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Understanding Projections onto Vectors in Multivariate Data Analysis

This section explores the concept of projecting data points onto vectors within multivariate data analysis. If ( X' ) represents an ( n times p ) data matrix and ( a ) is a ( p times 1 ) vector, the projection ( Y' = X'a ) yields coordinates of each observation along the vector ( a ). We illustrate this with an example where points such as ( x_1 = (1, 2)', x_2 = (2, 1)', x_3 = (-1, 1)' ) are projected onto the vector ( a = (3, -4)/5 ). The resulting coordinates describe their positions relative to the direction of the vector.

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Understanding Projections onto Vectors in Multivariate Data Analysis

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  1. Aside: projections onto vectors • If X is an np data matrix and a a p1 vector then Y= Xais the projection of X onto a • Values of Y give the coordinates of each observation along the vector a • Example: x1=(1, 2), x2=(2, 1), x3=(1,1) a=(3, 4)/5, • So Multivariate Data Analysis

  2. Aside: projections onto vectors x1 2  x3 x2   (0,0)  1 Multivariate Data Analysis

  3. Aside: projections onto vectors x1 2  x3 x2   (0,0)  1 a=(3,  4)/5 Multivariate Data Analysis

  4. Aside: projections onto vectors x1 2  x3 x2   y1 (0,0)  1 a=(3,  4)/5 Multivariate Data Analysis

  5. Aside: projections onto vectors x1=(1, 2) 2  x3 x2   y1 = (3142)/5 = 1 (0,0)  1 a=(3,  4)/5 Multivariate Data Analysis

  6. Aside: projections onto vectors x1 2  x3 x2   y1 = 1 (0,0)  1 y2 = 2/5 a=(3,  4)/5 Multivariate Data Analysis

  7. Aside: projections onto vectors x1 2  y3 = 7/5 x3 x2   y1 = 1 (0,0)  1 y2 = 2/5 a=(3,  4)/5 Multivariate Data Analysis

  8. Aside: projections onto vectors 2 y3 = 7/5   y1 = 1 (0,0)   1 y2 = 2/5 a=(3,  4)/5 Multivariate Data Analysis

  9. Aside: projections onto vectors 2      (0,0)   1 a=(3,  4)/5 Multivariate Data Analysis

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