: Y X
Y=a+bX+ε,
ε - , , . { Y i,X i, i=1,,n } ( n ≥ 2). , m , {cj,kj, j=1,,m} k = a×c+b +ε. a, b, , - ໖ ():
min ∑ (a+bX i Y i)2
a,b i=1,,n
( , , ,
min ∑ (kj (a×cj+b))2).
a,b j=1,,m
, :
∑ [2×(a+bX i Y i)× X i ] = 0, ∑ [2×(a+bX i Y i)] = 0
i=1,, n i=1,, n
(∑ [2×(a×cj+b kj)×cj ] = 0, ∑ [2×(a×cj+b kj)] = 0).
j=1,,m j=1,,m
,
b = (n ×∑ X i × Y i (∑ X i) × (∑ Y i)) / (n ×∑ X i 2-(∑ X i)2),
a = Y 0 b × X 0,
X 0=(1/ n) ×∑ X j, Y 0=(1/ n) ×∑ Y i.
, .
i= Xi-X 0, i= Y i-Y 0, , . , ( )
b = ∑ (X j X 0) × (Y j Y 0) / ∑ (X j X 0)2, a = Y 0 b × X 0
j=1,,n j=1,,n
( = ∑ (cj c0) × (k j k 0) / ∑ (cj c0)2, b = k0 × 0).
j=1,,m j=1,,m
X 1 Y 11 e 1
X= [ ], Y = [ ], s= [ ], e= [ ], Y^ = as + bX, e = Y - Y^
X n Y n1 e n
Y^-, s, X.
e s, X ( , - ):
s e=0, X e=0.
(nX2)
1 X 1 Y 1
X= [ ]; Y = [ ], β= [a,b] -
1 X n Y n
. :
X e=0
X (Y - X β)=0.
:
X Y - X X β=0.
(, -1 - )
X X)-1 X Y.
|
|
N ∑ X i ∑ Y i
β = (X X)-1 X Y = [ ] -1 [ ].
∑ X i ∑ X i2 ∑ X i Y i
( ) 1 (.[3, .41, . 2.9]).
(Y,X), 4- . Y,X , , , ( ).
/ | (Y) | (X) |
5+N1 | ||
32-N2 |
, N1,N2 - , (, , ).
( N1=N2=0).
Excel , .
/ , ( ) /腻
( )
/
, , , ( ):
Y = 79,95 1,63 X, R2=0,86.
(5,20) (0,23)
3- :
3.1. ? , ?
3.2. ?
3.3. ?
3.4. ?
3.5. - ?
3.6. ?
3.7. , ( )?