Principles of Mathematics in Operations Research

218 Solutions

3.2

V = A) + P\x + e =» E[y) =/3 0 + 0ix.

Data:

yi= Po + P\x\

2/2 = Po + P\x 1

Vm = Po + P\Xm

&

1 X\

1 x 2

_ J- Xm _

'/So'

=

"j/i"

2/2

_2/m_

«• A/3 = y.

The problem is to minimize SSE = ||y - Apf = EHifai - Po - Pixi)^2

The solution is to choose P = \Po'

Pi

such that Ap is as close as possible to y.

1 xi

1 xi

=* AJ A = m E

x»

I>i E^^2

(ATAy^1 =

, det(,4TA) = m ^ x^2 - (^ x*

Ex? -£*<

m£if - (E^)^2 L-E^i m

0 = (^)-M

T

j,

P =

(^1) E^f -E^ I I ••• I
*^1 *-2 ' ' •&m
2/1
2/2
0 = (ATA)-^1 ATy = m
Ex^2 -(Exi)
E2/i
Exi2/i
/?:
A) (A'M)- (^1) ^ =
TO
E
X
?- (E
x
)
2
E x 1 E 2/J - E xi E xi2/i
~ E a* E 2/« + "i E x,2/i
We know from statistics that
bbx
bbx
Pi = ^PL,Po = y-Pix,
where
Since
—^^1 , V = -, SSxy = ^2{xi-x)(yi-y), SSXX = ]P(x;-x)(x,-x).
bbxx = / \Xj — x) = y ^ xi — 2x y ^ Xi• mx -f ;^2