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PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.866, 0.5) direction and of 1 in the orthogonal direction. The vectors shown are the eigenvectors of the covariance matrix scaled by the square root of the corresponding eigenvalue, and shifted so their tails are at the mean.

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Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing.

The data is linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified.

The principal components of a collection of points in a real coordinate space are a sequence of $p$ unit vectors, where the $i$ -th vector is the direction of a line that best fits the data while being orthogonal to the first $i-1$ vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points.^[1]

Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science.

^ Jolliffe, Ian T.; Cadima, Jorge (2016-04-13). "Principal component analysis: a review and recent developments". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 374 (2065): 20150202. Bibcode:2016RSPTA.37450202J. doi:10.1098/rsta.2015.0202. PMC 4792409. PMID 26953178.