1 Matrices and Vectors and Lists

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1 University of Wollongong School of Mathematics and Applied Statistics STAT231 Probability and Random Variables 2014 Second Lab - Week 4 If you can t finish the log-book questions in lab, proceed at home. If you have questions ask your tutor or come to the consultation hours. Be reminded that logbook questions are an assessment task and that future assignments will rely on lab knowledge. 1 Matrices and Vectors and Lists The basic data object in R is a vector. Even scalars are vectors of length 1. There are several ways to create vectors. To create a vector you might use c() operator. The : operator creates sequences incrementing/decrementing by 1. If the step-size should be different, then seq() function can be used. To repeat numbers in a certain fashion you can use rep() by specifying arguments times and each. > b1<-c(0,0,0,1) > #1:n stands for seq(from=1,to=n,by=1) > b2<-1:10 > b3<-10:1 > b4<-seq(from=0,to=10,by=2) > b5<-rep(1:4,times=5) > b6<-rep(1:4,each=2) Matrices are created with the command > matrix(data = NA, nrow = 1, ncol = 1, byrow = FALSE, dimnames = NULL) where the first optional argument is a vector containing the data that is used to fill the matrix, nrow specifies the number of rows and ncol the number of columns. The default option byrow = FALSE is to fill column by column. The matrix multiplication operator in R is %*%. > #matrices > A1<-matrix(c(1,2,3,4,5,6,7,8),nrow=2,ncol=4,byrow=TRUE) > A2<-matrix(c(1,2,3,4,5,6,7,8),nrow=2,ncol=4) #byrow=false > A1>2 > #dimensions of A1 and A2 1

2 > dim(a1) > dim(a2) > #accessing the first row of matrix A1 > A1[1,] > #accessing the second column of matrix A1 > A1[,2] > #multiply matrices > A1%*%A2 #doesn't work because dimensions don't fit > A1%*%t(A2) #so only A1 times transpose of A2 works > #matrix times vector > A1%*%b1 > #or alternatively > A2%*%as.matrix(b1) > #element-wise multiplication > A1*A2 > #row-sum and column-sum > apply(a1,1,sum) > apply(a1,2,sum) A list is a convenient structure to store multiple objects simultaneously. These obsjetcs can be of different type and different length and structure: > n <- c(2, 3, 5) # object 1 > s <- c("aa", "bb", "cc", "dd", "ee") # object 2 > b <- c(true, FALSE, TRUE, FALSE, FALSE) # object 3 > x <- list(n, s, b, 3) # list x contains n, s, b To access the, say, first element and second element of a list you use double brackets [[]] : > x[[1]] [1] > x[[2]] [1] "aa" "bb" "cc" "dd" "ee" 2

3 You can also name elements and then refer to the name > # list x contains n, s, b, now named "n", "s" and "b.vector" > x <- list(n=n,s=s,b.vector=b,number=3) > # option 1: using the entry number > x[[1]] [1] > x[[3]] [1] TRUE FALSE TRUE FALSE FALSE > #option 2: using element's name > x$n [1] > x$b.vector [1] TRUE FALSE TRUE FALSE FALSE Do you remember the data-frame object from the lab of week 1? A dataframe has properties of a matrix and of a list. You can access the elements of the dataframe in the same way as with lists by calling its column name, but also by using the matrix notation that is used to access, say, the columns of a matrix. There is one fundamental difference between lists and dataframes. Each column/object of a dataframe needs to have the same length. A list doesn t have this requirement. A dataframe can also be easily converted into a matrix (if dataframe consists of numbers for example) using the command as.matrix(). 2 Some Control Structures in R 2.1 Loops R has a several types of loops, for example the for and while-loops. Examples are given below when computing the product of the numbers 1 to 10. > #loops > #for loop > f<-1 3

4 > index<-1:10 > #"for" loop executed as many times as length of index and > # i takes values of the index-set "index" > for(i in index){ + f<-f*i + #cat is here to produce output of current values of f + cat("current value of f at ",i,"th iteration: ",f,"\n") + } current value of f at 1 th iteration: 1 current value of f at 2 th iteration: 2 current value of f at 3 th iteration: 6 current value of f at 4 th iteration: 24 current value of f at 5 th iteration: 120 current value of f at 6 th iteration: 720 current value of f at 7 th iteration: 5040 current value of f at 8 th iteration: current value of f at 9 th iteration: current value of f at 10 th iteration: > f [1] > #while loop > f<-1 > i<-1 > while(i<=10){ + #executes the statements that follow until + # condition not met, here as long as i<= f<-f*i + cat("current value of f at ",i,"th iteration: ",f,"\n") + i<-i+1 + } current value of f at 1 th iteration: 1 current value of f at 2 th iteration: 2 current value of f at 3 th iteration: 6 current value of f at 4 th iteration: 24 current value of f at 5 th iteration: 120 current value of f at 6 th iteration: 720 current value of f at 7 th iteration: 5040 current value of f at 8 th iteration:

5 current value of f at 9 th iteration: current value of f at 10 th iteration: > f [1] > #easier > prod(1:10) [1] If/Then Statement The if/then statement in R has the following form. > if(cond){ + #evaluate expression if cond is true + }else{ + #evaluate expression if cond is false + } Alternatively you can use ifthen(cond,expr.true,expr.false). > x<-5 > if(x<0){cat("number <0\n")}else{cat("Number >=0\n")} Number >=0 > x<--3 > if(x<0){cat("number <0\n")}else{cat("Number >=0\n")} Number <0 > x <- c(6:-4) > for(i in x){ + ifelse(i >= 0, print(sqrt(i)), print(na)) + } [1] [1] [1] 2 5

6 [1] [1] [1] 1 [1] 0 [1] NA [1] NA [1] NA [1] NA 3 Functions Sometimes we want to calculate or apply an operation to many numbers or repeat an analysis which consist of many lines of code. In this case it is often easier to define functions that contain this complex code. When a function is defined we need to allocate a name. Once it is defined, the name can be applied to several data sets by only calling the name of the function and its arguments without needing to repeat numerous times the possibly lengthy code. > myfunction <- function(arg1, arg2,... ){ + statements + return(object) + } Here myfunction is an arbitrary chosen name, and arg1, arg2,... are arguments of the function. The definition needs to finish with returning of an object, here achieved by return(object). In Section 2.1 we calculated the product of the first 10 number, in fact that s the factorial function, in r implemented by factorial(). To illustrate functions in R we define the factorial functions in several ways. > factorial1<-function(n){return(prod(1:n))} > factorial2<-function(n){prod(1:n)} #without using return > # since prod(1:n) shows an output when entered in R console, it > # returns this output automatically without using return() > # to be "cleaner", using "return" is recommended > > #factorial3 calls itself, called recursion > factorial3<-function(n){ + cat("n=",n,"\n"); 6

7 + if(n==1){ + return(n) + }else{ + return(n*factorial3(n-1)) + }#end if + }#end function factorial 3 Now test by yourself and try to understand why and how these functions work! > #tests > factorial1(10) > factorial2(10) > factorial3(10) > factorial1(15) > factorial2(15) > factorial3(15) > #compare with default R-function > factorial(10) > factorial(15) In R you can only return one object not several. For example if you have a data set and you want to return ȳ and s 2, you need to set-up a list which then contains these two values as elements. > y<-seq(1,100,by=2) > bary.s2<-function(data){ + meany<-mean(data); + vary<-var(data) + list1<-list(bary=meany,s2=vary) #first element is called bary, 2nd s2 + return(list1) + } > #now apply function to data "y" > bary.s2(data=y) $bary [1] 50 $s2 [1] 850 7

8 > #or storing > test<-bary.s2(y) > #show bary=sample mean > test$bary [1] 50 > #show s2=sample var > test$s2 [1] 850 > #directly > mean(y) [1] 50 > var(y) [1] Hypergeometric Distribution Create the probability function (p.f.) f(y) and the cumulative distribution function (c.d.f.) F (y) for the hypergeometric distribution with parameters N = 42, K = 7,n = 21 with (notations from lecture). Recall (in case this has already been covered) that N is the population size, n the size of sample taken (sample size), and K is the number of objects with certain characteristic (e.g. white balls). First create a sequence of numbers from 0 to 7, the domain of Y. The R-function dhyper (p.f. of hypergeometric) uses different notations for parameters: > y<-0:7 #y<-sequence(0,7,by=1) > # m is the number of while balls (what we call K) > # n the number of black balls, therefore N=m+n and m=n-n > # k=21 (what we call usually n) > # The probability function in R gives > # the probability of having x while balls! > fy<- dhyper(x=y, m=7, n=42-7, k=21) 8

9 The Vector fy contains the 8 probabilities when referring to the events that the sample contains 0, 1,..., 7 white balls. Calculate the c.d.f. using cumsum(), a function for the cumulative sum, see help(cumsum): > FY<-cumsum(fY) > #alternatively use > FY1<-phyper(q=y, m=7, n=42-7, k=21) > #compare FY with FY1 > FY-FY1 [1] e e e e e-16 [6] e e e-16 Some plotting: > #first plain plots > plot(0:7,fy) > points(0:7,fy) > #doesn't look good fy :7 This looks rather non-informative, instead we draw lines using a histogram style, a stepfunction style and some coloring. > plot(0:7,fy,type="h",col="red",ylim=c(0,1)) #histogram type > points(0:7,fy,type="s",lty=1,col="blue") #step type > # needs legend, where you need to define also colors and line-type (lty) > legend("topleft",c("f(y)","f(y)"),col=c("red","blue"),lty=c(1,1)) 9

10 fy f(y) F(y) :7 There are many graphical parameters for functions like plot(), points(), lines(), etc. to enable changing almost every aspect of a plot, which makes R very powerful, but sometimes this is complicated finding the correct parameters for the desired outcome. 5 Simulating Random Numbers Before we continue enter > set.seed(studentid) where studentid is your actual UOW student ID. This command will ensure that every student creates unique random numbers, which guarantees that all of you have different results of your log-book questions. We first generate 150 random numbers from the uniform distribution in [0, 1]. Then we transform the data and obtain a vector Y1, which contains random numbers 0, 1,..., 7, in fact numbers from the hypergeometric distribution described above, achieved by the transformation inside the for-loop. > size<-150 > U<-runif(size) > Y1<-rep(0,size) > for(i in 1:size){ + Y1[i]<- min((0:7)[u[i]<=fy]) + }#end for loop > #to check what happens fix i, e.g. i<-10 10

11 > # enter 1) U[i], 2) FY, 3) U[i]<=FY > # then 4) (0:7)[U[i]<=FY] > > #simpler is to use R command > Y2<-rhyper(nn=size, m=7, n=42-7, k=21) > # Plot the histograms > hist(y1,breaks=seq(-0.5,7.5,by=1)) > hist(y2,breaks=seq(-0.5,7.5,by=1)) Histogram of Y1 Histogram of Y2 Frequency Frequency Y Y2 Next we use the simulated data to estimate the mean, the variance, P (Y = 3), P (2 Y 4) and P (1 Y 6) with the R-functions mean(), var(). > mean(y1) > mean(y2) > var(y1) > var(y2) > # Find > #P(Y=3) > Y1==3 #look at output and compare with Y1 > mean(y1==3) # sum(y1==3)/150 11

12 > mean(y2==3) # sum(y2==3)/150 > #P(2<=Y<=4) > Y1>=2 & Y1<=4 #look at output and compare with Y1 > mean(y1>=2 & Y1<=4) > mean(y2>=2 & Y2<=4) > #P(1<=Y<=6) > Y1>=1 & Y1<=6 #look at output and compare with Y1 > mean(y1>=1 & Y1<=6) > mean(y2>=1 & Y2<=6) To calculate an estimate for P (Y = 3), we first simulate the random numbers stored in Y1, then we check whether each number equals 3 by Y1==3, which results in a vector of size 150 of Boolean-type (TRUE/FALSE). Then the mean()-function is applied to this vector, however TRUE is converted to 1 and FALSE is converted to 0, hence mean() is applied to a vector of 1 s and 0 s. It counts the average proportion of observing a 3, which is an estimate for the probability of observing a 3. 12