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them so that the variation along each axis is now unity: . Note that the matrix, Z, like X and Y previously, contains two column vectors, z1 = (z11,z21, … , zn1)T and z2 = (z12,z22, … , zn2)T, both of length n.

Let be a set of points that lie on a circle of radius r centered on the origin, and let Z = [z1z2] be a ma-12 trix containing these points (the number of points is arbitrary and can be adjusted upward or downward to obtain any desired level of precision for drawing the circle or corresponding ellipse). We find the minimum radius r such that 90 percent of the transformed data points lie inside the circle. Computing and X = (y1 + μ1, y2+ μ2), where y1 and y2 are the columns of Y, each point in the circle is mapped back onto the original basis. The points corresponding to rows of X form an ellipse that encloses 90 percent of the original data points.

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