Using the tidyverse for more than data manipulation: estimating pi with Monte Carlo methods

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This blog post is an excerpt of my ebook Modern R with the tidyverse that you can read for
free here. This is taken from Chapter 5, which presents
the {tidyverse} packages and how to use them to compute descriptive statistics and manipulate data.
In the text below, I show how you can use the {tidyverse} functions and principles for the
estimation of \(\pi\) using Monte Carlo simulation.

Going beyond descriptive statistics and data manipulation

The {tidyverse} collection of packages can do much more than simply data manipulation and
descriptive statisics. You can use the principles we have covered and the functions you now know
to do much more. For instance, you can use a few {tidyverse} functions to do Monte Carlo simulations,
for example to estimate \(\pi\).

Draw the unit circle inside the unit square, the ratio of the area of the circle to the area of the
square will be \(\pi/4\). Then shot K arrows at the square; roughly \(K*\pi/4\) should have fallen
inside the circle. So if now you shoot N arrows at the square, and M fall inside the circle, you have
the following relationship \(M = N*\pi/4\). You can thus compute \(\pi\) like so: \(\pi = 4*M/N\).

The more arrows N you throw at the square, the better approximation of \(\pi\) you’ll have. Let’s
try to do this with a tidy Monte Carlo simulation. First, let’s randomly pick some points inside
the unit square:


n <- 5000

points <- tibble("x" = runif(n), "y" = runif(n))

Now, to know if a point is inside the unit circle, we need to check wether \(x^2 + y^2 < 1\). Let’s
add a new column to the points tibble, called inside equal to 1 if the point is inside the
unit circle and 0 if not:

points <- points %>% 
    mutate(inside = map2_dbl(.x = x, .y = y, ~ifelse(.x**2 + .y**2 < 1, 1, 0))) %>% 

Let’s take a look at points:


## # A tibble: 5,000 x 4
##        N       x      y inside
##    <int>   <dbl>  <dbl>  <dbl>
##  1     1 0.770   0.984       0
##  2     2 0.713   0.0107      1
##  3     3 0.303   0.133       1
##  4     4 0.618   0.0378      1
##  5     5 0.0505  0.677       1
##  6     6 0.0432  0.0846      1
##  7     7 0.820   0.727       0
##  8     8 0.00961 0.0758      1
##  9     9 0.102   0.373       1
## 10    10 0.609   0.676       1
## # ... with 4,990 more rows

The rowid_to_column() function, from the {tibble} package, adds a new column to the data frame
with an id, going from 1 to the number of rows in the data frame. Now, I can compute the estimation
of \(\pi\) at each row, by computing the cumulative sum of the 1’s in the inside column and dividing
that by the current value of N column:

points <- points %>% 
    mutate(estimate = 4*cumsum(inside)/N)

cumsum(inside) is the M from the formula. Now, we can finish by plotting the result:

ggplot(points) + 
    geom_line(aes(y = estimate, x = N), colour = "#82518c") + 
    geom_hline(yintercept = pi) +

In Chapter 6, we are going to learn all about {ggplot2}.

As the number of tries grows, the estimation of \(\pi\) gets better.

Using a data frame as a structure to hold our simulated points and the results makes it very easy
to avoid loops, and thus write code that is more concise and easier to follow.
If you studied a quantitative field in u8niversity, you might have done a similar exercise at the
time, very likely by defining a matrix to hold your points, and an empty vector to hold whether a
particular point was inside the unit circle. Then you wrote a loop to compute whether
a point was inside the unit circle, save this result in the before-defined empty vector and then
compute the estimation of \(\pi\). Again, I take this opportunity here to stress that there is nothing
wrong with this approach per se, but R, with the {tidyverse} is better suited for a workflow
where lists or data frames are the central objects and where the analyst operates over them
with functional programming techniques.

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