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   "source": [
    "> 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.\n"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "# 13.3. Simulating a Brownian motion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Let's import NumPy and matplotlib."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. We will simulate Brownian motions with 5000 time steps."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "n = 5000"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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   "outputs": [],
   "source": [
    "x = np.cumsum(np.random.randn(n))\n",
    "y = np.cumsum(np.random.randn(n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "k = 10  # We add 10 intermediary points between two \n",
    "        # successive points.\n",
    "# We interpolate x and y.\n",
    "x2 = np.interp(np.arange(n*k), np.arange(n)*k, x)\n",
    "y2 = np.interp(np.arange(n*k), np.arange(n)*k, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Now, we draw our points with a gradient of colors.\n",
    "plt.scatter(x2, y2, c=range(n*k), linewidths=0,\n",
    "            marker='o', s=3, cmap=plt.cm.jet,)\n",
    "plt.axis('equal');\n",
    "plt.xticks([]); plt.yticks([]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).\n",
    "\n",
    "> [IPython Cookbook](http://ipython-books.github.io/), by [Cyrille Rossant](http://cyrille.rossant.net), Packt Publishing, 2014 (500 pages)."
   ]
  }
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