Principal component analysis

Principal component analysis (PCA) is a <a href="/facts/Linear_map/Q1PBKNVV">linear</a> <a href="/facts/Dimensionality_reduction/XoRpcrB0">dimensionality reduction</a> technique with applications in <a href="/facts/Exploratory_data_analysis/8hvnf7JS">exploratory data analysis</a>, visualization and <a href="/facts/Data_Preprocessing/CS2W8m5B">data preprocessing</a>.
The data is <a href="/facts/Linear_map/Q1PBKNVV">linearly transformed</a> onto a new <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate system</a> 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 <a href="/facts/Real_coordinate_space/tEtvw03F">real coordinate space</a> are a sequence of 
 
 
 
 p
 
 
 {\displaystyle p}
 
 <a href="/facts/Unit_vector/GyaXneDn">unit vectors</a>, where the 
 
 
 
 i
 
 
 {\displaystyle i}
 
-th vector is the direction of a line that best fits the data while being <a href="/facts/Orthogonal/JVIjYPog">orthogonal</a> to the first 
 
 
 
 i
 −
 1
 
 
 {\displaystyle i-1}
 
 vectors. Here, a best-fitting line is defined as one that minimizes the average squared <a href="/facts/Perpendicular_distance/P0aEswE4">perpendicular</a> <a href="/facts/Distance_from_a_point_to_a_line/4mcAB59F">distance from the points to the line</a>. These directions (i.e., principal components) constitute an <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> in which different individual dimensions of the data are <a href="/facts/Linear_correlation/egkluAEm">linearly uncorrelated</a>. 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.
Principal component analysis has applications in many fields such as <a href="/facts/Population_genetics/v55qAnKu">population genetics</a>, <a href="/facts/Microbiome/lVjQnvpm">microbiome</a> studies, and <a href="/facts/Atmospheric_science/HKjTA3F4">atmospheric science</a>.

Principal component analysis open-in-new

Principal component analysis