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Resampling Methodswith R

Engin YILDIZTEPE, Ph.D.2017-2018 Fall

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Reference Books

• R manuals

• Braun W.J., Murdoch D.J., A First Course in StatisticalProgramming with R, Cambridge, 2009.

• Kabacoff, R.I., R in Action, 2011.

• Matloff, N., The Art of R Programming, 2011.

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Reference Books

• Efron B., Tibshirani, R.J., “An Introduction to the Bootstrap”, Chapman&Hall, 1993.

• Davison, A. C. & Hinkley, D. V., “Bootstrap Methods and theirApplication”, Cambridge University Press, 1997.

• Zieffler A.S., Harring, R.H., Long J.D., “Comparing GroupsRandomization and Bootstrap Methods Using R”, Wiley, 2011.

• Manly, B.F.J, “Randomization, Bootstrap and Monte Carlo Methods in Biology”, Chapman&Hall, 2007.

• Chihara, L., Hesterberg, T., Mathematical Statistics withResampling and R, 2011.

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Installation of R• R can be downloaded from

http://www.r-project.org/ .• It runs on:

– Microsoft Windows,– A wide variety of UNIX,– MacOS

• R Studiohttp://www.rstudio.com

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Getting Help• Online help• Help menu

– Search– Html help

• Manuals on R-Project web site

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Getting Help

> help()> help(mean)>?mean> help.search("mean")>RSiteSearch("mean")> apropos("mean")

>?Syntax>?Arithmetic

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Calculating with R• Prompt symbol (>)• R can be used as a calculator:

>2+2>[1] 4

> 2+2*3[1] 8> (2+2)*3[1] 12

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Operators: Arithmetic, Comparison, LogicThey are listed in precedence groups, from highest to lowest.

Operator Function

( , { Function calls and grouping expressions (respectively)[ , [[ Indexing

:: , ::: access variables in a name space$ , @ access named components, access slots

^ Exponentiation (right to left)- Unary minus: sequence operator

%any% Special operators (%%, %/% etc.)* , / Multiply, divide+ , - Add, substract (binary)

< ,> , <= , >= == , !=

Comparison operators (smaller than, bigger than, smaller orequal than, bigger or equal than, equal to, not equal to)

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Operators: Arithmetic, Comparison, LogicOperator Function

! Logical negation

& , && And

| , || Or~ As in formulas-> , ->> Rightward assignment= Assignment (right to left)<- ,<<- Assignment (right to left)

? Help (unary and binary)

Within an expression operators of equal precedence are evaluated from left to right.

>3 / 2 * 4

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Some functions%% Gives the reminder (modulus). Ex: 17%%5%/% Gives the integer part of a division. Ex: 17%/%5

seq(a,b) To generate a sequence of integers between two valuesEx: seq(1,10)

seq(1.575, 5.125, by=0.05)abs(x) Gives the absolute value of x. Ex: abs(-5)sqrt(x) Gives the square root value of x.

log, log10, log2

log computes logarithms, by default natural logarithms, log10 computes common (i.e., base 10) logarithms, and log2 computes binary (i.e., base 2) logarithms. The general form log(x, base) computes logarithms with base base.

exp exp computes the exponential functionround rounds the values in its first argument to the specified number of

decimal places (default 0). min, max Returns the maxima and minima of the input valueslength(x) Gets or sets the length of vectorsum(x) Gives sum of the elements in vector x 10

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Assignment

• x<-2 or x=2

• Variable names can be built from letters, digits andthe dot symbol. The limitation,• The name must not start with a digit or a dot

symbol followed by a digit.• Names are CASE-SENSITIVE

(X and x don’t refer to the same variable)

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Vectors• Vectors are variables that can be thought of as

contiguous cells containing data.• Cells are accessed through indexing operations

(square brackets) such as a[1].• R has six basic (‘atomic’) vector types:

logical, integer, real, complex, string (or character), raw

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Data is a Vector

• c, seq, and rep that are used to create vectors in various situation.

• The concatenation function c is used to define vectors.> a<-c(1,3,5,6)> a[1] 1 3 5 6• We can also concatenate vectors of more than one element

as in> b <- c(23, 44) > ab<-c(a, b)> ab[1] 1 3 5 6 23 44

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Data is a Vector

• We can make a copy> a1<-a

• Assign the first value to 0> a1[1]<-0

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Vectors• seq (“sequence”) is used for equidistant series of numbers> seq(1,10)[1] 1 2 3 4 5 6 7 8 9 10> seq(1,10,2)[1] 1 3 5 7 9> seq(1,2,0.3)[1] 1.0 1.3 1.6 1.9• rep (“replicate”) is used to generate repeated values.> rep(a,2)[1] 1 3 5 6 1 3 5 6> rep(a, each=2)[1] 1 1 3 3 5 5 6 6> rep(a,1:4)[1] 1 3 3 5 5 5 6 6 6 6> rep(1:4,a)[1] 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 15

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Exercises

• Arithmetic calculations:› 7 * 4 + 3 › 3 + 7 * 4 › (3 + 7) * 4 › 1:10 › 1:10*3 › 3^1:10

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Exercises

• Write out the required line of R code (Use the rep() and seq() functions):a)1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7

7 8 8 8 8

b)1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8

c) 0 -1 0 -1 1 0 1 0 2 1 2 1 3 2 3 2 4 3 4 3 5 4 5 4 6 5 6 5 7 6 7 6

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Exercises

• Write out the required line of R code:a) "A" "B" "C" "D" "E" "F" "G" "H" "I" "J" "K" "L" "M" "N"

"O" "P" "Q" "R" "S" "T" "U" "V" "W" "X" "Y" "Z"

b) 9 27 81 243 729 2187 6561 19683 59049

c) -1 0 1 8 27 64 125 216 343 512 729

• Calculate the mean value of a vector

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Logical vectors

• A logical vector can only takes the TRUE and FALSE values.

> x<-c(TRUE,TRUE,FALSE,TRUE,FALSE)> x[1] TRUE TRUE FALSE TRUE FALSE• You can also use the capital T and F.> x<-c(T,T,F,T,F)

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Logical vectors as subscript

• if a logical vector is used as a subscript vector…> y<-1:10> y[1] 1 2 3 4 5 6 7 8 9 10> x[1] TRUE TRUE FALSE TRUE FALSE

> y[x][1] 1 2 4 6 7 9

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Logical vectors Example:

a<-7:16b<-19:10> a[1] 7 8 9 10 11 12 13 14 15 16

> b[1] 19 18 17 16 15 14 13 12 11 10

> s<-a<b> s[1] TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE

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using negative sign (-) as subscript

>x<- 5:20>x[1] 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20> x[-1][1] 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

> x[-c(1,3,5)][1] 6 8 10 11 12 13 14 15 16 17 18 19 20

> y<-x[-c(1,length(x))]> y[1] 6 7 8 9 10 11 12 13 14 15 16 17 18 19

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Character vectors

› colors<-c("red", "yellow", "blue")› more.colors <- c(colors, "green", "magenta", "cyan")

• To take substrings, use substr(x, start, stop) :> substr(colors, 1, 2)[1] "re" "ye" "bl"• Use the paste()function for building up strings by

concatenation:> paste(colors, "flowers")[1] "red flowers" "yellow flowers" "blue flowers"

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Character vectors

> paste("several ", colors, "s", sep="")[1] "severalreds" "severalyellows" "severalblues"

> paste("I like", colors, collapse = ", ")[1] "I like red, I like yellow, I like blue"

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List of available objects• The list of available objects in the specified environment can

be viewed with ls().> ls()[1] "a" "df" "f" "m" "Orange" "y"

• The class of a object is viewed with class(objname).> class(a)[1] “integer"

• Unnecessary object is deleted from the workspace using rm command.> rm(f)> ls()[1] "a" "df" "m" "Orange" "y"

• How can you remove all the objects in the workspace ?25

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Matrices• matrix() function> m <- matrix(1:6)> m

[,1][1,] 1[2,] 2[3,] 3[4,] 4[5,] 5[6,] 6

> m <- matrix(1:6, nrow=2, ncol=3)>m

[,1] [,2] [,3][1,] 1 3 5[2,] 2 4 6

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Matrices

• byrow=T switch causes the matrix to be filled in a rowwise rather than columnwise

> m <- matrix(1:6, nrow=2, ncol=3, byrow=T)>m

[,1] [,2] [,3][1,] 1 2 3[2,] 4 5 6

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Matrices

> rownames(m)<-c("a","b")

> colnames(m)<-c("c1","c2","c3")

> m

c1 c2 c3

a 1 2 3

b 4 5 6

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Matrices• Accessing elements:

> m[1,2] # the value in the first row and second column[1] 3

• Accessing whole rows or columns> m[1, ][1] 1 3 5> m[ ,1][1] 1 2

> m["a",]c1 c2 c3 1 2 3 > m[,"c3"]a b 3 6 29

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Matrices• Extend the matrix by adding rows or columns:

> m1<-rbind(m,c(7,8,9))> m1

c1 c2 c3a 1 2 3b 4 5 6

7 8 9

> m2<-cbind(m,c(7,8))> m2

c1 c2 c3 a 1 2 3 7b 4 5 6 8

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Matrices• Some useful functions :

> dim(m)[1] 2 3

> ncol(m)[1] 3

> nrow(m)[1] 2

> length(m)[1] 6

> t(m)a b

c1 1 4c2 2 5c3 3 6

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Exercises

• Let m be a matrix like below:[,1] [,2] [,3] [,4]

[1,] 1 4 7 10[2,] 2 5 8 11[3,] 3 6 9 12

• What is the required R code to compute the sum of the first row?

• How to calculate the mean of the last column?

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Arrays • Arrays are a multidimensional extension of vectors. All of the objects of

an array must be of the same mode.• array() functiona <- array(data_vector, dim_vector)

> a <- array(1:24, dim=c(3,4,2))> a, , 1

[,1] [,2] [,3] [,4][1,] 1 4 7 10[2,] 2 5 8 11[3,] 3 6 9 12

, , 2

[,1] [,2] [,3] [,4][1,] 13 16 19 22[2,] 14 17 20 23[3,] 15 18 21 24

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Arrays• Accessing elements:

> a[1,2,1] # the value in the [1,2,1][1] 4

• Accessing whole rows or columns> a[1, , ]

[,1] [,2][1,] 1 13[2,] 4 16[3,] 7 19[4,] 10 22

> a[,1, ][,1] [,2]

[1,] 1 13[2,] 2 14[3,] 3 15

> a[ , ,1][,1] [,2] [,3] [,4]

[1,] 1 4 7 10[2,] 2 5 8 11[3,] 3 6 9 12

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Logical operations in R

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Logical operations in R> a <- c(TRUE, FALSE, FALSE, TRUE)> b <- c(13, 7, 8, 0)> b[a][1] 13 0

> sum(a)[1] 2

>!a[1] FALSE TRUE TRUE FALSE

• “If we attempt logical operations on a numerical vector, 0 is taken to be FALSE, and any nonzero value is taken to be TRUE:”

>a&b[1] TRUE FALSE FALSE FALSE 36

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Subset vectors&matrices using logical conditions> > b[1] 23 22 19 16 25 30Which values are greater than 20 ?

> b>20[1] TRUE TRUE FALSE FALSE TRUE TRUE

> b[b>20][1] 23 22 25 30

How many values are greater than 20 ?> sum(b>20) [1] 4

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Subset vectors&matrices using logical conditionsWhat are the indices of the values greater than twenty?> which(b > 20)[1] 1 2 5 6

We can also use “which” in indice to get values.> b[which(b > 20)][1] 23 22 25 30

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Data Frame

• A data frame is a collection of column vectors, a data table.

• Columns (variables) have names, and the data can be addressed by referencing to these names.

• Example:> df<-

data.frame(isim=c("ali","yeşim","murat", "hakan","gülay"),yas=c(21,20,22,21,19), bol=c("tarih","fizik","mat","kimya","ist"),puan=c(93,78,88,91,90))

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Data Frame> df

isim yas bol puan1 ali 21 tarih 932 yeşim 20 fizik 783 murat 22 mat 884 hakan 21 kimya 915 gülay 19 ist 90

> rownames(df)<-c("ogr1","ogr2","ogr3","ogr4","ogr5")> df

isim yas bol puanogr1 ali 21 tarih 93ogr2 yeşim 20 fizik 78ogr3 murat 22 mat 88ogr4 hakan 21 kimya 91ogr5 gülay 19 ist 90

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Data Frame

> class(df)[1] "data.frame"

> dim(df)[1] 5 4

> names(df)[1] "isim" "yas" "bol" "puan"

> rownames(df)[1] "ogr" "ogr2" "ogr3" "ogr4" "ogr5"

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Data Frame

• Extracting data from a data frame object, we can use indices or names.> df[ ,2][1] 21 20 22 21 19

> df[ ,"yas"][1] 21 20 22 21 19

> class(df[ ,2])[1] "numeric“

> class(df[ ,3])[1] "factor" 42

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Data Frame> df[2, ]

isim yas bol puanogr2 yeşim 20 fizik 78

> df["ogr2", ]isim yas bol puan

ogr2 yeşim 20 fizik 78

> df[2:4, ]isim yas bol puan

ogr2 yeşim 20 fizik 78ogr3 murat 22 mat 88ogr4 hakan 21 kimya 91

> class(df[2, ])[1] "data.frame" 43

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Data Frame

• A variable from a data frame (dataframe), which has some name (name) can be accessed through dataframe$name.> df$yas[1] 21 20 22 21 19

> df$yas[3][1] 22

> df[3,2][1] 22

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Sub-set elements of a Data Frame• Example:if you are interested in students who have the

puan values equal or over 90, > df$puan>=90[1] TRUE FALSE FALSE TRUE TRUE

• you can find out what are the indices of these student:> which(df$puan>=90)[1] 1 4 5

> sec<-which(df$puan>=90)> sec[1] 1 4 5

> df$puan[sec][1] 93 91 90

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Sub-set elements of a Data Frame

> df[sec,]isim yas bol puan

ogr ali 21 tarih 93ogr4 hakan 21 kimya 91ogr5 gülay 19 ist 90

> df2<-df[sec,]> df2

isim yas bol puanogr ali 21 tarih 93ogr4 hakan 21 kimya 91ogr5 gülay 19 ist 90

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Sub-set elements of a Data Frame• Sometimes using the function subset() may be a little easier

way to subset the data frame.

subset(dataframename, logical expression, select=expression)

> df2<-subset(df,puan>=90)> df2

isim yas bol puanogr ali 21 tarih 93ogr4 hakan 21 kimya 91ogr5 gülay 19 ist 90

> df2<-subset(df,puan>=90,select=c(isim,puan))> df2

isim puanogr ali 93ogr4 hakan 91ogr5 gülay 90 47

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Sub-set elements of a Data Frame

• By using negative sign (-) we can drop the variables other than some you specify.

> df3<-subset(df,puan>=90,select=c(-isim,-puan))> df3

yas bologr 21 tarihogr4 21 kimyaogr5 19 ist

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Sub-set elements of a Data Frame

• data() function• airquality dataset

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List Structures• We can store multiple data types in the same object with list

structures.> mylist =list(first=1:10,second=letters[1:10], third=matrix(1:12,3))> mylist

$first[1] 1 2 3 4 5 6 7 8 9 10

$second[1] "a" "b" "c" "d" "e" "f" "g" "h" "i" "j"

$third[,1] [,2] [,3] [,4]

[1,] 1 4 7 10[2,] 2 5 8 11[3,] 3 6 9 12

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List Structures

• The same result can be achieved using the names function after creating the (unnamed) list:> mylist = list(1:10, letters[1:10], matrix(1:12,3))> names(mylist) = c(“first”,”second”,”third”)

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List Structures - subscripting

• For lists, there is a subtle distinction between partof a list, and the object which that part of the list represents.> mylist[1]$first[1] 1 2 3 4 5 6 7 8 9 10

> mean(mylist[1])[1] NAWarning message:In mean.default(mylist[1]) :argument is not numeric or logical: returning NA

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List Structures - subscripting• R provides two convenient ways to resolve this issue:• Accessing the elements by name:

> mylist$first[1] 1 2 3 4 5 6 7 8 9 10> mean(mylist$first)[1] 5.5

• the double bracket subscript operator: (if the dollar sign notation would be inappropriate, for example, accessing elements through their index or through a namestored in a character variable)

> mylist[[1]][1] 1 2 3 4 5 6 7 8 9 10> mean(mylist[[1]])[1] 5.5

• mylist[["first"]] is the same as mylist$first53

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List Structures - Concatenating

• > mylist[4]<-list(matrix(1:10,5))

• > mylist2 <- c(mylist, listB, listC,…)

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Exercises

• Calculate the sum , for n = 100, 200.

• data() function

• Use attitute dataset• User summary() function

• Extract every column of attitute dataframe as a vector.

• airquality dataset• Extract every column of airquality dataframe as a

vector.

n

kk

1

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