Biplot

The biplot was introduced by K. Ruben Gabriel (1971).

Construction

A biplot is constructed by using the singular value decomposition (SVD) to obtain a low-rank approximation to a transformed version of the data matrix X, whose n rows are the samples (also called the cases, or objects), and whose p columns are the variables. The transformed data matrix Y is obtained from the original matrix X by centering and optionally standardizing the columns (the variables). Using the SVD, we can write Y = Σ_k=1,...pd_ku_kv_k^T;, where the u_k are n-dimensional column vectors, the v_k are p-dimensional column vectors, and the d_k are a non-increasing sequence of non-negative scalars. The biplot is formed from two scatterplots that share a common set of axes and have a between-set scalar product interpretation. The first scatterplot is formed from the points (d₁^αu_1i, d₂^αu_2i), for i = 1,...,n. The second plot is formed from the points (d₁^1−αv_1j, d₂^1−αv_2j), for j = 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.