Outline

In our last thrilling episode

Data frames, encore

Creating an example data frame

library(datasets)
states = data.frame(state.x77, Abbr=state.abb, 
                    Region=state.region, Division=state.division)

Here data.frame() is combining a pre-existing matrix (state.x77), a vector of characters (state.abb), and two vectors of qualitative categorical variables (called factors; state.region, state.division)

Column names are preserved, or guessed if not explicitly set

colnames(states)
##  [1] "Population" "Income"     "Illiteracy" "Life.Exp"   "Murder"    
##  [6] "HS.Grad"    "Frost"      "Area"       "Abbr"       "Region"    
## [11] "Division"
states[1,]
##         Population Income Illiteracy Life.Exp Murder HS.Grad Frost  Area
## Alabama       3615   3624        2.1    69.05   15.1    41.3    20 50708
##         Abbr Region           Division
## Alabama   AL  South East South Central

Data frame access

By row and column index:

states[49,3]
## [1] 0.7

By row and column names:

states["Wisconsin","Illiteracy"]
## [1] 0.7

Data frame access (continued)

All of a row:

states["Wisconsin",]
##           Population Income Illiteracy Life.Exp Murder HS.Grad Frost  Area
## Wisconsin       4589   4468        0.7    72.48      3    54.5   149 54464
##           Abbr        Region           Division
## Wisconsin   WI North Central East North Central

(Exercise: what is the class of states["Wisconsin",]?)

Data frame access (continued)

All of a column:

head(states[,3])
## [1] 2.1 1.5 1.8 1.9 1.1 0.7
head(states[,"Illiteracy"])
## [1] 2.1 1.5 1.8 1.9 1.1 0.7
head(states$Illiteracy)
## [1] 2.1 1.5 1.8 1.9 1.1 0.7

Data frame access (continued)

Rows matching a condition:

states[states$Division=="New England", "Illiteracy"]
## [1] 1.1 0.7 1.1 0.7 1.3 0.6
states[states$Region=="South", "Illiteracy"]
##  [1] 2.1 1.9 0.9 1.3 2.0 1.6 2.8 0.9 2.4 1.8 1.1 2.3 1.7 2.2 1.4 1.4

Replacing values

Parts or all of the data frame can be assigned to:

summary(states$HS.Grad)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   37.80   48.05   53.25   53.11   59.15   67.30
states$HS.Grad = states$HS.Grad/100
summary(states$HS.Grad)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.3780  0.4805  0.5325  0.5311  0.5915  0.6730
states$HS.Grad = 100*states$HS.Grad

with()

What percentage of literate adults graduated HS?

head(100*(states$HS.Grad/(100-states$Illiteracy)))
## [1] 42.18590 67.71574 59.16497 40.67278 63.29626 64.35045

with() takes a data frame and evaluates an expression “inside” it:

with(states, head(100*(HS.Grad/(100-Illiteracy))))
## [1] 42.18590 67.71574 59.16497 40.67278 63.29626 64.35045

Means that you can save writing states$xxx many times

Data arguments

Similar to the usage in with(), lots of functions take data arguments, and look variables up in that data frame:

plot(Illiteracy~Frost, data=states)

Conditionals and flow control

Have the computer decide what to do based on a condition. A mathematical example you all know: \[ |x| = \left\{ \begin{array}{cl} x & \mathrm{if}~x\geq 0 \\ -x &\mathrm{if}~ x < 0\end{array}\right. \] Another more complex one: \[ \psi(x) = \left\{ \begin{array}{cl} x^2 & \mathrm{if}~|x|\leq 1\\ 2|x|-1 &\mathrm{if}~ |x| > 1\end{array}\right. \]
(Exercise: plot \(\psi\) in R)

if()

Simplest conditional:

if (x >= 0) {
  x
} else {
  -x
}

Condition in if() needs to give one TRUE or FALSE value

else() clause is optional

Single line actions don’t need braces, as in:

if (x >= 0) x else -x

Nested if()

We can nest if() statements arbitrarily deeply:

if (x^2 < 1) {
  x^2 
} else {
  if (x >= 0) {
    2*x-1
  } else {
    -2*x-1
  }
}

This can get ugly though

switch()

We can simplify a nested if with switch(): give a variable to select on, then a value for each option

switch(type.of.summary,
       mean=mean(states$Murder),
       median=median(states$Murder),
       histogram=hist(states$Murder),
       "I don't understand")

(Exercise: set type.of.summary to, succesively, “mean”, “median”, “histogram”, and “mode”, and explain what happens)

Combining Booleans: && and ||

& work | like + or *: the combine terms elementwise

But flow control always wants just one Boolean value. Hence we can skip calculating what’s not needed

&& and || give just one Boolean, lazily:

(0 > 0) && (all.equal(42%%6, 169%%13))
## [1] FALSE
(0 > 0) && (MyCoolNewVariable == 0)
## [1] FALSE

(In both cases, R never evaluates the expression on the right)

In summary, use && and || for conditionals, & and | for subsetting

Iteration

Repeat similar actions multiple times, as in:

table.of.logarithms = vector(length=7,mode="numeric")
table.of.logarithms
## [1] 0 0 0 0 0 0 0
for (i in 1:length(table.of.logarithms)) {
  table.of.logarithms[i] = log(i)
}
table.of.logarithms
## [1] 0.0000000 0.6931472 1.0986123 1.3862944 1.6094379 1.7917595 1.9459101

for()

for (i in 1:length(table.of.logarithms)) {
  table.of.logarithms[i] = log(i)
}

for increments a counter (here i) along a vector (here 1:length(table.of.logarithms)). It loops through the body (the expression inside the braces) until it runs through the vector

We call this “iterating over the vector”

The body of the for() loop

Can contain just about anything, including:

Nested iteration example

c = matrix(0, nrow=nrow(a), ncol=ncol(b))
if (ncol(a) == nrow(b)) {
  for (i in 1:nrow(c)) {
    for (j in 1:ncol(c)) {
      for (k in 1:ncol(a)) {
        c[i,j] = c[i,j] + a[i,k]*b[k,j]
      }
    }
  }
} else {
  stop("matrices a and b non-conformable")
}

while(): conditional iteration

Babylonian method for finding square root of a number x:

while (abs(x - r^2) > 1e-06) {
  r = (r + x/r)/2
}

Condition in the argument to while() must be a single Boolean value (like if())

Body is looped over until the condition is FALSE (so can loop forever …)

Loop never begins unless the condition starts TRUE

for() versus while()

for() is better when the number of times to repeat (values to iterate over) is clear in advance

while() is better when you can recognize when to stop once you’re there, even if you can’t guess it to begin with

Every for() could be replaced with a while() (Exercise: show this)

while(TRUE) or repeat: unconditional iteration

while(TRUE) and repeat: both have the same function; just repeat the body indefinitely, unless something causes the flow to break

repeat {
  ans = readline("Who is the best Professor of Statistics at CMU? ")
  if (ans == "Tibs" || ans == "Tibshirani" || ans == "Ryan") {
    cat("Yes! You get an 'A'.")
    break
  }
  else {
    cat("Wrong answer!\n")
  } 
}

Note that break gets us out of the loop; also, next skips the rest of the body, and starts us at the top of the loop again

Avoiding iteration

R has many ways of avoiding iteration, by acting on whole objects.
This is called vectorization

Vectorized arithmetic

How many languages add 2 vectors:

c = vector(length(a))
for (i in 1:length(a)) {  c[i] = a[i] + b[i]  }

How R adds 2 vectors:

a+b

Also recall our for() loop for matrix multiplication, versus the simple a %*% b

Vectorized functions

Many functions are set up to vectorize automatically

abs(-3:3)
## [1] 3 2 1 0 1 2 3
log(1:7)
## [1] 0.0000000 0.6931472 1.0986123 1.3862944 1.6094379 1.7917595 1.9459101

See also apply() from last week

We’ll come back to this in great detail later

Vectorized conditions: ifelse()

ifelse(x^2 > 1, 2*abs(x)-1, x^2)

The first argument here is a Boolean vector. Then ifelse() picks from the second and third arguments as appropriate, when it encounters TRUE or FALSE

Summary

What is truth?

Advice: don’t play games here; be sure control expressions are getting Boolean values

Conversely, in arithmetic, FALSE is 0 and TRUE is 1

mean(states$Murder > 7)
## [1] 0.48

Babylonian method of root finding

Look this up for a proper historical account!

Given: \(x\), find \(\sqrt{x}\). Take a first guess \(r\); either \(r^2 > x\), \(r^2 < x\) or \(r^2 = x\)

Therefore, in the latter two cases, we can replace \(r\) with average of \(r\) and \(x/r\), and try again