Principles of Mathematics in Operations Research

Solutions 219

SSXX = ]P x^2 - 2mx^2 + mx^2 = y^x; 2 -2

mx ,

Pi —

-mSSxy __ - E x» E 2/i + TO ExiVi

which is dictated by the matrix equation above.

•v SS^ - x SSxy _ y E A ~ myx^2 - x ]T xtyi + myx^2

0o =

A> =

y E

x

l -

x

E ^2/i _ E ?/; E ^ - E ^ E ^2/*

A)

bbxx mbbxx

E ^i E 2/i - E Xi E a^iJ/i

"»!>?-Q><)

2

which is dictated by the matrix equation above.
We may use calculus to solve min SSE:

SSE = Hi/ - A/3||^2 = £(W - [A, + /3iXi])^2

SS£ = J2 Vf - 2 5Z Vifa - 2/3i X) ^* +

m/3

°

+ 2

^>^ I]

Xi

+ # 13

x

?-

dSSE

d(3o

-2 ^ i/j + 2m/3 0 + 2/3! ^ Xi = 0

^ „ E«• /3 2/i - /?i E ^t - „ -

0 = — —— = y - fax.

dSSE

dpi = -2 J2

x

iVi +

2

/3o J2

Xi

+

2

& Yl

x2

i = °

«• 53 aiij/i - (?/ - /3ix) ^ Xj - ft ^ x

2

= 0

a ExiVi -yYuxi Ea:»yi - mxv = ssxv

2-jXi ~~ x 2-jXi /_jXi ~ mx bbxx
As it can be observed above, the matrix system and the calculus mini-
mization yield the same solution!
Let the example data be (1,1), (2,4), (3,4), (4,4), (5,7). Then,

A/3 = y&

"1 r

12

13

14

15

r a 1

PO =

"1"

4

4

4

7

A^1 A = 5 15

15 55

1 3

3 11

, det{ATA) = 10.