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Biplots: quantitative data

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Abstract Biplots provide visualizations of two things, usually, but not necessarily, in two dimensions. This paper deals exclusively with biplots for quantitative data X; qualitative data or data in the form of counts will be addressed in a subsequent paper. Data X may represent either (1) a matrix with n rows representing samples/cases and columns representing p quantitative variables or (2) a two‐way table whose rows and columns both represent classifying variables. Data sets of both types (1) and (2) are considered. Plotting symbols are usually points (typically for samples and distinguished by shape and/or color) and lines (typically for variables which may be calibrated or treated as arrowed vectors). Furthermore, variables may be nonlinear in both regularity of calibration and/or curvature. Interpretation is through distance, inner‐products, and sometimes area. Biplots may be improved by judicious shifts of axes, by scaling and by rotation. Nearly always, biplots give approximations to X and measures, incorporated in the biplot, expressing the degree of approximation are discussed. These aspects are illustrated with reference to examples from principal component analysis, nonlinear biplots, biplots for biadditive models, canonical variate analysis and the analysis of distance between grouped samples. WIREs Comput Stat 2015, 7:42–62. doi: 10.1002/wics.1338 This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Statistical Graphics and Visualization
PCA biplot with alpha bags for each species superimposed. Panel (a) shows 95% bags; panel (b) shows 50% bags. By default when the number of samples is less than 10 a convex hull is drawn.
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Correlation monoplot of the data in Table .
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An orthogonal parallel translation of the biplot axes shown in Figure (a).
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The geometry for classical biplots.
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Biplot representations of the Ocotea data in Table . In (a) a plot is shown with calibrated biplot axes while (b) shows the axes in vectorized form. The surrounding infobox in (a) gives the quality of the approximation of each axis and the color‐coded points indicate how closely each point is approximated in the two‐dimensions shown.
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Biadditive biplot of the interaction matrix associated with Table where 1/2J is used for plotting the treatments and 1/2J for the blends. The biplot shows the blends as biplot axes calibrated to include the main effect for blends as black dots.
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Biadditive biplot of the interaction matrix Z associated with Table where 1/2J is used for plotting the treatments and 1/2J for the blends. The biplot shows the blends as biplot axes calibrated in terms of interactions with all axes intersecting at zero. Note that each axis passes through one of the blends.
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Analysis of distance (AoD) biplot of the Ocotea data where we have used Clarke distance. Superimposed onto the biplot are 95% bags. Note the small deviations from linearity of the axes and the irregular calibrations on the axes.
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Canonical variate analysis (CVA) biplot of Ocotea data. Means are exactly represented because the three centroids must fit exactly in two dimensions; alpha bags show degree of separation/overlap. Axis predictivities all equal unity, indicating that values for group means are accurately read off for all variables.
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Nonlinear biplot of Table data. Clarke distance is used with circle projection biplot axes constructed.
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Regression biplot of a nonmetric MDS approximation of the Clarke distance matrix calculated for the Table data.
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PCA biplot overlayed with the two‐dimensional density estimates for Oken, Opor, and Obul respectively. Axes shifted to peripheral positions using orthogonal parallel translation.
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Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification
Statistical and Graphical Methods of Data Analysis > Statistical Graphics and Visualization
Statistical and Graphical Methods of Data Analysis > Multivariate Analysis

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