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Andrews curves

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Andrews curves are examples of the space transformed visualization (STV) techniques for visualizing multivariate data, which represent k‐dimensional data points by a profile line (or curve) in two‐ or three‐dimensional space using orthogonal basis functions. Andrews curves are based on Fourier series where the coefficients are the observation's values. One advantage of the plot is based on the Parseval's identity (energy norm), which indicates that the information through transformation from the data space into the parameter space is preserved, and information that can be deduced in the hyperdimensional original space can be easily deduced in the two‐dimensional parameter space. This duality empowers the discovery of correlated records, clusters and outliers based on the curve's intersections, gaps and isolations, respectively. This article focuses on STV, in general, Andrews curves visualizations, in particular, and the effective use of these methods in the exploration of clusters, classes, and outliers. WIREs Comp Stat 2011 3 373–382 DOI: 10.1002/wics.160

Figure 1.

Andrews plot of four data points on a line in five dimensions. The curve's intersections correspond to the point–curve duality or highly correlated data. Note that a perfect curve intersection corresponds to a perfect correlation (±1) in the data.

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Figure 2.

Andrews plot for automobile data, using Fourier basis (Φλt = (1,2,3,4)) (top left), modified Fourier \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$({\boldsymbol{\Psi}}_{\lambda t},\lambda=(1,\sqrt{2},\sqrt{3},\sqrt{5}))$\end{document} (top right), Lagrange of degree 8 (bottom left) and PCW–Lagrange for consecutive pairs (bottom right). As we can see some bases can be better than others in revealing patterns and this is due to their mathematical properties and orientations in the hyperspace.

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Figure 3.

Three‐dimensional Andrews plot for automobile data from different perspectives. The three classes are brushed with distinct colors (red, green, and blue) and there is a projection that fully separates the classes and shows the twist of the curves through the changes in the angles.

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Figure 4.

Imaging the Andrews plot for the automobile data set. The classes have consistent pattern in the image of the original data (left), but the patterns are more consistent and easier to spot in the Andrews image (right). This is because we are not only viewing the data, but also viewing some of amenable views of their grand tour.

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Figure 5.

Enveloped Andrews plot for automobile data. Each class is represented by the CIE: \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$\bar{X}{\boldsymbol{\Phi}}_t\pm\alpha{\boldsymbol{\Phi}}^{\rm T}_t{\boldsymbol{\Sigma}}{\boldsymbol{\Phi}}_t,$\end{document} for α = (2,1,0.5,0). Note that the envelope sizes decrease as α decreases, and for α = 0, the envelope is a single profile representing the centroid of the class.

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Figure 6.

QGPCP approach applied to STV in order to reduce the clutter effects inherent in the plots for large data sets. The hidden pattern in this data, which is heavily cluttered in the top figures (top right and left), can be clearly seen in the bottom figures after applying the QGPCP technique (bottom left and right). QGPCP saturated the patterns with color intensity proportion to the profile's frequency in the plot. The pattern(s) reflects the compact housing located in certain area in California and can be distinguished by the latitude and longitude and with reasonable price range.

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