Multiple Regression

Forrest Young's Notes

Copyright © 1997-9 by Forrest W. Young.

Linear Combinations: Example

  • Much of multivariate analysis is concerned with linear combinations of variables. Consider the following three observations on two variables:

     
    X1
    X2
    obs1
    2
    4
    obs2
    5
    2
    obs3
    1
    2

  • We can create a new variable Y that is a linear combination of the two X's by multiplying each of the original variables by a constant and adding the products.

  • Let's say we arbitrarily select mutiplicands of 3 and 2. Then we have:

    • Y = 3(X1) + 2(X2)

    And we have

    Y = [14 19 7]

    Geometry of Linear Combinations

    Here is a schematic of the geometry of the linear combination above. Notice that the result of the linear combination Y, is shown, both in the schematic and in the table below it.

     
     
    X1
    X2
    		 Y
    obs1
    2
    4
    		 14
    obs2
    5
    2
    		 19
    obs3
    1
    2
    		 7