How to estimate PI when we only have R and the formula for the surface of a circle (Surface = PI * r * r)?

The estimation of this number has been one of the greatest challenge in the history of mathematics. PI is the ratio between a circle's circumference and diameter. The first estimation I have heard about has been done by an Egyptian 3700 years ago. His estimation was around 3.16. Which is fairly accurate for a 3700-year-old estimation.

In this post I propose an easy way to estimate PI. Though it is far from being the most efficient estimation ever made, it is a good opportunity to introduce Monte Carlo simulation in the blog.

Monte Carlo simulations are methods to estimate results by repeating a random process. The limit of this method is the source of randomness in the results. By construction of these methods, it cannot be mathematically proved, but only "confidence interval" results. Besides, most of these methods are very slow if we want to obtain a fairly accurate result. The optimization of Monte Carlo is an amazing area which is really useful in many disciplines. 

Again, the method I program here, is far from being efficient, it is only a basic of the Monte Carlo theory and I think one of the easiest illustrations.

Let's explain how it works.

As we know, if r is the radius of a circle, then the surface of this circle is PI * r * r. Therefore, if we know the radius of a circle, and if we can estimate the surface of this circle, we can estimate PI. How can we estimate the surface of a circle with a random method. The answer is found in the square! Yes, in the square. Especially in the following figure. 


A circle in a square is our solution. Indeed, this is very easy to know the surface of a square. It is the square of the length of one of its sides. Do you see where this is leading us? What if we were able to estimate the rate between the surface of the circle and the surface of the square? Then we would know the surface of the circle.

Indeed, 
surfaceCircle = (surfaceCircle / surfaceSquare) * surfaceSquare. 



Finally, we just need to estimate (surfaceCircle/surfaceSquare). And this is where Monte Carlo is useful. If we uniformly distribute k random variables in the square and count how many of them are in the circle, then we have a proxy of the quantity surfaceCircle/surfaceSquare.

We would obtain something like the next graph. Then the proportion between the number of blue spots and the total number of spots is equal to surfaceCircle/surfaceSquare.


I have done this experience with R. The next plot is the estimation of PI we can obtain with this method. The red line is the real value of PI. The black line is my estimator of PI according to the number of simulated random variables. As we can see, the accuracy tends to increase with the size of our sample. After 10,000 simulations, we obtain an estimation of PI of 3.1444 which is not so good. Indeed, you can already find in the Old Testament a better estimation of PI!



Even though it is a weak estimation, Monte Carlo simulation is a very powerful method. Indeed, if the circle is now a well known mathematical object, all the mathematical concepts are not as well understood. The advantage of Monte Carlo simulation is that we do not need any analytic study of the object. This is why Monte Carlo is used in many applied areas.

 The code (R):

simulation = function(long){
  c = rep(0,long)
  numberIn = 0
  for(i in 1:long){
    x = runif(2,-1,1)
    if(sqrt(x[1]*x[1] + x[2]*x[2]) <= 1){
      numberIn = numberIn + 1
    }
    prop = numberIn / i
    piHat = prop *4
    c[i] = piHat
  }
  return(c)
}

size = 1000
res = simulation(size)
ini = 1
plot(res[ini:size], type = 'l')
lines(rep(pi, size)[ini:size], col = 'red')
Posted by
Location: University of Cambridge, Cambridge CB2, UK
Labels: estimation of Pi Monte Carlo Monte Carlo simulation
0

Add a comment

The financial market is not only made of stock options. Other financial products enable market actors to target specific aims. For example, an oil buyer like a flight company may want to cover the risk of increase in the price of oil.

I found a golden website. The blog of Esteban Moro. He uses R to work on networks. In particular he has done a really nice code to make some great videos of networks. This post is purely a copy of his code. I just changed a few arguments to change colors and to do my own network.

3

As you have certainly seen now, I like working on artificial neural networks. I have written a few posts about models with neural networks (Models to generate networks, Want to win to Guess Who and Study of spatial segregation).

1

I already talked about networks a few times in this blog. In particular, I had this approach to explain spatial segregation in a city or to solve the Guess Who? problem. However, one of the question is how to generate a good network.

The function apply() is certainly one of the most useful function. I was scared of it during a while and refused to use it. But it makes the code so much faster to write and so efficient that we can't afford not using it.

1

Have you ever played the board game "Guess who?".

If you want to choose randomly your next holidays destination, you are likely to process in a way which is certainly biased. Especially if you choose randomly the latitude and the longitude.

4

My previous post is about a method to simulate a Brownian motion. A friend of mine emailed me yesterday to tell me that this is useless if we do not know how to simulate a normally distributed variable.

The Brownian motion is certainly the most famous stochastic process (a random variable evolving in the time). It has been the first way to model a stock option price (Louis Bachelier's thesis in 1900).

1

The merge of two insurance companies enables to curb the probability of ruin by sharing the risk and the capital of the two companies.

For example, we can consider two insurance companies, A and B.

How to estimate PI when we only have R and the formula for the surface of a circle (Surface = PI * r * r)?

The estimation of this number has been one of the greatest challenge in the history of mathematics. PI is the ratio between a circle's circumference and diameter.

I was in a party last night and a guy was totally drunk. Not just the guy who had a few drinks and speaks a bit too loud, but the one who is not very likely to remember what he has done during his night, but who is rather very likely to suffer from a huge headache today.

I'm very glad if you use or at least read sometimes my blog. However, I'd like to give you a list of wonderful blogs which are useful if you are interested in applied mathematics and programming for simulation.

My favorite one is written by Arthur Charpentier, a French mathematician.

I am currently doing an internship in England. Therefore, I keep alternating between French and English in my different emails and other forms of communication on the Internet. I have been surprised to see that some websites are able to recognize when I use French or when I use English.

The VIX (volatility index) is a financial index which measures the expectation of the volatility of the stock market index S&P 500 (SPX). The higher is the value of the VIX the higher are the expectations of important variations in the S&P 500 during the next month.

The Olympic Games have finished a couple of days ago. Two entire weeks of complete devotion for sport. Unfortunately I hadn’t got any ticket but I didn’t fail to watch many games on TV and internet.

Hello (New World!), 

My name is Edwin, I’m a 22 year-old French student in applied mathematics. In particular, I study probability, statistics and risk theory.

Blog Archive
Blog Archive
Translate
Translate
Loading