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Biplot or dual plot is an exploratory graph to present - as points or vectors - both the observations (sample) and the variables of the data. The axes are typically latent principal dimensions. Biplot is often used to depict principal component analysis, correspondence analysis, and other multivariate methods.

Biplot is an exploratory graph to present both the data points (sample) and the variables. There are several variations, and the mostly used version is the biplot for principal components.

The original matrix $X$ (dimension $n \times p$, $n$ observations, $p$ variables) is transformed to $Y$ by centering and/or standardizing the columns. Using the singular value decomposition (SVD), we can write $$Y = UDV^T=\sum_{k=1,...p}d_k\mathbf u_k\mathbf v_k^T,$$ where the $\mathbf u_k$ are $n$-dimensional column vectors, the $\mathbf v_k$ 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 ($d_1^\alpha u_{1i}, d_2^\alpha u_{2i}$), for $i = 1,...,n$. The second plot is formed from the points ($d_1^{1-\alpha}v_{1j}, d_2^{1-\alpha}v_{2j}$), for $j = 1,...,p$. The $\alpha$ can be set as 0, 0.5, or 1. [Wikipedia]

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