ps1_ex2_subm.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "ps1_ex2_subm.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "authorship_tag": "ABX9TyMhk2G5XQw9LqmoTjzBBF50",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/TobiasLaimer/ColabStuff/blob/master/ps1_ex2_subm.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "H89VCXquRLxH",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hDDE7Ysp5tkk",
        "colab_type": "code",
        "outputId": "d422807a-00cb-4c6e-e065-31192107e0eb",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        }
      },
      "source": [
        "#2.1  \"Plot f (x) = x2 for the interval (−5, 5)\"\n",
        "\n",
        "x= np.linspace(-5, 5, 1000) #get an array of 1000 equidistant values from the interval (−5, 5)\n",
        "y= x**2\n",
        "plt.plot (x,y) #plot f(x)"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f7545e86e80>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 3
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "fKt_z8YVGiWP",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#2.2 \"Generate a dataset of 100 points drawn at random from a 2D Gaussian with mean (1, 0.5) and standard deviation (0.4, 0.4)\"\n",
        "\n",
        "N=100 #our sample size\n",
        "mean = [1, 0.5]\n",
        "std = [0.4, 0.4] #parametrize 2D Gaussian with mean (1, 0.5) and standard deviation (0.4, 0.4)- easy to change our parameters for the generation of data points later if we like\n",
        "x = np.random.normal(mean[0], std[0], N) #get x data points, Python starts counting with 0\n",
        "y = np.random.normal (mean[1], std[1], N) #get y data points"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3yV0HUHO59IB",
        "colab_type": "code",
        "outputId": "aa7c3dda-cee2-4648-bba4-d9bb8ab280e8",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 337
        }
      },
      "source": [
        "#2.3 \"Now, highlight the mean and 1-σ contour lines in a plot with your data points\"\n",
        "\n",
        "#means\n",
        "plt.axvline(x.mean(), color='g') #plot mean of x\n",
        "plt.axhline(y.mean(), color='y') #plot mean of y. use different colors to minimise confusion\n",
        "\n",
        "#1-sigma ellipse\n",
        "def eigsorted(cov):\n",
        "    vals, vecs = np.linalg.eigh(cov)\n",
        "    order = vals.argsort()[::-1]\n",
        "    return vals[order], vecs[:,order]\n",
        "\n",
        "nstd = 1 #number of standard deviations\n",
        "ax = plt.subplot(111) #1x1 grid, first subplot\n",
        "\n",
        "cov = np.cov(x, y)\n",
        "vals, vecs = eigsorted(cov)\n",
        "theta = np.degrees(np.arctan2(*vecs[:,0][::-1]))\n",
        "w, h = 2 * nstd * np.sqrt(vals)\n",
        "ell = Ellipse(xy=(np.mean(x), np.mean(y)),\n",
        "              width=w, height=h,\n",
        "              angle=theta, color='red', linewidth=2.0)\n",
        "ell.set_facecolor('none')\n",
        "ax.add_artist(ell)\n",
        "\n",
        "\n",
        "plt.scatter(x, y) #now plot x,y and ell\n",
        "plt.show() #we see that there are considerably less than 68% of our sample inside the ellipse. This is expected since we were supposed to draw a 1-sigma error ellipse, not a confidence ellipse!\n",
        "\n",
        "\n",
        "#now, sanity checks:\n",
        "\n",
        "#did we really get good Gaussians?\n",
        "i=k=0\n",
        "for i in range(0,N):\n",
        "  if x.mean()-x.std()<x[i]<x.mean()+x.std():\n",
        "    k+=1\n",
        "  i+=1\n",
        "dev_1s= k/N #68% of the data points should be within one std of the mean- of course there is noise with only 100 data points.\n",
        "\n",
        "print (x.mean(), x.std(), y.mean(), y.std(), dev_1s, \" should be close to [1, 0.4, 0.5, 0.4, 0.68]\") #data are close to our specified Gaussian- of course there is noise with only 100 data points.\n",
        "\n",
        "#/end"
      ],
      "execution_count": 44,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:11: MatplotlibDeprecationWarning: Adding an axes using the same arguments as a previous axes currently reuses the earlier instance.  In a future version, a new instance will always be created and returned.  Meanwhile, this warning can be suppressed, and the future behavior ensured, by passing a unique label to each axes instance.\n",
            "  # This is added back by InteractiveShellApp.init_path()\n"
          ],
          "name": "stderr"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        },
        {
          "output_type": "stream",
          "text": [
            "1.0078542579966996 0.40742567329397095 0.5331515522027491 0.3859526627887832 0.68  should be close to [1, 0.4, 0.5, 0.4, 0.68]\n"
          ],
          "name": "stdout"
        }
      ]
    }
  ]
}