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  "body": "\n## Summary\n\nPrincipal component analysis replaces correlated variables by new uncorrelated variables, the principal components, ordered by decreasing variance.\nThe components are the coordinates along the [[eigenvectors|0N6960W0BV]] of the [[covariance matrix|0V7G6WMFK0]].\nKeeping only the first few components gives the best low-dimensional linear summary of the data in the least squares sense.\nIt is the most common method of dimension reduction.\n\n## Explanations\n\n### L1 human\n\nPicture a cloud of data points in the plane shaped like a tilted cigar.\nThe first principal component is the direction along the cigar, where the points spread the most.\nThe second is the direction across it, at a right angle to the first.\nIf two test scores have variances $2$ and $2$ and covariance $1$, the first direction is $(1, 1)/\\sqrt{2}$, along which the variance is $3$.\nDescribing each point by its position along the cigar alone loses little information.\n",
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