http://ablejec.nib.si/R/MVA09

Kanoniˇcna korelacija Uporaba R pri seminarju iz Multivariatne analize Podiplomski študij Statistika

A. Blejec andrej.blejec@nib.si

11. junij 2009 Povzetek

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Kazalo 1

Uvod

1

2

Korelacijska matrika 2.1 Trikotniki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2

3

Generiranje podatkov 3.1 Dopolnitev prikaza pairs . . . . . . . . . . . . . . . . . . . . . . . . .

3 5

4

Kategorizacija podatkov

6

5

Kanoniˇcna korelacija v R

7

6

Paket CCA

7

7

Primer iz pomoˇci za CCA

7

8

Simulirani podatki

1

10

Uvod

Dokumenti, ki opisujejo funkcije za generiranje in obvladovanje manjkajoˇcih podatkov si oglejte dokumente iz prejšnjih let. Na razpolago so na spletnem naslovu http://ablejec.nib.si/R/MVA.

2

Korelacijska matrika > R <- read.xls("../data/R.xls") > R <- as.matrix(R) > m <- dim(R)[1]

1

> m1 <- 3 > m2 <- m - m1 > R x1 x2 x3 y1 y2

x1 1.0 0.2 0.4 0.3 0.3

x2 0.2 1.0 0.3 0.4 0.4

x3 0.4 0.3 1.0 0.3 0.4

y1 0.3 0.4 0.3 1.0 0.3

y2 0.3 0.4 0.4 0.3 1.0

Vpis matrike > R <- c(1, 0.2, 0.4, 0.3, 0.3, 0.2, 1, 0.3, 0.4, 0.4, + 0.4, 0.3, 1, 0.3, 0.4, 0.3, 0.4, 0.3, 1, 0.3, 0.3, + 0.4, 0.4, 0.3, 1) > R <- matrix(R, 5, 5) > varNames <- c(paste("x", 1:3, sep = ""), paste("y", 1:2, + sep = "")) > varNames [1] "x1" "x2" "x3" "y1" "y2" > dimnames(R) <- list(varNames, varNames) > R x1 x2 x3 y1 y2

x1 1.0 0.2 0.4 0.3 0.3

x2 0.2 1.0 0.3 0.4 0.4

x3 0.4 0.3 1.0 0.3 0.4

y1 0.3 0.4 0.3 1.0 0.3

y2 0.3 0.4 0.4 0.3 1.0

> det(R) [1] 0.43118

Spremenimo eno od vrednosti > R[4, 5] <- (-0.3)

2.1

Trikotniki

> R[upper.tri(R)] [1]

0.2

0.4

0.3

0.3

0.4

0.3

0.3

0.4

> R[lower.tri(R)] <- NA > R x1 x2 x3 y1 y2

x1 x2 x3 y1 y2 1 0.2 0.4 0.3 0.3 NA 1.0 0.3 0.4 0.4 NA NA 1.0 0.3 0.4 NA NA NA 1.0 -0.3 NA NA NA NA 1.0

Takole lahko preslikamo zgornji trikotnik v spodnjega

2

0.4 -0.3

> R[lower.tri(R)] <- t(R)[lower.tri(R)] > R x1 x2 x3 y1 y2 x1 1.0 0.2 0.4 0.3 0.3 x2 0.2 1.0 0.3 0.4 0.4 x3 0.4 0.3 1.0 0.3 0.4 y1 0.3 0.4 0.3 1.0 -0.3 y2 0.3 0.4 0.4 -0.3 1.0

Funkcij za preverjanje simetriˇcnosti > is.symmetric <- function(x) { + identical(x, t(x)) + } > is.symmetric(R) [1] TRUE > U <- R > U[1, 2] <- 100 > U x1 x2 x3 y1 y2

x1 x2 x3 y1 y2 1.0 100.0 0.4 0.3 0.3 0.2 1.0 0.3 0.4 0.4 0.4 0.3 1.0 0.3 0.4 0.3 0.4 0.3 1.0 -0.3 0.3 0.4 0.4 -0.3 1.0

> is.symmetric(U) [1] FALSE

Narediti bi bilo dobro funkcijo (ˇce je še ni??) za spremebo zgornje/spodnje trikotne matrike v simetriˇcno.

3

Generiranje podatkov > library(MASS) > (Sigma <- matrix(c(10, 3, 3, 2), 2, 2)) [,1] [,2] [1,] 10 3 [2,] 3 2 > X <- mvrnorm(50, c(20, 20), Sigma, emp = T) > cov(X) [1,] [2,]

[,1] [,2] 10 3 3 2

> X1 <- mvrnorm(500, c(20, 20), Sigma) > cov(X1) [,1] [,2] [1,] 10.351849 3.216887 [2,] 3.216887 2.007822 > X <- mvrnorm(500, c(20, 20), Sigma, emp = T) > plot(X, xlim = c(10, 30), ylim = c(10, 30))

3

30 25 20

X[,2]

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●

10

15

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●

10

15

20

25

30

X[,1]

> set.seed(1230) > X <- mvrnorm(100, mu = rep(0, m), Sigma = R, emp = T)

> pairs(X)

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2 0 −2 1 2 −3

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0

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2

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y1

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−2

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−2

0

2

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1

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0

x3

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−2

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2

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1 2

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1

x2

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●

●

−1

0

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x1

−1

−3 ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ●● ●● ● ●● ● ● ●● ● ● ● ● ●●●● ● ●● ●● ● ● ●● ●● ● ● ● ●● ● ● ●● ● ●● ● ●

−2

2

1

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2

y2 −2

4

0 1 2

0

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−2

−2

0 1 2

3.1

Dopolnitev prikaza pairs

> panel.cor <- function(x, y, digits = 2, prefix = "", + cex.cor) { + usr <- par("usr") + on.exit(par(usr)) + par(usr = c(0, 1, 0, 1)) + r <- (cor(x, y, use = "pairwise.complete.obs")) + txt <- format(c(r, 0.123456789), digits = digits)[1] + txt <- paste(prefix, txt, sep = "") + if (missing(cex.cor)) + cex.cor <- 0.8/strwidth(txt) + text(0.5, 0.5, txt, cex = cex.cor * abs(r)) + }

> panel.abline <- function(x, y, col.ab = "red", lwd.ab, + ...) { + points(x, y) + if (missing(lwd.ab)) + lwd.ab <- 1 + abline(lm(y ~ x), col = col.ab, lwd = lwd.ab, ...) + }

> pairs(X, lower.panel = panel.cor, cex.cor = 5, upper.panel = panel.abline)

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1 2

0.40

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y1

0.30

●

−3

−1

0.30

2

● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ●●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ●●● ● ●● ● ● ●● ● ● ●●●●● ● ● ● ●● ● ● ● ●●● ●●● ●● ●● ●● ● ● ● ● ● ●● ●● ●●

1

● ●● ●● ● ● ●● ● ● ●●●● ● ● ● ● ● ● ●●● ●●● ● ●● ● ● ●●●●● ● ● ●●● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●●● ● ● ●● ● ● ● ● ● ●● ● ●● ●●●● ● ●

0

● ● ● ● ● ● ● ●● ● ● ● ●●●● ● ●●● ● ●●●●● ● ● ●● ● ● ● ● ● ●● ●●● ●● ●● ● ●● ● ●● ●●● ● ●●●● ● ● ● ● ● ●● ●●● ● ● ●●●● ● ● ● ● ● ● ●● ● ●●● ●● ● ● ● ● ●● ● ● ●

●

−2

● ● ●● ● ● ● ●●● ● ●● ● ● ●● ● ● ● ● ● ●● ● ●●●● ●● ●●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ●● ●

●

●

x3

0.30

1 2

● ● ●● ● ●●●● ●●● ● ● ●● ● ●● ● ●●● ● ● ● ● ● ● ● ●● ●● ● ● ●●●● ● ● ● ●● ●● ●● ● ●● ●● ● ● ●● ●●● ● ● ●●● ● ●● ● ●● ● ●●●●● ●●● ● ● ● ●● ●● ● ● ● ●

2

−2

0.40

●

−1

1

x2

●

0

−3

● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ●● ●● ● ● ●● ●● ● ●● ● ● ● ● ●●● ●● ● ● ●● ●● ● ● ●●● ● ●●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ●

0.20

0

2 ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ●● ● ●● ● ●●● ● ● ● ●● ● ●● ● ●● ●● ●● ● ● ● ●●● ● ● ● ● ●● ●●● ● ● ●● ●●● ●● ●● ● ● ● ●● ● ● ●● ● ●● ●

2

x1

0

●● ● ● ● ● ● ● ● ● ● ●●●● ●● ●●● ●●● ● ●● ●●● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ●●●● ● ●● ● ●● ● ●● ●● ● ● ● ● ●● ● ● ●●● ●● ● ●● ● ●● ●●● ●● ●● ● ●● ●

−2

−2

0.40

0.40

y2

−0.30

−2

0.30

0 1 2

●

−2

0

1

2

−2

0

1

2

−2

0 1 2

Shranimo generirane podatke za kasnejšo rabo > data <- X

5

4

Kategorizacija podatkov

Za kategorizacijo uporabimo funkcijo cut. Kot argument lahko vpišemo število enako širokih razredov ali pa meje (uporabno za neenako široke razrede). > x <- X[, 1] > range(x) [1] -2.267590 1.911885 > table(cut(x, 5)) (-2.27,-1.43] (-1.43,-0.597] (-0.597,0.241] 7 26 26 (1.08,1.92] 17

(0.241,1.08] 24

> table(cut(x, breaks = c(-3, -2, -1, 1, 2, 3))) (-3,-2] (-2,-1] (-1,1] (1,2] (2,3] 1 14 67 18 0

Razrez na intervale z enako frekvenco. > quantile(x) 0% 25% 50% 75% 100% -2.26759014 -0.81037719 0.06219079 0.77638817 1.91188462 > table(cut(x, quantile(x))) (-2.27,-0.81] (-0.81,0.0622] (0.0622,0.776] (0.776,1.91] 24 25 25 25 > quantile(x) 0% 25% 50% 75% 100% -2.26759014 -0.81037719 0.06219079 0.77638817 1.91188462 > table(cut(x, quantile(x, c(0.01, 0.05, 0.25, 0.75, 0.95, + 0.99)))) (-1.93,-1.54] (-1.54,-0.81] (-0.81,0.776] (0.776,1.67] (1.67,1.77] 4 20 50 20 4

Razrez vseh spremenljivk v matriki. > Y <- cut(X, 5) > str(Y) Factor w/ 5 levels "(-2.97,-1.83]",..: 3 3 3 2 3 2 3 2 2 3 ... > Y <- as.numeric(Y) > dim(Y) <- dim(X) > head(Y) [,1] [,2] [,3] [,4] [,5] [1,] 3 4 2 3 3 [2,] 3 1 4 2 3 [3,] 3 3 5 4 3 [4,] 2 3 2 3 2 [5,] 3 3 2 4 2 [6,] 2 2 3 2 3 > apply(X, 2, min) x1 x2 x3 y1 y2 -2.267590 -2.469638 -2.284331 -2.963278 -2.334583

6

5

Kanoniˇcna korelacija v R

V osnovnem paketu stats je za kanoniˇcno korelacijo predvidena funkcija cancor.

Vzemimo primer iz pomoˇci za cancor > pop <- LifeCycleSavings[, 2:3] > oec <- LifeCycleSavings[, -(2:3)] > cancor(pop, oec) $cor [1] 0.8247966 0.3652762 $xcoef [,1] [,2] pop15 -0.009110856 -0.03622206 pop75 0.048647514 -0.26031158 $ycoef [,1] [,2] [,3] sr 0.0084710221 3.337936e-02 -5.157130e-03 dpi 0.0001307398 -7.588232e-05 4.543705e-06 ddpi 0.0041706000 -1.226790e-02 5.188324e-02 $xcenter pop15 pop75 35.0896 2.2930 $ycenter sr dpi 9.6710 1106.7584

6

ddpi 3.7576

Paket CCA

Za razširitev funkcionalnosti lahko uporabimo funkcije iz paketa CCA (?). > library(CCA)

7

Primer iz pomoˇci za CCA > > > > > >

data(nutrimouse) X = as.matrix(nutrimouse$gene[, 1:10]) Y = as.matrix(nutrimouse$lipid) X <- pop Y <- oec res.cc <- cc(X, Y)

> names(res.cc) [1] "cor"

"names"

"xcoef"

"ycoef"

> plot(res.cc$cor, type = "b")

7

"scores"

0.4

0.5

0.6

res.cc$cor

0.7

0.8

●

●

1.0

1.2

1.4

1.6

1.8

2.0

Index

1.0

> plt.cc(res.cc)

Japan 2

South Rhodesia

1 0

Norway Canada Denmark Sweden United States Austria United Kingdom New Zealand Belgium Iceland

−2

−0.5

−1

Dimension 2

0.0

● ●

Germany Uruguay Greece Netherlands Spain South Africa Luxembourg Italy Portugal Malta SwitzerlandAustralia

Brazil India Korea Paraguay Zambia Chile China Turkey Honduras Malaysia Panama Guatamala Venezuela Peru Jamaica Nicaragua Bolivia Philippines Colombia Tunisia Ecuador Costa Rica Libya

France

−1.0

Dimension 2

0.5

Finland

Ireland −1.0

−0.5

0.0

0.5

1.0

−1.5

−1.0

Dimension 1

−0.5

0.0

Dimension 1

8

0.5

1.0

> names(matcor(X, Y)) [1] "Xcor"

"Ycor"

"XYcor"

> img.matcor(matcor(X, Y))

XY correlation

−1.0

−0.5

0.0

0.5

1.0

> img.matcor(matcor(X, Y), type = 2) X correlation

Y correlation

Cross−correlation

−1.0

−0.5

0.0

0.5

9

1.0

8

Simulirani podatki > X <- data[, 1:3] > Y <- data[, 4:5]

> img.matcor(matcor(X, Y))

XY correlation

â&#x2C6;&#x2019;1.0

â&#x2C6;&#x2019;0.5

0.0

0.5

1.0

> res.cc <- cc(X, Y)

> names(res.cc) [1] "cor"

"names"

"xcoef"

"ycoef"

> plot(res.cc$cor, type = "b")

10

"scores"

0.2

0.4

res.cc$cor

0.6

0.8

●

●

1.0

1.2

1.4

1.6

1.8

2.0

Index

1.0

> plt.cc(res.cc)

32 2

2

3

58

−2

−1

−0.5

0

Dimension 2

0.0

● ●

−1.0

Dimension 2

1

0.5

●

−1.0

−0.5

0.0

0.5

1.0

34 47 79 84 18 24 78 30 39 69 2825 88 6 11 43 76 31 16 83 10 62 66 37 14 82 23 98 60 12 75 94 67 93 97 7 9290 21 99 68 71 63 64 73 89 8 61 5322 100 35 4 72 9 49 80 40 45 8651 36 85 59 70 27 46 56 55 95 54 26 19 44 52 74 96 81 50 1387 41 333891 77 29 57 20 5 42 48 17 15 65 1 −2

−1

Dimension 1

0

Dimension 1

> plt.var(res.cc, 1, 2)

11

1

2

1.0 0.0

●

−0.5

●

−1.0

Dimension 2

0.5

●

−1.0

−0.5

0.0

0.5

Dimension 1

12

1.0

SessionInfo Windows XP (build 2600) Service Pack 3 • R version 2.8.0 (2008-10-20), i386-pc-mingw32

• Locale: LC_COLLATE=Slovenian_Slovenia.1250;LC_CTYPE=Slovenian_Slovenia.1250;LC_MON • Base packages: base, datasets, graphics, grDevices, methods, splines, stats, utils • Other packages: CCA 1.2, fda 2.1.2, fields 5.02, Hmisc 3.5-2, MASS 7.2-46, spam 0.15-4, xlsReadWrite 1.3.2, zoo 1.5-5 • Loaded via a namespace (and not attached): cluster 1.11.13, grid 2.8.0, lattice 0.17-22, tools 2.8.0

Revision Id: Sweave.Rnw 5 2008-11-20 09:24:44Z ablejec

13

(c) A. Blejec