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

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.

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