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

The {tidyverse} collection of packages can do much more than simply data manipulation anddescriptive statisics. You can use the principles we have covered and the functions you now knowto 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 thesquare will be (\pi/4). Then shot K arrows at the square; roughly (K\pi/4) should have falleninside the circle. So if now you shoot N arrows at the square, and M fall inside the circle, you havethe 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’stry to do this with a tidy Monte Carlo simulation. First, let’s randomly pick some points insidethe unit square:

1 2	library(tidyverse) library(brotools)

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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’sadd a new column to the points tibble, called inside equal to 1 if the point is inside theunit circle and 0 if not:

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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:

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points

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## # A tibble: 5,000 x 4 ## N x y inside ## ## 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 framewith an id, going from 1 to the number of rows in the data frame. Now, I can compute the estimationof (\pi) at each row, by computing the cumulative sum of the 1’s in the inside column and dividingthat by the current value of N column:

1 2	points <- points %>% mutate(estimate = 4*cumsum(inside)/N)

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

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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 easyto 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 thetime, very likely by defining a matrix to hold your points, and an empty vector to hold whether aparticular point was inside the unit circle. Then you wrote a loop to compute whethera point was inside the unit circle, save this result in the before-defined empty vector and thencompute the estimation of (\pi). Again, I take this opportunity here to stress that there is nothingwrong with this approach per se, but R, with the {tidyverse} is better suited for a workflowwhere lists or data frames are the central objects and where the analyst operates over themwith functional programming techniques.

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