Data Structures: Vectors

Enrico Toffalini

What Are Data Structures

Data structures, like vectors, matrices, dataframes, lists, are fundamental tools that allow you to organize and store complex information, so that they can be easily processes by functions (e.g., lm() function may fit a linear model on variables stored in a dataframe)

Most operations you will perform in R (e.g., processing data, fitting models, plotting outputs) are performed on these data structures

Vectors

Simple one-dimensional structures that store data of different types

Here is an actual example (of a numerical vector):

Vectors as 1-D Arrays

Vectors are just special cases of arrays

Create Vectors with c()

Vectors can easily be created using the c() base function, with a sequence of elements separated by commas “,”

Vectors can be of different types. The following example shows a character vector (note the quotes " " around objects):

Teachers = c("Pastore", "Kiesner", "Granziol", "Toffalini", 
             "Calignano", "Epifania", "Bastianelli")

or numeric:

Hours = c(10, 15, 20, 10, 15, 5, 15, 5)

Vectors Must be Homogeneous

Vectors must contain elements of the same type. If you mix types, R will automatically coerce the elements to a single type, which may lead to undesired results.

Therefore, avoid mixing data types! Example:

Hours = c(10, 15, 20, 10, 15, "tbd", 15, 5)
Hours
[1] "10"  "15"  "20"  "10"  "15"  "tbd" "15"  "5"

everything was coerced to become a character!

If needed, use NA (Not Available):

Hours = c(10, 15, 20, 10, 15, NA, 15, 5)
Hours # remains a numerical vector, NA does not affect type
[1] 10 15 20 10 15 NA 15  5

Vectors Must be Homogeneous

You may coerce a vector to be a particular type if needed

Hours = c(10, 15, 20, 10, 15, "tbd", 15, 5)
Hours
[1] "10"  "15"  "20"  "10"  "15"  "tbd" "15"  "5"  
as.numeric(Hours)
Warning: NAs introduced by coercion
[1] 10 15 20 10 15 NA 15  5

But be careful! Elements that cannot be coerced to the target type, will be replace with NA

Hours = c("10", "15,", "20", " 10", "15 ", "tbd", "15.", "5_")
as.numeric(Hours)
Warning: NAs introduced by coercion
[1] 10 NA 20 10 15 NA 15 NA

Indexing Vectors

Select/extract elements with INDEXING using square brackets []:

Hours = c(10, 15, 20, 10, 15, 5, 15, 5)
Hours[4] # a single element
[1] 10
Hours[5:7] # a range of elements
[1] 15  5 15
Hours[c(1,3,6)] # specific elements
[1] 10 20  5

Know the length of a vector using the length() function, and use it:

length(Hours)
[1] 8
Hours[length(Hours)] # use it to extract the last element
[1] 5

Indexing Vectors

Negative indexing

You can use the minus sign - to select all elements except some from a vector. (This method is also applicable to dataframes)

Hours = c(10, 15, 20, 10, 15, 5, 15, 5)
Hours[-4] # ALL BUT a single element
[1] 10 15 20 15  5 15  5
Hours[-c(5:7)] # ALL BUT a range of elements
[1] 10 15 20 10  5
Hours[-c(1,3,6)] # ALL BUT specific elements
[1] 15 10 15 15  5
Hours[-length(Hours)] # ALL BUT the last element
[1] 10 15 20 10 15  5 15

Logical Indexing

Often, you’ll need to extract values from a vector based on specific logical conditions. Here’s an example:

Hours = c(10, 15, 20, 10, 15, 5, 15, 5)
Hours[Hours >= 15] # extract only values greater than or equal to 15
[1] 15 20 15 15

This is called logical indexing because you are selecting elements based on a logical vector (i.e., a sequence of TRUE, FALSE):

Hours >= 15 # the logical vector actually inside the square brackets
[1] FALSE  TRUE  TRUE FALSE  TRUE FALSE  TRUE FALSE

Also, you can use a vector to extract values from another vector:

Teachers[Hours >= 15]
[1] "Kiesner"     "Granziol"    "Calignano"   "Bastianelli"

Indexing and Assignment

With indexing, you can not only select, but also assign or modify elements in a vector:

Hours = c(10, 15, 20, 10, 15, 5, 15, 5)

Hours[1] = 0 # assign a new value
Hours[3] = Hours[3]+50 # modify an existing element
Hours
[1]  0 15 70 10 15  5 15  5

You can even assign values outside the current range of the vector. But what happens?

Hours[20] = 5
Hours
 [1]  0 15 70 10 15  5 15  5 NA NA NA NA NA NA NA NA NA NA NA  5

Operating on Vectors

you can simultaneously apply an operation to a whole vector, like

Hours = c(10, 15, 20, 10, 15, 5, 15, 5)
Hours / 5
[1] 2 3 4 2 3 1 3 1

Of course, this is useful when you want to save the result as a new vector:

ECTS = Hours / 5

Similarly, you can apply functions to all elements of a vector:

sqrt(Hours) # computes square root of each element
[1] 3.162278 3.872983 4.472136 3.162278 3.872983 2.236068 3.872983 2.236068
log(Hours) # computes the natural logarithm of each element
[1] 2.302585 2.708050 2.995732 2.302585 2.708050 1.609438 2.708050 1.609438

Summary Statistics on Vectors

A whole vector may serve to compute summary statistics, for example using functions such as mean(), sd(), median(), quantile(), max(), min():

mean(Hours) # returns the average value (mean) of the vector
[1] 11.875
sd(Hours) # returns the Standard Deviation of the vector
[1] 5.303301
median(Hours) # returns the median value of the vector
[1] 12.5

Summary Statistics on Vectors

A whole vector may serve to compute summary statistics, for example using functions such as mean(), sd(), median(), quantile(), max(), min():

quantile(Hours, probs=c(.25, .50, .75)) # returns desired quantiles
  25%   50%   75% 
 8.75 12.50 15.00 
max(Hours) # returns largest value
[1] 20
min(Hours) # returns smallest value
[1] 5

Summary Statistics - Managing Missing (NA) Values

All of the previous summary statistics will fail if there is even a single NA value:

Hours = c(10, 15, 20, 10, 15, NA, 15, 5)

mean(Hours) # a single NA value implies that the average is impossible to determine
[1] NA
quantile(Hours, probs=c(.25, .75)) # quantile() will even return an Error
Error in quantile.default(Hours, probs = c(0.25, 0.75)): missing values and NaN's not allowed if 'na.rm' is FALSE

You can easily manage missing values by adding the na.rm=TRUE argument:

mean(Hours, na.rm=TRUE) # NA values are ignored 
[1] 12.85714
quantile(Hours, probs=c(.25, .75), na.rm=TRUE) # NA values are ignored 
25% 75% 
 10  15

Replacing NA With the Average Value

Replacing a missing value with the average across valid values is risky, as it may alter many other summary statistics, but it is a good example for understanding different concepts seen so far:

Hours = c(10, 15, 20, 10, 15, NA, 15, 5)

# compute the average value ignoring NAs, and put it wherever 
# there is a NA value in the vector
Hours[is.na(Hours)] = mean(Hours, na.rm=TRUE)

# now let's inspect the updated content of the vector
Hours
[1] 10.00000 15.00000 20.00000 10.00000 15.00000 12.85714 15.00000  5.00000
# by the way... na.rm=TRUE is no longer needed now, as NA is no longer there
mean(Hours)
[1] 12.85714

Frequency Counts

Another useful summary statistic is the frequency count, which shows how often each unique value appears in a vector. You can use the table() function to calculate frequencies easily:

type = c("METHODOLOGY", "METHODOLOGY", "PROGRAMMING", "SOFT SKILLS", "SOFT SKILLS", 
         "METHODOLOGY", "SOFT SKILLS", "METHODOLOGY", "PROGRAMMING")
table(type)
type
METHODOLOGY PROGRAMMING SOFT SKILLS 
          4           2           3

Be careful: R is case sensitive!

type = c("METHODOLOGY", "methodology", "PROGRAMMING", "SOFT SKILLS", "SOFT SKILLS", 
         "METHODOLOGY", "SOFT SKILLS", "METHODOLOGY", "Programming")
table(type)
type
methodology METHODOLOGY Programming PROGRAMMING SOFT SKILLS 
          1           3           1           1           3