回帰分析の理論とその応用に関する研究

1
− −
.
(
)
.
2
.
.
R
2
2.1
n
y, x j (j = 1, . . . , k), ε
y i = β 0 + β 1 x i1 + · · · + β k x ik + ε i
i = 1, . . . , n
. β 0 , β 1 , . . . , β k
. n × 1
Y n × (k + 1)
X (k + 1) × 1 β
n × 1 ε
([4], [5] ).
Y = Xβ + ε
2009SE291
:
p(x i ) y i x i p(x i )
p(x i ) =
exp(g(x i))
1 + exp(g(x i )) = 1
1 + exp(−g(x i ))
x ij p
−∞ < x ij < +∞, 0 < p < 1
([3], [6] )
3.2 ()
2 p p/(1 − p)
.
log p/(1 − p) = log p − log(1 − p)
.
([3], [6] )
3.3
.
x ([3] )
3.4
exp(β 0 + β 1 x 1 )
1 + exp(β 0 + β 1 x 1 ) = 1 2 =⇒ x 0.5 = − β 0
β 1
β n
2.2
ε i y i
.
1. E(ε i ) = 0, i = 1, 2, . . . , n ()
2. var(ε i ) = σ 2 , ()
3. cov(ε i , ε j ) = 0, i ≠ j ()
4ε i ∼ N(0, σ 2 I), i = 1, 2, . . . , n ()
1 3 1 4
([4], [5]
)
3
3.1
g(x i ) = β 0 + β 1 x i1 + β 2 x i2 + · · · +
β k x ik x i = (x i1 , . . . , x ik ) t
Y = (Y 1 Y 2 . . . Y n ) t y = (y 1 y 2 . . . y n ) t x i = (x i1
x i2 . . . x ik ) t Y
y
L(β) =
=
n∏
p(x i ) yi (1 − p(x i )) 1−yi
i=1
n∏
( exp(p(xi ))
) yi
(
1
1 + exp(p(x i )) 1 + exp(p(x i ))
i=1
) 1−yi
L(β) L(β)
Newton-Raphson
([6] ).
3.3
2
0

(
)
([2], [6] )
4
4.1 1
2012
108
([1] [7] ).
1
P P P
() 1.150 0.023 0.273 0.529 0.375 0.106
0.326 0.024 − − − −
1.583 0.036 1.708 0.035 − −
1.178 0.082 − − − −
− − 0.880 0.137 − −
− − 1.709 0.179 − −
− − 0.363 0.105 0.542 0.016
−0.450 0.003 −0.197 0.114 − −
−1.781 0.016 −1.136 0.037 −2.441 0.004
−0.740 0.005 − − −1.044 0.0004
− − − − 0.962 0.044
4.2 1
shouhai
0.4 0.6 0.0 0.2 0.8 1.0
Giants
Swallows
Tigers
Carp
Baystars
0 5 10 15
tokuten
1
shouhai
0.4 0.6 0.0 0.2 0.8 1.0
Giants
Swallows
Tigers
Carp
Baystars
0 5 10 15
sitten
2
5
3 .
1
.
. 2
.
1
.
4.3 2
3 ()4
6 ()7 ()
AIC ([1]
[7] )
4.4 2
5
[1] : 2010
2011
[2] : , 2012
[3] : , 2009
[4] Rencher A.C and Schaalje G.B:
Linear Models in StatisticsJohn Wiley & Sons
Inc2007
[5] : 1979
[6] :
−SAS −
1996
[7] nikkansports.com
http://www.nikkansports.com/