Ex:            Find a multiple regression (i.e., hyperplane fit) to the following data:

            (3, 4, 27)      (5, 5, 15)      (2, 6, 23)      (1, 3, 9)        (4, 2, 1)

Sol'n:       The hyperplane is defined as follows:

We write matrix terms for the data and hyperplane fit:

                

The matrix equation for the data and hyperplane fit are as follows:

A pseudoinverse of X yields the least-squares solution:

where

Not'n:      XT ≡ X transpose

 

Plugging the (x1, x2) coordinates into the hyperplane equation yields the following estimated y values:

or

The errors are found by subtracting y values from estimated y values:

Note:       The errors sum to zero, but this is not always the case.

In the plot, below, the estimates are shown as circles: