhaps a few different linear combinations of these will capture most of the information in X. For any vector α ∈ R the random vector z = Xα ∈ R will have iid entries zi whose means and variances are
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