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Biplots: qualititative data

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A previous paper, Biplots: Quantitative data, dealt exclusively with biplots for quantitative data. This paper is mainly concerned with qualitative data or data in the form of counts. Qualitative data can be nominal or ordinal, and it is usually reported in a coded numerical form. In the analysis of qualitative data, many methods can be grouped as quantification methods (e.g., categorical principal component analysis, correspondence analysis, multiple correspondence analysis, homogeneity analysis): transforming qualities into quantitative values that may then be treated with quantitative methods. All the features of quantitative biplots are found in qualitative biplots, but calibrated interpolation axes become labeled category‐level points and calibrated prediction axes become prediction regions. Interpretation remains in terms of distance, inner products, and sometimes area. WIREs Comput Stat 2016, 8:82–111. doi: 10.1002/wics.1377 This article is categorized under: 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 > Analysis of High Dimensional Data
Biplot of Table data using a dissimilarity coefficient for binary variables. Left panel: Biplot based on the extended matching coefficient (EMC). Right panel: A zoomed version of the category‐level points (CLPs) enclosed in the dashed square shown in the left panel.
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Prediction regions for the category‐level points (CLPs) shown in Figure . For comparison, the projected CLPs are also shown and are seen not to generate nearest‐neighbor regions in the biplot diagrams.
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The multiple correspondence analysis (MCA) biplot of the data in Table using the principal components analysis (PCA) of . The row points (the gray filled circles) are plotted using the second and third columns of Z0 = UΣ and the column points (the filled squares) are plotted as the projected positions of the category‐level points (CLPs) i.e., the second and third columns of Z = p− 1L− 1/2V, ensuring that every sample point is at the centroid of its category quantifications.
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Figure with the axes translated to more convenient peripheral positions.
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Quantifications obtained in the Categorical principal components analysis (CatPCA) biplot of Figures (blue: unconstrained with instances of nonordinality) and (magenta: ordinal constraints, showing some instances of ties between consecutive points) and (yellow: smoothed monotonic spline fitted to ordered categories) All quantification methods give similar results, apart from the unordered quantifications of the variable VesL whose second level is not consistent with ordinality or even with increasing monotonicity; perhaps an insistence on constrained monotonicity is not advisable for this variable.
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Categorical principal components analysis (CatPCA) biplot of Table data. Similar to Figure but all variables are treated as ordinal categorical. Samples are plotted and color‐coded according to their group membership. The category‐level points (CLPs) associated with a specific categorical variable are connected with a gray line. Each gray line is drawn with a varying thickness that represents the ordinality of the category levels. The grouping of the samples is shown more clearly by shifting the lines away from the origin (shown as a cross) using orthogonal parallel translation. Category levels attained by each sample are predicted according to the nearness property.
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Biadditive biplot of the quantified matrix E.
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Categorical principal components analysis (CatPCA) biplot of Table data. All variables are treated as nominal categorical. Samples are plotted and color‐coded according to their group membership. The category levels associated with a specific categorical variable are connected with a color‐coded line. The color‐coding of each connecting line is used to predict the category level of any projected sample. To see the grouping of the samples more clearly, the lines have been shifted away from the origin (shown as a cross) using orthogonal parallel translation.
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The predictive biplots for the variable NumVes. Plotting symbols of samples are color‐coded according to their category levels (with group‐means in black) and with shapes according to their respective group memberships. (Reprinted with permission from Ref . Copyright 2014 Elsevier)
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Triplot. The five levels of filler are shown by the points P1, P2, P3, P4, and P5. Twelve axes give the jk levels of the other two factors. Each axis has a single positive calibration point denoting a triadditive interaction of 10 units of wear resistance. Projection circles aid the estimation of the interaction at the points where the kth circle cuts each of the axes. Interactions are positive or negative, depending on whether they occur on the same or opposite side of the origin as the jk label. (Reprinted with permission from Ref . Copyright 2014 Elsevier)
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CA Model: Pearson Residuals Case A: 1/2; 1/2
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The quantified samples underlying the Multiple correspondence analysis (MCA) monoplot in Figure plotted by using the first two columns of (I‐N)YV after discarding the trivial singular vector.
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Principal components analysis (PCA) biplot of Y = R− 1XC− 1/2 and Y = R− 1/2XC− 1 where X is represented by the two‐way contingency Table . The left panel shows the rows of Table plotted as points in a PCA of Y = R− 1XC− 1/2 while the right panel shows the columns of Table plotted as points in a PCA biplot of the transpose of Y = R− 1/2XC− 1.
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Principal components analysis (PCA) biplot of the matrix [G1z1, G2z2, …, Gpzp] by first finding the quantification vector z as in Figure and then fitting the quantifications obtained for each categorical variable by monotone regression. The ordered quantifications were then used to calculate the matrix [G1z1, G2z2, …, Gpzp] visualized as a PCA biplot.
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Principal components analysis (PCA) biplot of the matrix [G1z1, G2z2, …, Gpzp] constructed by plugging Eq. into [G1z1, G2z2, …, Gpzp].
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Multiple correspondence analysis (MCA) monoplot of the normalized Burt matrix L− 1/2GGL− 1/2 = ()(ΣV′). category‐level points (CLPs) plotted as first two columns of (after discarding the column associated with the singular value of unity).
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Statistical and Graphical Methods of Data Analysis > Multivariate Analysis
Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data

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