RPR-OBJECTS-Vectors

R scalars and vectors

 

Keywords:  Types of R objects: scalars, vectors and matrices

 

 

Caution!

This unit is under development. There is some contents here but it is incomplete and/or may change significantly: links may lead to nowhere, the contents is likely going to be rearranged, and objectives, deliverables etc. may be incomplete or missing. Do not work with this material until it is updated to "live" status.

 

Abstract

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This unit ...

Prerequisites

You need to complete the following units before beginning this one:

• RPR-Syntax_basics

 

Objectives

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Outcomes

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Deliverables

• Time management: Before you begin, estimate how long it will take you to complete this unit. Then, record in your course journal: the number of hours you estimated, the number of hours you worked on the unit, and the amount of time that passed between start and completion of this unit.
• Journal: Document your progress in your Course Journal. Some tasks may ask you to include specific items in your journal. Don't overlook these.
• Insights: If you find something particularly noteworthy about this unit, make a note in your insights! page.

 

Evaluation

Evaluation: NA

This unit is not evaluated for course marks.

 

Contents

Data items

R objects can be composed from different kinds of data according to the type and number of "atomic" values they contain:

• Scalar data are single values;
• Vectors are ordered sequences of scalars, they must all have the same "data type" (e.g. numeric, logical, character ...);
• Matrices are vectors for which one or more "dimension(s)" have been defined;
• Data frames are spreadsheet-like objects, columns are like vectors and all columns must have the same length, but within one data frame, columns can have different data types;
• Lists are the most general collection of data items, the can contain items of any type and kind, including matrices and lists.

 

Scalar data

Scalars are single numbers, the "atomic" parts of more complex datatypes. Of course we can work with single numbers in R, but under the hood they are actually vectors of length 1. (More on vectors in the next section). We have encountered scalars above, e.g. with the use of constants and their assignment to variables. To round this off, here are some remarks on the types of scalars R uses, and on coercion between types, i.e. casting one datatype into another. The following scalar types are supported:

• Boolean constants: TRUE and FALSE. This type has the "mode" logical";
• Integers, floats (floating point numbers) and complex numbers. These types have the mode numeric;
• Strings. These have the mode character.

Other modes exist, such as list, function and expression, all of which can be combined into complex objects.

The function mode() returns the mode of an object and typeof() returns its type. Also class() tells you what class it belongs to.

typeof(TRUE)
class(3L)
mode(print)

I have combined these information functions into a single function, objectInfo() which gets loaded and defined with the BasicSetup script so you can experiment with it. We can use objectInfo() to explore how R objects are made up, by handing various expressions as arguments to the function. Many of these you may not yet recognize ... bear with it though:

#Let's have a brief look at the function itself: typing a function name without its parentheses returns the source code for the function:
objectInfo

# Various objects:

#Scalars:
objectInfo( 3.0 )    # Double precision floating point number
objectInfo( 3.0e0 )  # Same value, exponential notation

objectInfo( 3 )   # Note: integers are double precision floats by default.
objectInfo( 3L )  # If we really want an integer, we must use R's
                  # special integer notation ...
objectInfo( as.integer(3) )  # or explicitly "coerce" to type integer...

# Coercions: For each of these, first think what result you would expect:
objectInfo( as.character(3) )  # Forcing the number to be interpreted as a character.
objectInfo( as.numeric("3") )   # character as numeric
objectInfo( as.numeric("3.141592653") )  # string as numeric. Where do the
                                         # non-zero digits at the end come from?
objectInfo( as.numeric(pi) )    # not a string, but a predefined constant
objectInfo( as.numeric("pi") )  # another string as numeric ... Ooops -
                                # why the warning?
objectInfo( as.complex(1) )

objectInfo( as.logical(0) )
objectInfo( as.logical(1) )
objectInfo( as.logical(-1) )
objectInfo( as.logical(pi) )      # any non-zero number is TRUE ...
objectInfo( as.logical("pie") )   # ... but not non-numeric types.
                                  # NA means "Not Available".
objectInfo( as.character(pi) )    # Interesting: the conversion eats digits.

objectInfo( Inf )                # Larger than the largest representable number
objectInfo( -Inf )               # ... or smaller
objectInfo( NaN )                # "Not a Number" is numeric
objectInfo( NA )                 # "Not Available" - i.e. missing value is
                                 # logical

# NULL
objectInfo( NULL )     # NULL is nothing. Not 0, not NaN,
                       # not FALSE - nothing. NULL is the value that is
                       # returned by expressions or
                       # functions when the result is undefined.

objectInfo( as.factor("M") )     # factor
objectInfo( Sys.time() )         # time
objectInfo( letters )            # inbuilt
objectInfo( 1:4 )                # numeric vector
objectInfo( matrix(1:4, nrow=2)) # numeric matrix
objectInfo( data.frame(arabic = 1:3,                           # dataframe
                       roman = c("I", "II", "III"),
                       stringsAsFactors = FALSE))
objectInfo( list(arabic = 1:7, roman = c("I", "II", "III")))   # list

# Expressions:
objectInfo( 3 > 5 ) # Note: any combination of variables via the logical
                    # operators ! == != > < >= <= | || & and && is a
                    # logical expression, with values TRUE or FALSE.
objectInfo( 3 < 5 )
objectInfo( 1:6 > 4 )

objectInfo( a ~ b )              # a formula
objectInfo( objectInfo )         # this function itself

 

Sometimes (but rarely) you may run into a distinction that R makes regarding integers and floating point numbers. By default, if you specify e.g. the number 2 in your code, it is stored as a floating point number. But if the numbers are generated e.g. from a range operator as in 1:2 they are integers! This can give rise to confusion as in the following example:

a <- 7
b <- 6:7
str(a)             # num 7
str(b)             # int [1:2] 6 7
a == b[2]          # TRUE
identical(b[2], a) # FALSE ! Not identical! Why?
                   # (see the str() results above.)

# If you need to be sure that a number is an
# integer, write it with an "L" after the number:
c <- 7L
str(c)             # int 7
identical(b[2], c) # TRUE

 

Vectors

Since we (almost) never do statistics on scalars, R obviously needs ways to handle collections of data items. In its simplest form such a collection is a vector: an ordered list of items of the same type. Vectors are created from scratch with the c() function which concatenates individual items into a list, or with various sequencing functions. Vectors have properties, such as length; individual items in vectors can be combined in useful ways. All elements of a vector must be of the same type. If they are not, they are coerced silently to the most general type (which is often character). (The actual hierarchy for coercion is raw < logical < integer < double < complex < character < list ).

# The c() function concatenates elements into a vector
c(2, 4, 6)


#Create a vector and list its contents and length:
f <- c(1, 1, 3, 5, 8, 13, 21)
f
length(f)

# Often, for teaching code, I want to demonstrate the contents of an object after
# assigning it. I can simply wrap the assignment into parentheses to achieve that.
# Parentheses return the value of whatever they enclose. So ...
a <- 17
# ... assigns 17 to the variable "a". But this happens silently. However ...
( a <- 17 )
# ... returns the result of the assignment. I will use this idiom often.

( f <- c(1, 1, 3, 5, 8, 13, 21, 34, 55, 89) )


# Coercion:
# all elements of vectors must be of the same mode
c(1, 2.0, "3", TRUE)  # trying to get a vector with mixed modes ...
[1] "1"    "2"    "3"    "TRUE"

# ... shows that all elements are silently being coerced
# to character mode. The emphasis is on _silently_. This might
# be unexpected, for example if you are reading numeric data
# from a text-file but someone has entered a " " for a missing
# value...



# Various ways to retrieve values from the vector:

# Extracting by index ...
f[1] # "1" is first element.
f[length(f)] # length() is the index of the last element.

# With a vector of indices ...
1:4 # This is the range operator
f[1:4] # using the range operator (it generates a sequence and returns it in a vector)
f[4:1] # same thing, backwards
seq(from=2, to=6, by=2) # The seq() function is a flexible, generic way to generate sequences
seq(2, 6, 2) # Same thing: arguments in default order
f[seq(2, 6, 2)]

# since a scalar is a vector of length 1, does this work?
5[1]


# ...using an index vector with positive indices
a <- c(1, 3, 4, 1) # the elements of index vectors must be
                   # valid indices of the target vector.
                   # The index vector can be of any length.
f[a] # In this case, four elements are retrieved from f[]

# Negative indices omit elements ...
# ...using an index vector with negative indices

# If elements of index vectors are negative integers,
# the corresponding elements are excluded.
( a <- -(1:4) ) # Note that this is NOT the same as -1:4

f[a] # Here, the first four elements are omitted from f[]
f[-((length(f)-3):length(f))] # Here, the last four elements are omitted from f[]

# Extracting with a logical vector...
f > 4 # A logical expression operating on the target vector
      # returns a vector of logical elements. It has the
      # same length as the target vector.
f[f > 4]; # We can use this logical vector to extract only
          # elements for which the logical expression evaluates as TRUE.
          # This is sometimes called "filtering".
# Note: the logical vector is aligned with the elements of the original
# vector. You can't retrieve elements more than once, as you could
# with index vectors. If the logical vector is shorter than its target
# it is "recycled" to the full length.



# Example: extending the Fibonacci series for three steps.
# Think: How does this work? What numbers are we adding here and why does the result end up in the vector?
( f <- c(f, f[length(f)-1] + f[length(f)]) )
( f <- c(f, f[length(f)-1] + f[length(f)]) )
( f <- c(f, f[length(f)-1] + f[length(f)]) )


# Some more thoughts about "["
# "[" is not just a special character, it is an operator. It
# operates on whatever it is attached to on the left. We have attached it
# to vectors above, but we can also attach it directly to function
# expressions, if the function returns a vector. For example, the
# summary() function returns some basic statistics on a vector:
summary(f)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
   1.00    5.00   21.00   75.69   89.00  377.00

# This is a vector of six numbers:
length(summary(f))

# We can extract e.g. the median like so:
summary(f)[3]

# ... or the boundaries of the interquartile range:
summary(f)[c(2, 5)]

# Note that the elements that summary() returns are "named".
# "Names" are attributes.
objectInfo(summary(f))

# The names() function can retrieve (or set) names:
names(summary(f))

# ... which brings us to yet another way to extract elements from Vectors:

# Extracting named elements...
# If the vector has named elements, vectors of names can be used exactly like
# index vectors:

summary(f)["Median"]
summary(f)[c("Max", "Min")]  # Oooops - I mistyped. But you can fix the expression, right?

Many operations on scalars can be simply extended to vectors and R computes them very efficiently by iterating over the elements in the vector.

f
f+1
f*2

# computing with two vectors of same length
f  # the Fibonacci numbers you have defined above
( a <- f[-1] )  # like f, but omitting the first element
( b <- f[1:(length(f)-1)] ) # like f, but shortened by the least element
c <- a / b # the "golden ratio", phi (~1.61803 or (1+sqrt(5))/2 ),
           # an irrational number, is approximated by the ratio of
           # two consecutive Fibonacci numbers.
c
abs(c - ((1+sqrt(5))/2)) # Calculating the error of the approximation, element by element

What could possibly go wrong?...

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When a number is not a single number ...

One of the "warts" of R is that some functions substitute a range when they receive a vector of length one. Most everyone agrees this is pretty bad. This behaviour was introduced when someone sometime long ago thought it would be nifty to save two keystrokes. This has caused countless errors, hours of frustration and probably hundreds of undiscovered bugs instead. Today we wouldn't write code like that anymore (I hope), but the community believes that since it's been around for so long, it would probably break more things if it's changed. Two functions to watch out for are sample() and seq(); other functions include diag() and runif().

Consider:

x <- 8; sample(6:x)
x <- 7; sample(6:x)
x <- 6; sample(6:x)  # Oi!

# also consider
x <- 6:8; seq(x)
x <- 6:7; seq(x)
x <- 6:6; seq(x)    # Oi vay!

Wherever this misbehaviour is a possibility - i.e. when the number of elements to sample from is variable and could be just one, for example in some simulation code - you can write a replacement function like so...

safeSample <- function(x, size, ...) {
	# Replace the sample() function to ensure sampling from a single
	# value gives that value with probability p == 1.
        # Respect additional arguments if present.
    if (length(x) == 1 && is.numeric(x) && x > 0) {
    	if (missing(size)) size <- 1
        return(rep(x, size))
    } else {
        return(sample(x, size, ...))
    }
}

Don't be discouraged though: such warts are rare in R.

 

 

Matrices

If we need to operate with several vectors, or multi-dimensional data, we make use of matrices or more generally k-dimensional arrays R. Matrix operations are very similar to vector operations, in fact a matrix actually is a vector for which the number of rows and columns have been defined. Thus matrices inherit the basic limitation of vectors: all elements have to be of the same type.

The most basic way to define matrix rows and columns is to use the dim() function and specify the size of each dimension. Consider:

( a <- 1:12 )
dim(a) <- c(2,6)
a
dim(a) <- c(2,2,3)
a

dim() also allows you to retrieve the number of rows resp. columns a matrix has. For example:

dim(a)    # returns a vector
dim(a)[3]  # only the third value of the vector

If you have a two-dimensional matrix, the function nrow() and ncol() will also give you the number of rows and columns, respectively. Obviously, dim(a)[1] is the same as nrow(a).

As an alternative to dim(), matrices can be defined using the matrix() or array() functions (see there), or "glued" together from vectors by rows or columns, using the rbind() or cbind() functions respectively:

( a  <- 1:4 )
( b  <- 5:8 )
( m1 <- rbind(a, b) )
( m2 <- cbind(a, b) )
( m  <- cbind(m2, c = 9:12) )  # naming a column :c" while cbind()'ing it

Addressing (retrieving) individual elements or slices from matrices is simply done by specifying the appropriate indices, where a missing index indicates that the entire row or column is to be retrieved. This is called "subsetting" or "subscripting" and is one of the most important and powerful aspects of working with R.

Explore how you extract rows or columns from a matrix by specifying them. Within the square brackets the order is [rows, columns]

m[1,] # first row
m[, 2] # second column
m[3, 2] # element at row == 3, column == 2
m[3:4, 1:2] # submatrix: rows 3 to 4 and columns 1 to 2

More on subsetting below.

Note that R has numerous functions to compute with matrices, such as transposition, multiplication, inversion, calculating eigenvalues and eigenvectors and more.

 

 

Further reading, links and resources

 

Notes

 

Self-evaluation

 

 

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If in doubt, ask! If anything about this learning unit is not clear to you, do not proceed blindly but ask for clarification. Post your question on the course mailing list: others are likely to have similar problems. Or send an email to your instructor.

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About ...
 
Author:

Boris Steipe <boris.steipe@utoronto.ca>

Created:

2017-08-05

Modified:

2017-08-05

Version:

0.1

Version history:

• 0.1 First stub

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