HW_2D_RandomWalks_CW.ipynb

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  "metadata": {
    "colab": {
      "name": "HW_2D RandomWalks_CW",
      "provenance": [],
      "collapsed_sections": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/cw805/MATH-242/blob/main/HW_2D_RandomWalks_CW.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1sz37RJiHzE2"
      },
      "source": [
        "# 2D Random Walks Practice Problems\n",
        "\n",
        "**Learning Goals:** Gain proficiency with Python coding. Explore properties of 2-dimensional random walks.\n",
        "\n",
        "**Your task:** Complete the problem below.\n",
        "\n",
        "**Due:** Wednesday, April 29, at 5:00pm. Do the following to submit your work.\n",
        "\n",
        "1. Click on *Share* in the upper right corner of this window. Make sure your notebook is viewable by Prof. Wright and the course grader(s).\n",
        "2. In the *Get link*, click *Copy link*.\n",
        "3. Go to the [2D Random Walks](https://mdl.stolaf.edu/mod/assign/view.php?id=191525) assignment on Moodle. Paste the link to your notebook in the text field for the assignment submission. This will allow Prof. Wright or either of the course graders to view your Colab notebook.\n",
        "\n",
        "**Grading:** The problem below will be graded on a Satisfactory/Unsatisfactory scale. You will earn a Satisfactory grade by using Python to produce an appropriate plot showing how the average distance from the origin depends on the number of steps.\n",
        "\n",
        "**Need help?** Review the notes from class and Python documentation links on the course web site. If you still need help, email Prof. Wright. Don't forget there are also evening help sessions &mdash; the help session schedule is at the top of the course website.\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import random\n",
        "import math\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "metadata": {
        "id": "0exNwUGU9JTq"
      },
      "execution_count": 127,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Problem: 2D distance from origin\n",
        "\n",
        "What is the average distance from the origin after $n$ steps? Simulate many 2D simple symmetric random walks for $n$ steps and compute their average distance from the origin after step $n$. Do this for many values of $n$ and make a plot showing how the average distance from the origin depends on $n$. Summarize your observations."
      ],
      "metadata": {
        "id": "N7X8PLavhIq9"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Generate a 2D random Walk"
      ],
      "metadata": {
        "id": "6eKCRQVMXXca"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def randomWalk2D(numSteps):\n",
        "  # define the possible moves at each step\n",
        "  dirs = np.array( [ [1,0], [0,1], [-1,0], [0,-1] ] )\n",
        "\n",
        "  # set up a matrix of zeros to store the locations of the random walk\n",
        "  locations = np.zeros( (numSteps+1, 2) )\n",
        "\n",
        "  # take steps and store locations\n",
        "  for i in range(1, numSteps+1):\n",
        "    r = random.randrange(4)   # random integer 0, 1, 2, 3\n",
        "    move = dirs[r]            # random move\n",
        "    locations[i] = locations[i-1] + move  # store the next location\n",
        "  \n",
        "  return locations"
      ],
      "metadata": {
        "id": "tG6nj3DxXbrS"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "randomWalk2D(4)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Lh4ORFayXTKJ",
        "outputId": "0e5433d0-4ad3-462d-8675-b4ca07ad50f2"
      },
      "execution_count": 189,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[0., 0.],\n",
              "       [0., 1.],\n",
              "       [1., 1.],\n",
              "       [1., 0.],\n",
              "       [2., 0.]])"
            ]
          },
          "metadata": {},
          "execution_count": 189
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Find Distance from Origin after n steps\n"
      ],
      "metadata": {
        "id": "tuC1h3JPXN5k"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def distance(numStep):  \n",
        "   \n",
        "  locations = randomWalk2D(numStep) \n",
        "  \n",
        "  # get the x and y coordinates of each position\n",
        "  x = locations[:,0]\n",
        "  y = locations[:,1]\n",
        " \n",
        "  # calculate distance from origin for all points\n",
        "  distance = np.sqrt(x ** 2 + y ** 2)    \n",
        "\n",
        "  # return the point furthest from origin\n",
        "  return max(distance)"
      ],
      "metadata": {
        "id": "ahmvJYrKXjFZ"
      },
      "execution_count": 174,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "distance(100)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "TwS8H2dsa02s",
        "outputId": "eae484a8-42ad-406b-9300-45ba3fef2b65"
      },
      "execution_count": 178,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "11.180339887498949"
            ]
          },
          "metadata": {},
          "execution_count": 178
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "def avgDist(numSteps, numWalks):\n",
        " \n",
        "  total = 0\n",
        "  \n",
        "  for i in range(numWalks):\n",
        "    total += distance(numSteps)\n",
        "\n",
        "  return total/numWalks"
      ],
      "metadata": {
        "id": "VpqalaXqY75f"
      },
      "execution_count": 185,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "avgDist(1000,50)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "-fMgEevQZVg5",
        "outputId": "332d9efa-8e8e-461d-f43c-8c1655311b22"
      },
      "execution_count": 186,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "39.06817590493895"
            ]
          },
          "metadata": {},
          "execution_count": 186
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# get a list of average distances\n",
        "avgDistList = [avgDist(n, 500) for n in range(2,100)]"
      ],
      "metadata": {
        "id": "cm3OE22TbD00"
      },
      "execution_count": 200,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(range(2,200), avgDistList)\n",
        "plt.xlabel(\"number of steps\")\n",
        "plt.ylabel(\"average distance\");"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "8b2rYax1bLrU",
        "outputId": "42b34056-9656-444d-c273-602f4b89c851"
      },
      "execution_count": 198,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Similar to what we found in 1D random walks, the plot looks like a squared root function, we can try to square the average distances"
      ],
      "metadata": {
        "id": "wE1r-Ae9cm8X"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "squaredDist = [d**2 for d in avgDistList]\n",
        "\n",
        "plt.plot(range(2,100), squaredDist)\n",
        "plt.xlabel(\"number of steps\")\n",
        "plt.ylabel(\"average diameter squared\");"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "vFqBFiRtcM02",
        "outputId": "c01cdb2d-444d-40ba-cf3e-8b955cad5dfb"
      },
      "execution_count": 201,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "The plot is also linear! The higher the number of steps, the further the distance because it has more time to go further away from the origin. Also, the rate of increase in average distances seems slower in 2D random walks"
      ],
      "metadata": {
        "id": "lo4gzX8Dc_3o"
      }
    }
  ]
}