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Biplots: the joy of singular value decomposition

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Abstract The biplot is a generalization of a scatterplot for two variables to the case of many variables. Instead of having samples represented as points with respect to two perpendicular axes, as in a bivariate scatterplot, there are as many axes as variables pointing in different directions. Samples are then perpendicularly projected onto axes to obtain approximate values of the data. The word ‘approximate’ is important, because it is not possible to represent data on many variables exactly by this procedure, but the biplot arranges the axes to display the data as accurately as possible, usually by least‐squares fitting. The ‘bi’ in biplot refers to the rows and columns of a multivariate data matrix, where the rows are usually cases and the columns are variables. Biplots are almost always displayed in a two‐dimensional plot but can just as well be displayed in three‐dimensions, with more accurate data representation, using suitable graphical software, for example dynamic rotation or conditioned plots. The usual linear biplot, using least‐squares approximation, relies analytically on the singular value decomposition, which in turn can be thought of as a two‐sided regression problem. Biplot geometry underlies many classical multivariate procedures, such as principal component analysis, simple and multiple correspondence analysis, discriminant analysis, and other variants of dimension reduction methods such as log‐ratio analysis. WIREs Comput Stat 2012 doi: 10.1002/wics.1200 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Dimension Reduction Statistical Learning and Exploratory Methods of the Data Sciences > Exploratory Data Analysis Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Statistical Graphics and Visualization

Regression biplot of four economic variables CPI, UNE, PRC, and UN%, as linear functions of two other variables, INP and BOP. All variables are prestandardized. The 27 countries are shown in a scatterplot with respect to INP, BOP. The four response variables are depicted by their gradient vectors, that is, vectors with coefficients equal to the respective regression coefficients. The dashed lines perpendicular to the vector PRC indicate contours of its regression plane corresponding to estimated values of −2, −1, 0, 1, and 2 standard deviations of PRC, increasing from bottom right to top left in the direction of the gradient vector of PRC. Notice separate scales for countries and variables.

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Biplot of Table 1 using scaling Eq. (4). The percentage of the countries' variance explained is 63.3%.

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Correlation biplot of data in Table 1, with respect to the first two principal component axes. The percentage of the variables' variance explained is 63.3%.

[ Normal View | Magnified View ]

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Statistical and Graphical Methods of Data Analysis > Dimensional Reduction
Statistical and Graphical Methods of Data Analysis > Statistical Graphics and Visualization
Statistical and Graphical Methods of Data Analysis > Multivariate Analysis
Statistical Learning and Exploratory Methods of the Data Sciences > Exploratory Data Analysis

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