R R-console R R Rscript R-console GUI 1
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- みいか さわなか
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1 November 2015
2 R R-console R R Rscript R-console GUI 1
3 2 X Y
4 x = [x 1, x 2,..., x n ] y = [y 1, y 2,..., y n ] n x 1, x 2,... x = x 1 x 2. x n, y = T x = [x 1, x 2,..., x n ] T y 1 y 2. y n
![.., y n ] n x 1, x 2,.](/docs-images/47/20506032/images/page_4.jpg ".. x = x 1 x 2. x n, y = T .")
5 2 x, y
6 x = 1 n (x 1 + x x n ) ȳ = 1 n (y 1 + y y n )
7 ? 3 x 1, x 2, x 3 µ 3 S S = (x 1 µ) 2 + (x 2 µ) 2 + (x 3 µ) 2 S µ µ µ
8 S µ ds dµ = 2(µ x 1) + 2(µ x 2 ) + 2(µ x 3 ) = 0 µ = 1 3 (x 1 + x 2 + x 3 )
9 s 2 x = 1 n ( (x1 x) 2 + (x 2 x) (x n x) 2) s 2 y = 1 n ( (y1 ȳ) 2 + (y 2 ȳ) (y n ȳ) 2) x i x
10 n 21
11 n n = 21, p = 0.4
12 n n = 1, p = 0.4 p = 0, 1/2, 1
13 s xy = 1 n ((x 1 x)(y 1 ȳ) + + (x n x)(y n ȳ)) x y x i x, y i ȳ
14 Y y I II IV III I, III II, IV x X
15 r xy = r xy = I II I II Y Y IV III IV III X X
16 (1 ) r xy = s xy s x s y r xy = 1 y = ax + b (a > 0) r xy = 1 y = ax + b (a < 0) r xy 0
17 X Y x, y
18 x = [x 1, x 2,..., x 10 ] y = [y 1, y 2,..., y 10 ] x 1, x 2,... p x 1 = [x 1,1, x 1,2,..., x 10,p ]
![.., y 10 ] x 1, x 2,.](/docs-images/47/20506032/images/page_18.jpg ".. p x 1 = [x 1,1, x 1,2,.")
19 (Raw) x R1, x R2,... x x = x R x = [x R1 x, x R2 x,...] y s 2 x = 1 n (x2 1 + x x 2 n) s xy = 1 n (x 1y 1 + x 2 y x n y n )
![..] y s 2 x = 1 n (x2 1 + x 2 2 +](/docs-images/47/20506032/images/page_19.jpg "+ x 2 n) s xy = 1 n (x 1y 1 + x 2")
20 x 2 = (x, x) = x 1 x 1 + x 2 x x n x n s 2 x = 1 n x 2, s x = 1 n x x : x 2 (x, y) = x 1 y 1 + x 2 y x n y n s xy = 1 (x, y) n
21 a, b ( a, b) = a b cos θ (x2,y2) b θ (x1,y1) a O ( a, b) = x 1 x 2 + y 1 y 2
22 r xy = s xy s x s y = (x, y) x y = cos θ 2 θ cos θ r xy < 0 r xy = 0 r xy > 0
23 I x y z = x y z (x, y) (x, y ) O : AA 1 = I
24 y 1 = a 11 x 1 + a 12 x 2 + b 1 y 2 = a 11 x 1 + a 12 x 2 + b 2 y 1 y 2 = a 11 a 12 a 21 a 22 x 1 x 2 + b 1 b 2 x y = Ax + b
25 x = A 1 (y b) y = ax + b x = a 1 (y b)
26 X, Y E[X + Y ] = E[X] + E[Y ] () V [X ± Y ] = V [X] + V [Y ] (X, Y ) σx+y 2 = σx 2 + σy 2 X, Y, Z Y, Z X + Y X + Z
27 X σ X Y σ Y X + Y σ X+Y σ XY ρ XY X σ X Z σ Z X + Z σ X+Z σ XZ ρ XZ X Y σ X+Y = σ X + σ Y X Z σx+z 2 = σx 2 + σz 2
28 Y Z X X 1 X Y 2 X Y
29 1 (xi,yi) Y (xi,a+bxi) (x2,y2) h2 h1 (x1,y1) hi y = a + b x X h i a, b (ax + b ) b = s xy s 2 x, a = ȳ b x
30
31 y = a + b 1 x 1 + b 2 x 2 b 1, b 2 b 1 b 2 = s2 x 1 s x1 x 2 s x1 x 2 s 2 x 2 1 s x 1 y s x2 y s x1 x 2
32 ( p ) y = a + b 1 x 1 + b 2 x b p x p b 1, b 2,... b 1 b 2. b p = s x1 x 1 s x1 x 2 s x1 x p s x2 x 1 s x2 x 2 s x2 x p s xp x 1 s xp x 2 s xp x p 1 = s x1 y s x2 y. s xp y x 1 n
33 R xy10.dat 2 10 X Y DT <- read.table("xy10.dat",header=true) postscript("images/lmxy10.ps", horizontal=false, height=5.2,width=6,onefile=true) result = lm(y ~ X, data = DT) # summary(result) # plot(y ~ X, data = DT) # abline(result) #
34 DT <- read.table("xy10.dat",header=true) xy10.dat DT X Y 1 X, 2 Y postscript("lmxy10.ps", horizontal=false, height=5.2,width=6,onefile=true) Postscript (2.54 cm) result = lm(y ~ X, data = DT) (linear model) result
35 lm Y ~ X, data = DT DT X Y summary(result) result plot(y ~ X, data = DT) DT X Y abline(result)
36 R
37 Call: lm(formula = Y ~ X, data = DT) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) X *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: 3.54 on 8 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: on 1 and 8 DF, p-value:
38 Residuals: Coefficients: Intercept, X Estimate Std. Error t value t Pr(> t ), p ( p) Residual standard error: Multiple R-squared: ( ) 2 Adjusted R-squared: ( ) 2 ( ) F-Statistic: F p
39 R BodyScore.txt Bust West Hip Weight ( 30 ) Bust, West, Hip Weight
40 BS <- read.table("bodyscore.txt",header=t) cor(bs) ## Bust West Hip Weight Bust West Hip Weight
41 cor(bs) pairs(bs) ( [2] 7 )
42 BS.fit <- lm(weight ~ Bust + West + Hip, data = BS) # + summary(bs.fit) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) * Bust West * Hip * --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 26 degrees of freedom Multiple R-squared: 0.554,Adjusted R-squared: F-statistic: on 3 and 26 DF, p-value: 8.859e-05 West, Hip Bust
43
44 p x 1, x 2,..., x p m z 1, z 2,..., z m (m p) z 1 = c 11 x 1 + c 12 x c 1p x p z 2 = c 21 x 1 + c 22 x c 2p x p... z m = c m1 x 1 + c m2 x c mp x p m z i (1 i m) p
45 x 1, x 2,..., x p R = r x1 x 1 r x1 x 2 r x1 x p r x2 x 1. r x2 x 2. r x2 x p.... r xp x 1 r xp x 2 r xp x p
46 SO2 Neg.Temp Manuf Pop Wind Precip Days Phoenix Little Rock San Francisco Denver Hartford Wilmington Washington Jacksonville Miami Atlanta Charleston Milwaukee SO2: SO 2 Neg.Temp: Manuf: Pop:, Wind: (MPH), Precip: (inch), Days:
47
48 D <- read.table("usair.txt",header=t) VD <- D[,-1] # SO2 cor(vd) # VD.pc <- princomp(vd,cor=t) # cor=t summary(vd.pc,loading=t) # Neg.Temp Manuf Pop Wind Precip Days Neg.Temp Manuf Pop Wind Precip Days
49 Importance of components: Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Standard deviation Proportion of Variance Cumulative Proportion Comp.6 Standard deviation Proportion of Variance Cumulative Proportion (components) % 3
50 ( ) Loadings: Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Neg.Temp Manuf Pop Wind Precip Days
51 (PC1, PC2, PC3) postscript("pc1vspc2.ps",horizontal=f, onefile=true) par(pty = "s") # plot(vd.pc$scores[,1],vd.pc$scores[,2], #1 2 ylim = range(vd.pc$scores[,1]), # y PC1 xlab = "PC1", ylab = "PC2", type = "n", lwd = 2) # type="n" text(vd.pc$scores[,1], VD.pc$scores[,2], # labels = abbreviate(row.names(d)),cex = 0.7,lwd=2) dev.off() ## PC1 PC2 ##
52 PC1 vs. PC2 Chcg
53 PC1 vs. PC3 Chcg
54 PC2 vs. PC3 Phnx
55 VD.pc str(vd.pc) # VD.pc VD.pc$scores # 6 VD.pc$scores[,1:3] # 3 Comp.1 Comp.2 Comp.3 Phoenix Little Rock San Francisco Denver Hartford Wilmington Washington Jacksonville Miami Atlanta Charleston Milwaukee
56 3 SO2 postscript("3pcvsso2.ps",horizontal=f, height=5.2,width=6, onefile=true) par(mfrow = c(1,3)) # 3 plot(vd.pc$scores[,1], VD[,1], xlab = "PC1", ylab="so2") plot(vd.pc$scores[,2], VD[,1], xlab = "PC2", ylab="so2") plot(vd.pc$scores[,3], VD[,1], xlab = "PC3", ylab="so2") ## VD.pc x ## VD[,1] SO2
57 : PC1 SO2
58 ## PC1,2,3 pclm <- lm(d$so2 ~ VD.pc$scores[,1] + VD.pc$scores[,2] + VD.pc$scores[,3]) summary(pclm) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-12 *** VD.pc$scores[, 1] e-05 *** VD.pc$scores[, 2] VD.pc$scores[, 3] Signif. codes: 0 *** ** 0.01 * Residual standard error: on 37 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: on 3 and 37 DF, p-value:
59
60 p z 1, z 2,..., z p (p ) z 1 = a 11 f 1 + a 12 f a 1r f r + u 1 v 1 z 2 = a 21 f 1 + a 22 f a 2f f r + u 2 v 2... z p = a p1 f 1 + a p2 f a pf f r + u p v p f 1, f 2,... : v 1, v 2,... : a i,j : h 2 j = a 2 j1 + + a 2 jr : j u j : j h 2 j + u 2 j = 1 ( )
61 f 1, f 2,... v i (i = 1, 2,..., ) v i, v j (i j) f 1, f 2,..., v 1, v 2,... 1 f f i, f j (i j) ( )
62 10 3 x y z x y z f x y z z x = af + u x v x z y = bf + u y v y z z = cf + u z v z z x, z y, z z
63 a, b, c r xy = ab, r yz = bc, r xz = ac 33 a = rxy r xz r yz = b = 0.910, c = a 2, b 2, c 2 u 2 x = 0.094, u 2 y = 0.172, y 2 z = 0.314
64 x y z x y z a = 1.2, b = 0.7, c = 0.5 u 2 x = 0.44, u 2 y = 0.51, u 2 z =
65 z 1 = a 11 f 1 + a 12 f a 1r f r + u 1 v 1 z 2 = a 21 f 1 + a 22 f a 2f f r + u 2 v 2... z p = a p1 f 1 + a p2 f a pf f r + u p v p
66 Z = AF + UV R F Z = (AR)(R 1 F ) + UV R 1 3
67 32 ( ) m0 m25 m50 m75 w0 w25 w50 w75 Algeria Cameroon Madagascar Mauritius Reunion Seychelles South Africa(C) South Africa(W) Tunisia United States (66) United States (NW66) United States (W66) United States (67) Argentina Chile Columbia Ecuador
68 1 life <- read.table("../data/chap4lifeexp.txt",header = T) # life.fa1 <- factanal(life, factors = 1, method = "mle") # 1 life.fa1 #
69 1 Call: factanal(x = life, factors = 1, method = "mle") Uniquenesses: # m0 m25 m50 m75 w0 w25 w50 w Loadings: # Factor1 # 1 m m m m w w w w Factor1 SS loadings Proportion Var Test of the hypothesis that 1 factor is sufficient. #! The chi square statistic is on 20 degrees of freedom. The p-value is 1.88e-24 # p
70 life.fa2 <- factanal(life, factors = 2, method = "mle") life.fa2 life.fa3 <- factanal(life, factors = 3, method = "mle") life.fa3 # Test of the hypothesis that 2 factors are sufficient. The chi square statistic is on 13 degrees of freedom. The p-value is 1.91e-05 # 3 Test of the hypothesis that 3 factors are sufficient. The chi square statistic is 6.73 on 7 degrees of freedom. The p-value is # p
71 3 Uniquenesses: m0 m25 m50 m75 w0 w25 w50 w Loadings: Factor1 Factor2 Factor3 m m m m w w w w Factor1 Factor2 Factor3 SS loadings Proportion Var Cumulative Var Test of the hypothesis that 3 factors are sufficient. The chi square statistic is 6.73 on 7 degrees of freedom. The p-value is 0.458
72 scores <- scores factanal(life, factors = 3, method = "mle", scores = "regression")$scores Factor1 Factor2 Factor3 Algeria Cameroon Madagascar Mauritius Reunion Seychelles South Africa(C) South Africa(W) Tunisia Canada
73 plot(scores[,1], scores[,2], type = "n", xlab = " 1", ylab = " 2") text(scores[,1],scores[,2],labels=row.names(life), cex = 1.1, lwd=2)
74
75
76 I X1 X2 Y Med G Med G Med G Hi G Hi G Lo G Hi G Hi G X1, X2 X1 Lo, Med, Hi 3 X2 G1 G4 4 Y
77 Med, G I X1, X2, I R I
78 X = Med G2 < G3 < G4 Lo < Med < Hi Lo G4 Y I
79 R D <- read.table("numerizei.txt",header=t) # ## Y X1 X2 (lm) result <- lm(y ~ X1 + X2, data = D) summary(result) # predict(result) #
80 predict I summary Residuals: e e e e e e e e e e e e e e e e-02
81 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) ** X1Lo ** X1Med * X2G * X2G X2G ** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 2 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: on 5 and 2 DF, p-value:
82 predict Med G2 9.88
83
84 2 (x 1, y 1 ), (x 2, y 2 ) d = (x 1 x 2 ) 2 + (y 1 y 2 ) 2 2 (x 1, y 1, z 1 ), (x 2, y 2, z 2 ) d = (x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 ) 2 n ( ) x 1, x 2,... x p x 1, x 2 d = (x 1,1 x 1,2 ) 2 + (x 2,1 x 2,2 ) (x n,1 x n,2 ) 2
85 d = x 1 x 1 2( )
86 m0 m25 m50 m75 w0 w25 w50 w75 Algeria Cameroon Madagascar Mauritius Reunion Seychelles South Africa(B) South Africa(W) Tunisia Canada Costa Rica Dominican Rep Ecuador mxx, wxx xx ( )
87 R ## life <- read.table("lifeexp.txt",header=t) ## country country <- row.names(life) ## dist <- dist(life) postscript("lifeexp.ps",horizontal=f, width=7, height=7,onefile=true) ## plot(hclust(dist, method = "complete"), labels = country, # xlab = " ", ylab = " ", main = " ")
88
89
90 TibetScull.txt Type Length Breadth Height Fheight Fbreadth Type
91 NewScull.txt Length Breadth Height Fheight Fbreadth A B library(mass) # MASS DT <- read.table("tibetscull.txt",header=t) dis <- lda(type ~ Length + Breadth + Height + Fheight + Fbreadth, data = DT, prior = c(0.5,0.5)) # ##. Type newscull <- read.table("newscull.txt",header=t) predict(dis, newdata = newscull) #
92 A, B , $class [1] 1 2 Levels: 1 2 $posterior 1 2 A B
93 1 ( ) Musicchoice.txt χ 2 ## MData MData <- read.table("musicchoice.txt",header=t) ## chisq.test(mdata)
94 Pearson s Chi-squared test data: MData X-squared = , df = 6, p-value = p = χ (df) = 6 0.5% α ν = ν = ν =
95 [1] CRAN ( R [2] R ABC A.(2010) R [3] R S-PLUS B. (2007) [4] R (2009) R [5] ( ),, (1985) ( ) [6] ( )
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Title 講義のーと : データ解析のための統計モデリング Author(s) 久保, 拓弥 Issue Date 2008 Doc URL http://hdl.handle.net/2115/49477 Type learningobject Note この講義資料は, 著者のホームページ http://hosho.ees.hokudai.ac.jp/~kub ードできます Note(URL)http://hosho.ees.hokudai.ac.jp/~kubo/ce/EesLecture20
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回帰分析 単回帰 麻生良文 単回帰モデル simple regression model = α + β + u 従属変数 (dependent variable) 被説明変数 (eplained variable) 独立変数 (independent variable) 説明変数 (eplanator variable) u 誤差項 (error term) 撹乱項 (disturbance term)
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講義のーと : データ解析のための統計モデリング. 第２回
Title 講義のーと : データ解析のための統計モデリング Author(s) 久保, 拓弥 Issue Date 2008 Doc URL http://hdl.handle.net/2115/49477 Type learningobject Note この講義資料は, 著者のホームページ http://hosho.ees.hokudai.ac.jp/~kub ードできます Note(URL)http://hosho.ees.hokudai.ac.jp/~kubo/ce/EesLecture20
96 7 1m =2 10 7 N 1A 7.1 7.2 a C (1) I (2) A C I A A a A a A A a C C C 7.2: C A C A = = µ 0 2π (1) A C 7.2 AC C A 3 3 µ0 I 2 = 2πa. (2) A C C 7.2 A A
7 Lorentz 7.1 Ampère I 1 I 2 I 2 I 1 L I 1 I 2 21 12 L r 21 = 12 = µ 0 2π I 1 I 2 r L. (7.1) 7.1 µ 0 =4π 10 7 N A 2 (7.2) magnetic permiability I 1 I 2 I 1 I 2 12 21 12 21 7.1: 1m 95 96 7 1m =2 10 7 N
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