PCA: reduce high dimensional containing a high number of dimensions/features per observation

• transform data into a new coordinate system (a system uses one or more numbers to uniquely determine the position of the points)
• e.g. many studies use the first two principal components to plot data in two dimensionals

Principal components: a collection of points in real coordinatie space (real coordinate space of dimension is the set of te n-tuples of real numbers) are a sequence of p unit vectors

• i-th vector is the direction of a line that best fit the data while being orthogonal to the first i-1 vectors.
• best-fitting line: miniimize the average squared perpendicular distance from the points to the line

<aside> 💫 PCA is the process of computing the principal components and using them to perfrom a change of basis on the data, sometimes usimng only the first few principal components and ignore the rest.

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First principal components of a set of p variables: presumed to be jointly normally distributed, is the derived varaible formed as a linear combination of the orginal variables that explains the most variance.

Second principal components: explains the most variance is what’s left once the effect of first components is removed. (we may proceed thorugh p iterations until all variance is explained).

• pricinpal components: eigenvectors of the data’s covariance matrix,