表紙／目次

2013 9 27

PC 25 ii

20 30 5 9 11 Lagrage x y y=ax+b 10 1 4 (2009) 21) (1)~(3) 9, 10, 11 2013 9 27 iii

1 1 2 Ca 2+ 2 (1)Ca 2+ (2) (3) 3 11 (1) (2) 4 14 (1) (2) (3) 5 21 (1) (2) 6 25 (1) (2) (3) (4) (5) (6) (7) 7 36 (1) (2) ( ) 8 39 (1) (2) (3) iv

9 48 (1) (2) a j (j=1,2,, ) (3) (4) (5) (6) 10 67 (1) 1-1. 3 1-2. 6 8 1-3. L(III) 1-4. Jørgese 1-5. Jørgese (2) Jørgese-Kawabe REE 2-1. REE artifact 2-2. REE 11 93 (1) x y=ax+b (2) y x y=ax+b 2-1. Lagrage 2-2. 2-3. Lagrage 2-4. Newto 2-5 a, b a, b 2-6 a, b (1) 66 (2) 78 (3)-1 X Y major axis 110 (3)-2 111 (3)-3 Wedt-2 112 v

1 113 2 Jacobi 116 3 122 4 135 138 140 vi

1 (1-1) PC 6 6 sigificat figures (1) Theory of experimetal errors 18 systematic errors Ca 2+ EDTA EDTA 5.0 +5.0 99.99 Z 2+ Z 2+ EDTA Z 2+ EDTA 1

radom errors (1-2) 0 (1-3) (Gaussia distributio) 0 2 18 19 18 19 2 Ca 2+ 2

Ca 2+ EDTA Ca 2+ EDTA EDTA Ca 2+ EDTA 1 EDTA Ca 2+ 1 EDTA (V EDTA ) EDTA (M EDTA ) (V sample ) Ca 2+ (M ca ) M ca (V EDTA ) (M EDTA ) (V sample ) 2-1 NN Ca 2+ EDTA EDTA Ca 2+ Ca 2+ Ca 2+ Ca 2+ EDTA Ca 2+ Ca 2+ EDTA 1) 2) Ca 2+ 3

(=0.02ml) (=0.02ml) 0 3) Ca 2+ 2+ Ca 2+ EDTA 2+ CN - EDTA CN - 2+ Ca 2+ KCN Fe 3+ Fe 2 KCN 4) KCN HCN 2+ KCN 1 2 3 4 5 6 4

) 25ml b 150ml c) 20 25ml d) ph e f g (100ml, 500ml) h a) 0.01M EDTA b) NN c) ph12 b) d) (100ml ) e) KCN 10wt f) 10wt 1) 0.01M EDTA 2) 20ml 25ml 3 1 ml KCN 4 ph12 4 ml, 5 NN 0.1g 6 EDTA 0.01M EDTA EDTA 7) 0.01MEDTA (V EDTA ) 2-1 Ca 2+ (M ca ) 5

Ca 2+ (M ca ) EDTA Ca 2+ 0 (0.02ml) 10ml 0.2 10ml 2ml 10ml 0.1ml 0.1ml 1/10 1/10 6 0.1mg 6

10g 0.1mg 1 M ca (V EDTA ) (M EDTA ) (V sample ) 2-1 (2-1) (V EDTA ) (M EDTA ) (V sample ) (2-1) y x y=ax+b (2 2) b (2 2) 7

(Method of least squares) 18 200 EDTA EDTA EDTA 8

(i) 0.1mg 10g 0.1mg 10g g (ii) 200 9

iii 3 10

3 (1) 20 0.1ml 1/10 1/10 3-1 S h d S h d 1/2 S h d d h h/2 3-1 S h d S h d h d 11

d 3~5mm Ar 1 h 10mm 0.1 h 3 d 3~5mm d 2 10 S h d (3 ) (2 ) (2 ) S 2 d d 1 d 3 S 3 0.1 2 (2) CRT M M 279.48 m 3-2 A M 0.05ppm B M 0.03ppm C 12

A: M(0.05 ppm) B: M(0.03ppm) C: 3-2 M B C C (Noise) S sigal 2S S/N=2 3-2 M 0.03ppm 10 0.003ppm 0.003ppm. 2 13

4 accidetal or radom error mistake 2 - (6) (1) x 1, x 2, " " " ", x m " i, " i = x i # m, (i =1, 2, $ $ $, ) (4-1) " i (4-1) x i = m + " i (i =1, 2, # # #, ) (x i ) 4-1 4-1 14

0 m (4-1) " i " i = x i # m 4-1 x i " # i = x i $ m m " i = 0 =0 (ε i ) 4-2 x i i i i i i p i p i = i / 4-2 4-1 (1/) p i = i / 4-1 i " p i = i / i i " =1 (4 1) 4-2 4-2 i p i = i / d", f (") d" f (")d" p i = i / f ("). (4 1) 15

+$ % f (")d" =1 (4 2) #$ f (") (Probability desity fuctio) f (") (2) 1/2 1/2 m ( ) " " = #$ + (m % ) # (%$) = (2 % m) #$ (4 3) P P = m C " (1#1/2) m# " (1/2) = m C " (1/2) m (4 4) +1) ( - -1) " +1, P +1, " +1 = ( +1) #$ + (m % %1) # (%$) = (2 % m + 2) #$ (4 5) P +1 = m C +1 " (1#1/2) m##1 " (1/2) +1 = m C +1 " (1/2) m (4 6) m C = m!/[(m " )!!] 3 (4 3) (4 5) (4 4) (4 6) " +1 = " + 2# (4 7) 16

P +1 P = m " +1 (4 8) " f (") (4 7) (4 8) " # " = (2 $ m) % & " +1 # " + 2$ (4 9) P +1 P = f (" + 2#) f (") (4 10) = m $ +1 f (" + 2#) = ( m $ +1 ) % f (") (4 11) f (" + 2#) $ f (") f (" + 2#) + f (") = ( m $ $ $1) /(m +1 +1 m $ 2 $1 +1) = = m $ 2 m +1 m (4 10) (4 11) (4 12) m (4 12) (m " 2) (4 9) (4 12) (2 " m) = "( # $ ) (4 13) f (" + 2#) $ f (") f (" + 2#) + f (") = $( " m# ) (4 14) f (" + 2#) Taylor (4 14) f (") f (" + 2#) = f (") + (2#) $ f '(") + (1/2!) $ (2#) 2 $ f ''(") + $ $ f (" + 2#) $ f (") % (2#) & f '(") f (" + 2#) + f (") $ 2 f (") (4 14) f '(") f (") = # " m$ (4-15) 2 1 f " df = # $ m% 2 " d$ 17

ε2 log f = 2mδ + C 2 f f (") = A # exp($ "2 2m% 2 ) (4-16) A (4 2) 1/(2m" 2 ) # h 2 (4-17) f (") = A # exp($h 2 " 2 ) (4-18) A (4 2) +$ % f (")d" = A % exp(#h 2 " 2 )d" =1 #$ +$ #$ A +$ % exp("h 2 # 2 )d# = & h "$ 1 2 A = h " f (") = h # $ exp(%h 2 " 2 ) (4-19) h h 2 "1/(2m# 2 ) (4-17) m 1/2 (Gaussia distributio) (Normal distributio), 4-3 18

h h (4 2) h h h Precisio costat h / " "=0 4 3 (4-19) h (4 2) h h (4-19) 3 4 (3) (4 19) X 100 X ph 19

-log 10 X X X 20

5 (1) f (ε) = h π exp( h 2 ε 2 ) (5 1) 2 (mea) (expectatio) (variace) x (radom variable) P(x) E(x) σ 2 (x), E(x) σ 2 (x) + x P(x)dx (5-2) + [x E(x)] 2 P(x)dx (5-3) x (5 1) h E(ε) + ε f (ε)dε = h ε exp( h 2 ε 2 )dε = 0 (5-4) π + (5 1) f (ε) ε = 0 f (ε) = f ( ε) ε f (ε) = ε f ( ε) ε ε ε f ( ε) = ε f (ε) ε f (ε) f( ) 0 0 σ 2 (ε) + [ε E(ε)] 2 f (ε)dε = ε 2 f (ε)dε = h ε 2 exp( h 2 ε 2 )dε (5-5) π + + ε 2 exp( h 2 ε 2 )dε = π 2h 3 + 21

1 h 2 1 2 (5-5) + σ 2 (ε) = ε 2 f (ε)dε = 1 2h (5-6) 2 0 σ 2 (ε) =1/(2h 2 ) (5-1) f (ε) h (5-6) σ 2 (ε) σ 2 f (ε) = h π exp( h 2 ε 2 ) = 1 ε2 exp( 2π σ 2σ ) (5-7) 2 (5-7) x m x m m =0 (5-2) x m=0 m x x g(x) (5-7) g(x) = 1 (x m)2 exp[ ] (5-8) 2π σ 2σ 2 x m (5-7) (5-7) g(x) (5-8) m 2 x (5-7) (5-8) 3 4 x σ 2 (x) (5-6) h σ 2 (x) σ 2 (x) σ(x) σ(x) Stadard deviatio σ(x) root mea square error 22

(2) x g(x) x x 1 x x 2 P(x 1 x x 2 ) g(x) P(x 1 x x 2 ) = x2 1 (x m)2 exp[ ]dx (5-9) π σ 2σ 2 x1 t = (x m) /σ t 1 = (x 1 m) /σ, t 2 = (x 2 m) /σ (5-9) P(t 1 t t 2 ) = Φ(t) = t 2 t 1 t2 1 2π exp( t 2 2 )dt 1 2π exp( t 2 2 )dt t1 1 2π exp( t 2 2 )dt = Φ(t 2 ) Φ(t 1 ) (5-10) t 1 2π exp( t 2 2 )dt (5-11), Error fuctio Gauss Error itegral of Gauss (5-11) 3 (5-10) t = (x m) /σ =0 =1 N(0,1) = 1 2π exp( t 2 2 ) (5-12) N (Noral distributio) t = (x m) /σ =m 2 23

N(m, σ 2 ) = 1 (x m)2 exp[ ] (5-13) π σ 2σ 2 t = (x m) /σ N(m, σ 2 ) N(0,1) =m 2 x =m 2 x m σ x m + σ t = (x m) /σ P(m σ x m + σ) = Φ(1) Φ( 1) 0.683 (5-14-1) P(m 2σ x m + 2σ) = Φ(2) Φ( 2) 0.955 (5-14-2 P(m 3σ x m + 3σ) = Φ(3) Φ( 3) 0.997 (5-14-3) m m ± σ 68 m ± 2σ 95 ( 5-15) m ± 3σ 99 N(m, σ 2 ) =m 2 =m 2 24

6 ( ) (1) (x 1, x 2, """", x ) m (x 1, x 2, """", x ) the most probable value x i ( i =1, 2, """", ) N(m," 2 ) x i ( i =1, 2, """", ) N(m," 2 ) (m, " 2 ) Maximum likelihood method " f (x; ") (x 1, x 2, """", x ) (6-1) L("; x 1, x 2,# ##, x ) $ f (x 1 ; ") # f (x 2 ; ") # f (x 3 ; ") # # # # f (x ; ") (6-2) (6-1) (x 1, x 2, """", x ) L("; x 1,x 2,###,x ) " " L("; x 1,x 2,###,x ) L("; x 1,x 2,###,x ) (Likelihood fuctio) 25

L("; x 1,x 2,###,x ) L =logl L("; x 1,x 2,###,x ) L'("; x 1,x 2,###, x ) $ log[ f (x 1 ; ") # f (x 2 ; ") # f (x 3 ; ") # # # # f (x ; ")] (6-3) L("; x 1,x 2,###,x ) L("; x 1,x 2,###,x ) (6-1) (x 1, x 2, """", x ) (6-4) (Maximum liklihood estimate) (x 1, x 2, """", x ) N(m, " 2 ) # = log f (x i ; ") "L(#; x 1,x 2,$$$,x ) "# L("; x 1,x 2,###,x ) = $ = ( (6-5) (6-6) (6-6) m 0 m (6-7) m m ˆ = 0 1 logl("; x 1,x 2,###,x ) = # log( " logl "m = 1 %(x # 2 i $ m) = 0 2% #& exp[' (x i ' m)2 2& 2 ] 1 2" #$ ) # exp[% 1 &(x 2$ 2 i % m) 2 ] #(x i " m ˆ ) = 0 #(x i " m ˆ ) = # x i " $ m ˆ = 0 1 2$ #% ) & 1 '(x 2% 2 i & m) 2 (6-7) 26

ˆ m = 1 " x i (6-8) " 2 " ˆ 2 (6-6) logl " 2 m (6-8) 0 " logl(#; x 1,x 2,$$$,x ) "(% 2 ) 1 "[$ log( = 2& $% )] "[' 1 ((x 2% 2 i ' m) 2 ] + = 0 "(% 2 ) "(% 2 ) (6-9) 2 " log( (6-9) (6-10) " 2 " ˆ 2 (6-10) 2" 4 m (6-8) m ˆ 1 2# "$ ) = " log[(2# "$ 2 ) %1/ 2 ] = % 2 " log(2# "$ 2 ) " "(# 2 ) [ $ log( 1 2% $# )] = " "(# 2 ) [& 2 $ log(2% $# 2 )] = & 2# 2 " "(# 2 ) [$ 1 %(x 2# 2 i $ m) 2 ] = 1 %(x 2# 4 i $ m) 2 " logl(#; x 1,x 2,$$$,x ) "(% 2 ) "# 2 + $ (x i " m ˆ ) 2 = 0 ˆ " 2 = 1 $ ˆ " 2 (x i # m ˆ ) 2 = & 2% + 1 '(x 2 2% 4 i & m) 2 = 0 ˆ " 2 (6-11) 27

(x 1, x 2, """ x ) m ˆ " ˆ 2 (6-8) (6-11) m ˆ = 1 " x i = x ˆ " 2 = 1 $ (x i # m ˆ ) 2 = 1 $ (x i # x ) 2 (2) x i (i =1, 2, " " ", N) N x i N(m i, " 2 i ) y = c 0 + c 1 x 1 + c 2 x 2 + """ + c N x N (6-12) y N(m y, " 2 y ) c 0, c 1, c 2, """", c N y m y = c 0 + c 1 m 1 + c 2 m 2 + """ + c N m N (6-13-1) " 2 y = c 2 1 " 2 1 + c 2 2 " 2 2 + ### + c 2 2 N " N (6-13-1) 1989 3 晛 (1973) 5 (3) 1 1 N ( m ˆ = x ) ( " ˆ 2 ) 28

N N m ˆ = x σ ˆ 2 m ˆ = x = 1 " x i = 1 x + 1 1 x + #### + 1 2 x (6-14). x i (i =1,2,"""",) N(m, " 2 ) m ˆ = x m ˆ = x E(x ) m ˆ = x " 2 (x) E(x ) = m + m + """" + m = m (16-15-1) (16-15-2) m ˆ = x N(m, " 2 /) (i) " 2 (x ) = " 2 + " 2 2 + # ### + " 2 2 = " 2 2 (ii), lim "# $ 2 (x ) = lim "# ($ 2 /) = 0 (6-16) 0 (i) ubiased character (ii) cosistecy (6-16) (4) 29

2 (6-17) s 2 s (6-17) s 2 (6-18) s 2 (6-19) m 0= -m+m (6-19) (6-19) " ˆ 2 = s' 2 = (1/) $ (x i # x ) 2 m(s 2 ') = (1/N) "[s' 2 (1) + s' 2 (2) + """ + s' 2 (N)] = 1 N s' 2 (k) (6-20) 2 2 (6-20) (6-18) N # s' 2 = (1/)# (x i " x ) 2 = (1/)#[(x i " m) + (m " x )] 2 = (1/)# (x i " m) 2 + (2 /)(m " x )#(x i " m) + (1/)# (m " x ) 2 (2 /)(m " x )#(x i " m) = 2(m " x ) $ (1/)# (x i " m) (1/)# (m " x ) 2 = (m " x ) 2 k=1 = 2(m " x ) #[(1/) $ x i " m] = " 2m(m " x ) 2 s' 2 = (1/)# (x i " x ) 2 = (1/)# (x i " m) 2 " (m " x ) 2 = $ 2 " (m " x ) 2 N m(s 2 ') = 1 " s' 2 (k) = 1 " N N [# 2 $ (m $ x k ) 2 ] k=1 = " 2 # 1 N N $ k=1 N k=1 (m # x k ) 2 30

(16-15-2), (6-21) m (ubiased estimate) s 2 m(s 2 ') = " 2 # " 2 1/ 1/(-1) s 2 = (6-17) (6-22) ( 1) s2 ' (6-23) s 2 N m(s 2 ) (6-23) m(s 2 ) = ( 1) m(s2 ') (6-21) m(s 2 ') = [( 1) /] σ 2 (6-24) (6-22) /(-1) (6-23) s 2 ( #1) = $" 2 " ˆ 2 = s' 2 = 1 $ (x i # x ) 2 1 s 2 " ( #1 ) $ (x # x i ) 2 m(s 2 ) = " 2 ˆ " 2 # s 2 = 1 ( $1) % (x i $ x ) 2 m(s 2 ') " # 2 m(s 2 ') < " 2 (6-25) 31

s 2 s 2 (-1) =1 =1 s 2 =0 s 2 s 2 (-1) m (6-25) " 2 (x ) = " 2 / " ˆ 2 (x) = s2 = 1 ( #1) $ % (x # x i )2 (16-15-2) (6-26) (5) (i) m ˆ m ˆ = (1/)" x i = x (ii) ˆ " 2 = s 2 = i 1 ( #1) $ % (x # x i )2 i " ˆ ˆ " = s = 1 ( #1) $ % (x # x i )2 i (root mea square error of a idividual measuremet) (iii) x " ˆ 2 (x) " ˆ 2 (x) = s2 = 1 ( #1) $ % (x # x i )2 i 32

" ˆ (x) " ˆ (x) = s = 1 ( #1) $ % (x i # x ) 2 i (root mea square error for the average) (6) 2 1/100 M Ca 2+ f=1.2988 1.2988x10-2 M Ca 2+ EDTA EDTA ml s = No. 1 10.68 +0.051 0.0026 2 10.53-0.091 0.0098 3 10.60-0.029 0.0008 4 10.59-0.039 0.0015 5 10.70 +0.071 0.0050 6 10.54-0.089 0.0079 7 10.68 +0.051 0.0026 8 10.62-0.009 0.0001 9 10.67 +0.041 0.0017 10 10.63 +0.001 0.0000 11 10.69 +0.061 0.0037 12 10.62-0.009 0.0001 (=12) x =10.629 #(x i " x ) 2 = 0.0358 1, ( "1) # $ (x " x i )2 = 0.057 (ml) s/ = 0.016 (ml) (I) x i (ml) x i " x (x i " x ) 2 10.63 ± 0.06 ml ( =12) 33

II 10.63 ± 0.02 ml ( =12) (I) (II) EDTA (II) (I) (7) 1/ " ˆ 2 (x) = s2 = 1 ( #1) $ % (x # x i )2 (6-26) 0 s 2 1/ (a) 1 mm (b) 1/10 1 mm 2 ( ) 2 ( ) 2 200 (6-26) 1/ 34

(6-26) (6-26) PC 1 (6-26) (6-26) 10 8 9 10 11 12 35

7 i) ii) (1) (x 1, x 2, " " ", x i, " ", x ) x i " i i " i x i, m N(m," 2 i) m L(m; x 1, x 2, " "", x, # 1, # 2, " " ", # ) m logl = log{ " 1 exp[& (m & x i )2 ]} 2# $% i 2% 2 i x i " i x i " i 36

= " log( 1 ) & 2# $% i " [(m & x i )2 ] 2% 2 i (7-1) m m logl m " logl "m = #2 $ ( m # x i ) = # 2% $ (m # x i ) = 0 2 i % 2 i (7-2) m" ( 1 ) = " x i # 2 i # 2 i m m ˆ ˆ m = " [( 1 # 2 i ) $ x i ]/" ( 1 ) (7-2) (7-3) 1/" 2 i (weight) w i = 1 " 2 i ˆ m = "(w # x ) i i (7-4) (7-5) m ˆ w i =1/" 2 i (7-5) x i /" w i # 2 i x i w i (2) ( m ˆ ) ( m ˆ ) (7-5) m ˆ = w 1 # x 1 + w 2 " w # x 2 + # # # + w i " w i " w i # x (7-6) (x 1, x 2, " " ", x i, " ", x ) a i = w i /" w i (7-7) 37

x i, m N(m," 2 i) ( m ˆ ) 6 (2) w m ˆ = a 1 x 1 + a 2 x 2 + +a x = ( i w ) m = m (7-8) j σ 2 ( ˆ m ) = (a 1 ) 2 σ 2 1 + (a 2 ) 2 σ 2 2 + +(a ) 2 σ 2 j= w = ( i 1 1 ) 2 σ 2 i = ( w i w )2 (1/σ 2 i) 2 σ 2 i = ( i w )2 (1/σ 2 i) i j=1 j=1 1 = (1/σ 2 i) =1/ (1/σ 2 i) (7-9) [ (1/σ 2 i)] 2 j=1 i (7-3) (7-9) σ 2 i σ 2 m ˆ = ( 1 ) x σ 2 i / ( 1 ) = ( 1 i σ 2 i σ ) x 2 i /( /σ 2 ) = 1 x i = x σ 2 ( m ˆ ) =1/ (1/σ 2 i) = σ 2 (16-15-1) (16-15-2) (7-3) (7-9) (7-3) (7-9) j=1 i 38

8 Ca 2+ ( 2) 2 Ca 2+ (M ca ) Ca 2+ 1 M ca (V EDTA ) (M EDTA ) (V sample ) 2-1 (V EDTA ) (M EDTA ) (V sample ) 2-1 Ca 2+ (M ca ) Propagatio of errors 2-1 (1) (x 1, x 2,, x ) y (x 1, x 2,, x ) y = f (x 1, x 2,, x ) (8-1) (8-1) (x 1, x 2,, x ) Taylor y = f (x 1, x 2,, x ) + f x 1 (x 1 x 1 ) + f x 2 (x 2 x 2 ) + + f x (x x ) + higher terms (8-2) (x 1, x 2,, x ) f ( f ) x1 = x x i x 1, x 2 = x 2,, x = x (8-3) i (8-1) (x 1, x 2,, x ) y y = f (x 1, x 2,, x ) (8-4) (8-2) y y y f x 1 (x 1 x 1 ) + f x 2 (x 2 x 2 ) + + f x (x x ) (8-5) 39

(y y ) 2 ( f x 1 ) 2 (x 1 x 1 ) 2 + ( f x 2 ) 2 (x 2 x 2 ) 2 + +( f x ) 2 (x x ) 2 +2( f x 1 )( f x 2 )(x 1 x 1 )(x 2 x 2 ) + (8-6) x i m (8-6) m. m x i k x i, k y m k y y k x i x j k ( f x 1 )( f x 2 )(x 1, k x 1 )(x 2, k x 2 ) m m (8-6) ( ) m m m 1 m (y y k )2 1 m m ( f ) 2 (x 1, k x 1 ) 2 + 1 m x 1 m ( f ) 2 (x 2, k x 2 ) 2 + x 2 k=1 + 1 m k=1 m k=1 ( f x ) 2 (x, k x ) 2 k=1 + 2 m m ( f )( f )(x 1, k x 1 )(x 2, k x 2 ) + (8-7) x 1 x 2 k=1 (8-3) m 1 m (y y k )2 ( f ) 2 1 m x 1 m (x x 1, k 1 )2 + ( f ) 2 1 m x 2 m (x x 2, k 2 )2 + k=1 k=1 +( f ) 2 1 m x m (x x, k )2 k=1 m +( f )( f ) 2 x 1 x 2 m (x x )(x x ) 1, k 1 2, k 2 + (8-8) k=1 (1/m) m m 1 m (y y k )2 σ 2 (y), k=1 m k=1 1 m (x x i, k i )2 σ 2 (x i ) (i =1, 2,, ) (8-9) k=1 40

m 1 m (x x )(x x ) 1, k 1 2, k 2 cov(x 1, x 2 ) (8-10) k=1 x 1 x 2 Covariace X Y Correlatio coefficiet ρ X X i (, 2,., ), Y Y i (, 2,., ) Cov(X, Y) 1 (X i X )(Y i Y ) (8-11-1) ρ(x,y) Cov(X,Y) σ(x) σ(y) (8-11-2) (8-11-2) (8-8) m 1 m (x x )(x x ) 1, k 1 2, k 2 cov(x 1, x 2 ) = ρ(x 1, x 2 ) σ(x 1 ) σ(x 2 ) (8-12) k=1 (8-8) σ 2 (y) ( f x 1 ) 2 σ 2 (x 1 ) + ( f x 2 ) 2 σ 2 (x 2 ) + +( f x ) 2 σ 2 (x ) +2( f x 1 )( f x 2 ) ρ(x 1, x 2 ) σ(x 1 ) σ(x 2 ) + (8-13) y = f (x 1, x 2,, x ) (x 1, x 2,, x ) y y (2) (8-11-1) (8-11-2) X Y, ρ(x,y) Cov(X,Y) X X i (, 2,., ), Y Y i (, 2,., ) pairs (X 1, Y 1 ), (X 2, Y 2 ),, (X, Y ) (8-11-1) Cov(X, Y) 1 (X i X )(Y i Y ) 41

T P 30 T P pair data 30 Cov(T,P) = 1 30 30 (T i T ) (P i P ) 8 1 Y (a) (X i X )(Y i Y ) > 0 (X i,y i ) (X,Y ) X (b) (X i X )(Y i Y ) < 0 (c) Y (X,Y ) (X,Y ) (X i,y i ) X 8 1. X Y (a) X Y (X i X )(Y i Y ) > 0 b X Y (X i X )(Y i Y ) < 0 c (X i X )(Y i Y ) X Y 42

8 1 (a) (X i X )(Y i Y ) Cov(X,Y) = 1 (X i X )(Y i Y ) > 0 =1 X Y b (X i X )(Y i Y ) Cov(X,Y) = 1 (X i X )(Y i Y ) < 0 =1 c (X i X )(Y i Y ) > 0 (X i X )(Y i Y ) < 0 Cov(X,Y) = 1 (X i X )(Y i Y ) 0 =1 (8-14) Cov(X, X) Cov(Y, Y) Cov(X,X) = 1 =1 (X i X )(X i X ) = σ 2 (X) X X X ρ(x,y) Cov(X,Y) σ(x) σ(y) Cov(X,Y) = 0 ρ(x,y) = 0 (8-13) X Y Y = a + b X Y i = a + bx i Y = (1/) Y i = (1/) (a + bx i ) = a + bx Cov(X,Y) = (1/) (X i X )(Y i Y ) = (1/) (X i X )(a + bx i a bx ) = (1/) b(x i X )(X i X ) = b σ 2 (X) 43

σ 2 (Y) = b 2 σ 2 (X) ρ(x,y) Cov(X,Y) σ(x) σ(y) = b σ 2 (X) σ(x) b σ(x) = b b =1 (b > 0) or -1(b < 0) b>0 = + b<0 1 +1 (8-13) σ 2 (y) ( f x 1 ) 2 σ 2 (x 1 ) + ( f x 2 ) 2 σ 2 (x 2 ) + +( f x ) 2 σ 2 (x ) + 2( f x 1 )( f x 2 ) ρ(x 1, x 2 ) σ(x 1 ) σ(x 2 ) + (8-13) ρ(x 1, x 2 ) = 0 pairs (i j) ρ(x i, x j ) = 0 σ 2 (y) ( f x 1 ) 2 σ 2 (x 1 ) + ( f x 2 ) 2 σ 2 (x 2 ) + +( f x ) 2 σ 2 (x ) (8-15) ρ(x i, x j ) = 0 f (x 1, x 2,, x ) (x 1, x 2,, x ) (8-15) (1) f = a x ± b y ± c z a, b, c x, y, z σ(x), σ(y), σ(z) 44

σ 2 ( f ) = a 2 σ 2 (x) + b 2 σ 2 (y) + c 2 σ 2 (z) σ( f ) = a 2 σ 2 (x) + b 2 σ 2 (y) + c 2 σ 2 (z) a = b =1, c = 0 σ(x ± y) = σ 2 (x) + σ 2 (y) (8-16) (2) f = x y z σ 2 ( f ) = ( f x )2 σ 2 (x) + ( f y )2 σ 2 (y) + ( f z )2 σ 2 (z) = ( y z )2 σ 2 (x) + ( x z )2 σ 2 (y) + ( x y z 2 )2 σ 2 (z) = ( x y ) 2 [ σ 2 (x) + σ 2 (y) + σ 2 (z) ] z x 2 y 2 z 2 σ( f ) = ( x y ) z σ 2 (x) x 2 + σ 2 (y) y 2 + σ 2 (z) z 2 z =1 σ(x y) = (x y ) ( σ(x) x ) 2 + ( σ(y) ) 2 (8-17) y y=1 σ(x /z) z y σ( x y ) = ( x y ) (σ(x) ) 2 + ( σ(y) ) 2 (8-18) x y (8-16), (8-17), (8-18) Covariace matrix (8-13) (8-15) 45

(x i =1,2,,) x i, x j cov(x i, x j ) C(x) cov(x 1, x 1 ) cov(x 1,x 2 ) cov(x 1,x ) cov(x 2, x 1 ) cov(x 2,x 2 ) C(x)= cov(x, x 1 ) cov(x,x ) (8-19) cov(x i, x i ) = σ 2 (x i ) σ 2 (x i ) cov(x i, x j ) = cov(x j, x i ) σ 2 (x 1 ) cov(x 1, x 2 ) cov(x 1, x ) cov(x C(x)= 2,x 1 ) σ 2 (x 2 ) cov(x,x 1 ) σ 2 (x ) (8-20) (8-13) T T = ( f x 1, f x 2,, f x ) (8-21-1) T T T T T = f x 1 f x 2 f x (8-21-2) (8-13) σ 2 (y) T T C(x) T σ 2 (y) = T(1,) C(x)(,) T T (,1) = T C(x) T T (8-22) ρ(x 1,x 2 ) cov(x 1,x 2 ) /[σ(x 1 ) σ(x 2 )] ρ(x 1,x 2 ) σ(x 1 ) σ(x 2 ) = cov(x 1,x 2 ) (8-13) cov(x i, x j ) 2 46

(8-22) (8-20) cov(x i,x j ) = cov(x j,x i ), i j 0 (8-15) σ 2 (y) 47

9 S.G. (2012) 18 (1795 ) (1) (9-1) t y y = a 1 f 1 (t) + a 2 f 2 (t) + +a j f j (t) + + a f (t) (9-1) a j (j=1,2,, ) f j (t) (j=1,2,, ) t y=a 1 +a 2 t y (9-1) t=t i y=y i ˆ y i ε i (9-1) ˆ y i = a 1 f 1 (t i ) + a 2 f 2 (t i ) + +a j f j (t i ) + + a f (t i ) + ε i (9-2) y i = a 1 f 1 (t i ) + a 2 f 2 (t i ) + + a j f j (t i ) + + a f (t i ) (9-3) ˆ y i = y i + ε i (9-4) y i, ε i y ˆ i ε i y ˆ i ε i = y ˆ i y i ε i, g(ε i ) = 1 exp[ ( y ˆ y i i )2 ] = 2π σ i 2(σ i ) 2 1 2π σ i exp[ (ε i )2 2(σ i ) 2 ] (9-5) E(ε i ) = 0, σ 2 (ε i ) = (σ i ) 2 ˆ y i (i =1, 2,, m) m, (9-2)~(9-4) y ˆ 1 [a 1 f 1 (t 1 ) + a 2 f 2 (t 1 ) + + a j f j (t 1 ) + +a f (t 1 )] = ε 1 y ˆ 2 [a 1 f 1 (t 2 ) + a 2 f 2 (t 2 ) + + a j f j (t 2 ) + +a f (t 2 )] = ε 2 y ˆ i [a 1 f 1 (t i ) + a 2 f 2 (t i ) + + a j f j (t i ) + +a f (t i )] = ε i (9-6) y ˆ m [a 1 f 1 (t m ) + a 2 f 2 (t m ) + + a j f j (t m ) + +a f (t m )] = ε m 48

ˆ y i (i =1, 2,, m) a j (j=1,2,, ) m (9-7) (9-6) m ε i (i =1, 2,, m) ε i (i =1, 2,, m), L a j (j=1,2,, ) a j (j=1,2,, ) 0 a j (j=1,2,, ) (9-5) ε i (i =1, 2,, m) m L = Π g(ε m i) = Π { 1 exp[ (ε i )2 ]} (9-8) 2 2π σ i 2(σ i ) m 1 logl = ( ) 1 m (ε i ) 2 (9-9) 2π σ i 2 (σ i ) 2 ε i = y ˆ i y i = y ˆ i [a 1 f 1 (t i ) + a 2 f 2 (t i ) + + a j f j (t i ) + +a f (t i )] a j (j=1,2,, ) logl logl / a j = 0 (j =1,2,, ) (9-10) a j (j=1,2,, ) (9-9) logl m (ε i ) 2 (σ i ) mi. (9-11) 2 (9-9) logl ˆ y i (σ i ) 2 (σ y ) 2 m (ε i ) 2 m m = (σ (σ i ) 2 y ) 2 (ε i ) 2 = (σ y ) 2 ( y ˆ i y i ) 2 mi. ˆ y i y i = a 1 f 1 (t i ) + a 2 f 2 (t i ) + + a j f j (t i ) + +a f (t i ) 49

m ( y ˆ i y i ) 2 mi. (9-12) (9-11) (9-12) a j (j=1,2,, ) 0 a j (j=1,2,, ) (2) a j (j=1,2,, ) (9-9) logl a j (j=1,2,, ) m 1 logl = ( ) 1 m (ε i ) 2 (9-9) 2π σ i 2 (σ i ) 2 ε i = ˆ y i y i = ˆ y i [a 1 f 1 (t i ) + a 2 f 2 (t i ) + + a j f j (t i ) + +a f (t i )] (9-6) logl / a j = 0 [ a j m 1 (σ i ) 2 { ˆ y i [a 1 f 1 (t i ) + a 2 f 2 (t i ) + a j f j (t i ) + a f (t i )]} 2 = 0 (9-13) (9-13) a j (j=1,,) m 1 (σ i ) 2 { ˆ y i [a 1 f 1 (t i ) + a 2 f 2 (t i ) + a j f j (t i ) + a f (t i )]} f j (t i ) = 0 m m y ˆ i 1 f (σ i ) 2 j (t i ) = (σ i ) [a 2 1 f 1 (t i ) + a 2 f 2 (t i ) + a j f j (t i ) + a f (t i )] f j (t i ) (9-14) m m y ˆ i a f j (t i ) = {[ k f k (t i ) ] f j (t i )} (9-15) 2 σ i k=1 σ i 2 Z j S j Z j = S j m m y ˆ Z j = i a f j (t i ), S j = {[ k f k (t i ) ] f j (t i )} (9-16) 2 σ i j (j=1,2,., ) logl / a j = 0 (9-16) a j (j=1,2,, ) X(,1) m k=1 σ i 2 50

( y ˆ i : i =1,2,, m) Y(m,1) m 1 (σ i ) 2 G(m,m) X(,1) = a 1 a 2 a 2 1/σ 1 0 0 2 0 1/σ G(m,m) = 2 0 2 0 0 1/σ m y ˆ 1 y ˆ 2 (9-17), Y(m,1) = ˆ f j (t i ) C(m,) f 1 (t 1 ) f 2 (t 1 ) f (t 1 ) f C(m,) = 1 (t 2 ) f 2 (t 2 ) f (t 2 ) f 1 (t m ) f 2 (t m ) f (t m ) y m (9-18) (9-19) (9-20) C(m,) C T (,m) f 1 (t 1 ) f 1 (t 2 ) f 1 (t m ) f C T (,m) = 2 (t 1 ) f 2 (t 2 ) f 2 (t m ) f (t 1 ) f (t 2 ) f (t m ) 2 y ˆ 1 /σ 1 2 y ˆ G(m,m) Y(m,1) = GY(m,1) = 2 /σ 2 2 y ˆ m /σ m (9-21) (9-22) (9-16) Z j C T (,m) GY(m,1) j C T (,m) GY(m,1) = C T GY(,1) = m m m ( y ˆ i /σ 2 i ) f 1 (t i ) ( y ˆ i /σ 2 i ) f j (t i ) ( y ˆ i /σ 2 i ) f (t i ) (9-23) 51

(9-16) m m y ˆ Z j = i a f j (t i ) = S j = {[ k f k (t i ) ] f j (t i )} (9-16) 2 σ i S j y ˆ i a k f k (t i ) k=1 σ i 2 j k=1 σ i 2 C(m,) X(,1) = CX(m,1) CX(m,1) = k=1 k=1 f k (t 1 )a k = f k (t m )a k k=1 k=1 a k f k (t 1 ) a k f k (t m ) G(m m) G(m,m) CX(m,1) = GCX(m,1) = C T (,m) C T (,m) GCX(m,1) = C T GCX(,1) = m m m k=1 k=1 2 a k f k (t 1 ) /σ 1 2 a k f k (t m ) /σ m f 1 (t i ){ a k f k (t i ) /σ 2 i } k=1 f j (t i ){ a k f k (t i ) /σ 2 i } k=1 f (t i ){ a k f k (t i ) /σ 2 i } k=1 j m S j = {[ k=1 a k f k (t i ) σ i 2 ] f j (t i )} (9-16),(9-23),(9-24) C T GY(,1)= C T GCX(,1) (9-24) (9-17) X(,1) C T GCX(,1)= C T GY(,1) (9-25) C T GC (,m)(m,m)(m,)=(,) (C T GC) -1 52

X(,1)= (C T GC) -1 C T GY(m,1) (9-26) C T GC(,) (9-26) a j (j=1,2,, ) (3) C T GC(,) G(m,m) C(m,) = GC(m,) = f 1 (t 1 ) σ 1 2 f 2 (t 1 ) σ 1 2 f j (t 1 ) σ 1 2 f (t 1 ) σ 1 2 f 1 (t i ) f 2 (t i ) f j (t i ) f (t i ) 2 σ i 2 σ i 2 σ i f 2 (t m ) f 2 j (t m ) f 2 (t m ) 2 σ m σ m σ m σ i 2 f 1 (t m ) σ m 2 (9-27) GC(m,) C T (.m) C T GC(,) (i,j) m (C T G C) ij = (C T 2 ) ik (GC) kj = f i (t k ) f j (t k ) /σ k (j,i) k=1 k=1 m (C T G C) ji = (C T 2 ) jk (GC) ki = f j (t k ) f i (t k ) /σ k k=1 k=1 m m (9-28) (C T G C) ij = (C T G C) ji (9-29) C T GC(,) C T GC 0 det C T G C 0 (C T G C) -1 A(,) A -1 AX=E X A adja A deta A A -1 =adja/ deta (9-30) adja adjoit of A A adja (,) i,j (adja) ij =(-1) i+j (M) ji (9-31) 53

(M) ji A(,) j i (-1,-1) i,j j,i =2 A = a 11 a 12, X = x 11 x 12, AX = E = 1 0 a 21 a 22 x 21 x 22 0 1 X A -1 (9-30) deta= A ( 1) 2 a 22 A A 1 = ( 1) 3 a 21 A ( 1) 3 a 12 A = 1 a 22 a 12 ( 1) 2 a 11 A a 21 a 11 A A -1 X AX = 1 a 11 a 22 a 12 a 21 a 11 a 12 + a 11 a 12 = 1 A 0 = E A a 21 a 22 aa 21 a 22 a 21 a 12 + a 11 a 22 A 0 A AX=E (9-32) (9 32) X A=E A -1 deta= A =0 A -1 >2 (9-30) (9-30) 54

(4) C T GC(,)=B(,) B(,) 4-1. B(,) x=(x 1, x 2,, x ) T 0 Bx = λx (9-33) B(,) x=(x 1, x 2,, x ) T (9-33) B x x E (,) (9-33) Bx = λex x (B λe)x = 0 (9-34) x=(x 1, x 2,, x ) T (9-34) B λe b 11 λ b 12 b 1 F(λ) = B λe = b 21 b 22 λ b 2 b 1 b 2 b λ = 0 (9-35) F( )=0 ( i :,2,,) B F( ) k (9-33) Bx = λ k x (9-36) b 11 x 1 + b 12 x 2 + +b 1 x = λ k x 1 b 21 x 1 + b 22 x 2 + +b 2 x = λ k x 2 (9-37) b 1 x 1 + b 2 x 2 + +b x = λ k x 0 x=(x 1, x 2,, x ) T k (9-37) k x=(x 1, x 2,, 55

x ) T (9-33) cx 4-2. x=(x 1, x 2,, x ) T x i (k k k x ) i (k x ) (k i x ) (k i / ( x ) i ) 2 l i,k (9-38) 2 2 (l i,k ) 2 = x (k) i / ( x (k) i ) 2 (k [1/ ( x ) i ) 2 (k ] ( x ) i ) 2 =1 (9-38) 4-3. k l k (,1) Bl k λ k El k = 0 ( λ 1, l 1 ) ( λ 2, l 2 ) Bl 1 = λ 1 l 1, Bl 2 = λ 2 l 2 (9-39) 0 56

l 1 l 1 = l j1 l j 2 j=1 1 l 12 l 22 = (l 11,l 21,,l,1 ) = l T 1 l 2 (9-40) l 2 (9-40) l 1 T l 2 B B=B T, (Bl 1 ) T l 2 (Bl 1 ) T l 2 = (λ 1 l 1 ) T l 2 = λ 1 (l 1 T l 2 ) (9-41) (Bl 1 ) T l 2 (Bl 1 ) T l 2 = (l 1 ) T B T l 2 = (l 1 ) T (B l 2 ) = (l 1 ) T (λ 2 l 2 ) = λ 2 (l 1 T l 2 ) (9-42), λ 1 (l 1 T l 2 ) = λ 2 (l 1 T l 2 ) (λ 1 λ 2 )(l 1 T l 2 ) = 0 λ 1 λ 2 (l 1 T l 2 ) = 0 4-4. ( k ) (l k ) (,1) l i,k (9-38) (l k ) k=1,2, (,) L ( ) = L = l 1 l 2 l l 1,1 l 1,2 l 1, l 2,1 l 2,2 l 2, l,1 l,2 l, L, A (9-43) A T A = E (9-44) A A (i) A T = A 1, (ii) A T A = AA T, (iii) deta = ±1 (9-45) 57

(i) A A 1 A = A 1 A = E A T A = E (ii) (i) A A 1 A = A 1 A = E (iii) det A = det A T, det(a T A) = det(a T ) det(a) AT A = E det(a T A) = det(a T ) det(a) = [det(a)] 2 = det(e) =1 L (9 44) L T L (L T L) ij (L T L) ij = l ki l kj = 0 (i j), 1(i = j) k =1 i j i = j L T L = E (9-46) L (9 41) (9-45) L 4-5. B(,) ( i ) (9-47) λ 1 0 0 0 λ Λ = 2 0 0 0 0 λ (9-47) Bx λex = 0 Bl k λ k El k = 0 B L B BL = LΛ (9-48) Bl k = λ k El k b 11 b 12 b 1 l 11 l 12 l 1 b 1k l k1 b k 1k l k 2 b k 1k l k k b BL = 21 b 22 b 2 l 21 l 22 l 2 b = 2k l k1 b k 2k l k 2 b k 2k l k k b 1 b 2 b l 1 l 2 l b k l k k1 b k l k k 2 b k l k k 58

l 11 l 12 l 1 λ 1 0 0 λ 1 l 11 λ 2 l 12 λ l 1 l 21 l 22 l 2 LΛ = 0 λ 2 0 = λ 1 l 21 λ 2 l 22 λ l 2 l 1 l 2 l 2 0 0 λ 2 λ 1 l 1 λ 2 l 2 λ l Bl k = λ k El k (9-48) (9-48), L L L L T = E, L 1 = L T (9-49) BL = LΛ L T BL L T = B = LΛ L T B 1 = (LΛ L T ) 1 = (L T ) 1 (LΛ) 1 = (L T ) 1 (Λ) 1 (L) 1 = L(Λ) 1 L T (9-50) ( ) -1 B B (9-50) =0 Schmidt >2 Schmidt (2006) 22) 5-2. 5-3 (2012) 23) 9-4 =2 3 PC 59

PC (1969) 19), (1987) 20) (1971) 18) (5) σ 2 ( ˆ y i ) = σ i 2 σ 2 ( ˆ y i ) = σ i 2 (i) σ 2 ( ˆ y i ) = σ i 2 (9-26) a j (j=1,2,, ) X(,1) X(,1)= (C T GC) -1 C T GY(,1) (9-26) (C T GC) -1 (.) C T (.m) G (m,m) Y (m,1) Y (C T GC) -1 C T G (,m) (9-26) P(,m) (C T GC) -1 C T G(,m) (9-51) a 1 a 2 a j a X=PY (9-52) p 11 p 12 p 1i p 1m y ˆ 1 p 21 p 22 p 2m ˆ y 2 = p j1 p j 2 p ji p jm ˆ y i ˆ p 1 p 2 p i p m Y ˆ y i a j m y m (9-53) a j = p j i y ˆ i (9-54) ˆ y i 60

a j m σ 2 (a j ) = ( p j i ) 2 σ 2 ( y ˆ i ) = (p j i ) 2 2 σ i σ 2 2 ( y ˆ i ) = σ i (9-55) a j m (9-55) (ii) σ 2 ( ˆ y i ) = σ i 2 σ 2 2 ( y ˆ i ) = σ i ˆ y i G 1/σ 2 ( ˆ y i ) =1/σ i 2 G E E (9-25) G C T GCX(,1)= C T GY(,1) C T CX(,1)= C T Y(,1) (9-56) (9-25) ε i = ˆ y i y i = ˆ y [a 1 f 1 (t i ) + a 2 f 2 (t i ) + + a f (t i )] (9-6) a j (j=1,2,, ) (9-6) y i (obs) = ˆ y i y i (calc) = [a 1 f 1 (t i ) + a 2 f 2 (t i ) + + a f (t i )] (9-57) y i (obs) y i (calc) = ε i m m/(m-) s 2 σ 2 (y) (9-55) a j (9-55) G E m σ 2 (a j ) = ( p j i ) 2 σ 2 i = σ 2 (y) (p j i ) 2 (9-58) σ 2 (y) m (i) (ii) 61

(6) (i) y = a 1 + a 2 x a 1, a 2 y(obs) x σ(y) y(obs) x y(obs) σ(y) 0 0.9 0.3 5 4.3 0.4 10 6.5 0.5 15 10.3 0.6 0.9 1/(0.3) 2 0 0 0 4.3 Y =, X = a 1, G = 0 1/(0.4) 2 0 0 6.5 a 2 0 0 1/(0.5) 2 0 10.3 0 0 0 1/(0.6) 2 1 0 1 5 C =, C T = 1 1 1 1 1 10 0 5 10 15 1 15 C T GY = C T GCX X = (C T GC) -1 C T GY 0.9/(0.3) 2 C T GY = 1 1 1 1 4.3/(0.4) 2 = 91.4861 0 5 10 15 6.5/(0.5) 2 823.5417 10.3/(0.6) 2 62

1/(0.3) 2 0 C T GC = 1 1 1 1 1/(0.4) 2 5/(0.4) 2 24.1389 112.9167 = 0 5 10 15 1/(0.5) 2 10/(0.5) 2 112.9167 1181.2500 1/(0.6) 2 15/(0.6) 2 1 1181.25 112.9167 (C T GC) -1 = 15763.8945 112.9167 24.1389 = 7.493389 10 2 7.162995 10 2 7.162995 10 2 1.531278 10 2 (9-32) X = (C T GC) -1 C T GY = 7.493389 10 2 7.162995 10 2 91.4861 = 0.95638 = a 1 7.162995 10 2 1.531278 10 2 823.5417 0.60576 a 2 P = (C T GC) -1 C T G 1/(0.3) 2 0 0 0 C T G = 1 1 1 1 0 1/(0.4) 2 0 0 0 5 10 15 0 0 1/(0.5) 2 0 0 0 0 1/(0.6) 2 = 1/(0.3)2 1/(0.4) 2 1/(0.5) 2 1/(0.6) 2 0 5/(0.4) 2 10/(0.5) 2 15/(0.6) 2 0.8326 0.2445 0.01322 0.090308 P = 0.07959 0.003084 0.03260 0.043906 4 X = PY σ 2 (a 1 ) = ( p 1i ) 2 σ 2 (y i ) = 0.074193, σ(a 1 ) = 0.2737 4 σ 2 (a 2 ) = ( p 2i ) 2 σ 2 (y i ) = 0.0015313, σ(a 2 ) = 0.03913 y = (0.96 ± 0.27) + (0.606 ± 0.039)x 63

(ii) y = a 1 + a 2 x a 1, a 2 y(obs) x y(obs) x y(obs) 0 0.9 5 4.3 10 6.5 15 10.3 X = (C T GC) -1 C T GY G=E X = (C T C) -1 C T Y a 1, a 2 y(calc) y(obs) - y(calc) σ 2 (y) σ 2 (a 1 ), σ 2 (a 2 ) X = (C T C) -1 C T Y 1 0 C T C = 1 1 1 1 1 5 = = 4 30 0 5 10 15 1 10 30 350 1 15 0.9 (C T C) -1 = 1 350 30 0.7 0.06 =, C T Y = 1 1 1 1 4.3 = 22.0 500 30 4 0.06 0.008 0 5 10 15 6.5 241 10.3 X = a 1 0.7 0.06 = 22.0 = 0.940 0.06 0.008 241 0.608 a 2 y=0.940+0.608x 64

x y(obs) y(calc) = y(obs)- y(calc) 0 0.9 0.94-0.04 5 4.3 3.98 +0.32 10 6.5 7.02-0.52 15 10.3 10.06 +0.24 4 1 s 2 = [y i (obs) y i (calc)] 2 = 0.216 (4 2) 2 2 (y )/ (y 2) 1/(4-2) σ(y) = s 2 = 0.465 X = (C T C) -1 C T Y X = PY 0.7 0.06 P = (C T C) -1 C T = 1 1 1 1 0.7 0.4 0.1 0.2 = 0.06 0.008 0 5 10 15 0.06 0.02 0.02 0.06 a 1 a 2 0.7 0.4 0.1 0.2 = 0.06 0.02 0.02 0.06 y 1 y 2 y 3 y 3 σ 2 (a 1 ) = {(0.7) 2 + (0.4) 2 + (0.1) 2 + ( 0.2) 2 }σ 2 (y) = 0.1512, σ(a 1 ) = 0.389 σ 2 (a 2 ) = {( 0.06) 2 + ( 0.02) 2 + (0.02) 2 + (0.06) 2 }σ 2 (y) =1.728 10 3, σ(a 2 ) = 0.04157 y = (0.94 ± 0.39) + (0.608 ± 0.042)x (i) y = (0.96 ± 0.27) + (0.606 ± 0.039)x 65

(1) 40 20 30 R 24) R Califoria Berkeley 66

10 9 y = a 1 f 1 (t) + a 2 f 2 (t) + + a f (t) =5 lathaide tetrad effect 11 (1) L(III) L(III) 30 25) 1-1. 3 Rare earth elemets REE 3 4, 5, 6 Sc, Y, La~Lu 17 Z=57~71 La~Lu 15 (lathaides L) (REE) (L) La~Lu 15 [Xe](4f q ) q=0 La 3+ (Z=57) q=14 Lu 3+ (Z=71) 4f Z 4f 67

3 4f L(III) 1-2. 6 8 L(III) 6 L 2 O 3 (cub), 8 LF 3 (rhm) (10-1) L(III) 6, 8 L(III) L 2 O 3 L-O 6 LF 3 (rhm) L-F 8 6 8 6 NaCl (10-1) 25) L(III) [Xe](4f q ) 4f [Ar](3d q ) 68

4f [Xe] 5s 5p 3d [Xe](4f q ) [Ar](3d q ). d f d f L(III) [Xe](4f q ) 1-3. L(III) 4f L(III) atomic-like L(III) 4f L(III) L(III) (4f 4f) L(III) ephelauxetic effect 4f Racah parameters [Xe](4f q ) ( ) (4f 4f) 69

LF 3 LCl 3 L(III) L 2 O 3 L 2 S 3 L(III) L(III) L(III) tetrad effect Gd Gd break L(III) L(III). L(III) L 2 O 3 (cub) L(III) O LF 3 (rhm) L(III) F (1/2)L 2 O 3 (cub)=l 3+ (g)+(3/2)o 2- (g) (10-2) LF 3 (rhm)= L 3+ (g)+3f - (g) (10-3) 0K 1 25, L(III) O L(III) F M r M/r 9~10 M(1-1/)/r r (10-2) (10-3) L 3+ (g) L(III) L 3+ (g) L 3+ (g) L(III) (1/2)L 2 O 3 (cub) LF 3 (rhm) L(III) L 3+ (g) L 2 O 3 L-O 70

6 LF 3 (rhm) L-F 8 L 3+ (g) L 2 O 3 L(III) 4f LF 3 L(III) L(III) (4f 4f) L(III) [Xe](4f q ) Al(III) Al 2 O 3, AlF 3 (1/2)Al 2 O 3 (s)= Al 3+ (g)+(3/2)o 2- (g) (10-4) AlF 3 (s)=al 3+ (g)+3f - (g) (10-5) Al 3+ [Ne] Al Sc, Y Sc 3+, Y 3+ [Ar], [Kr] [Xe](4f q ) L(III) [Ar](3d q ) 6 8 L(III) 30) 1992 20 L(III) 25) 1-4. Jørgese Peppard et al.(1969) 26) L(III) logk d La-Ce-Pr-Nd Pm-Sm-Eu-Gd Gd-Tb-Dy-Ho Er-Tm-Yb-Lu logk d L(III) logk d logk d = ΔG/(RT) L(III) L(III) Jørgese(1970) 28) Nuget(1970) 29) Jørgese(1962) 27) 71

Peppard et al.(1969) Jørgese(1962) 27) Refied spi-pairig eergy theory (RSPET) L (I 3 ) L(III) I 3 L 2+ L 3+ +e - (10-6) e - I 3 [Xe](4f q ) [Xe](4f q-1 ) (10-7) La 2+ Gd 2+ L 2+ [Xe](4f q ) 10-1 La 2+ [Xe](5d4f 0 ) Gd 2+ [Xe](5d4f 7 ) (10-7) La 2+ [Xe](4f) Gd 2+ [Xe](4f 8 ) Lu 2+ [Xe](4f 14 6s) Lu 2+ L 2+ [Xe](4f q ) L 3+ [Xe](4f q-1 ) [Xe](4f 14 ) Lu 3+ (10-7) q 1~14 La~Yb I 3 10-1 L 3+ [Xe](4f q ) L 2+ La 2+ Gd 2+ [Xe](4f q+1 ) q 0~13 [Xe](4f q+1 ) [Xe](4f q ) (10-7 ) Jørgese(1962) (10-7) (10-7) I 3 (10-7) ΔE 1 (q)=e(4f q-1 ) E(4f q ) (10-8) [Xe] Jørgese(1962) L(III) L(III) L(III) + e - = L(II) (10-9) (10-6) I 3 72

L(III) ( I 3 ) Jørgese (1962) (10-8) [Xe](4f q ) E [(4f) q, LS(max), lowest J] = E [(4f) q, LS(max)] + p(s, L, J) 4f = W + qw + (1/2)q(q-1) [ E 0 + (9/13)E 1 ] + (9/13)(S) E 1 + m(l)e 3 + p(s, L, J) 4f (10-10) LS(max) [Xe](4f q ) (S) (L) LS (10-10) p(s, L, J) 4f W [Xe] W [Xe] 4f (1/2)q(q-1) 4f q [E 0 +(9/13)E 1 ] 4f Cofiguratio average eergy (9/13)(S)E 1 m(l)e 3 LS (S) m(l) LS L S E 1 E 3 (10-10) 1 5 LS E 0 E 1 E 2 E 3 4 E 2 p(s, L, J) 4f J J p(s, L, J) (S, L, J) 4f (10-10) [Xe] (4f q ) 1 S L J Hud (10-10) Hud 25 (Slater-Codo-Racah Theory) Jørgese(1962) L 3+ 73

(S L J) (S) m(l) p(s, L, J) 10-1, 10-1 (10-8) ΔE 1 (q)=e(4f q-1 ) E(4f q ) (10-10) I 3 4f q E(4f q-1 ) E(4f q ) ΔE 1 (q)= I 3 =E(4f q-1 ) E(4f q ) = {W + (q-1)w + (1/2)(q-1)(q-2) [E 0 + (9/13)E 1 ] + (9/13)(S) E 1 + m(l)e 3 + p(s, L, J) 4f } q-1 {W + qw + (1/2)q(q-1) [ E 0 + (9/13)E 1 ] + (9/13)(S) E 1 + m(l)e 3 + p(s, L, J) 4f } q Jørgese(1962) L 3+ L 2+ W c, W 0, (E 0 E 1 E 3 ), 4f ΔE 1 (q) = I 3 ( W ) (q-1) [E 0 + (9/13)E 1 ] + (9/13){(S) q-1 (S) q }E 1 + {m(l) q-1 m(l) q }E 3 + {p(s, L, J) q-1 p(s, L, J) q} 4f (10-11), (-W 0 )>0 2 W E* (q-1) (-W 0 )=W+E*(q-1) (10-12) 4f [E 0 + (9/13)E 1 ] A (10-13) Jørgese(1962) I 3 ΔE 1 (q) = I 3 =E(4f q-1 ) E(4f q ) W+(E*-A)+ (9/13){(S) q-1 (S) q }E 1 + {m(l) q-1 m(l) q }E 3 + {p(s, L, J) q-1 p(s, L, J) q} 4f (10-14) {(S) q-1 (S) q } N(S) q, {m(l) q-1 m(l) q } M(L) q (10-14-1) {p(s, L, J) q-1 p(s, L, J) q} P(S, L, J) q (10-14) ΔE 1 (q) = I 3 =E(4f q-1 ) E(4f q ) W+(E*-A)+ (9/13) N(S) q E 1 + M(L) q E 3 + P(S, L, J) q 4f (10-14-2), 10-1 Jørgese(1962) (10-14-2) L(III) 74

I 3 Refied spi-pairig eergy theory (RSPET) (10-14-2) I 3 Peppard (1969) L(III) logk d (10-14) E(4f q-1 ) E(4f q ) L 3+ E(4f q ) Jørgese(1962) L 3+ ΔE 2 (q) E(4 f q )aq E(4 f q )org. = ΔW C + q ΔW 0 + (1/2)q(q 1)[E 0 + (9/13)ΔE 1 ] +(S) (9 /13)ΔE 1 + m(l) ΔE 3 + p(s,l,j) Δζ 4 f (10-15) Δ L(III) (9/13)(S)ΔE 1 + m(l)δe 3 0 10-1 (S) m(l) p(s, L, J) Δ 4 Δ 4f 0 10-1 p(s, L, J) p(s, L, J)Δ 4f (9/13)(S)ΔE 1 + m(l)δe 3 Δ 4f 0 (9/13)(S)ΔE 1 + m(l)δe 3 Peppard et al. (1969) Jørgese(1970) Nuget (1970) 10-1 (S) m(l) Slater-Codo-Racah L 3+ E(4f q ) ΔE 2 (q) (10-15) q ΔE 1 (q) = I 3 (10-14) Jørgese(1962) Peppard et al. (1969), logk d = ΔG/(RT), Jørgese RSPET 1-5. Jørgese (10-15) ΔE 2 (q) q ΔE 1 (q) = I 3 (10-14) ΔE 2 (q) (10-15) 75

Jørgese(1970) Nuget (1970) Jørgese(1962) (10-12) (10-13) (10-15) 30) (10-15) ΔE 2 (q) q (Z-S 4f ) ΔE 2 (q) q 2 S 4f S 4f q ΔE 2 (q) Jørgese (Kawabe,1992) 30). (10-15) ΔW C ΔW C ΔW C = 0 (10-16). W 0 (Z S 4 f ) 2 E 0, E 1, E 3 (Z S 4 f ) (10-17) ζ 4 f (Z S 4 f ) 4 (S 4f ) L 3+ ζ 4f S 4f = 32, Z S 4f = q + 25 (10-18) 30) L 3+ q ΔW 0 = (C W + C' W q + C'' W q 2 + )(q + 25) 2 ΔE 0 + (9 /13)ΔE 1 = (C a + C' a q + C'' a q 2 + )(q + 25) ΔE 1 = (C 1 + C' 1 q + C'' 1 q 2 + )(q + 25) (10-19) ΔE 3 = (C 3 + C' 3 q + C'' 3 q 2 + )(q + 25) Δζ 4 f = (C ls + C' ls q + C'' ls q 2 + )(q + 25) 4 4 76

q ΔW 0 + (1/2)q(q 1)[E 0 + (9 /13)ΔE 1 ] = q(q + 25)(a + bq + cq 2 + ) cq 2 (10-15) ΔE 2 (q) ΔY(obs) ΔY(obs) = A 0 (q) + ΔE 2 (q) = A 0 (q) + q(a + bq)(q + 25) + (9 /13)(S)C 1 (q + 25) +m(l)c 3 (q + 25) + p(s,l,j)c ls (q + 25) 4 (10-20) A 0 (q) ΔE 2 (q) L 3+ A 0 (q) A 0 Δ 4f 0 C ls = 0 30) L 3+ ΔY(obs) ΔH 0 f (A) ΔH 0 f (B) = A 0 + q(a + bq)(q + 25) +(9 /13)(S)C 1 (q + 25) + m(l)c 3 (q + 25) (10-21) 3 (A 0, a, b, C 1, C 3 ) ΔE 1 =C 1 (q+25), ΔE 3 =C 3 (q+25) (10-22) E 1 E 3 (10-22) E 1 E 3 L(III) 10 2 (10-21) Jørgese Jørgese-Kawabe 9 y = a 1 " f 1 (t) + a 2 " f 2 (t) + " " " + a " f (t) (10-21) (q+25) Δ(Y) obs /(q + 25) = A 0 /(q + 25) + q(a + bq) + (9/13)(S)C 1 + m(l)c 3 77

=5 (A 0, a, b, C 1, C 3 ). Peppard et al.(1969) L(III) logk d logk d = ΔG/(RT) ΔG (10-21) (Kawabe ad Masuda, 2001) 31). (2) 30 10-(1)1-1, 1-2 20 (3) 11 78

10-1. (4f) q (S) (9/13) E 1 + m(l) E 3 p(s, L, J) 4f (S), m(l), p(s, L, J) L 2+ L 3+ (4f) q (S) m(l) p(s, L, J) --- La 3+ (4f) 0 1 S 0 0 0 0 (La 2+ ) Ce 3+ (4f) 1 2 F 5/2 0 0-2 Ce 2+ Pr 3+ (4f) 2 3 H 4-1 -9-3 Pr 2+ Nd 3+ (4f) 3 4 I 9/2-3 -21-7/2 Nd 2+ Pm 3+ (4f) 4 5 I 4-6 -21-7/2 Pm 2+ Sm 3+ (4f) 5 6 H 5/2-10 -9-3 Sm 2+ Eu 3+ (4f) 6 7 F 0-15 0-2 Eu 2+ Gd 3+ (4f) 7 8 S 7/2-21 0 0 (Gd 2+ ) Tb 3+ (4f) 8 7 F 6-15 0-3/2 Tb 2+ Dy 3+ (4f) 9 6 H 15/2-10 -9-5/2 Dy 2+ Ho 3+ (4f) 10 5 I 8-6 -21-3 Ho 2+ Er 3+ (4f) 11 4 I 15/2-3 -21-3 Er 2+ Tm 3+ (4f) 12 3 H 6-1 -9-5/2 Tm 2+ Yb 3+ (4f) 13 2 F 7/2 0 0-3/2 Yb 2+ Lu 3+ (4f) 14 1 S 0 0 0 0 (La 2+ ) (Gd 2+ ) (4f) q (5d), (5d)(4f) 7 L 2+ L 3+ (4f) q [Xe] 79

0 (S) -10-20 0-10 -20 m(l) 0-2 -4 p(l,s,j ) q 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 10-1. (4f) q J [(S)(9 /13)E 1 + m(l)e 3 + p(l,s,j)ζ 4 f ] (S) m(l) p(l,s,j ). (S) m(l) 80

10-2. LF 3 L(OH) 3 LCl 3 L(OH) 3 (10-21) Jorgese-Kawabe (10-21) 1 2 (10-21) 3 4 LF 3 LCl 3 L 3+ (E 1, E 3 ) L(OH) 3 L(OH) 3 +3F - = LF 3 +3(OH) - ΔH r L 3+ (E 1,E 3 ) 4f 4f L 3+ (1/2)L 2 O 3 (s)=l 3+ (g)+(3/2)o 2- (g) L 3+ (g) O 2- LO 3 (s) L 3+ 6 ΔH r L 3+ 8 LF 3 (s)=l 3+ (g)+3f - (g) ΔH r (2) Jørgese-Kawabe REE Jørgese Jørgese-Kawabe (10-22) ΔY(obs) = A 0 + q(a + bq)(q + 25) + (9/13)(S)C 1 (q + 25) + m(l)c 3 (q + 25) (10-22) 81

ΔY 0 L(III) ΔH r, ΔG r, logk d L 3+ L(III) L 3+ L 3+ L(III) L L L(III) L 3+ 10-2 LF 3 (rhm) L(OH) 3 (hex) LCl 3 (hex) L(III) L L(III) (10-22) L 2-1. REE artifact SiO 2, REE REE REE REE REE log(ree sample /REE chodrite ) Peppard et al. (1969) REE MORB( ) (Kawabe et al., 2008) 32) log(ree sample /REE chodrite ) Jørgese-Kawabe (10-22) 82

La~Ce La~Ce L L McLea (1994) 33) ad, McLea ad Taylor (2012) 34) REE REE REE REE artifact McLea REE REE REE 2-2. REE [lathaite (La, Ce) 2 (CO 3 ) 3 8H 2 O] [kimuraite CaY 2 (CO 3 ) 4 6H 2 O] Nagashima et al. (1986) 35), Akagi et al. (1993 ad 1996) 36, 37), Graham et al (2007) 38) Y REE(III) REE ID-TIMS ICP-AES ICP-MS ID-TIMS REE (Pr, Tb, Ho, Tm) ICP-AES ICP-MS REE (Pr, Tb, Ho, Tm) 83

REE Akagi et al. (1993 ad 1996) REE Jørgese-Kawabe (10-22) REE 2-2-a Graham et al. (2007) 38) Whitiaga Akagi et al. (1994) 37) 10-3. Whitiaga 38) ( ) 37) B CI REE 39) A/B REE Whitiaga REE 84

10-3 B REE CI REE 39) REE 10-3 Whitiaga REE Whitiaga (A) ICP-MS Tm REE B ID-TIMS REE (Pr, Tb, Ho, Tm) Ce La Pr Ce Ce Ce(IV) REE(III) REE(III) REE (A) Tm Er Yb REE (A) (Ce, Tm) Jørgese-Kawabe (10-22) 41) (B), Ce REE (Pr, Tb, Ho, Tm) Ce, Pr, Tb, Ho, Tm (10-22) 5 REE B 10-3 (A/B) Jørgese-Kawabe Ce REE (Pr, Tb, Ho, Tm) (A) (B) REE (B) (A/B) [(La, Ce) 2 (CO 3 ) 3 8H 2 O] REE REE La, Ce 10 (A/B) REE log(a/b) REE 85

log(a/b) (10-22) log(a/b) 10-3 REE Jørgese-Kawabe 2-2-b REE Akagi et l. (1993) 36) 3 REE ID-MS ICP-MS REE 10-4. REE Akagi et l. (1993) 36) 10-3 86

kimuraite-a (middle layer ier layer) kimuraite-b 10-4 10-5 kimuraite-a, -B kimuraite-a (middle layer ier layer) 10-5 ICP-MS ICP-AES REE (Kimuraite-NU) 40) 10-5. REE ICP-MS Akagi et l. (1993) 36) B Ce 10-3 10-4 10-5 REE 87

Type : Kimuraite-A(middle layer), Kimuraite-B Type II: Kimuraite-A(ier layer), Kimuraite-NU I Ce Jørgese-Kawabe (10-22) II (La~Pr) 10-6. Kimuite-NU 40) II REE I Kimuraite-A(middle layer) II La~Pr (kimuraite-b) Kimuraite-A(middle layer) 88

Kimuraite-NU 40) II Ca EPMA (Ca+Y) REE 10 10-6 I Kimuraite-A(middle layer) II Kimuraite-NU REE II La~Pr I REE 10-3 REE REE Akagi et al. (1993) 36) Kimuraite-A(ier layer) Kimuraite-NU Kimuraite-NU Jørgese-Kawabe (10-22) 10-5 10-6 Akagi et al. (1993) 36) Kimuraite-A(middle layer), Kimuraite-B, Kimuraite-A(ier layer) 3 REE Kimuraite-A(ier layer) Kimuraite-NU Nagashima et al. (1986) 35) REE I 10-7 A Jørgese-Kawabe (10-22) Ce Nagashima et al. (1986) 10-7 (A) (B), (C) REE (B) REE (C) Jørgese-Kawabe (10-22) 89

10-7 Nagashima et al. (1986) 35) REE A Kimuraite-A(middle layer) B C Ce Jørgese-Kawabe (10-22) 5 REE 3 Jørgese-Kawabe (10-22) 2 (La~Pr) 3 Jørgese-Kawabe L(III) 90

4f L(III) Jørgese-Kawabe 2-2-c 10-7 (A) B C Peppard et al. (1969), logk d =log(ree org /REE aq ) 2 REE 41) 10-7(A) REE REE (B) (A) REE (B), (C), (C) Peppard et al. (1969) logk=log(ree kimuraite /REE lathaite ) K 41) REE (A) REE Jørgese-Kawabe (10-22) (9/13)(S)C 1 (q+25)+m(l)c 3 (q+25), (C) C 1 C 3 A 0 +q(a+bq)(q+25) (10-22) (A) REE 91

Jørgese-Kawabe (10-22) Jørgese-Kawabe (10-22) REE 10 92

11 2 9 (x,y) x y=ax+b 9 (a±σ a ) (b±σ b ) a b σ a σ b σ a σ b (a±σ a ) (b±σ b ) 9 y x x 9 (x,y) y=ax+b two-error regressio treatmet or method (1) x y=ax+b 9 (2), y x,,2,.,, y=ax+b (a±σ a ) (b±σ b ) (x i, y i ) y ε j ε i (x i, y i ) y=ax+b x 11 y=ax+b ε i y i 93

9 (ii) (a±σ a ) (b±σ b ) 11 y y 1 (ax 1 + b) = ε 1 y 2 (ax 2 + b) = ε 2..... y i (ax i + b) = ε i..... (11-1) y (ax + b) = ε (ε i ) 2 mi. a [ (ε i) 2 ] = 0 (11-2) b [ (ε i) 2 ] = 0 (11-3) 2(y i ax i b)( x i ) = 0 2(y i ax i b)( 1) = 0 2 y i x i + a (x i ) 2 + b x i = 0 (11-4-1) y i + a x i + b 1= 0 (11-4-2) (11-4-2) b = 1 y i a x i (11-5) b (11-4-1) a (x i y i ) (1/)( x i )( y i ) a = (x i ) 2 (1/)( x i ) 2 (11-6) 94

(x i y i ) (1/)( x i )( y i ) = (x i y i ) {(1/)( x i )(1/)( y i )} ( (1/)( x i ) = x, (1/)( y i ) = y ) = (x i y i ) x y = (x i y i ) x y + { x y + x y } ( { x y + x y } = 0. 1 = ) = (x i y i ) x y i x i y + x y = (x i x ) (y i y ) (11-7) (11-6) =1 (x i ) 2 (1/)( x i ) 2 = (x i ) 2 (1/ 2 )( x i ) 2 =1 = (x i ) 2 (x) 2 = (x i ) 2 (x) 2 + { (x) + (x)} (x i ) 2 2 (x) 2 + (x) 2 = (x i ) 2 2 x i x + (x) 2 = (x i x ) 2 (11-8) (11-6) b (11-5) a = ˆ a (x i x ) (y i y ) a a ˆ =1 = (x i x ) 2 (11-9-1) b b ˆ = y a ˆ x (11-9-2) a a ˆ, b b ˆ σ 2 (a), σ 2 (b) y σ 2 (y) 1 1 1 σ 2 (y) = (ε i ) 2 = (y i m ˆ i ) 2 = {y i ( b ˆ + a ˆ x i )} 2 (11-10) ( 2) ( 2) ( 2) m ˆ i = b ˆ + a ˆ x i y (-2) y (-2) =1 =1 95

(11-10) (11-9-2) ˆ b 1 σ 2 (y) = {y i ( ˆ 1 b + a ˆ x i )} 2 = ( 2) ( 2) 1 = ( 2) {y i y a ˆ (x i x )} 2 {(y i y ) 2 2a ˆ (y i y )(x i x ) + a ˆ 2 (x i x ) 2 } a (11-9-1) 1 σ 2 (y) = {(y i y ) 2 a ˆ (y i y )(x i x )} (11-11) ( 2) a y i (11-11) σ 2 (y) σ 2 (a) (11-6) x c (11-8) c = (x i x ) 2 (11-12) (11-6) a a a ˆ = (1/c) { x i y i (1/)( x i )( y i )} = (1/c) { = (1/c) (x i y i x y i ) = (1/c) (x i x )y i x i y i x y i } y i (1/c) (x i x ) σ 2 (a) = {(1/c) (x i x )} 2 σ 2 (y) = σ 2 (y) (x c 2 i x ) 2 (11-12) i= = σ 2 (y) / (x i x ) 2 (11-13) σ 2 (b) (11-9-2) ˆ b = y ˆ a x b ˆ = y a ˆ x = (1/) y i ˆ a x i= 96

σ 2 (b) = (1/ 2 ) σ 2 (y) + (x) 2 σ 2 (a) = (1/) σ 2 (y) + i (x) 2 σ 2 (y) = { 1 (x i x ) 2 + (x) 2 } σ 2 (y) (11-14) (x i x ) 2 x y=ax+b x, y i ˆ a = i i (x i x )(y i y ) (x i x ) 2 (11-15-1) ˆ b = y ˆ a x (11-15-2) 1 σ 2 (y) = {(y i y ) 2 a ˆ (y i y )(x i x )} (11-15-3) ( 2) σ 2 ( ˆ a ) = i 1 σ 2 (y) (11-15-4) (x i x ) 2 σ 2 ( b ˆ ) = { 1 + (x) 2 } σ 2 (y) (11-15-5) (x i x ) 2 i a b σ 2 (y),σ 2 ( a ˆ ),σ 2 ( b ˆ ) 2 DEVSQ = (x i x ), AVERAGE = (1/) xi i i COUNT=, COVAR=(1/) (x i x )(y i i y ) ˆ a, ˆ b, σ( ˆ a ), σ( ˆ b ) 97

(2) y x y=ax+b 9 y x x 9 (x,y) y=ax+b y x y=ax+b Lagrage two-error regressio treatmet 42 43 44) York(1966) 42) (2-1) Lagrage X i Y i (, 2, 3,.., ) y i =ax i +b X i, Y i (X i, Y i ) (, 2, 3,.., ) (x i, y i ) (, 2, 3,.., ) S a, b 42) S = {ω(x i )(X i x i ) 2 + ω(y i )(Y i y i ) 2 } mi. (11-16) ω(x i ) ω(y i ) (X i, Y i ) y (x i, y i ) y=ax+b (X i, Y i ) x 11-2 (X i, Y i ) (, 2, 3,.., ) y i =ax i +b (Y i -y i )/(X i - x i )=(-1/a)ω(X i )/ω(y i ) 98

(residual) y x 11-2 a ω(x i )/ω(y i ) (Y i - y i )/ (X i - x i )=(-1/a)ω(X i )/ω(y i ) (11-17) Lagrage (2-2) (11-16) ω(x i ) ω(y i ) (X i, Y i ) (, 2, 3,.., ) a, b (11-16) (X i, Y i ) y i =ax i +b (11-18) (x i, y i ) (, 2, 3,.., ) S S S(x 1, x 2,.., x, y 1, y 2,.., y ) (δx i, δy i ) (, 2, 3,.., ) S δs (X i, Y i ) (, 2, 3,.., ) a, b (11-16) δs = [( S )δx i + ( S )δy i ] = 0 x i y i δs = {ω(x i )( 2)(X i x i )δx i + ω(y i )( 2)(Y i y i )δy i } = 2 {ω(x i )(x i X i )δx i + ω(y i )(y i Y i )δy i } = 0 2 {ω(x i )(x i X i )δx i + ω(y i )(y i Y i )δy i } = 0 (11-19) (x i, y i ) (, 2, 3,.., ) (11-18) y i =ax i +b δy i = x i δa + aδx i + δb x i δa + aδx i δy i + δb = 0 (11-20) a, b, x i, y i 0 λ i 0 λ i (x i δa + aδx i δy i + δb) = 0 (11-21) 99

(11-21) 0 λ i (x i δa + aδx i δy i + δb) = 0 (11-22) (11-19) (11-22) 0 {ω(x i )(x i X i )δx i + ω(y i )(y i Y i )δy i + λ i (x i δa + aδx i δy i + δb) } = 0 (11-23) {ω(x i )(x i X i ) + aλ i }δx i + {ω(y i )(y i Y i ) λ i }δy i + (λ i x i )δa + λ i δb = 0 (11-24) 0 δx i, δy i, δa, δb 0 ω(x i )(x i X i ) + aλ i = 0 (11-25-1) ω(y i )(y i Y i ) λ i = 0 (11-25-2) (λ i x i ) = 0 (11-25-3) λ i = 0 (11-25-4) a, b, x i, y i 2+2 (11-25-1) (11-25-4) 2+2 a, b, x i, y i λ i Lagrage λ i Lagrage (udetermied multipliers) S. (1976) 18 (11-16) (2-2) 11-2 (11-25-1) (11-25-2) 100

Y i y i λ = i /ω(y i ) X i x i ( aλ i ) /ω(x i ) = 1 ( a) ω(x i) ω(y i ) (11-26) (11-17) 11-3 (X i, Y i ) y i =ax i +b (x i, y i ) (x i, y i ) (11-26) (Y i - y i )/ (X i - x i )=(-1/a)ω(X i )/ω(y i ) (x i, y i ) A ω(x i ) x y 9 B ω(x i ) ω(y i ) (-1/a) y=ax+b ω(x i ) ω(y i )=1 major axis method 42) C ( 7) ω(x i ) 1/σ 2 (x), ω(y i ) 1/σ 2 (y) D ω(y i ) A (x i, y i ) B y y=ax+b D C (X i, Y i ) x 11-3 (X i, Y i ) y i =ax i +b (x i, y i ) (x i, y i ) (X i, Y i ) (-1/a)ω(X i )/ω(y i ) A ω(x i ) x y 9 B ω(x i ) ω(y i ) (-1/a) y=ax+b C D ω(y i ) y x 101

y x ω(y i ) =1/σ 2 (y) σ 2 (y) 0 ω(y i ) y A ω(x i ) = ω(x i ) =1/σ 2 (x) σ 2 (x) 0, ω(x i ) (2-3) Lagrage (11-25-1) (11-25-2) (x i, y i ) (11-18) y i =ax i +b (x i, y i ) Y i + λ i /ω(y i ) = a[x i λ i a /ω(x i )] + b λ i = ax i + b Y i a 2 /ω(x i ) +1/ω(Y i ) = W i (ax i + b Y i ) (11-27) W i = ω(x i ) ω(y i ) a 2 ω(y i ) + ω(x i ) (11-28) (11-25-3) (11-25-4) i (11-25-3) λ i x i = λ i [X i λ i a /ω(x i )] = λ i X i (λ i ) 2 a /ω(x i ) = 0 11-27 W i (ax 2 i + bx i Y i X i ) + W 2 i (ax i + b Y i ) 2 ( a) /ω(x i ) = 0 (11-25-4) λ i = W i (ax i + b Y i ) (11-29) = 0 (11-30) (11-30) b W i b = W i Y i W i a X i W i Y i aw i X i b = W i (11-31) 102

b (11-29) a Y = W i Y i, X = W i X i W i W i (11-32) Y i X i b = Y ax (11-33) (11-29) b a W i [ax 2 i + (Y ax )X i Y i X i ] + W 2 i [ax i + (Y ax ) Y i ] 2 ( a) /ω(x i ) = 0 (11-34) W i [ax i 2 + (Y ax )X i Y i X i ] = W i [ax i (X i X ) (Y i Y )X i ] = W i X i [a(x i X ) (Y i Y )] = W i X i (au i V i ) (11-35) U i X i X, V i Y i Y (11-36) (X,Y ) x y W i 2 [ax i + (Y ax ) Y i ] 2 ( a) /ω(x i ) = W i 2 [a(x i X ) (Y i Y )] 2 ( a) /ω(x i ) = W i 2 (au i V i ) 2 ( a) /ω(x i ) = W 2 i (a 2 U 2 i 2aU i V i + V 2 i )( a) /ω(x i ) (11-37) (11-34) 103

[W i (au i V i )] X i + [W i (au i V i )] 2 ( a) /ω(x i ) = 0 (11-38) [W i (au i V i )] 1 [W i (au i V i )] 2 2 0 [W i (au i V i )] (X,Y ) W i (11-35) (11-37) a a 2 2 W a 3 i ω(x i ) U 2 W i + 2a 2 i ω(x i ) U iv i +a[ W i 2 W i X i U i ω(x i ) V 2 i ] W i X i V i = 0 (11-39) X i 3 5 X i (= U i + X ) 3 4 aw i X i U i W i X i V i = aw i (U i + X )U i W i (U i + X )V i 0 = aw i U 2 i + aw i X U i W i U i V i W i X V i 2 = aw i U i W i U i V i + aw i X U i W i X V i 2 = aw i U i W i U i V i + W i X (au i V i ) W i X (au i V i ) = W i X [a(x i X ) (Y i Y )] = W i X (ax i Y i + Y ax ) (b = Y ax ) = W i X (ax i + b Y i ) = X W i (ax i + b Y i ) = X W i i W i i (11-38) (-1) W i (ax i + b Y i ) = X ( W i i )(ax + b Y ) = 0 11-40 2 2 2 W a 3 i ω(x i ) U 2 W i 2a 2 i ω(x i ) U V 2 W i i a[ W i U i i ω(x i ) V 2 i ] + W i U i V i = 0 (11-41) 104

a a 3 (11-28) W i a a (2-4) Newto (11-41) f(a), f (a) = 0 (11-42). a 0 (11-45) y 9 (11-41) a 0 Taylor f (a) = f (a 0 ) + f '(a 0 ) (a a 0 ) + (1/2) f ''(a 0 ) (a a 0 ) 2 +... f (a) = 0 = f (a 0 ) + f '(a 0 ) (a a 0 ) a = a 0 f (a 0 ) / f '(a 0 ) (11-43) a a 1 a 1 = a 0 f (a 0 ) / f '(a 0 ) k a k +1 = a k f (a k ) / f '(a k ) (11-44) a (11-44) Newto Newto-Raphso 20) (11-41) f(a) f (a) f '(a k ) f (a k ) f (a k 1 ) a k a k 1 (11-44) a k +1 a k [ a k a k 1 f (a k ) f (a k 1 ) ] f (a k) = a f (a ) a f (a ) k 1 k k k 1 f (a k ) f (a k 1 ) b (11-44 ) (11-33) 105

(2-5) a, b a, b a, b (11-38) [W i (au i V i )] X i + [W i (au i V i )] 2 ( a) /ω(x i ) = 0 (11-38) [W i (au i V i )] W i (X,Y ) 1 2 (11-40) W i (au i V i ) X = 0 [W i (au i V i )] (X i X ) = W i (au i V i ) U i 0 a a W i U 2 i W i V i U i a ( W i V i U i ) /( W i U 2 i ) (11-45) b (11-33) b = Y ax b = Y X ( W i V i U i ) /( W i U 2 i ) (11-46) a, b b = Y ax b a a b σ 2 (b) = ( b a )2 σ 2 (a) = (X ) 2 σ 2 (a) (11-47). X, Y a a (11-16) S S = {ω(x i )(X i x i ) 2 + ω(y i )(Y i y i ) 2 } (11-16) (11-25-1, -2) x i y i i (11-27) S = W i (ax i + b Y i ) 2 = W i (au i V i ) 2 = (ε i ) 2 (11-48) b = Y ax b (ε i ) 2 106

S = (ε i ) 2 = [ W i (au i V i )] 2 (11-49) ε i = W i (au i V i ) (11-50) S S = (ε i ) 2 = ( 2){ (ε i ) 2 }/( 2) = ( 2)σ 2 (ε) (11-51) 1 σ 2 (ε) = S /( 2) = ( 2) W i(au i V i ) 2 (11-52) (11-50) δε i = ( ε i / a)δa = [ W i (au i V i ) a = [ W i U i + (au i V i ) ( W ) i ]δa a W i U i δa + (au i V i ) ( W ) i ]δa a a δε i = W i U i δa (δε i ) 2 = W i U 2 i (δa) 2 δε i = ε i ε i, ε i 0 δε i ε i (δε i ) 2 (ε i ) 2 = ( 2)σ 2 (ε) = S = W i U 2 i (δa) 2 = (δa) 2 2 W i U i δa = a a (δa) 2 = (1/)(a a ) 2 1= (1/) (a a ) 2 (1/) (a i a ) 2 σ 2 (a) (11-54) σ 2 2 (a) S / W i U i = W i (au i V i ) 2 2 / W i U i σ 2 (ε) σ 2 (a) (11-53) (11-54) (11-55) (11-56) 107

σ 2 2 (a) {( 2) / W i U i } σ 2 (ε) (11-57-1) 1 σ 2 (ε) ( 2) { W 2 iu i }σ 2 (a) (11-57-2) σ 2 (a), σ 2 (b) York (1966) York (1966) σ 2 (a), σ 2 (b) S σ 2 (a) ( 2) / W 2 iu i = W i (au i V i ) 2 /{( 2) W i U 2 i } (11-58) σ 2 2 (b) { W i X i / W i } σ 2 (a) (11-59) i. York(1966) y=a+bx (11-58), (11-59) y=ax+b York (1966) (11-58) (11-56) σ 2 (a) 1/( 2) (11-47) σ 2 (b) = (X) 2 σ 2 (a) York (1966) (11-59) (2-6) a, b (11-56) ε i a. (11-50) δε i = [ W i U i + (au i V i ) ( W ) i ]δa a (δε i ) 2 = [ W i U i + (au i V i ) ( W i ) ] 2 (δa) 2 (11-60) a ( 2)σ 2 (ε) = S = (ε i ) 2 = σ 2 (a) [ W i U i + (au i V i ) ( W i ) a σ 2 (a) = S [ W i U i + (au i V i ) ( W i ) a ] 2 ] 2 (11-61) 108

W i a = [ 2a ω(x i ) ](W i) 3 / 2 (11-62) (11-61) (11-50) ε i W i U i b (11-47) b = Y ax (11-33) δb = ( b )δa = ( Y a a X a X a )δa (11-63) σ 2 (b) = ( b a )2 σ 2 (a) = ( Y a X a X a )2 σ 2 (a) (11-64) Y = ( W i Y i i ) / W i i, X = ( W i X i i ) / W i i (11-32) W i W i a Y a = Y W i, i W i a X a = X W i (11-64) i W i a Y = [Y i ( W i ) ( W W i i i Y i )]/( W i i ) 2 = V i /( W i i ) i (11-65) X = [X i ( W i ) ( W W i i i X i )]/( W i i ) 2 = U i / W i i (11-66) i W i a = W 2 i /ω(x i ) (11-67) ( Y a X a X a ) (11-61) σ2 (a) σ 2 (b) ( Y a X a X a ) = (1/ W i i) [ W i ω(x i ) (au V )] i i i X 2 = {X [ W 2 i ω(x i ) (au i V i ) /( W i i i )]} 109

σ 2 (b) = ( Y a X a X a )2 σ 2 (a) = {X [ W 2 i ω(x i ) (au V ) i i i /( W i i )]} 2 σ 2 (a) (11-68) X (11-47) (11-68) York(1966) 42) two-error regressio method York(1969) 43) σ 2 (y) σ 2 (x) Brooks et al. (1972) 44, ω(x i ) 1/σ 2 (x), ω(y i ) 1/σ 2 (y) 11-3 C σ 2 (a), σ 2 (b) 11 (2) York(1966) 42) Brooks et al. (1972) 44) 40 3 (3)-1 X Y major axis 1974 X Y X Y y X Y 11 3 major axis X Y 110

(3)-2 Chemical Abstract Chem. Ab. Brooks et al. (1972) 44) Realistic use of two-error regressio treatmets as applied to 45) Google two-error regressio Net 46) Net 111

3 40 (3)-3 Wedt-2 Brooks et al. (1972) 44) York(1966) 42) 11-(2-5) Brooks et al. (1972) Wedt-2 Brooks et al. (1972) 44) Wedt Busesastalt für Bodeforshug Wedt-2 Wedt Wedt-2 FORTRAN Wedt X Y Wedt-2 figure captio Wedt-2 York York Wedt-2 40 York Wedt-2 10 (2) 30 112

113

+ I = exp( x 2 ) dx,. I 2 = + exp( x 2 )dx exp( y 2 )dy = exp[ (x 2 + y 2 )]dxdy (A1-1) + + + x y r θ x=r cosθ y= r siθ ( A1-1) r = (x 2 + y 2 ) 1/ 2 xy 0 r <, 0 θ < 2π rdθdr dxdy=rdθdr A1 1 Jacobo J y dθ θ rdθ dr rdθ dy dr θ dx O x A1-1 dxdy = J drdθ J = (x,y) (r,θ) = x r y r x θ = y θ (A1-1) + + cosθ rsiθ siθ rcosθ = r(cos2 θ + si 2 θ) = r J = r (A1-2) I 2 = exp[ (x 2 + y 2 )]dxdy = exp( r 2 )rdrdθ 2π r= 0 0 = exp( r 2 )rdr dθ = 2π exp( r 2 )rdr. (A1-3) r= 0 2π 0 r= 0 113

r 2 = u du /dr = 2r I = I 2 = 2π exp( r 2 )rdr = π exp( u)du = π[ e u ] 0 = π. (A1-4) r= 0 u= 0 + exp( x 2 )dx = π. (A1-5) exp( x 2 ) x=0 (0, ) (A1-5) + exp( x 2 )dx = π 2. (A1-6) 0 ( 1-6) exp( x 2 ), I 0 = + 0 exp( αx 2 )dx u = α x du = α dx (1-6) I 0 = + exp( αx 2 )dx = 1 2 π α 0 exp( αx 2 ) x, I 1 = + 0 (A1-7) x exp( αx 2 )dx,, u = αx 2 du /dx = 2α x I 1 = + x exp( αx 2 )dx = 1 + 2α exp( u)du = 1 2α 0 0 (A1-8) exp( αx 2 ) x 2, I 2 = I 3 = + + 0 x 2 exp( αx 2 )dx x 3 exp( αx 2 )dx, I 4 = x 4 exp( αx 2 )dx,... 0 + 0 114

I = + 1 x exp( αx 2 )dx = x ( 2α x ) d[exp( α x 2 )] dx (A1-9) dx + 0 (A1-9) 0 I = ( 1 + 1 2α ){[x dx 1 exp( α x 2 )] + 0 ( dx ) exp( α x 2 )dx} 2 0 I = ( 1 2α ) + 0 x 2 exp( α x 2 )dx = ( 1 2α ) I 2 (A1-10) (A1-7) (A1-8) I 0 I 1 (A1-10) I = ( 1 2α ) I 2 I ( 2) + I 0 = e α x 2 dx = ( 1 2 )(π α )1/ 2 0 + I 1 = x e α x 2 dx = 1 2α 0 + I 2 = x 2 e α x 2 dx = π 1/ 2 0 4α 3 / 2 + I 3 = x 3 e α x 2 dx = 1 2α 2 0 I 4 = x 4 e α x 2 dx = + 0 3π 1/ 2 8α 5 / 2 (A1-11) 115

Jacobi x, y (u,v) x = p(u,v), y = q(u,v) (A2 1) x, y f (x, y) x, y (A2 1) u, v u, v dxdy dudv x = x(u) x dx = (dx /du)du u (dx /du) (A 2 1) (u,v) u = p 1 (x, y), v = q 1 (x, y) (A 2-2) u v xy v u A2-1 x y (u,v) (u,v) y M 4 u+du u dσ M 1 M 2 M 3 v v+dv O A2-1. (u,v) x 116

A2-1 (u,v) dσ M 1 M M M dσ M 1 M M M p(u,v), q(u,v) M 1 x 1 = p(u,v), y 1 = q(u,v) M 2 M 3 M 4 x 2 = p(u + du,v) = p(u,v) + p(u,v) u y 2 = q(u + du,v) = q(u,v) + q(u,v) u du du x 3 = p(u + du,v + dv) = p(u,v) + p(u,v) u y 3 = q(u + du,v + dv) = q(u,v) + q(u,v) u x 4 = p(u,v + dv) = p(u,v) + p(u,v) v dv du + p(u,v) v du + q(u,v) v dv (A 2-3) dv y 4 = q(u,v + dv) = q(u,v) + q(u,v) v M 1 M M M dv y dσ M 1 a 1 θ a 2 M 3 O M 2 A2-2 x M 1 M M A2-2 M 2 117

M 1 a 1 M M a 2 dσ = a 1 a 2 siθ (A2-4) 2 (dσ) 2 = (a 1 ) 2 (a 2 ) 2 si 2 θ = (a 1 ) 2 (a 2 ) 2 (a 1 ) 2 (a 2 ) 2 cos 2 θ (A2-5) a b a b cosθ = (a,b) (A2-5) (dσ) 2 = (a 1,a 1 ) (a 1.a 2 ) (a 2,a 1 ) (a 2,a 2 ) (A2-6) a 1 = a 11 i + a 21 j, a 2 = a 12 i + a 22 j x y (A2-6) (dσ) 2 = (a 11 )2 + (a 21 ) 2 a 12 a 11 + a 22 a 21 a 12 a 11 + a 22 a 21 (a 12 ) 2 + (a 22 ) 2 = (a 11 ) 2 (a 22 ) 2 + (a 21 ) 2 (a 12 ) 2 2(a 12 )(a 11 )(a 22 )(a 21 ) = a 2 11 a 12 (A2-7) a 21 a 22 dσ = a 11 a 12 a 21 a 22, (A2-8) M 2 M 1 a 1 M M a 2 (A2-3) a 11 = x 1 x 2 = p(u,v) u a 12 = x 3 x 2 = p(u,v) v du, dv, a 21 = y 1 y 2 = q(u,v) u a 22 = y 3 y 2 = q(u,v) v dv, du, (2-8) dσ = p(u,v) u q(u,v) u du du p(u,v) dv q(u,v) v dv v = p(u,v) u q(u,v) u p(u,v) v dudv q(u,v) v (A2-9) (2-9) dσ /(dudv) Jacobi 118

(Jacobia determiat) Jacobia determiat p dσ = (p,q) ( (u,v) dudv = u ) ( q u ) ( p v ) ( q v ) dudv (A2-10) (2-10) (p,q) (u,v) J J (A2 10) f (x, y)dσ = ( p,q) F(u,v) (u,v) dudv (A2-11) σ σ (A2 1) x = p(u,v), y = q(u,v) (2-11) (p,q) (u,v) ( p,q) (u,v) (x,y) x = x(u) (u,v) x dx = (dx /du)du u (dx /du), (A2-11) ( p,q) (u,v) = (x,y) (u,v) = J f = f (x, y,z,...), g = g(x, y,z,...), h = h(x,y,z,...),... (A2-12) dfdgdh = ( f,g,h,...) (x, y,x,...) dxdydz (A2-13) Jacobi (,) 119

J ( f,g,h,...) (x, y,z,...) = ( f x ) ( g x ) ( f y ) ( g y ) ( f z ) ( g z ) ( h x ) ( h y ) ( h z ) (A2-14) (A2-13) Jacobi (x,y) dxdy (r,θ) dxdy = r drdθ Jacobi x = r cosθ y = r siθ (A2-15) J (x, y) (r,θ) = x r y r x θ = y θ cosθ rsiθ siθ rcosθ = rcos2 θ + rsi 2 θ = r (A2-16) dxdy = J drdθ = (x, y) drdθ = r drdθ (r,θ) (A2-17) (x,y,z) 120

dxdydz (r,θ,ϕ) dxdydz = r 2 siθ drdθdϕ Jacobi (x,y,z) (r,θ,ϕ) x = r siθ cosϕ y = r siθ siϕ z = r cosθ (A2-18) A2-3 z θ (x,y,z), (r,θ,ϕ) O ϕ r y x A2-3 (A2-18) (2-13) Jacobi J = (x, y,z) (r,θ,ϕ) = x r y r z r x θ y θ z θ x ϕ y = ϕ z ϕ siθ cosϕ r cosθ cosϕ rsiθsiϕ siθsiϕ r cosθsiϕ rsiθ cosϕ cosθ rsiθ 0 = r 2 siθ (A2-19) (A2 13) dxdydz = J drdθdϕ = (x, y,z) (r,θ,ϕ) drdθdϕ = r2 siθ drdθdϕ 121

1 a 1, 2 a 2, 3 a 3, 1 2 3 A3 r a r 1 2 (-1) r [-(r-1)] ( 1) ( 2) [ (r 1)] = ( 1) ( 2) [ (r 1)] ( r) [ (r + 1)] 2 1 ( r) [ (r + 1)] 2 1 =! ( r)! P r (A3-1) (permutatio) r= r=0 0! 1, P 0 1 (A3-2) 122

r I II 1 2 3 r a 1 a 2 a 2 a r II A3-2 r r r P r r II II r II II r r P r I II r P r α r!= P r (A3-3) r 123

α = P r r! =! ( r)!r! = C r (A3-4) (combiatio) (radom variable) x N (average, mea) x =< x >= 1 N N x i (A3-5) (stadard deviatio) 2 (variace) N N σ 2 = 1 (x i x ) 2 =< (x i x ) 2 > (A3-6) N < > x j f j p j x j f j x j N N N N j x j p j N p j = f j /N. N' x j p j = x j f j N j=1 N' j=1 = 1 N N' j=1 x j f j = 1 N N x i = x x j p j (expectatio) (A3-6) x j p j 124

σ 2 = 1 N N N' (x i x ) 2 = 1 f j (x j x ) 2 = (x j x ) 2 ( f j N N ) = (x x j )2 j=1 N' σ 2 = (x j x ) 2 p j = [(x j ) 2 2x j x + x 2 ] N' j=1 j=1 N' N' N' j=1 p j = (x j ) 2 p j 2x x j p j + (x) 2 p j j=1 j=1 j=1 N' = (x j ) 2 p j 2(x ) 2 + (x) 2 = (x j ) 2 p j (x) 2 j=1 j=1 x j p j x = 1 N N N' x i = x j p j =< x > (A3-7) j=1 N σ 2 = 1 (x i x ) 2 = (x j x ) 2 p j = (x j ) 2 p j (x) 2 =< x 2 > < x > 2 (A3-8) N N' j=1 N' j=1 N' N' N' j=1 p j p p = q P() P() (biomial distributio) P() = N C p q (N ) = N!!(N )! p (1 p) (N ) (A3-9) N (N-) N C = N!!(N )! p q (N ) P() (A3-9) 125

(biomial theorem) (a + b) = r= 0! r!( r)! a r b r = a + a 1 b + ( 1) 2! a 2 b 2 + ( 1)( 2) 3! a 3 b 3 + + ab 1 + b (a + b) (A3-9) (p + q), (p + q) N N N N! (p + q) N =!(N )! p q (N ) = P() (A3-10) = 0 = 0 (p + q) = p + (1 p) = 1 N P() = 1 (A3-11) = 0 N P() (A3-9) < >= < >= = P() = = N =1 N = 0 N = 0 N!!(N )! p q (N ) N! ( 1)!(N )! p q (N ) = Np N =1 (N 1)! ( 1)!(N )! p 1 q (N ) N 1 (N 1)! = Np!(N 1)! p q (N 1) = Np(p + q) N 1 = Np (A3-12) = 0 < >= N p σ 2 () N' (A3-8) σ 2 = (x j ) 2 p j (x) 2 j=1 N σ 2 () = 2 P() (< >) 2 = 0 (A3-13) 126

N 2 P() p ( p ) p p, p ) p [p (p p ] = p p (p ) = 2 p = 0 N N N! 2 P() = 2 = 0 = 0 =!(N )! p q (N ) = N = 0 N = 0 N!!(N )! p p [ p p ]p q (N ) N!!(N )! ( 2 p )q (N ) (p + q) = 1 p p [p N p ] N!!(N )! p q (N ) = p p ( p N p ) P() = 0 = 0 = p p (p )(p + q)n p = p p [pn(p + q)n 1 ] = p[n(p + q) N 1 + pn(n 1)( p + q) N 2 ] = p p [pn(p + q)n 1 ] = pn + p 2 N(N 1) (A3-14) (A3-12) < >= Np N σ 2 () = 2 P() (< >) 2 = pn + p 2 N(N 1) p 2 N 2 = 0 = pn p 2 N = Np(1 p) = Npq (A3-15) N < >= σ 2 () < >= = Np, σ 2 () = Npq = Np(1 p) (A3-16) σ() < > = σ() = q Np (A3-17) 127

p q N σ() / < > 1/ N N 1/ N (A3-16), (A3-17), p=q=1/2 N=10 20 σ() / = 10 10 N~10 20!= 1 2 3... l!= l1+ l2 + l3+...+ l (A3-18) A3-3 l x l x 1 2 3 4 5 6 7 8 x A3-3 128

y lx >>1 l!= l xdx = [x l x x] 1 = l + 1 l (A3-19) 1 (A3-19) l 2π l!= l xdx l + l 2π (A3-20) 1 Γ( + 1) =! x Γ(x + 1) 2πx x x e x (A3-19) (A3-20) (Stirlig) (A3-19) (A3-20) =10, 50, 60 =10 l10!= 15.1044, 10l10 10 = 13.025, 10l10 10 + l 2π 10 = 15.095, =50 l50!= 148.478 50l50 50 = 145.6011, 50l50 50 + l 2π 50 = 148.476, =60 l60!= 188.628 60l60 60 = 185.661, 60l60 60 + l 2π 60 = 188.627 (A3-20), =10 =50 0.06 ~ 0.001% (A3-19) =50 1.9%, =60 1.6 =1,000 0.06%, =10,000 0.01 (A3-20) (A3-19) (A3-20) (A3-20) (A3 19) (A3 20) (A3 20) = l 2π l + l 2π (A3-21) 129

3-3), N (N>>1 >>1) (A3-9) P() = N C p q (N ) = N!!(N )! p (1 p) (N ) (A3-9) N>>1 >>1 3-4) lp() = ln! l! l(n )!+ l p + (N )l(1 p) (A3-22) (A3 19) l! l (1 p) = q lp() = N ln l (N )l(n ) + l p + (N )lq (A3-23) (N, p q p) P() (A3-23) lp() (N ) p = l + l(n ) + l p lq = l q (A3-24) P() lp() lp() (N ) p q (N ) p = l = 0 q = 1 (A3-25) (N ) p = (1 p) = N p 3-3) (A3-12) =< >= Np 130

P() lp() =< >= Np P() lp() = =< >= Np (A3-26) x = + x P() = P( + x) lp() = lp( + x) = Np x lp( + x) = lp( ) + x( lp ) = + 1 2 x 2 ( 2 lp 2 ) = +... (A3-27) (A3-25) ( l P ) = = 0 x x (A3-24) 2 lp() 2 = 1 1 (N ) = N (N ) = Np ( 2 l P N ) 2 = = Np(N Np) = 1 Np(1 p) = 1 Npq (A3-28) 3-3) P() 2 Npq = Np(1 p) = σ 2 (A3-15) (A3-27) lp() = lp( + x) lp( ) 1 2σ 2 x 2 (A3-29) lp() = lp( + x) lp( ) + le ( 1 2σ x 2 ) ( 1 2 = l{p( ) e 2σ 2 x 2 ) } 131

P() = P( + x) = P( ) e ( 1 2σ x 2 ) 2 (A3-30) (A3-11) + P() = 1 = P( ) e ( 1 2σ x 2 ) + ( 1 2 dx = P( ) e 2σ x 2 ) 2 dx all + 1 P( ) e ( 2σ x 2 ) 2 dx = 1 (A3-31) + I 0 = e α x 2 dx = ( 1 2 )(π α )1/ 2 0 α = 1/(2σ 2 ) + e ( 1 2σ x 2 ) 2 dx = 2 e ( αx 2 ) dx = 2I 0 = ( π α )1/ 2 = (2σ 2 π) 1/ 2 = 2π σ (A3-31) + 0 P( ) = 1 2πσ 2 (A3-32) (A3-30) P() = 1 2πσ 2 e( 1 2σ 2 x 2 ) = 1 2πσ 2 e[ ( ) 2 2σ 2 ] (A3-33) = =< >= Np σ 2 = Npq = Np(1 p) (Normal distributio ) (Gaussia distributio) N(m, ) 132

N Normal distributio (A3-33) x ϕ(x) = 1 2πσ 2 e[ (x m) 2 2σ 2 ] (A3-34) N(m, ) x (x, x + dx) ϕ(x)dx x a x b Φ(a x b) Φ(a x b) = b ϕ(x)dx = a b 1 ] 2σ 2 dx (A3-35) a 2πσ 2 e[ (x m) 2 m σ x m + σ 68, m 2σ x m + 2σ 95.4%, m 3σ x m + 3σ 99.7% A3 m ϕ(x) ±σ ±2σ ±3σ x A3-4 b m + bσ ϕ(x)dx = m + aσ 1 ] 2σ 2 dx (A3-36) m + bσ m + aσ 2πσ 2 e[ (x m ) 2 133

t = x m σ (A3-37) dx = σ dt m + bσ ϕ(x)dx = m + aσ = m + bσ m + aσ 1 1 2π σ 2πσ 2 e[ (x m ) 2 2σ 2 t 2 2 ) σ dt ] dx b e( = 1 b a 2π a e ( t 2 2 ) dt N(0,1) = (A3-38) 1 t 2 2π e( 2 ) N(m, ), (A3-37) N(0, 1) x t 1 2 2 ) dt (A3-39) 0 2π e( m σ x m + σ 68 N(0,1) = 1 t 2 2π e( 2 ) (A3-39) 0 x. (A3-39) 7). 134

Γ(x) Γ(x) = e t t x 1 dt (A4-1) t= 0 x p Γ( p) = e t t p 1 dt = (t p 1 )( e t )'dt t= 0 t= 0 = t p 1 ( e t [ )] t= 0 u (dv /dx)dx = u v v (du /dx)dx + ( p 1) e t t p 2 dt = (p 1) e t t p 2 dt = (p 1)Γ(p 1) (A4-2) t= 0 t= 0. Γ(1) = Γ( p) = ( p 1)Γ( p 1) (A4-3) e t dt = ( e t )'dt t= 0 t= 0 = [ e t ] t= 0 = 0 ( 1) = 1 Γ(1) = 1 (A4-4) Γ(1/2) = e t t 1/ 2 dt t = x 2 (A1-7) t= 0 e x 2 x 1 2x dx = 2 e x 2 dx = π 0 0 Γ(1/2) = π (A4-5) (A4-3) (A4-4) Γ(2) = Γ(1), Γ(3) = 2Γ(2) = 2 1, Γ(4) = 3 Γ(3) = 3! 135

Γ( + 1) =! (A4-5) (A4-3) (A4-5) Γ(3/2) = (1/2)Γ(1/2) = (1/2) π Γ(5 /2) = (3/2)Γ(3/2) = (3/2) (1/2) π Γ(7 /2) = (5 /2)Γ(5 /2) = (5 /2) (3/2) (1/2) π Γ( 2 + 1 ) = ( 2 2( 1) + 1 )Γ( 2 2( 1) + 1 ) 2 = = = (2 1) (2 3) 7 5 3 1 2 π (2)(2 1) (2 2) (2 3) 7 6 5 4 3 2 1 2 (2)(2 2)(2 4) 6 4 2 (2)! 2 2 ()( 1)( 2) 3 2 1 π π = (2)! 2 2! π (A4-6) (A4-5) Γ( + 1) =! x x!= Γ(x + 1) 2πx x x e x [1+ 1 12x + 1 288x 2 + ] (A4-7) (A4-7) 136

l(!) l() 3 137

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Bevigto ad Robiso (2002) 1989 141