Assignments. Math 338 Lab 1: Introduction to R. Atoms, Vectors and Matrices

Transcription

1 Assignments Math 338 Lab 1: Introduction to R. Generally speaking, there are three basic forms of assigning data. Case one is the single atom or a single number. Assigning a number to an object in this case is quite trivial. All we need is to use < or = for assigning a number or an atom to a character. In the following, > refers to the prompt in R. The second form is the vector form. In this form, we assign a name to an array of numbers. This can be done with the command c which stands for concatenation. The interesting fact is that we can call any member of the vector or we can replace that member with a new member or to perform various arithmetic operations on that vector, as shown below. Finally, the third form of storing data is to put them in a matrix form. The command is matrix. First we need to input the data set of interest, followed by telling R the dimensionality of the matrix that needs to be specified. For example, we can put an array of 9 numbers into a matrix with 3 rows and 3 columns. We demonstrate all of these below. Atoms, Vectors and Matrices (a) Atoms: > sam=2 > sam [1] 2 > sam+sam [1] 4 > (2*sam*2)/2 [1] 4 > sam^(1/3) [1] > sqrt(sam) [1] > abs(-sam) [1] 2 1

2 (b) Vectors > class.age=c(35,35,36,37,37,38,38,39,40.5,43,44,44.5,50,19) > class.age [1] > class.age[3] [1] 36 > class.age[1:5] [1] > class.age[-5] [1] > class.age[-c(2,7)] [1] > class.age*2 [1] > sqrt(class.age) [1] [14] > class.age^(-1) [1] [11] > class.age*class.age [1] > class.age^2 [1] > mean(class.age) 2

3 [1] > median(class.age) [1] 38 > class.age=(class.age)/2 > class.age [1] > class.age=class.age*2 Often it is useful to create an empty vector. Here is the way this is done: > hi=numeric(10) > hi [1] Vectors do not have to be numerical. We can create a vector of characters: > hi=c("hello","whasup","longday") > hi [1] "hello" "whasup" "longday" Later, it becomes useful to ask R the length of a vector: > length(class.age) [1] 14 (c) Matrices > sam = matrix(nrow=3,ncol=4) > sam [,1] [,2] [,3] [,4] [1,] NA NA NA NA [2,] NA NA NA NA [3,] NA NA NA NA > sam = matrix(c(1,2,3,4,5,6,7,8,9,10,11,12),nrow=3,byrow=t) 3

4 > sam [,1] [,2] [,3] [,4] [1,] [2,] [3,] > sam<-matrix(c(1,2,3,4,5,6,7,8,9,10,11,12),nrow=3,byrow=f) > sam [,1] [,2] [,3] [,4] [1,] [2,] [3,] > sally =c(1,2,3,4,5,6,7,8,9,10,11,12) > sam=matrix(sally,nrow=3,byrow=t) > sam [,1] [,2] [,3] [,4] [1,] [2,] [3,] > v1=c(1,2,3,4) > v2=c(5,6,7,8) > v3=c(9,10,11,12) > sam=matrix(c(v1,v2,v3),nrow=3,byrow=t) > sam [,1] [,2] [,3] [,4] [1,] [2,] [3,]

5 > sam[1,] [1] > sam[,2] [1] > sam[1,3] [1] 3 sam[3,]<-v2 > sam [,1] [,2] [,3] [,4] [1,] [2,] [3,] > sam[1,]<-log(v1) > sam [,1] [,2] [,3] [,4] [1,] [2,] [3,] (d) Lists R provides a powerful additional storing function called list. The importance of list is in that we can store various objects of different natures such as matrices, vectors, or atoms into a unique space, followed by calling different parts of that object separately. Let s assume that we would like to store the following three object into a list-object called sam: > s1=3 > s2=seq(1,10,2) > s3=matrix(c(1:9),nrow=3) > s1 [1] 3 > s2 5

6 [1] > s3 [,1] [,2] [,3] [1,] [2,] [3,] > s<-list(s1,s2,s3) > s [[1]] [1] 3 [[2]] [1] [[3]] [,1] [,2] [,3] [1,] [2,] [3,] > s[[1]] [1] 3 > s[[2]] [1] > s[[3]] [,1] [,2] [,3] [1,] [2,] [3,] The for loop Often times, it becomes necessary to repeat certain calculations a number of times. This is done in R using a simple command called for. Here are some examples: 6

7 > for(i in 1:3) + { + print("sam") + } [1] "sam" [1] "sam" [1] "sam" > s=matrix(c(1,2,3,4,5,6,7,8,9),nrow=3,byrow=t) > for(i in 1:3) + print(s) [,1] [,2] [,3] [1,] [2,] [3,] [,1] [,2] [,3] [1,] [2,] [3,] [,1] [,2] [,3] [1,] [2,] [3,] or: > for(i in 1:3) + { print(s)} [,1] [,2] [,3] [1,] [2,] [3,] [,1] [,2] [,3] [1,] [2,] [3,] [,1] [,2] [,3] [1,] [2,] [3,]

8 > for(i in 1:3) +{ + print(s[i,]) + } [1] [1] [1] Functions In principle, there are two sorts of functions in R. The most common and useful ones are the library functions or the already written commands. For example, mean and sd are commands that calculate the average and the standard deviation of an object, say a vector respectively. Here are a couple of examples: > s2<-seq(1,10,2) > s2 [1] > mean(s2) [1] 5 > var(s2) [1] 10 > sd(s2) [1] > median(s2) [1] 5 Second type of functions are those that the users of R create. These functions will remain in the command memory of the software unless you delete them or overwrite them. Expectedly, the command to create a function is f unction. Here is an example of a function that gets a matrix, and calculates the standard deviation divided by the mean of its rows σ. This measurement is called the coefficient µ of variation. Note that in writing this function, I use the commands mean, and sd. 8

9 In general, any time you are not sure what an R command does or to learn about its specifics, just type a question mark, followed by the command in the prompt. > m.cv<-function(mat) { u=nrow(mat) t=numeric(u) for(i in 1:u) { t[i]<-sd(mat[i,])/mean(mat[i,]) } return(t) } > sm3<-matrix(c(1:9),nrow=3) > sm3 [,1] [,2] [,3] [1,] [2,] [3,] > m.cv(sm3) [1]

10 Visualizing Data: Pie Charts, Stem plots, Histograms Categorical Data For categorical data, we keep track of counts or relative frequencies of each group. Therefore, a schematic presentation should reflect the percentage of occurrences in each category. This is usually done via two types of graphs: 1- Pie charts, and 2- Barplots. Both graphs are easy to create in R. An important issue here is that in most cases, it would make sense to label the categories. We show you how to do this below. Example 1. The counts and the percentages of the marital status of American women was collected by the Current Population Survey in 1995 as following: Marital Status Count (millions) Percent Never Married Married Widowed Divorced Here are the commands to provide the pie-chart for these data (figure 1): > married<-c(43.9,116.7,13.4,17.6) > married.code<-as.factor(c(1,2,3,4)) > pie(married,married.code) Alternatively, we could label each piece of pie by creating a factor vector that contains the names of each pie (figure 2): > married.code<-c("never married","married","widowed","divorced") > pie(married,married.code) To create a barplot for the married data, it is sufficient to execute the following function (figure 3): > married<-c(22.9,60.9,7,9.2) > barplot(married,names.arg=married.code) 10

11 Figure 1: Pie chart for the Married data. Stemplots and Histograms For quantitative data, stemplots and histograms are the useful visual tools. Example 2. Let s revisit the class-age data we introduced previously. To create the stemplot for these data, we can do the following: > class.age<-c(35,35,36,37,37,38,38,39,40.5,43,44,44.5,50,19) > class.age [1] > stem(class.age) The decimal point is 1 digit(s) to the right of the

12 never married married divorced widowed Figure 2: Pie chart for the Married data with labels. 5 0 > stem(class.age,scale=2) The decimal point is 1 digit(s) to the right of the > test<-c(0,0.01,0.22,0.34,0.36,0.31,0.36,0.45,0.4,0.55,0.65) > stem(test) 12

13 never married married widowed divorced Figure 3: Barplot for the Married data with labels. The decimal point is 1 digit(s) to the left of the > stem(test,scale=2) The decimal point is 1 digit(s) to the left of the

14 To create a histogram for the class-age data, it is sufficient to use the hist command (figure 4): > hist(class.age) Histogram of class.age Frequency class.age Figure 4: The histogram of class.age. We can make the bars finer. Here is a simple trick (figure 5): > b1<-seq(15,50,3) > b1 [1] > b1<-seq(15,50,3)+2 > hist(class.age,breaks=b1) 14

15 Histogram of class.age Frequency class.age Figure 5: The histogram of class.age with finer classes. Measuring Center: Quartile The Mean, The Median, and the The measures of centrality play a fundamental role in understanding the statistical distributions. The chief important ones are the mean, the median, and the other quantiles. > mean(class.age) [1] > median(class.age) [1] 38 > quantile(class.age) 0% 25% 50% 75% 100% > quantile(class.age,prob=0.66) 15

16 66% Comparing Mean and Median The Symmetric Case For symmetric distributions such as the one in figure 6, the median and the mean are close to each other. Histogram of n1 Frequency n1 Figure 6: A symmetric distribution. Mean= , Median= For distributions that are skewed to the left such as the one in figure 7, the mean is smaller than the median (why?) Finally, for the right-skewed distributions, the mean is larger than the median (figure 8). Measuring Spread: The Standard Deviation The standard deviation reflects the number of units away from the mean. For example, the standard deviation for the data in figures 6, 7, and 8 are 10.03, 0.19, and 0.19 respectively. To calculate the variance and the standard deviation for the class.age data, we can type: 16

17 Histogram of n3 Frequency n3 Figure 7: Left-skewed Distribution. Mean= 0.75, Median= > var(class.age) [1] > sd(class.age) [1] Project 1. Visualizing Grades. First, read the file grades.txt from the webpage. To do this, run the following code in R: > grades=read.table(" This will generate a data frame of the size called grades in R for you. Rows are students, and columns represent the Verbal SAT score, the Math SAT score, and the GPA for each student. To examine the dimensionality of this object you can type: > dim(grades) [1] Which confirms what we planned initially. Now, we are at a position to answer the following questions: 17

18 Histogram of n5 Frequency n5 Figure 8: Right-skewed Distribution. Mean= 0.24, Median= (1.) Create barplots, dotplots, stemplots, and histograms for the three variables of interest. To make life easier, use the command attach to make the data file grades your working data. Next, proceed by just typing the name of the column of interest. Here is what I mean: > attach(grades) > GPA [1] [19] [37] [55] [73] [91] (2.) Calculate the min, the max, the mean, the median, first quartile, third quartile, and the standard deviation of each variable. A good chunk of that information may be obtained through using the command summary: > summary(gpa) 18

19 Min. 1st Qu. Median Mean 3rd Qu. Max (3.) Report your findings in detail. Compare the verbal scores with the math scores. Comment on the symmetry, measures of centrality, measures of spread, and the potential outliers in each distribution. Make sure to comment on the statistical features of the GPA as well. Boxplots Boxplots are efficient tools for representing data distributions. The five number summary can be traced on a boxplot. Additionally, we can figure-out outliers with boxplots. Remember the three distributions in figures 6, 7 and 8. Note that these distributions are symmetric, left skewed and right skewed respectively. Here is how I created figure 9 below: > n1=rnorm(100000,2,3) > n2=rpois(100000,3) > n3=rbeta(100000,12,3) > par(mfrow=c(3,2)) > hist(n1) > boxplot(n1) > hist(n2) > boxplot(n2) > hist(n3) > boxplot(n3) Note that the command par(mfrow) creates a 3 by 2 grid in the graphic area. The side-by-side boxplots are very helpful in comparing two or more distributions. For example figure 2 shows a side-by-side boxplots for the two skewed distributions of figure 1. > boxplot(n1,n2) 19

20 Histogram of n1 Frequency n1 Histogram of n2 Frequency n2 Histogram of n3 Frequency n3 Figure 9: Histograms along with boxplots for the three simulated datasets. 20

21 Figure 10: Side-by-side boxplots for the two of the distributions of figure 1. 21