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R Under development (unstable) (2015-09-06 r69303) -- "Unsuffered Consequences"Copyright (C) 2015 The R Foundation for Statistical ComputingPlatform: x86_64-pc-linux-gnu (64-bit)R is free software and comes with ABSOLUTELY NO WARRANTY.You are welcome to redistribute it under certain conditions.Type 'license()' or 'licence()' for distribution details.R is a collaborative project with many contributors.Type 'contributors()' for more information and'citation()' on how to cite R or R packages in publications.Type 'demo()' for some demos, 'help()' for on-line help, or'help.start()' for an HTML browser interface to help.Type 'q()' to quit R.> #### "All the examples" from ./R-intro.texi> #### -- in a way that this should be(come) an executable script.>> options(digits=5, width=65)##--- for outputs !> options(stringsAsFactors=TRUE) ## factory-fresh defaults> options(useFancyQuotes=FALSE) ## avoid problems on Windows>> ## 2. Simple Manipulations>> x <- c(10.4, 5.6, 3.1, 6.4, 21.7)> assign("x", c(10.4, 5.6, 3.1, 6.4, 21.7))> c(10.4, 5.6, 3.1, 6.4, 21.7) -> x> .Last.value[1] 10.4 5.6 3.1 6.4 21.7> 1/x[1] 0.096154 0.178571 0.322581 0.156250 0.046083>> y <- c(x, 0, x)> v <- 2*x + y + 1Warning message:In 2 * x + y :longer object length is not a multiple of shorter object length> ##- Warning message:> ##- longer object length> ##- is not a multiple of shorter object length in: 2 * x + y>> sqrt(-17)[1] NaNWarning message:In sqrt(-17) : NaNs produced> ##- [1] NaN> ##- Warning message:> ##- NaNs produced in: sqrt(-17)>> sqrt(-17+0i)[1] 0+4.1231i>> ###-- 2.3 .. regular sequences>> 1:30[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20[21] 21 22 23 24 25 26 27 28 29 30>> n <- 10>> 1:n-1[1] 0 1 2 3 4 5 6 7 8 9> 1:(n-1)[1] 1 2 3 4 5 6 7 8 9> 30:1[1] 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11[21] 10 9 8 7 6 5 4 3 2 1>> seq(2,10)[1] 2 3 4 5 6 7 8 9 10> all(seq(1,30) == seq(to=30, from=1))[1] TRUE>> seq(-5, 5, by=.2) -> s3> s4 <- seq(length=51, from=-5, by=.2)> all.equal(s3,s4)[1] TRUE>> s5 <- rep(x, times=5)> s6 <- rep(x, each=5)>> temp <- x > 13>> z <- c(1:3,NA); ind <- is.na(z)>> 0/0[1] NaN> Inf - Inf[1] NaN>> labs <- paste(c("X","Y"), 1:10, sep="")> labs[1] "X1" "Y2" "X3" "Y4" "X5" "Y6" "X7" "Y8" "X9" "Y10">> x <- c(z,z-2)#-- NOT in texi ; more interesting> y <- x[!is.na(x)]>> (x+1)[(!is.na(x)) & x>0] -> z> z[1] 2 3 4 2>> x <- c(x, 9:12)# long enough:> x[1:10][1] 1 2 3 NA -1 0 1 NA 9 10>> c("x","y")[rep(c(1,2,2,1), times=4)][1] "x" "y" "y" "x" "x" "y" "y" "x" "x" "y" "y" "x" "x" "y" "y"[16] "x">> y <- x[-(1:5)]> y[1] 0 1 NA 9 10 11 12>> fruit <- c(5, 10, 1, 20)> names(fruit) <- c("orange", "banana", "apple", "peach")> fruitorange banana apple peach5 10 1 20>> lunch <- fruit[c("apple","orange")]> lunchapple orange1 5>> x[1] 1 2 3 NA -1 0 1 NA 9 10 11 12> x[is.na(x)] <- 0> x[1] 1 2 3 0 -1 0 1 0 9 10 11 12>> y <- -4:9> y[y < 0] <- -y[y < 0]> all(y == abs(y))[1] TRUE> y[1] 4 3 2 1 0 1 2 3 4 5 6 7 8 9>> ###--------------->> z <- 0:9> digits <- as.character(z)> digits[1] "0" "1" "2" "3" "4" "5" "6" "7" "8" "9"> d <- as.integer(digits)> all.equal(z, d)[1] TRUE>> e <- numeric()> e[3] <- 17> e[1] NA NA 17>> alpha <- 10*(1:10)> alpha <- alpha[2 * 1:5]> alpha[1] 20 40 60 80 100>> winter <- data.frame(temp = c(-1,3,2,-2), cat = rep(c("A","B"), 2))> wintertemp cat1 -1 A2 3 B3 2 A4 -2 B> unclass(winter)$temp[1] -1 3 2 -2$cat[1] A B A BLevels: A Battr(,"row.names")[1] 1 2 3 4>> ###------------ Ordered and unordered factors --------> state <- c("tas", "sa", "qld", "nsw", "nsw", "nt", "wa", "wa",+ "qld", "vic", "nsw", "vic", "qld", "qld", "sa", "tas",+ "sa", "nt", "wa", "vic", "qld", "nsw", "nsw", "wa",+ "sa", "act", "nsw", "vic", "vic", "act")> statef <- factor(state)> statef[1] tas sa qld nsw nsw nt wa wa qld vic nsw vic qld qld sa[16] tas sa nt wa vic qld nsw nsw wa sa act nsw vic vic actLevels: act nsw nt qld sa tas vic wa>> levels(statef)[1] "act" "nsw" "nt" "qld" "sa" "tas" "vic" "wa">> incomes <- c(60, 49, 40, 61, 64, 60, 59, 54, 62, 69, 70, 42, 56,+ 61, 61, 61, 58, 51, 48, 65, 49, 49, 41, 48, 52, 46,+ 59, 46, 58, 43)>> incmeans <- tapply(incomes, statef, mean)> incmeansact nsw nt qld sa tas vic wa44.500 57.333 55.500 53.600 55.000 60.500 56.000 52.250>> stderr <- function(x) sqrt(var(x)/length(x))>> incster <- tapply(incomes, statef, stderr)> incsteract nsw nt qld sa tas vic wa1.5000 4.3102 4.5000 4.1061 2.7386 0.5000 5.2440 2.6575>> ##> z <- 1:1500> dim(z) <- c(3,5,100)>>> x <- array(1:20,dim=c(4,5)) # Generate a 4 by 5 array.> x[,1] [,2] [,3] [,4] [,5][1,] 1 5 9 13 17[2,] 2 6 10 14 18[3,] 3 7 11 15 19[4,] 4 8 12 16 20>> i <- array(c(1:3,3:1),dim=c(3,2))> i # @code{i} is a 3 by 2 index array.[,1] [,2][1,] 1 3[2,] 2 2[3,] 3 1>> x[i] # Extract those elements[1] 9 6 3>> x[i] <- 0 # Replace those elements by zeros.> x[,1] [,2] [,3] [,4] [,5][1,] 1 5 0 13 17[2,] 2 0 10 14 18[3,] 0 7 11 15 19[4,] 4 8 12 16 20>> n <- 60> b <- 5 ; blocks <- rep(1:b, length= n)> v <- 6 ; varieties <- gl(v,10)>> Xb <- matrix(0, n, b)> Xv <- matrix(0, n, v)> ib <- cbind(1:n, blocks)> iv <- cbind(1:n, varieties)> Xb[ib] <- 1> Xv[iv] <- 1> X <- cbind(Xb, Xv)>> N <- crossprod(Xb, Xv)> table(blocks,varieties)varietiesblocks 1 2 3 4 5 61 2 2 2 2 2 22 2 2 2 2 2 23 2 2 2 2 2 24 2 2 2 2 2 25 2 2 2 2 2 2> all(N == table(blocks,varieties))[1] TRUE>> h <- 1:17> Z <- array(h, dim=c(3,4,2))> ## If the size of 'h' is exactly 24> h <- rep(h, length = 24)> Z. <- Z ## the result is the same as> Z <- h; dim(Z) <- c(3,4,2)> stopifnot(identical(Z., Z))>> Z <- array(0, c(3,4,2))>> ## So if @code{A}, @code{B} and @code{C} are all similar arrays> ## <init>> A <- matrix(1:6, 3,2)> B <- cbind(1, 1:3)> C <- rbind(1, rbind(2, 3:4))> stopifnot(dim(A) == dim(B),+ dim(B) == dim(C))> ## <init/>> D <- 2*A*B + C + 1>> a <- 1:9> b <- 10*(1:3)>> ab <- a %o% b> stopifnot(ab == outer(a,b,"*"),+ ab == outer(a,b))>> x <- 1:10> y <- -2:2> f <- function(x, y) cos(y)/(1 + x^2)> z <- outer(x, y, f)>>> d <- outer(0:9, 0:9)> fr <- table(outer(d, d, "-"))> plot(as.numeric(names(fr)), fr, type="h",+ xlab="Determinant", ylab="Frequency")>> ##>> B <- aperm(A, c(2,1))> stopifnot(identical(B, t(A)))>> ## for example, @code{A} and @code{B} are square matrices of the same size> ## <init>> A <- matrix(1:4, 2,2)> B <- A - 1> ## <init/>>> A * B[,1] [,2][1,] 0 6[2,] 2 12>> A %*% B[,1] [,2][1,] 3 11[2,] 4 16>>> ## <init>> x <- c(-1, 2)> ## <init/>> x %*% A %*% x[,1][1,] 7>> x %*% x[,1][1,] 5> stopifnot(x %*% x == sum(x^2))>> xxT <- cbind(x) %*% x> xxT[,1] [,2][1,] 1 -2[2,] -2 4> stopifnot(identical(xxT, x %*% rbind(x)))>> ## crossprod ... (ADD)>> ## diag ... (ADD)>> ## linear equations ... (ADD)>> ## solve ... (ADD)>> ## eigen:> ## a symmetric matrix @code{Sm}> ## <init>> Sm <- matrix(-2:6, 3); Sm <- (Sm + t(Sm))/4; Sm[,1] [,2] [,3][1,] -1 0 1[2,] 0 1 2[3,] 1 2 3> ## </init>> ev <- eigen(Sm)>> evals <- eigen(Sm)$values>> ## SVD .....>> ## "if M is in fact square, then, ..."> ## <init>> M <- cbind(1,1:3,c(5,2,3))> X <- cbind(1:9, .25*(-4:4)^2)> X1 <- cbind(1:7, -1)> X2 <- cbind(0,2:8)> y <- c(1:4, 2:6)> ## </init>>> absdetM <- prod(svd(M)$d)> stopifnot(all.equal(absdetM, abs(det(M))))# since det() nowadays exists>> ans <- lsfit(X, y)>> Xplus <- qr(X)> b <- qr.coef(Xplus, y)> fit <- qr.fitted(Xplus, y)> res <- qr.resid(Xplus, y)> ##>> X <- cbind(1, X1, X2)>> vec <- as.vector(X)> vec <- c(X)>> statefr <- table(statef)> statefrstatefact nsw nt qld sa tas vic wa2 6 2 5 4 2 5 4> statefr <- tapply(statef, statef, length)> statefract nsw nt qld sa tas vic wa2 6 2 5 4 2 5 4>> factor(cut(incomes, breaks = 35+10*(0:7))) -> incomef> table(incomef,statef)statefincomef act nsw nt qld sa tas vic wa(35,45] 1 1 0 1 0 0 1 0(45,55] 1 1 1 1 2 0 1 3(55,65] 0 3 1 3 2 2 2 1(65,75] 0 1 0 0 0 0 1 0>> ###--- @chapter 6. Lists and data frames>> Lst <- list(name="Fred", wife="Mary", no.children=3,+ child.ages=c(4,7,9))> Lst$name[1] "Fred"> Lst$wife[1] "Mary"> Lst$child.ages[1][1] 4> stopifnot(Lst$name == Lst[[1]], Lst[[1]] == "Fred",+ Lst$child.ages[1] == Lst[[4]][1], Lst[[4]][1] == 4+ )>> x <- "name" ; Lst[[x]][1] "Fred">> ## @section 6.2 Constructing and modifying lists>> ##<init>> Mat <- cbind(1, 2:4)> ##</init>> Lst[5] <- list(matrix=Mat)>> ## @section 6.3 Data frames>> accountants <- data.frame(home=statef, loot=incomes, shot=incomef)> ## MM: add the next lines to R-intro.texi !> accountantshome loot shot1 tas 60 (55,65]2 sa 49 (45,55]3 qld 40 (35,45]4 nsw 61 (55,65]5 nsw 64 (55,65]6 nt 60 (55,65]7 wa 59 (55,65]8 wa 54 (45,55]9 qld 62 (55,65]10 vic 69 (65,75]11 nsw 70 (65,75]12 vic 42 (35,45]13 qld 56 (55,65]14 qld 61 (55,65]15 sa 61 (55,65]16 tas 61 (55,65]17 sa 58 (55,65]18 nt 51 (45,55]19 wa 48 (45,55]20 vic 65 (55,65]21 qld 49 (45,55]22 nsw 49 (45,55]23 nsw 41 (35,45]24 wa 48 (45,55]25 sa 52 (45,55]26 act 46 (45,55]27 nsw 59 (55,65]28 vic 46 (45,55]29 vic 58 (55,65]30 act 43 (35,45]> str(accountants)'data.frame': 30 obs. of 3 variables:$ home: Factor w/ 8 levels "act","nsw","nt",..: 6 5 4 2 2 3 8 8 4 7 ...$ loot: num 60 49 40 61 64 60 59 54 62 69 ...$ shot: Factor w/ 4 levels "(35,45]","(45,55]",..: 3 2 1 3 3 3 3 2 3 4 ...>> ## ..........>> ###--- @chapter 8. Probability distributions>> ## 2-tailed p-value for t distribution> 2*pt(-2.43, df = 13)[1] 0.030331> ## upper 1% point for an F(2, 7) distribution> qf(0.01, 2, 7, lower.tail = FALSE)[1] 9.5466>> attach(faithful)> summary(eruptions)Min. 1st Qu. Median Mean 3rd Qu. Max.1.60 2.16 4.00 3.49 4.45 5.10>> fivenum(eruptions)[1] 1.6000 2.1585 4.0000 4.4585 5.1000>> stem(eruptions)The decimal point is 1 digit(s) to the left of the |16 | 07035555558818 | 00002223333333557777777788882233577788820 | 0000222337880003577822 | 000233557802357824 | 0022826 | 2328 | 08030 | 732 | 233734 | 25007736 | 000082357738 | 233333558222557740 | 000000335778888800223355557777842 | 0333555577880023333355557777844 | 0222233555778000000002333335777888846 | 000023335770000002357848 | 0000002233580033350 | 0370>> hist(eruptions)>> ## <IMG> postscript("images/hist.eps", ...)> # make the bins smaller, make a plot of density> hist(eruptions, seq(1.6, 5.2, 0.2), prob=TRUE)> lines(density(eruptions, bw=0.1))> rug(eruptions) # show the actual data points> ## dev.off() <IMG/>>> plot(ecdf(eruptions), do.points=FALSE, verticals=TRUE)>> ## <IMG> postscript("images/ecdf.eps", ...)> long <- eruptions[eruptions > 3]> plot(ecdf(long), do.points=FALSE, verticals=TRUE)> x <- seq(3, 5.4, 0.01)> lines(x, pnorm(x, mean=mean(long), sd=sqrt(var(long))), lty=3)> ## dev.off() <IMG/>>> par(pty="s") # arrange for a square figure region> qqnorm(long); qqline(long)>> x <- rt(250, df = 5)> qqnorm(x); qqline(x)>> qqplot(qt(ppoints(250), df = 5), x, xlab = "Q-Q plot for t dsn")> qqline(x)>> shapiro.test(long)Shapiro-Wilk normality testdata: longW = 0.979, p-value = 0.011>> ks.test(long, "pnorm", mean = mean(long), sd = sqrt(var(long)))One-sample Kolmogorov-Smirnov testdata: longD = 0.0661, p-value = 0.43alternative hypothesis: two-sidedWarning message:In ks.test(long, "pnorm", mean = mean(long), sd = sqrt(var(long))) :ties should not be present for the Kolmogorov-Smirnov test>> ##@section One- and two-sample tests>> ## scan() from stdin :> ## can be cut & pasted, but not parsed and hence not source()d> ##scn A <- scan()> ##scn 79.98 80.04 80.02 80.04 80.03 80.03 80.04 79.97> ##scn 80.05 80.03 80.02 80.00 80.02> A <- c(79.98, 80.04, 80.02, 80.04, 80.03, 80.03, 80.04, 79.97,+ 80.05, 80.03, 80.02, 80, 80.02)> ##scn B <- scan()> ##scn 80.02 79.94 79.98 79.97 79.97 80.03 79.95 79.97> B <- c(80.02, 79.94, 79.98, 79.97, 79.97, 80.03, 79.95, 79.97)>> ## <IMG> postscript("images/ice.eps", ...)> boxplot(A, B)> ## dev.off() <IMG/>>> t.test(A, B)Welch Two Sample t-testdata: A and Bt = 3.25, df = 12, p-value = 0.0069alternative hypothesis: true difference in means is not equal to 095 percent confidence interval:0.013855 0.070183sample estimates:mean of x mean of y80.021 79.979>> var.test(A, B)F test to compare two variancesdata: A and BF = 0.584, num df = 12, denom df = 7, p-value = 0.39alternative hypothesis: true ratio of variances is not equal to 195 percent confidence interval:0.12511 2.10527sample estimates:ratio of variances0.58374>> t.test(A, B, var.equal=TRUE)Two Sample t-testdata: A and Bt = 3.47, df = 19, p-value = 0.0026alternative hypothesis: true difference in means is not equal to 095 percent confidence interval:0.016691 0.067348sample estimates:mean of x mean of y80.021 79.979>> wilcox.test(A, B)Wilcoxon rank sum test with continuity correctiondata: A and BW = 89, p-value = 0.0075alternative hypothesis: true location shift is not equal to 0Warning message:In wilcox.test.default(A, B) : cannot compute exact p-value with ties>> plot(ecdf(A), do.points=FALSE, verticals=TRUE, xlim=range(A, B))> plot(ecdf(B), do.points=FALSE, verticals=TRUE, add=TRUE)>> ###--- @chapter Grouping, loops and conditional execution>>> ###--- @chapter Writing your own functions>>> ###--- @chapter Statistical models in R>>> ###--- @chapter Graphical procedures>> ###--- @appendix A sample session>> ## "Simulate starting a new R session, by> rm(list=ls(all=TRUE))> set.seed(123) # for repeatability>> if(interactive())+ help.start()>> x <- rnorm(50)> y <- rnorm(x)> plot(x, y)> ls()[1] "x" "y"> rm(x, y)> x <- 1:20> w <- 1 + sqrt(x)/2> dummy <- data.frame(x = x, y = x + rnorm(x)*w)> dummyx y1 1 -0.065612 2 2.438533 3 2.539674 4 3.304915 5 2.984446 6 5.899827 7 5.176768 8 3.973239 9 8.0494310 10 12.3720611 11 9.4705512 12 13.6609913 13 8.4654414 14 13.8404915 15 16.5252316 16 16.9034617 17 17.3235318 18 16.0001519 19 16.2984120 20 16.68585> fm <- lm(y ~ x, data=dummy)> summary(fm)Call:lm(formula = y ~ x, data = dummy)Residuals:Min 1Q Median 3Q Max-3.540 -1.103 -0.054 1.152 3.262Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) -0.5431 0.8902 -0.61 0.55x 0.9653 0.0743 12.99 1.4e-10 ***---Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Residual standard error: 1.92 on 18 degrees of freedomMultiple R-squared: 0.904, Adjusted R-squared: 0.898F-statistic: 169 on 1 and 18 DF, p-value: 1.39e-10> fm1 <- lm(y ~ x, data=dummy, weight=1/w^2)> summary(fm1)Call:lm(formula = y ~ x, data = dummy, weights = 1/w^2)Weighted Residuals:Min 1Q Median 3Q Max-1.3205 -0.4492 -0.0088 0.5088 1.2656Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) -0.6155 0.6513 -0.94 0.36x 0.9721 0.0664 14.64 1.9e-11 ***---Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Residual standard error: 0.72 on 18 degrees of freedomMultiple R-squared: 0.922, Adjusted R-squared: 0.918F-statistic: 214 on 1 and 18 DF, p-value: 1.94e-11> attach(dummy)The following object is masked _by_ .GlobalEnv:x> lrf <- lowess(x, y)> plot(x, y)> lines(x, lrf$y)> abline(0, 1, lty=3)> abline(coef(fm))> abline(coef(fm1), col = "red")> detach()# dummy>> plot(fitted(fm), resid(fm),+ xlab="Fitted values",+ ylab="Residuals",+ main="Residuals vs Fitted")> qqnorm(resid(fm), main="Residuals Rankit Plot")> rm(fm, fm1, lrf, x, dummy)>>> filepath <- system.file("data", "morley.tab" , package="datasets")> if(interactive()) file.show(filepath)> mm <- read.table(filepath)> mmExpt Run Speed001 1 1 850002 1 2 740003 1 3 900004 1 4 1070005 1 5 930006 1 6 850007 1 7 950008 1 8 980009 1 9 980010 1 10 880011 1 11 1000012 1 12 980013 1 13 930014 1 14 650015 1 15 760016 1 16 810017 1 17 1000018 1 18 1000019 1 19 960020 1 20 960021 2 1 960022 2 2 940023 2 3 960024 2 4 940025 2 5 880026 2 6 800027 2 7 850028 2 8 880029 2 9 900030 2 10 840031 2 11 830032 2 12 790033 2 13 810034 2 14 880035 2 15 880036 2 16 830037 2 17 800038 2 18 790039 2 19 760040 2 20 800041 3 1 880042 3 2 880043 3 3 880044 3 4 860045 3 5 720046 3 6 720047 3 7 620048 3 8 860049 3 9 970050 3 10 950051 3 11 880052 3 12 910053 3 13 850054 3 14 870055 3 15 840056 3 16 840057 3 17 850058 3 18 840059 3 19 840060 3 20 840061 4 1 890062 4 2 810063 4 3 810064 4 4 820065 4 5 800066 4 6 770067 4 7 760068 4 8 740069 4 9 750070 4 10 760071 4 11 910072 4 12 920073 4 13 890074 4 14 860075 4 15 880076 4 16 720077 4 17 840078 4 18 850079 4 19 850080 4 20 780081 5 1 890082 5 2 840083 5 3 780084 5 4 810085 5 5 760086 5 6 810087 5 7 790088 5 8 810089 5 9 820090 5 10 850091 5 11 870092 5 12 870093 5 13 810094 5 14 740095 5 15 810096 5 16 940097 5 17 950098 5 18 800099 5 19 810100 5 20 870> mm$Expt <- factor(mm$Expt)> mm$Run <- factor(mm$Run)> attach(mm)> plot(Expt, Speed, main="Speed of Light Data", xlab="Experiment No.")> fm <- aov(Speed ~ Run + Expt, data=mm)> summary(fm)Df Sum Sq Mean Sq F value Pr(>F)Run 19 113344 5965 1.11 0.3632Expt 4 94514 23629 4.38 0.0031 **Residuals 76 410166 5397---Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1> fm0 <- update(fm, . ~ . - Run)> anova(fm0, fm)Analysis of Variance TableModel 1: Speed ~ ExptModel 2: Speed ~ Run + ExptRes.Df RSS Df Sum of Sq F Pr(>F)1 95 5235102 76 410166 19 113344 1.11 0.36> detach()> rm(fm, fm0)>> x <- seq(-pi, pi, len=50)> y <- x> f <- outer(x, y, function(x, y) cos(y)/(1 + x^2))> oldpar <- par(no.readonly = TRUE)> par(pty="s")> contour(x, y, f)> contour(x, y, f, nlevels=15, add=TRUE)> fa <- (f-t(f))/2> contour(x, y, fa, nlevels=15)> par(oldpar)> image(x, y, f)> image(x, y, fa)> objects(); rm(x, y, f, fa)[1] "f" "fa" "filepath" "mm" "oldpar"[6] "w" "x" "y"> th <- seq(-pi, pi, len=100)> z <- exp(1i*th)> par(pty="s")> plot(z, type="l")> w <- rnorm(100) + rnorm(100)*1i> w <- ifelse(Mod(w) > 1, 1/w, w)> plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+",xlab="x", ylab="y")> lines(z)>> w <- sqrt(runif(100))*exp(2*pi*runif(100)*1i)> plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+", xlab="x", ylab="y")> lines(z)>> rm(th, w, z)> ## q()>>