2変量データの共分散・相関係数・回帰分析
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- あおい しばもと
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1 2, 1, Excel 2, Excel ( ) L04 2 I(2017) 1 / 24 2 I L04( Wed) : Time-stamp: Tue 23:02 JST hig
2 L03-Q1 L03-Q2 Quiz : 1.6m, m 2, 0.05m. L03-Q3 Quiz : Sx 2 = 4, S x = 2. z = (87 90)/2 = 1.5. w = ( 1.5) = 35. x = 90, ( ) L04 2 I(2017) 2 / 24
3 Excel ( ) L04 2 I(2017) 3 / 24
4 (x, y). x, y. x y ( ) z J Div n = 18( ). ( ) x y z (x, y) =( (cm), (kg)), ( ( ), (m 2 ), (, ), (, ).... ( ) L04 2 I(2017) 4 / 24
5 2 2 = 5.2.2? ( ) L04 2 I(2017) 5 / 24
6 2 2 x:, y ( ) y \x ( ) L04 2 I(2017) 6 / 24
7 Excel ( ) L04 2 I(2017) 7 / 24
8 Y X 2 2 Y X Y X Y X Y X r = 0.99 r = 0.55 r = 0 r = 0.55 r = 0.99 : x y : x y / : / r: r xy.. ( ) L04 2 I(2017) 8 / 24
9 2 2 I x x = 1 N x Sx 2 = 1 N y, Sy 2. N i=1 x i N (x i x) 2 = 1 N i=1 N (x i x)(x i x) i=1 (covariance) x, y C xy = 1 N N (x i x) (y i y) i=1 : C xy = S xy, x S 2 x = S xx, y S 2 y = S yy. ( ) L04 2 I(2017) 9 / 24
10 2 2 p.110 Y (,+) (+,+) Y の平均値 (, ) (+, ) X Xの平均値 (+, ) = (x i x, y i y ). / / (?) ( ) ( ) L04 2 I(2017) 10 / 24
11 2 2 I p.111 x, y 1 5.4(p.112),,. (correlation coefficient) x, y r = C xy S x S y ( ) L04 2 I(2017) 11 / 24
12 2 2 1 r (p.114) r = 0 ( ) r = ±1 / y x (p.115) r x, y 1 5.6(p.114) ( ) L04 2 I(2017) 12 / 24
13 2 2 L04-Q1 Quiz( ( )) (xg, ycm) 1 x, y 2 x, y., y 122 = 5 = 4.94(cm). x(g) y(cm) ( ) L04 2 I(2017) 13 / 24
14 Excel ( ) L04 2 I(2017) 14 / 24
15 (regression), = =1 2 (x, y) r = ±1 (x, y) ( ) y = ax + b! a, b. shoot.received FK y: ( ) x: ( )? x y ( ) L04 2 I(2017) 15 / 24
16 d 2 n n L(a, b) = d 2 i = (y i (ax i + b)) 2 L Y a = L b i=1 i=1 = 0 a, b. I X ( ) L04 2 I(2017) 16 / 24
17 2 L(a, b) = N(1 r 2 )Sy. 2 ( ) L04 2 I(2017) 17 / , (5.11) x i, y i (i = 1,..., n) x, y, S x, S y, r., y = r S y S x (x x) + y = ax + b. a = r Sy S x = Cxy, b = ( (x, y) ) Sx 2 a: (x 1 y ) r 2 : ( ) 5.2.4
18 2 I, 2S x, 2S y Sy S x?,, 0, r. (x, y) (m,kg). r.. y (kg). r Sy(kg) S x(m) x(m) + b(kg), S x /S y. ( ) L04 2 I(2017) 18 / 24
19 2 L04-Q2 Quiz( ) 2 (x, y). x x 9 y y 4 x s 2 x 49 y s 2 y 36 x, y s xy 25 (x, y) n 16,, x, y.. ( ) L04 2 I(2017) 19 / 24
20 2 Excel Excel ( ) L04 2 I(2017) 20 / 24
21 2 Excel R... SPSS. Excel,. Office365. Excel. >Excel 2016 > > > Excel > > OK. ( ) L04 2 I(2017) 21 / 24
22 2 Excel (Excel) I,,, > > average, > > var.p, stdev.p, mode > > > median, quartile, > > frequency + > > = > >, > > >, covar=covariance.p, correl > > linest > > >, =, n 1 n, var.p. ( ) L04 2 I(2017) 22 / 24
23 2 Excel Excel, 1 ( ).,,. 2 (n ), S xy r xy S xx S yx S xy S yy, r xx r xy r yx r yy. S yy r yy, y = x S xy, r., S yy = Sy, 2 r yy = 1. n 3 n n. R = r R2 = r 2 = b X 1 = a n 3 (x1, x 2,..., x n 1, y), X 2,. ( ) L04 2 I(2017) 23 / 24
24 2 Excel , trial. Excel : ,,, ( ) trial , ( ) (1-539) 4(1-502), Math - (1-614) 1.4, 2.1, 2.2, 2.3. ( ) L04 2 I(2017) 24 / 24
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データの分布
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B L11(2017-07-03 Mon) : Time-stamp: 2017-07-03 Mon 11:04 JST hig,,,.,. http://hig3.net ( ) L11 B(2017) 1 / 28 L10-Q1 Quiz : 1 6 6., x[]={1,1,3,3,3,8}; (. ) 2 x = 0, 1, 2,..., 9 10, 10. u[]={0,2,0,3,0,0,0,0,1,0};
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! L02(2013-10-02 Wed) 1 (median),, 2 (mode),, 3 (mean),, http://hig3net () L02!(2013) 1 / 24 : 1 : 2 Excel and more () L02!(2013) 2 / 24 : () L02!(2013) 3 / 24 : L01-Q5 Quiz : 1 Q2= (60 + 70)/2 = 65 Q1
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yoshi@image.med.osaka-u.ac.jp http://www.image.med.osaka-u.ac.jp/member/yoshi/ II Excel, Mathematica Mathematica Osaka Electro-Communication University (2007 Apr) 09849-31503-64015-30704-18799-390 http://www.image.med.osaka-u.ac.jp/member/yoshi/
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