Page 372 - Fundamentals of Probability and Statistics for Engineers

Linear Models and Linear Regression 355

If we let
2 3 2 3 2 3
1 x 11 x 21 x m1 y 1 e 1
6 1 x 12 x 22 7 6 y 2 7 6 7
. . . . . 7; . 7; . 7;
6 x m2 7 6 7 6 e 2 7
. . . .
C ˆ 6 y ˆ 6 e ˆ 6
4 . . . . . . 5 4 . . 5 4 . . 5
1 x 1n x 2n x mn y n e n
and
2 3
0
1
6 7
. 7;
6 7
q ˆ 6
. . 5
4
m
Equation (11.47) can be represented by vector–matrix equation:
y ˆ Cq ‡ e: 11:48†
Comparing Equation (11.48) with Equation (11.12) in simple linear regression,
we see that the observed regression equations in both cases are identical except
that the C matrix is now an n (m ‡ 1) matrix and q is an (m ‡ 1)-dimensional
vector. Keeping this dimension difference in mind, the results obtained in the
case of simple linear regression based on Equation (11.12) again hold in the
multiple linear regression case. Thus, without further derivation, we have for
^
the solution of least-square estimates q of q [see Equation (11.15)]

^ T 1 T 11:49†
q ˆ C C† C y:

1
T
The existence of matrix inverse (C C) requires that there are at least (m ‡ 1)
distinct sets of values of (x 1i , x 2i ,..., x mi ) represented in the sample. It is noted
T
that C C is a (m ‡ 1) (m ‡ 1) symmetric matrix.

Example 11.6. Problem: the average monthly electric power consumption (Y )

at a certain manufacturing plant is considered to be linearly dependent on the
average ambient temperature (x 1 ) and the number of working days in a month
(x 2 ). Consider the one-year monthly data given in Table 11.4. Determine the
least-square estimates of the associated linear regression coefficients.

Table 11.4 Average monthly power consumption y (in thousands of kwh), with

number of working days in the month, x 2 , and average ambient temperature, x 1 , (in F)
for Example 11.6
x 1 20 26 41 55 60 67 75 79 70 55 45 33
x 2 23 21 24 25 24 26 25 25 24 25 25 23
y 210 206 260 244 271 285 270 265 234 241 258 230

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