## 10.2 Loops

In addition to the conditional statements in Section 10.1, another important type of statements that controls the flow of programs is loops.

### 10.2.1for loops with known sequence

The first type of loops is to repeat a segment of codes with a known sequence. Let’s look at its syntax first.

for (val in val_seq){
statement1
statement2
}

The for loop will sequentially assign each value of the val_seq to val and run the body of the loop.

For example, if you want to find the first ten values of the Fibonacci sequence. It is known that the first two values in a Fibonacci sequence is $$F_1 = 0$$ and $$F_2 = 1$$. Then, we have $$F_3 = F_1 + F_2 = 0 + 1 = 1$$ and in general $$F_i = F_{i-2} + F_{i-1}$$.

fib_seq <- rep(0,10)   #initialize the sequence
fib_seq[1] <- 0        #initialize the first element
fib_seq[2] <- 1        #initialize the second element
for(i in 3:10){        #loop i from 3 to 10
fib_seq[i] <- fib_seq[i-2] + fib_seq[i-1] #compute the i-th element of fib_seq as the sum of the previous two elements
}
fib_seq                #print the sequence
#>  [1]  0  1  1  2  3  5  8 13 21 34

Here, we first initialize the desired sequence to be a zero vector of length 10. Then, initialize the first element to be 0 and second element of 1. After that, the for loop is will initially set i to the value of 3, and go over all the values in the sequence 3:10. At the end, we get the length-10 Fibonacci sequence.

### 10.2.2 Combine the conditional statements with loops

In the body of the loop, you usually want to do different tasks depending on the conditions. In this case, you can add conditional statements in the body of the loop. Let’s print out all the numbers in fib_seq and add the information whether it is even or odd, and count the number of even values and odd values.

count_even <- count_odd <- 0  #initialize the counts of even and odd values to be 0
for(fib_val in fib_seq){  #go over all the values in fib_seq
if(fib_val %% 2 == 0){  #check if the current value is even
cat(fib_val, ": even\n")        #if so, print the value
count_even <- count_even + 1    #then, add 1 to the even count
} else{
cat(fib_val, ": odd\n")        #if not, print the value
count_odd <- count_odd + 1    #then, add 1 to the odd count
}
}
#> 0 : even
#> 1 : odd
#> 1 : odd
#> 2 : even
#> 3 : odd
#> 5 : odd
#> 8 : even
#> 13 : odd
#> 21 : odd
#> 34 : even
cat("There are ", count_even, " even values.\n")   #print the count of even values
#> There are  4  even values.
cat("There are ", count_odd, " odd values.\n")   #print the count  of odd values
#> There are  6  odd values.

It is worth noting that for this task, we can actually avoid the loop using vectorize operations on the vector.

even_ind <- fib_seq %% 2 == 0     #check whether each value is even
even_odd <- ifelse(even_ind, "even", "odd") #convert the logical vector to a character vector
cat(paste0(fib_seq, ": ", even_odd, "\n")) #create the string and print
#> 0: even
#>  1: odd
#>  1: odd
#>  2: even
#>  3: odd
#>  5: odd
#>  8: even
#>  13: odd
#>  21: odd
#>  34: even
cat("There are ", sum(even_ind), " even values.\n")   #print the count of even values
#> There are  4  even values.
cat("There are ", sum(!even_ind), " odd values.\n")   #print the count  of odd values
#> There are  6  odd values.

### 10.2.3 Control Loop Flow With next and break

Inside the if and else clause, you can use next and break to further control the flow.

The next function goes directly to the next loop cycle, while break jumped out of the current loop.

Let’s see an example of next. For each value of the fib_seq, if it is even, we will skip to the next value.

for(fib_val in fib_seq){  #go over all the values in fib_seq
if(fib_val %% 2 == 0){  #check if the current value is even
next
}
cat(fib_val, ": odd\n")        #if not, print the value
}
#> 1 : odd
#> 1 : odd
#> 3 : odd
#> 5 : odd
#> 13 : odd
#> 21 : odd

Now, suppose you want to find the first value in fib_seq that is larger than 4. The break can be used in this case.

for(fib_val in fib_seq){  #go over all the values in fib_seq
if(fib_val > 4){  #check if the current value is larger than 4
cat(fib_val, " is the first value larger than 4\n")
break
}
}
#> 5  is the first value larger than 4
cat(fib_val) #check the current fib_val
#> 5

You can see that the fib_val is 5, indicating that we have breaked from the loop since it is larger than 4.

### 10.2.4 Nested Loops

In addition to a single loop, it is also common to put one loop inside another one, named the nested loop.

Let’s say we want to create a matrix A of dimension $$5\times 5$$ where each element $$A_{ij} = i+j$$. To create such a matrix, we can write a nested loop over $$i$$ and $$j$$.

A <- matrix(0, 5, 5) #initialize the matrix A
for (i in 1:5)       #loop over index i
for (j in 1:5){    #loop over index j
A[i, j] <- i + j #set the (i, j)-th element of A
}
A
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    2    3    4    5    6
#> [2,]    3    4    5    6    7
#> [3,]    4    5    6    7    8
#> [4,]    5    6    7    8    9
#> [5,]    6    7    8    9   10

Let’s try another example where we want to create the correlation matrix in the so-called AR(1) model. In particular, the corresponding matrix $$A$$ is of dimension $$p\times p$$ where $$A_{ij} = \rho^{|i-j|}$$.

Let’s say an example of $$p = 4$$ and $$\rho = 0.5$$

p <- 4
A <- matrix(0, p, p) #initialize the matrix A
rho <- 0.5
for (i in 1:p)       #loop over index i
for (j in 1:p){    #loop over index j
A[i, j] <- rho^{abs(i - j)} #set the (i, j)-th element of A
}
A
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.50 0.25 0.125
#> [2,] 0.500 1.00 0.50 0.250
#> [3,] 0.250 0.50 1.00 0.500
#> [4,] 0.125 0.25 0.50 1.000

### 10.2.5 Nested Loop via the outer() Function

It turns out for the two examples we showed in the last part, you can use avoid the loop by the outer() function.

outer(1:5, 1:5, "+")
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    2    3    4    5    6
#> [2,]    3    4    5    6    7
#> [3,]    4    5    6    7    8
#> [4,]    5    6    7    8    9
#> [5,]    6    7    8    9   10
outer(1:4, 1:4, function(i, j){0.5^{abs(i-j)}})
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.50 0.25 0.125
#> [2,] 0.500 1.00 0.50 0.250
#> [3,] 0.250 0.50 1.00 0.500
#> [4,] 0.125 0.25 0.50 1.000

Let’s see another example of using outer().

outer(1:5, 1:5, function(x,y){
paste(x,"+",y,"=",x+y)})
#>      [,1]        [,2]        [,3]        [,4]        [,5]
#> [1,] "1 + 1 = 2" "1 + 2 = 3" "1 + 3 = 4" "1 + 4 = 5" "1 + 5 = 6"
#> [2,] "2 + 1 = 3" "2 + 2 = 4" "2 + 3 = 5" "2 + 4 = 6" "2 + 5 = 7"
#> [3,] "3 + 1 = 4" "3 + 2 = 5" "3 + 3 = 6" "3 + 4 = 7" "3 + 5 = 8"
#> [4,] "4 + 1 = 5" "4 + 2 = 6" "4 + 3 = 7" "4 + 4 = 8" "4 + 5 = 9"
#> [5,] "5 + 1 = 6" "5 + 2 = 7" "5 + 3 = 8" "5 + 4 = 9" "5 + 5 = 10"

Finally, the outer() function can also be applied on matrices, which will lead to a higher dimensional array.

A <- matrix(1:4, 2, 2)
outer(A, A, "-")       #a 4-dimensional array.
#> , , 1, 1
#>
#>      [,1] [,2]
#> [1,]    0    2
#> [2,]    1    3
#>
#> , , 2, 1
#>
#>      [,1] [,2]
#> [1,]   -1    1
#> [2,]    0    2
#>
#> , , 1, 2
#>
#>      [,1] [,2]
#> [1,]   -2    0
#> [2,]   -1    1
#>
#> , , 2, 2
#>
#>      [,1] [,2]
#> [1,]   -3   -1
#> [2,]   -2    0