# 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:

library(tidyverse) library(brotools)

n <- 5000 set.seed(2019) 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))) %>% rowid_to_column("N")

Let’s take a look at `points`

:

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) + theme_blog()

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.

Hope you enjoyed! If you found this blog post useful, you might want to follow

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