Biplots are a type of exploratory graph used instatistics, a generalization of the simple two-variablescatterplot.A biplot overlays ascore plot with aloading plot.A biplot allows information on bothsamples and variables of adata matrix to be displayed graphically. Samples are displayed as points while variables are displayed either as vectors, linearaxes or nonlinear trajectories. In the case of categorical variables,category level points may be used to represent the levels of a categorical variable. Ageneralised biplot displays information on both continuous and categorical variables.
The biplot was introduced byK. Ruben Gabriel (1971).[1]
A biplot is constructed by using thesingular value decomposition (SVD) to obtain alow-rank approximation to a transformed version of the data matrixX, whosen rows are the samples (also called the cases, or objects), and whosep columns are the variables. The transformed data matrixY is obtained from the original matrixX by centering and optionally standardizing the columns (the variables). Using the SVD, we can writeY = Σk=1,...pdkukvkT;, where theuk aren-dimensional column vectors, thevk arep-dimensional column vectors, and thedk are a non-increasing sequence of non-negativescalars. The biplot is formed from two scatterplots that share a common set of axes and have a between-setscalar product interpretation. The first scatterplot is formed from the points (d1αu1i, d2αu2i), fori = 1,...,n. The second plot is formed from the points (d11−αv1j, d21−αv2j), forj = 1,...,p. This is the biplot formed by the dominant two terms of the SVD, which can then be represented in a two-dimensional display. Typical choices of α are 1 (to give a distance interpretation to the row display) and 0 (to give a distance interpretation to the column display), and in some rare cases α=1/2 to obtain a symmetrically scaled biplot (which gives no distance interpretation to the rows or the columns, but only the scalar product interpretation). The set of points depicting the variables can be drawn as arrows from the origin to reinforce the idea that they represent biplot axes onto which the samples can be projected to approximate the original data.
