**Higher Order Functions**, and kindly contributed to R-bloggers)

There is a celebrated Commodore 64 program that randomly prints outs `/`

and `\`

characters and fills the screen with neat-looking maze designs. It is just one

line of code, but there is a whole book written about

it.

```
10 PRINT CHR$(205.5+RND(1)); : GOTO 10
```

The basic idea, from my reading of the code, is that `CHR$(205)`

is `\`

,

`CHR$(206)`

is `/`

, and the program randomly selects between the two by adding a

random number to 205.5. Endlessly looping over this command fills the screen

with that pleasing maze pattern.

In R, we could replicate this functionality with by randomly sampling the

slashes:

```
sample_n_slashes <- function(n) {
sample(c("/", "\\"), size = n, replace = TRUE)
}
withr::with_options(
list(width = 40),
cat(sample_n_slashes(800), sep = "", fill = TRUE)
)
#> //////\/\\/\/\/\/\\//\\\\//\//\/\//\///\
#> \\\/\//\\/\/////////////\\/\//\\\/\//\\\
#> /\\/\/\//\/\/\\/\\/\/\//\\\\//\/\\\/\///
#> \\//\/\\\/\///\\/\/\/////\//\///\/\/\//\
#> /\/\//\///\//\//\//\/\//\\/\\\\\/\\\//\/
#> //\\\\\//////\//\//\/\//\\////\/\\\/\/\/
#> \\////\/\\/\////\//////\\/\\/////\/////\
#> /\/\/\/\/\\///\/\\\\/\/\\//\/\\/\\\\\\//
#> //\\\/\///\///\/\\\//\//\\\\//\\/\\/\/\/
#> /\\/\//\\//\/\\\\\/\\\/\\\/\/\/\/\////\/
#> /\//\\//\////\\\///\//\/\\\/\\///\//\\\\
#> /\//\\////\/\\\//\\\//\/\///\\\\\/\/\\//
#> \\////\\/\\\\\\///\///\//\\\/\\\\//\////
#> \\\///\/\//\//\/\//\/\/\\\/\/\\///\/////
#> \\/\\//\\\\\\///\\\\/\\/\\//\//\\\/\\/\/
#> ////\\\////\/\\\\\/////\/\\\////\\///\\\
#> \//\\\//\///\/\\\\//\\\////\\//\\/\/\//\
#> \/\//\\//\\///\\\\\\//\///\/\\\\\\\\/\\/
#> ///\/\//\\/\/\\\\\\\\/\///\//\\///\\//\\
#> /////\\///\/\\/\/\\//\\//\/\\/\//\//\\\\
```

where `withr::with_options()`

lets us temporarily change the print width and

`cat()`

concatenates the slashes and prints out the characters as text.

We can also make this much prettier by drawing the patterns using ggplot2.

## Drawing line segments with ggplot2

Instead of writing out slashes, we will draw a grid of diagonal line segments,

some of which will be flipped at random. To draw a segment, we need a starting

*x*–*y* coordinate and an ending *x*–*y* coordinate. `geom_segment()`

will

connect the two coordinates with a line. Here’s a small example where we draw

four “slashes”.

```
library(ggplot2)
library(dplyr)
data <- tibble::tribble(
~row, ~col, ~x_start, ~x_end, ~y_start, ~y_end,
1, 1, 0, 1, 0, 1,
1, 2, 1, 2, 1, 0, # flipped
2, 1, 0, 1, 1, 2,
2, 2, 1, 2, 1, 2)
ggplot(data) +
aes(x = x_start, xend = x_end, y = y_start, yend = y_end) +
geom_segment()
```

The programming task now is to make giant grid of these slashes. Let’s start

with an observation: To draw two slashes, we needed three *x*

values: 0, 1, 2. The first two served as segment starts and the last two

as segment ends. The same idea applies to the *y* values. We can generate a

bunch of starts and ends by taking a sequence of steps and removing the first

and last elements.

```
# We want a 20 by 20 grid
rows <- 20
cols <- 20
x_points <- seq(0, 1, length.out = cols + 1)
x_starts <- head(x_points, -1)
x_ends <- tail(x_points, -1)
y_points <- seq(0, 1, length.out = rows + 1)
y_starts <- head(y_points, -1)
y_ends <- tail(y_points, -1)
```

Each `x_starts`

–`x_ends`

pair is a column in the grid, and each

`y_starts`

–`y_ends`

is a row in the grid. To make a slash at each

row–column combination, we have to map out all the combinations of the rows

and columns. We can do this with `crossing()`

which creates all *crossed*

combinations of values. (If it helps, you might think of *crossed* like crossed

experiments or the

Cartesian cross product of

sets.)

```
grid <- tidyr::crossing(
# columns
data_frame(x_start = x_starts, x_end = x_ends),
# rows
data_frame(y_start = y_starts, y_end = y_ends)) %>%
# So values move left to right, bottom to top
arrange(y_start, y_end)
# 400 rows because 20 rows x 20 columns
grid
#> # A tibble: 400 x 4
#> x_start x_end y_start y_end
#>
```
#> 1 0 0.05 0 0.05
#> 2 0.05 0.1 0 0.05
#> 3 0.1 0.15 0 0.05
#> 4 0.15 0.2 0 0.05
#> 5 0.2 0.25 0 0.05
#> 6 0.25 0.3 0 0.05
#> 7 0.3 0.35 0 0.05
#> 8 0.35 0.4 0 0.05
#> 9 0.4 0.45 0 0.05
#> 10 0.45 0.5 0 0.05
#> # ... with 390 more rows

We can confirm that the segments in the grid fill out a plot. (I

randomly color the line segments to make individual ones visible.)

```
ggplot(grid) +
aes(
x = x_start, y = y_start,
xend = x_end, yend = y_end,
color = runif(400)) +
geom_segment(size = 1) +
guides(color = FALSE)
```

Finally, we need to flip slashes at random. A segment becomes flipped if the

`y_start`

and `y_end`

are swapped. In the code below, we flip the slash in each

row if a randomly drawn number between 0 and 1 is less than .5. For style, we

also use `theme_void()`

to strip away the plotting theme, leaving us with just

the maze design.

```
p_flip <- .5
grid <- grid %>%
arrange(y_start, y_end) %>%
mutate(
p_flip = p_flip,
flip = runif(length(y_end)) <= p_flip,
y_temp1 = y_start,
y_temp2 = y_end,
y_start = ifelse(flip, y_temp2, y_temp1),
y_end = ifelse(flip, y_temp1, y_temp2)) %>%
select(x_start:y_end, p_flip)
ggplot(grid) +
aes(x = x_start, y = y_start, xend = x_end, yend = y_end) +
geom_segment(size = 1, color = "grey20")
last_plot() + theme_void()
```

Now, we wrap all these steps together into a pair of functions.

```
make_10_print_data <- function(rows = 20, cols = 20, p_flip = .5) {
x_points <- seq(0, 1, length.out = cols + 1)
x_starts <- head(x_points, -1)
x_ends <- tail(x_points, -1)
y_points <- seq(0, 1, length.out = rows + 1)
y_starts <- head(y_points, -1)
y_ends <- tail(y_points, -1)
grid <- tidyr::crossing(
data.frame(x_start = x_starts, x_end = x_ends),
data.frame(y_start = y_starts, y_end = y_ends))
grid %>%
arrange(y_start, y_end) %>%
mutate(
p_flip = p_flip,
flip = runif(length(y_end)) <= p_flip,
y_temp1 = y_start,
y_temp2 = y_end,
y_start = ifelse(flip, y_temp2, y_temp1),
y_end = ifelse(flip, y_temp1, y_temp2)) %>%
select(x_start:y_end, p_flip)
}
draw_10_print <- function(rows = 20, cols = 20, p_flip = .5) {
grid <- make_10_print_data(rows = rows, cols = cols, p_flip = p_flip)
ggplot(grid) +
aes(x = x_start, y = y_start, xend = x_end, yend = y_end) +
geom_segment(size = 1, color = "grey20")
}
```

## Now the fun part: custom flipping probabilities

We can vary the probability of flipping the slashes. For example, we can use the

density of a normal distribution to make flipping more likely for central values

and less likely for more extreme values.

```
xs <- seq(0, 1, length.out = 40)
p_flip <- dnorm(seq(-4, 4, length.out = 40))
ggplot(data.frame(x = xs, y = p_flip)) +
aes(x, y) +
geom_line() +
labs(
x = "x position",
y = "p(flipping)",
title = "normal density")
# We repeat p_flip for each row of the grid
draw_10_print(rows = 40, cols = 40, p_flip = rep(p_flip, 40)) +
theme_void()
```

We can use the cumulative density of the normal distribution so that

flipping becomes more likely as *x* increases.

```
xs <- seq(0, 1, length.out = 40)
p_flip <- pnorm(seq(-4, 4, length.out = 40))
ggplot(data.frame(x = xs, y = p_flip)) +
aes(x, y) +
geom_line() +
labs(
x = "x position",
y = "p(flipping)",
title = "cumulative normal")
draw_10_print(rows = 40, cols = 40, p_flip = rep(p_flip, 40)) +
theme_void()
```

The Cauchy distribution is said to have “thicker” tails than the normal

distribution, so here it shows more flips on the left and right extremes.

```
xs <- seq(0, 1, length.out = 40)
p_flip <- dcauchy(seq(-4, 4, length.out = 40))
ggplot(data.frame(x = xs, y = p_flip)) +
aes(x, y) +
geom_line() +
labs(
x = "x position",
y = "p(flipping)",
title = "Cauchy density")
draw_10_print(rows = 40, cols = 40, p_flip = rep(p_flip, 40)) +
theme_void()
```

The exponential distribution is a spike that quickly peters out. We can make a

probability “bowl” by splicing an exponential and a reversed exponential

together.

```
# Use flipped exponential densities as probabilities
p_flip <- c(dexp(seq(0, 4, length.out = 20)),
dexp(seq(4, 0, length.out = 20)))
ggplot(data.frame(x = xs, y = p_flip)) +
aes(x, y) +
geom_line() +
labs(
x = "x position",
y = "p(flipping)",
title = "exponential + flipped exponential")
draw_10_print(rows = 40, cols = 40, p = rep(p_flip, 40)) +
theme_void()
```

We might have the probabilities increase by 10% from row to row. In the code

below, I use a simple loop to boost some random probability values by 10% from

row to row. This gives us nice streaks in the grid as a column starts to flip

for every row.

```
boost_probs <- function(p_flip, nrows, factor = 1.1) {
output <- p_flip
for (i in seq_len(nrows - 1)) {
p_flip <- p_flip * factor
output <- c(output, p_flip)
}
output
}
draw_10_print(cols = 40, rows = 40, p = boost_probs(runif(40), 40, 1.1)) +
theme_void()
```

The probabilities can be anything we like. Here I compute the frequency of

English alphabet letters as they appear in *Pride and Prejudice* and based the

flipping probability on those values.

```
char_counts <- janeaustenr::prideprejudice %>%
tolower() %>%
stringr::str_split("") %>%
unlist() %>%
table()
letter_counts <- char_counts[letters] %>% as.vector()
p_letter <- letter_counts / sum(letter_counts)
ggplot(data.frame(x = letters, y = p_letter)) +
aes(x, y, label = x) +
geom_text() +
labs(
x = NULL,
y = "p(letter)",
title = "letter frequencies in Pride and Prejudice")
```

```
draw_10_print(cols = 26, rows = 80, p = rep(p_letter, 80)) +
theme_void()
```

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**Higher Order Functions**.

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