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Monte Carlo methods: basic introduction

This notebook is an element of the free risk-engineering.org courseware. It can be distributed under the terms of the Creative Commons Attribution-ShareAlike licence.

Author: Eric Marsden eric.marsden@risk-engineering.org

This notebook contains an introduction to use of Python and the NumPy library for simple Monte Carlo simulations. We illustrate the method by estimating the value of $\pi$. See the associated course materials for background material and to download this content as a Jupyter/Python notebook.

A Monte Carlo simulation simply means executing the same experiment (in our case, a computer model) with randomly selected inputs a large number of times, then extracting some summary statistic from the sum of outcomes. Typical steps are:

  1. Define a domain of possible inputs. This simulated “universe” should be similar to the universe whose behavior we wish to describe and investigate.

  2. Generate inputs randomly from a probability distribution over the domain. These inputs should be generated so that their characteristics are similar to the real universe we are trying to simulate (in particular, dependencies between the inputs should be represented).

  3. Perform a deterministic computation on the inputs.

  4. Aggregate the results to obtain the output of interest (typically histograms, summary statistics, confidence intervals).

Estimating the value of pi

Consider the largest circle which can be fit in the square ranging on $\mathbb{R}^2$ over $[-1, 1]^2$. The circle has a radius 1, and area $\pi$. The square has an area of $2^2$ = 4. The ratio between their areas is thus $\pi/4$.

We can approximate the value of π using a Monte Carlo method using the following procedure:

  • draw the square over $[-1,1]^2$ then draw the largest circle that fits inside the square
  • randomly scatter a large number $N$ of grains of rice over the square
  • count how many grains fell inside the circle
  • the count divided by $N$ and multiplied by 4 is an approximation of $\pi$

We can simulate this procedure in NumPy by drawing random numbers from a uniform distribution between -1 and 1 to represent the $x$ and $y$ positions of our grains of rice, and checking whether the point is within the circle using Pythagoras’ theorem. This procedure is an adaptation of what’s called Buffon’s needle problem, after the 18th century French mathematician the Count of Button. It belongs to a topic called geometric probability.

We start by importing the numpy library, which makes it possible to use functions and variables from the library, prefixed by numpy.

In [1]:
import numpy
import matplotlib.pyplot as plt
%config InlineBackend.figure_formats=["png"]
In [2]:
N = 10_000
inside = []
for i in range(N):
    x = numpy.random.uniform(-1, 1)
    y = numpy.random.uniform(-1, 1)
    if numpy.sqrt(x**2 + y**2) < 1:
        inside.append((x, y))

We can plot the points that we have marked as being inside the circle, to check that we indeed have an approximation to a circle.

In [3]:
plt.figure(figsize=(6, 6))
plt.scatter([x[0] for x in inside], [x[1] for x in inside], marker=".", alpha=0.5);

Finally, we can check our estimate of $\pi$ as the count divided by N and multiplied by 4:

In [4]:
4 * len(inside)/float(N)

Speed of convergence

Theory tells us that the convergence of a Monte Carlo procedure is related to the square root of the number of iterations. This means that the estimation error should decrease proportionally to the inverse of the square root of $N$, the number of iterations.

In [5]:
N = 1_000_000
inside = 0
approximation = numpy.empty(N)
for i in range(N):
    x = numpy.random.uniform(-1, 1)
    y = numpy.random.uniform(-1, 1)
    if numpy.sqrt(x**2 + y**2) < 1:
        inside += 1
    approximation[i] = 4 * inside/float(i+1)
plt.xlabel("Number of iterations")
plt.ylabel("Current approximation")
# plot the exact value of pi in red
plt.axhline(numpy.pi, color="r", alpha=0.5);

Exercise: let’s check that the error decreases proportionally to $\frac{1}{\sqrt{N}}$.

In [6]:
plt.loglog(numpy.arange(N), numpy.abs(approximation-numpy.pi))
plt.loglog(1/numpy.sqrt(numpy.arange(N)+1), color="r", alpha=0.5)
plt.xlabel("Number of iterations")
plt.ylabel("Estimation error");

Exercise: use a similar Monte Carlo procedure in dimension 3 to estimate $\pi$ (your analysis will be based on the volume of a sphere of radius 1, instead of the area of a circle of radius 1). Recall that the volume of a sphere of radius $r$ is $\frac{4}{3}\pi r^3$.


There are other approaches to estimating the value of pi using computational methods, such as the “method of exhaustion” developed by early Greek mathematicians and used by Archimedes to estimate pi (see an online simulation).

In [ ]: