#!/usr/bin/env python
# coding: utf-8

# > This is one of the 100 recipes of the [IPython Cookbook](http://ipython-books.github.io/), the definitive guide to high-performance scientific computing and data science in Python.
# 

# # 13.3. Simulating a Brownian motion

# 1. Let's import NumPy and matplotlib.

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import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
get_ipython().run_line_magic('matplotlib', 'inline')


# 2. We will simulate Brownian motions with 5000 time steps.

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n = 5000


# 3. We simulate two independent one-dimensional Brownian processes to form a single two-dimensional Brownian process. The (discrete) Brownian motion makes independent Gaussian jumps at each time step (like a random walk). Therefore, we just have to compute the cumulative sum of independent normal random variables (one for each time step).

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x = np.cumsum(np.random.randn(n))
y = np.cumsum(np.random.randn(n))


# 4. Now, to display the Brownian motion, we could just do `plot(x, y)`. However, the result would be monochromatic and a bit boring. We would like to use a gradient of color to illustrate the progression of the motion in time. Matplotlib forces us to use a small hack based on `scatter`. This function allows us to assign a different color to each point at the expense of dropping out line segments between points. To work around this issue, we interpolate linearly the process to give the illusion of a continuous line.

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k = 10  # We add 10 intermediary points between two 
        # successive points.
# We interpolate x and y.
x2 = np.interp(np.arange(n*k), np.arange(n)*k, x)
y2 = np.interp(np.arange(n*k), np.arange(n)*k, y)


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# Now, we draw our points with a gradient of colors.
plt.scatter(x2, y2, c=range(n*k), linewidths=0,
            marker='o', s=3, cmap=plt.cm.jet,)
plt.axis('equal');
plt.xticks([]); plt.yticks([]);


# > You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).
# 
# > [IPython Cookbook](http://ipython-books.github.io/), by [Cyrille Rossant](http://cyrille.rossant.net), Packt Publishing, 2014 (500 pages).