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</p> " data-medium-file="https://i1.wp.com/dogdogfish.com/wp-content/uploads/2014/05/rubbish.png?fit=300%2C226" data-large-file="https://i1.wp.com/dogdogfish.com/wp-content/uploads/2014/05/rubbish.png?fit=676%2C509" class="size-medium wp-image-169" src="https://i1.wp.com/dogdogfish.com/wp-content/uploads/2014/05/rubbish.png?resize=300%2C226" alt="A Circle in a Square" srcset="https://i1.wp.com/dogdogfish.com/wp-content/uploads/2014/05/rubbish.png?w=812 812w, https://i1.wp.com/dogdogfish.com/wp-content/uploads/2014/05/rubbish.png?resize=300%2C226 300w" sizes="(max-width: 300px) 100vw, 300px" data-recalc-dims="1" />

A Circle in a Square

Brilliant – was it worth it? Probably not. But there you have it – with that set up we can now start the Monte Carlo bit. We’ll throw darts at that picture randomly; you’d expect the number of darts in the circle to be proportional to the area of the circle and the number of darts in the square to be proportional to the area of the square. Using that fact and the formulae for the areas of a circle and a square you can estimate Pi using the ratio of darts in the circle and in the square.

Sound good? It’s fairly easy to run this in Python and graph the output using Matplotlib. You’ll see I’ve used Object Oriented Python for this particular exercise, I don’t really know why. Especially because I had a chance to use inheritance and didn’t. Well done me. I’ve let everybody down. Anyway – this is the code I came up with and the graph below shows what I ended up with:

 #!/usr/bin/python    import numpy as np  import math  import matplotlib.pyplot as plt    """  Calculate pi using Monte-Carlo Simulation  """    """  First - the maths:  A circle has area Pi*r^2  A square wholly enclosing above circle has area 4r^2  If we randomly generate points in that square we'd expect the ratio of points in the square/points in the circle to equal the area of the square divided by the circle.  By that logic n_in_sq/n_in_cir = 4/Pi and so Pi = (4 * n_in_cir)/n_in_sq  """    class pi_calculator(object):        def __init__(self, radius, iterations):          self.radius = radius          self.iterations = iterations          self.square = square(radius)          self.circle = circle(radius)        def scatter_points(self):          for _ in range(self.iterations):              point_x, point_y = ((2*self.radius) * np.random.random_sample(2)) - self.radius              self.square.increment_point_count(point_x, point_y)              self.circle.increment_point_count(point_x, point_y)        def return_pi(self):          return (4.0*self.circle.return_point_count())/self.square.return_point_count()        def calculate_accuracy(self, calc_pi):          absolute_error = math.pi - calc_pi          percent_error = 100*(math.pi - calc_pi)/math.pi          return (absolute_error, percent_error)        def return_iterations(self):          return self.iterations    class square(object):        def __init__(self, radius):          self.radius = radius          self.lower_x = -radius          self.lower_y = -radius          self.upper_x = radius          self.upper_y = radius          self.point_count = 0          def in_square(self, point_x, point_y):          return (self.upper_x > point_x > self.lower_x) and (self.upper_y > point_y > self.lower_y)          def increment_point_count(self, point_x, point_y, increment = 1):          if self.in_square(point_x, point_y):              self.point_count += increment        def return_point_count(self):          return self.point_count    class circle(object):        def __init__(self, radius):          self.radius = radius          self.point_count = 0        def in_circle(self, point_x, point_y):          return point_x**2 + point_y**2 < self.radius**2        def increment_point_count(self, point_x, point_y, increment=1):          if self.in_circle(point_x, point_y):              self.point_count += increment        def return_point_count(self):          return self.point_count      if __name__ == '__main__':      axis_values = []      pi_values = []      absolute_error_values = []      percent_error_values = []      for _ in range(1,3000,30):          pi_calc = pi_calculator(1, _)          pi_calc.scatter_points()          print "Number of iterations: %d    Accuracy: %.5f" % (pi_calc.return_iterations(), math.fabs(pi_calc.calculate_accuracy(pi_calc.return_pi())[0]))          axis_values.append(_)          pi_values.append(pi_calc.return_pi())          absolute_error_values.append(math.fabs(pi_calc.calculate_accuracy(pi_calc.return_pi())[0]))          percent_error_values.append(math.fabs(pi_calc.calculate_accuracy(pi_calc.return_pi())[1]))        improvement_per_iteration = [absolute_error_values[index] - absolute_error_values[index-1] for index, value in enumerate(absolute_error_values) if index > 0]      fig = plt.figure()      fig.suptitle('Calculating Pi - Monte Carlo Method')      ax1 = fig.add_subplot(2,2,1)      ax2 = fig.add_subplot(2,2,2)      ax3 = fig.add_subplot(2,2,3)      ax4 = fig.add_subplot(2,2,4)      plt.subplots_adjust(wspace=0.3, hspace=0.3)      ax1.set_xticklabels([str(entry) for entry in axis_values[::len(axis_values)/5]], rotation=30, fontsize='small')      ax1.set_xlabel('Iterations')      ax1.set_ylabel('Calculated value of Pi')      ax1.plot(pi_values, 'k')      ax1.plot([math.pi for entry in axis_values], 'r')      ax2.set_ylabel('Absolute error')      ax2.set_xticklabels([str(entry) for entry in axis_values[::len(axis_values)/5]], rotation=30, fontsize='small')      ax2.set_xlabel('Iterations')      ax2.plot(absolute_error_values, 'k', label="Total Error")      ax3.set_ylabel('Absolute percentage error (%)')      ax3.set_xticklabels([str(entry) for entry in axis_values[::len(axis_values)/5]], rotation=30, fontsize='small')      ax3.set_xlabel('Iterations')      ax3.plot(percent_error_values, 'k', label="Percent Error")      ax4.set_ylabel('Absolute improvement per iteration')      ax4.set_xticklabels([str(entry) for entry in axis_values[::len(axis_values)/5]], rotation=30, fontsize='small')      ax4.set_xlabel('Iterations')      ax4.plot(improvement_per_iteration, 'k', label="Absolute change")      plt.savefig('pi_calculation.png')      plt.show()      

giving us:

Monte Carlo estimation of Pi - an Investigation

Monte Carlo estimation of Pi – an Investigation

I can only apologise for any dodgy code in there – in my defence, it was early in the morning. As you can see, it only takes around 100 ‘darts thrown at the board’ to start to see a reasonable value for Pi. I ran it up to about 10,000 iterations without hitting any significant calculation time. The fourth graph doesn’t really show anything interesting – I just couldn’t think of anything to put there.

That’ll do for now – I built something that’ll stream tweets on the Scottish Independence Referendum but don’t know what to do with it yet; there’ll likely be some sort of blog post. There’s a chance I’ll do some sentiment analysis but I’m not sure yet.

When you play the Game of Thrones, you win or you die.

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  1. Hi,

    Your drawing of a ‘circle in a square’ is not actually a circle in a square.
    You need to force both axis to be proportional using the aspect argument:
    axis = fig.add_subplot(1,1,1, aspect=’equal’)

    Cheers

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