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Projections of Multivariate Data onto Vectors

This article explores the concept of projecting multivariate data onto vectors using a data matrix ( X' ) and a vector ( a ). We define ( Y' = X'a ) and discuss how the values in ( Y' ) represent the coordinates of each observation along the vector ( a ). Through illustrative examples, we compute projections for given observations ( x_1, x_2, x_3 ) and demonstrate how to derive their projections onto the specified vector ( a ), enhancing understanding of data representation in multivariate analysis.

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Projections of Multivariate Data onto Vectors

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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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