csu_ph315_pytut.ipynb

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      "name": "ph315pythontutorial.ipynb",
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      "include_colab_link": true
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    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
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      "source": [
        "<a href=\"https://colab.research.google.com/github/theowecker/csuph315pytut/blob/main/csu_ph315_pytut.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8MkWXbG3YfC9"
      },
      "source": [
        "# 1. Welcome to PH 315 for Spring 2021! \r\n",
        "For most of our data analysis in this course, we'll be using Python. Python is a multi-purpose programming language that is open-source and free to use. It is commonly used in a wide variety of industries and is similar to Matlab. In this course, we'll be using Google Colab as a free browser based version of a Python environment. You'll be able to upload data to Google Drive and import it into these Colab notebooks when you need to analyze it.\r\n",
        "\r\n",
        "We're not going to cover a lot of basic Python nitty-gritty details (like data types) and will instead focus on the tools you'll need for this class. We will then provide a framework that you will finish for each lab. If you'd like more info on basic Python, we'll provide some links to resources or feel free to ask! "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "czmI87pVatKB"
      },
      "source": [
        "# 2. Basic Introduction to Python, Numpy, and Indexing\r\n",
        "For a lot of our work in Python, we'll need to import some specific packages like Numpy and SciPy. These are libraries of functions that make our lives a lot easier and make complex mathematical operations simple. To import something we simply use the import command and specify what we want to name it as.  Once you import it once in your notebook, you don't need to do it again, you can just use the package in lower blocks of code.\r\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "S3aj97_gYhZY"
      },
      "source": [
        "import numpy as np"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OnBqsEwLxCQy"
      },
      "source": [
        "We can also import a single function from a library like the genfromtxt function from the SciPy or MatPlotLib libraries."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "n3U_XXXTxFQ7"
      },
      "source": [
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "53--5SFk8htp"
      },
      "source": [
        "To run these blocks of code just press the play button in the [ ]. You can make a new Colab document with all of your code blocks in it."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LUQp05Y_x1eg"
      },
      "source": [
        "Now we'll learn how to use Numpy to create, organize, and utilize arrays in a few ways that will be helpful in your experiments."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "NzfW1phS1b3D",
        "outputId": "977fcd21-6bd4-448f-dabb-8a2158e1e2c9"
      },
      "source": [
        "m = np.array([[1,2], [3,4]]) # create a 2x2 array \r\n",
        "\r\n",
        "\r\n",
        "m = np.append(m, [[5,6],[7,8]], axis=0) # append another array to the original one. Be careful which axis you append it to.\r\n",
        "print('m = ',m)  # print the array to see what it looks like.  You can also just write print(m) for a faster look\r\n",
        "print('\\nthe shape of the m is:',m.shape)  "
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "m =  [[1 2]\n",
            " [3 4]\n",
            " [5 6]\n",
            " [7 8]]\n",
            "\n",
            "the shape of the m is: (4, 2)\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4OtUODNG6JXw"
      },
      "source": [
        "\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "**Exercise:** Create two 3x3 arrays using both np.array and using np.append to append 3 vectors together.  Add a code block below this text block to do this. \n",
        "\n",
        "\n",
        "---\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zMvBqtHU4LNX"
      },
      "source": [
        "An important thing we will need to learn for this class is how to index arrays. In Python, indices start at 0. To find the first value of the array above we can use"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "o-48120n41-o",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "728a7be3-3201-4452-8399-cb904a11f17b"
      },
      "source": [
        "print('The i=1, j=1 component of the array is:', m[0,0]) # where print tells Python to display the answer and array[0,0] selects the i=1,j=1 component"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "The i=1, j=1 component of the array is: 1\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZMvyM-uP5Ih_"
      },
      "source": [
        "\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "**Exercise:** Print the [i=3, j=2] and [i=1, j=2] components of your array using indexing. Then print the first row using just one line.  \n",
        "\n",
        "\n",
        "---\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LHX7P1MH5i6T"
      },
      "source": [
        "You can also call the last element in an array by using -1."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "qPUEIoGj9ive",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "887b66f2-723b-4ffe-f061-00cdcf8fc0b2"
      },
      "source": [
        "m[-1,-1] # where [-1,-1] refers to the last element in each column and row"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "8"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Itv3xbD89x5y"
      },
      "source": [
        "We might also want to \"slice\" arrays, i.e. take only a certain subset of the array. For example, you might want to only fit a certain range of your data \r\n",
        "(perhaps excluding the first data point). The syntax in Python for this is: `[first index (inclusive): last index (exclusive)]`.\r\n",
        "\r\n",
        "\r\n",
        "Let's make a new array **b** that we can illustrate this with. Below, b has 6 rows and 4 columns.  When we slice out the first row, we will call it a new array, b2, which will have 5 rows and 4 columns. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ySrXjQ7XAixt",
        "outputId": "bee3cee1-5f03-4c1d-af68-befb7e782216"
      },
      "source": [
        "b = np.array([[0,0,0,0], [5,6,7,8], [9,10,11,12], [5,6,7,8], [4,2,1,8], [1,2,3,8]]) # array from a data set\r\n",
        "print('b = ',b)\r\n",
        "print('\\nthe shape of b is:',b.shape,'\\n\\n')  \r\n",
        "\r\n",
        "b2 = b[1:6] # this slice says that we don't want to include the 0th row.  Notice that we had to use 6 as the upper index, even though the index only goes up to 5\r\n",
        "print('b2 =',b2)\r\n",
        "print('\\nthe shape of b2 is:',b2.shape)   "
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "b =  [[ 0  0  0  0]\n",
            " [ 5  6  7  8]\n",
            " [ 9 10 11 12]\n",
            " [ 5  6  7  8]\n",
            " [ 4  2  1  8]\n",
            " [ 1  2  3  8]]\n",
            "\n",
            "the shape of b is: (6, 4) \n",
            "\n",
            "\n",
            "b2 = [[ 5  6  7  8]\n",
            " [ 9 10 11 12]\n",
            " [ 5  6  7  8]\n",
            " [ 4  2  1  8]\n",
            " [ 1  2  3  8]]\n",
            "\n",
            "the shape of b2 is: (5, 4)\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Malgv1_9Bmd_"
      },
      "source": [
        "We can also slice specific parts (like rows or columns from a data table) from the above matrix. Let's pick out the 5x2 matrix from the end of **b2**, and put it in an array called **b3**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "5uyra3rfBrWj",
        "outputId": "2bd6d0eb-187c-4ea0-ea0c-5299f3dfe069"
      },
      "source": [
        "b3 = b2[:,2:5] # notice the \":\" operator by itself is used to select all the elements\r\n",
        "print('b3 = ',b3)\r\n",
        "print('\\nthe shape of b3 is:',b3.shape)   "
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "b3 =  [[ 7  8]\n",
            " [11 12]\n",
            " [ 7  8]\n",
            " [ 1  8]\n",
            " [ 3  8]]\n",
            "\n",
            "the shape of b3 is: (5, 2)\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AWjrJbQYRL97"
      },
      "source": [
        "For experiments in this course (and beyond), data is often saved in a table format where each column stores the data for a different variable.  You might then use slicing to get individual columns of the array to use for plotting, fitting, etc.  We can also select certain columns from an array with this syntax. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "SKFrOijH8pZp",
        "outputId": "1ef65557-608d-45d0-c994-01b12ebb5193"
      },
      "source": [
        "b4 = b[:,[0, 3]] # we only want to select the 1st and 4th column from b\r\n",
        "print('b4 = \\n',b4)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "b4 = \n",
            " [[ 0  0]\n",
            " [ 5  8]\n",
            " [ 9 12]\n",
            " [ 5  8]\n",
            " [ 4  8]\n",
            " [ 1  8]]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SalTCKLtCEL3"
      },
      "source": [
        "\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "**Exercise:** Create a matrix like above and slice out the 2nd column.  Make a code block below to do this.\n",
        "\n",
        "---\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sIm5ZUvmEaDH"
      },
      "source": [
        "# 3. Importing Data Files\r\n",
        "In this course, I suggest that data files are put into experiment folders on your Google Drive like this:\r\n",
        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      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yMe6_-nWH9C8"
      },
      "source": [
        "This allows for easier import into Google Colab. \n",
        "\n",
        "Every time you open your Colab notebook, you will have to mount your Google Drive, but this is relatively straightforward using the code below.  Although you can load the files manually, doing it this way (through Drive) saves you the hassle of finding your data on your local machine and importing it by hand every time you open the notebook."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "jiV3sajM4s2l",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "2d112b3d-2e04-4deb-ad6c-0a885016f241"
      },
      "source": [
        "from google.colab import drive\r\n",
        "drive.mount('/content/drive')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mounted at /content/drive\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PtOdxCm84uUD"
      },
      "source": [
        "These commands will mount your Google Drive on the virtual Google Colab machine and allow you to import files. Once the drive is mounted, you can use various Numpy commands to read in files and store the data in them as Numpy arrays. \r\n",
        "\r\n",
        "For example, you can use np.genfromtxt which will create an array from a .txt file. Here's an example from the Compton Effect lab."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "vdBN6CTU89-A"
      },
      "source": [
        "# Depending on your Google Drive file structure you may need to change the filepath variable\r\n",
        "filepath= '/content/drive/MyDrive/Colab Notebooks/Compton/' # defining a variable as the main part of the path\r\n",
        "\r\n",
        "norod_25deg = np.genfromtxt(filepath+'th25 deg no rod.txt',dtype=float,delimiter=',',skip_header=1) \r\n",
        "withrod_25deg = np.genfromtxt(filepath+'th25 deg with rod.txt',dtype=float,delimiter=',',skip_header=1)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xmv0SwgaL9Yt"
      },
      "source": [
        "The structure of your Google Drive will determine what file path you need to input. You can also just use the full file path in genfromtxt instead of defining a filepath variable, as below."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "oAhMoWMTMHC9"
      },
      "source": [
        "norod_25deg = np.genfromtxt('/content/drive/MyDrive/Colab Notebooks/Compton/th25 deg no rod.txt', dtype=float, delimiter=',', skip_header=1)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RRnZezQR6RTP"
      },
      "source": [
        "The Numpy function genfromtxt (used above) is very useful and flexible for loading data files.   You will use it extensively in this course.  \r\n",
        "\r\n",
        "There are a few options that should be specified when loading a file using this function, such as:\r\n",
        "\r\n",
        "`dtype` is used to specify what the data type is.\r\n",
        "\r\n",
        "`delimiter` is used to tell Python what the file uses to separate different columns.  Often it is a comma (as in .csv files), but sometimes it will be a space or a tab (which you would denote by ' ' or '\\t', respectively, in this function).\r\n",
        "\r\n",
        "`skip_header` allows you to skip a specified number of rows at the beginning of a file.  This is useful if you have information in the file header which is not data (such as column labels, or other information that is automatically included in your datafile)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "EvXunatL4H8v"
      },
      "source": [
        "Now let's practice with the testdata.csv that is included. This data looks a lot like the alpha counts from the Radioactivity lab. First we'll need to import it. Your filepath will vary depending on where the test file is in your Google Drive. \r\n",
        "\r\n",
        "If you open up the testdata.csv with Notepad or Excel you will also notice there are 22 rows of useless header information and 1 empty row at the bottom of the sheet (which we can call the footer). So, we will use skip_header and skip_footer to have Python ignore these rows. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "8nJooyI-5sZH",
        "outputId": "c212c3c9-c659-4271-f850-277f625fc83c"
      },
      "source": [
        "radon = np.genfromtxt('/content/drive/MyDrive/testdata.csv', dtype=float, delimiter=',', skip_header=22, skip_footer=1)\r\n",
        "print(radon)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[0.000e+00       nan 5.500e+01 ...       nan       nan       nan]\n",
            " [1.000e+00       nan 5.600e+01 ...       nan       nan       nan]\n",
            " [2.000e+00       nan 5.700e+01 ...       nan       nan       nan]\n",
            " ...\n",
            " [1.020e+03       nan 6.600e+01 ...       nan       nan       nan]\n",
            " [1.021e+03       nan 6.900e+01 ...       nan       nan       nan]\n",
            " [1.022e+03       nan 7.200e+01 ...       nan       nan       nan]]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0FmKEoQO7lWa"
      },
      "source": [
        "\r\n",
        "\r\n",
        "---\r\n",
        "\r\n",
        "\r\n",
        "**Exercise:** Create a new array with the two columns of data by excluding the nan columns.\r\n",
        "\r\n",
        "---\r\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DrB_-PjLu9nj"
      },
      "source": [
        "# 4. Plotting and Data Visualization\r\n",
        "Data visualization is extremely important in science. Python makes it easy to make quick plots to see what your data actually look like. This will be really useful for making sure your experimental set-up is working like you think it is or checking to see if you're getting useful results.  Also, you will need to be able to make nice plots for your lab reports.\r\n",
        "\r\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "joY_Dsp626GE"
      },
      "source": [
        "Most Python plotting is done using `matplotlib.pyplot` .   This package allows you to make plots in a very similar way as you would in Matlab. Let's start with a simple plot and see how it works.\n",
        "\n",
        "pyplot was imported imported at the beginning of this notebook, and was called `plt`.  All the plotting functions will need a `plt.` in front since they are from this package. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 312
        },
        "id": "g9Bcg-kw3JLs",
        "outputId": "a25b0151-cd48-4571-af3c-6512103c5f62"
      },
      "source": [
        "data = [1, 2, 3, 4]\r\n",
        "plt.plot(data, 'r-', linewidth=2.0) # r- specifies that we want a red line as the formatting \r\n",
        "plt.xlabel('time (s)') # generates an x-label for the plot\r\n",
        "plt.ylabel('height (m)') # generates a y-label for the plot\r\n",
        "plt.title('Not Fun Title') # makes a title"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.0, 'Not Fun Title')"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 15
        },
        {
          "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": "markdown",
      "metadata": {
        "id": "jjQ5cM233_6E"
      },
      "source": [
        "Notice how x-values were not specified, so values of x = [0,1,2,3] were automatically generated. However, you can also specify those values and let's also save this graph to our files on the left side-bar. \r\n",
        "**Remember** to download your graphs as .jpgs or .pngs when you're done making them, if you want to save them for your lab report.   You can either use `savefig` (as below), or you can right-click to download."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        },
        "id": "ZiP0z9lK44vJ",
        "outputId": "432ff72f-b461-46ea-bbe1-70c79066f531"
      },
      "source": [
        "x = [1, 2, 3, 4] # specific x values\r\n",
        "y = [25, 16, 9, 4]\r\n",
        "plt.plot(x,y, 'bo') # bo specifies that we want blue dots\r\n",
        "plt.xlabel('time (s)')\r\n",
        "plt.ylabel('height (m)')\r\n",
        "plt.title('Fun Title')\r\n",
        "plt.savefig('fundata.png') # saves the graph to your Google Colab files"
      ],
      "execution_count": null,
      "outputs": [
        {
          "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": "markdown",
      "metadata": {
        "id": "thd35--irCNC"
      },
      "source": [
        "Plots can also easily be saved by right-clicking on a plot and saving the graph as an image. Remember to save your graphs to a flash-drive or the Cloud (such as your Google Drive) so you can access them later."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0o3ZF_Gl-vYL"
      },
      "source": [
        "We can also easily plot two lines on the same graph. Let's use Numpy to make a range of evenly spaced values from 0 to 10 seconds in 0.5 second steps. We also can change the y-scale to a log scale which is useful for visualizing some  features in data sets in this class."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "4pNNcMCI-ygr",
        "outputId": "4a37b8f7-db46-4732-c557-e3a75fc17eaf"
      },
      "source": [
        "t = np.arange(0., 10., 0.5) # evenly sampled time at 200ms intervals\r\n",
        "\r\n",
        "plt.plot(t, t**2, 'bs', label='$t^2$') # plotting t vs. t^2 using a label for a legend\r\n",
        "plt.plot(t, t**3, 'g--', label='$t^3$') # plotting t vs. t^3 using a label for a legend\r\n",
        "\r\n",
        "# The use of $ $ around an expression tells Python that it is Tex script\r\n",
        "\r\n",
        "plt.yscale('log') #change the y scale to log\r\n",
        "plt.legend()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7fc8ea793ef0>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 20
        },
        {
          "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": "markdown",
      "metadata": {
        "id": "RWE5LC2iIAeS"
      },
      "source": [
        "\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "**Exercise:** Create a plot of $x$ vs. $x^2$ and $x$ vs. $x^4$ on the same graph from 0 to 5 with a title, proper axis labels, with a legend, and with different color dots. Then save the plot, and upload it to your PH315 Google Drive folder.\n",
        "\n",
        "\n",
        "---\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "T2YUG-wv386b"
      },
      "source": [
        "Now here is a more complicated example of plotting and sub-plotting with error bars which will hopefully be useful when you are making your own plots (adapted from [matplotlib's documentation](https://matplotlib.org/3.3.3/contents.html)).  The use of errobars on plots is very important in this class. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 558
        },
        "id": "VBudifg1DJp-",
        "outputId": "adbf08fe-2fc1-433a-9e7b-1cde31bbec22"
      },
      "source": [
        "import numpy as np\r\n",
        "import matplotlib.pyplot as plt\r\n",
        "\r\n",
        "# Example data\r\n",
        "x = np.arange(0.1, 4, 0.5)\r\n",
        "y = np.exp(-x)\r\n",
        "\r\n",
        "# Example variable error bar values\r\n",
        "yerr = 0.01 + 0.1*np.sqrt(x)\r\n",
        "xerr = 0.1 + yerr\r\n",
        "\r\n",
        "# First illustrate basic pyplot interface, using defaults where possible.\r\n",
        "plt.figure()\r\n",
        "plt.errorbar(x, y, xerr=0.2, yerr=0.4) # this is creating constant error bars\r\n",
        "plt.title(\"Constant errorbars, 0.2 in x, 0.4 in y\")\r\n",
        "\r\n",
        "# Now creating a 2x2 grid of subplots below the above plot\r\n",
        "fig, axs = plt.subplots(nrows=2, ncols=2, sharex=True) # you can insert a figsize=(x,y) to make your plots bigger or smaller here\r\n",
        "# example: fig, axs = plt.subplots(2,1, figsize=(5,5))\r\n",
        "\r\n",
        "# This line of code can be used to\r\n",
        "# format the subplots by adding or removing around the subplots. Play with it if you'd like!\r\n",
        "# plt.subplots_adjust(left=None, right=None, bottom=None, top=1, wspace=None, hspace=0.8) \r\n",
        "\r\n",
        "ax = axs[0,0] # choose the i=1,j=1 subplot\r\n",
        "ax.errorbar(x, y, yerr=yerr, fmt='o')\r\n",
        "ax.set_title('Vert. symmetric')\r\n",
        "\r\n",
        "ax = axs[0,1] # choose the i=1,j=2 subplot\r\n",
        "ax.errorbar(x, y, xerr=xerr, fmt='o')\r\n",
        "ax.set_title('Hor. symmetric')\r\n",
        "\r\n",
        "ax = axs[1,0] # choose the i=2,j=1 subplot\r\n",
        "ax.errorbar(x, y, yerr=[yerr, 2*yerr], xerr=[xerr, 2*xerr], fmt='--o')\r\n",
        "ax.set_title('H, V asymmetric')\r\n",
        "\r\n",
        "ax = axs[1,1] # choose the i=2,j=2 subplot\r\n",
        "ax.errorbar(x, y, 2*xerr, 2*yerr, fmt='--o')\r\n",
        "ax.set_title('2*xerr, 2*yerr')\r\n",
        "\r\n",
        "fig.suptitle('Variable errorbars') # creating a title for the subplots\r\n",
        "\r\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "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": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 4 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zuhz5qXKIhFM"
      },
      "source": [
        "\r\n",
        "\r\n",
        "---\r\n",
        "\r\n",
        "\r\n",
        "**Exercise:** Using your previous code, create constant error bars of xerror = 0.5 and yerror = 1 for $x$ vs. $x^2$ \r\n",
        "\r\n",
        "and xerror = 1 yerror = 2 for $x$ vs. $x^4$ in two subplots.\r\n",
        "\r\n",
        "\r\n",
        "---\r\n",
        "\r\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "x_5uJ_Ddvjsm"
      },
      "source": [
        "# 5. Curve Fitting\r\n",
        "\r\n",
        "We will commonly need to fit functions/curves to data in this class. Two common curve fitting methods in Python are Numpy's polyfit (np.polyfit) and SciPy optimize's curve_fit (scipy.optimize.curve_fit).  Polyfit only works for polynomials, while SciPy optimize lets you define arbitrary fitting functions (e.g. Gaussian, exponential, or whatever)\r\n",
        "\r\n",
        "Here is a simple example of Numpy's polyfit on a set of data.  Notice how the parameters extracted from fitting the line (the slope and intercept) have uncertainties associated with them.  This is true for *all fits* and is extremely important in this class.  We will talk about it in detail in the second week. You will need to know how to access the uncertainties, as below, so you can report them in your lab reports."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 350
        },
        "id": "aQDpHXdNEF9z",
        "outputId": "230b1d25-c8c9-4fa2-8ad8-5b3ec60241c9"
      },
      "source": [
        "N = 10\r\n",
        "\r\n",
        "x = np.random.random(N)\r\n",
        "y = np.random.random(N)\r\n",
        "\r\n",
        "# Use Polyfit to create a matrix of fitted parameters, z,\r\n",
        "# and a covariance matrix which we can square root to get the standard deviation/error of the fit\r\n",
        "z, cov = np.polyfit(x, y, 1, cov=True) # The 1 designates a degree-one polynomial, a line.  Cov is the covariance matrix, which will be used to get the uncertainties.\r\n",
        "stdev = np.sqrt(np.diag(cov)) # the standard deviations (uncertainties) is the sqrt of the covariance diagonals\r\n",
        "\r\n",
        "# Now we can print out the fitting parameters of the line (m and b) and the error of each parameter\r\n",
        "print(z)\r\n",
        "print(stdev)\r\n",
        "\r\n",
        "print(\"\\n\") # putting \\n in a print string will tell Python to make a new line\r\n",
        " \r\n",
        "# Using print to display the fitting parameters and their errors meaningfully\r\n",
        "print(\"The slope of the line is %1.3f +/- %1.3f and the y-intercept is %1.3f +/- %1.3f\" % (z[0], stdev[0], z[1], stdev[1]))\r\n",
        "\r\n",
        "plt.plot(x,y,'ro') # plot the x and y data\r\n",
        "trend = np.poly1d(z)\r\n",
        "plt.plot(x,trend(x)); # plot the trend line\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[0.09929407 0.32593933]\n",
            "[0.27942503 0.18965764]\n",
            "\n",
            "\n",
            "The slope of the line is 0.099 +/- 0.279 and the y-intercept is 0.326 +/- 0.190\n"
          ],
          "name": "stdout"
        },
        {
          "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": "markdown",
      "metadata": {
        "id": "u4B2tFooOQNc"
      },
      "source": [
        "\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "**Exercise:** Create an array of linear-ish data and plot it using polyfit. Also, feel free to try plotting quadratic data using polyfit.\n",
        "\n",
        "\n",
        "---\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "i5nW9k2PFe5i"
      },
      "source": [
        "Polyfit works great for lines and low-degree polynomials. However, for more compliated fitting, we'll utilize scipy.optimize. For example, if we want to fit a Gaussian to some data. \r\n",
        "\r\n",
        "First we must define the function we wish to fit. Let's try to fit an exponential: $y = ae^{bx}$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "qq6BxUECGJva"
      },
      "source": [
        "from scipy.optimize import curve_fit # import the curve fit module from scipy\r\n",
        "\r\n",
        "# Now define our exponential\r\n",
        "def exponential(x, a, b):\r\n",
        "  return a*np.exp(b*x)\r\n",
        "  "
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ueUKf0fBGzxA"
      },
      "source": [
        "Now we create a dummy dataset to fit with the function."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 280
        },
        "id": "_ngEt9nlGzFh",
        "outputId": "23a25005-7221-48c7-d559-1a61524f0961"
      },
      "source": [
        "x = np.linspace(start=1, stop=6, num=50)\r\n",
        "# To create our y data, we have to use x and specify a and b\r\n",
        "# In this case, a=0.5, b=1.5\r\n",
        "y = exponential(x, 1.5, 1.5)\r\n",
        "\r\n",
        "# I also am going to create noise (I could do this manually but this is faster)\r\n",
        "# This will make the data set imperfect\r\n",
        "noise = 1000*np.random.normal(size=y.size)\r\n",
        "y = y + noise\r\n",
        "\r\n",
        "# Now plot the data\r\n",
        "plt.plot(x, y, 'ro')\r\n",
        "\r\n",
        "plt.xlabel('x')\r\n",
        "plt.ylabel('y');"
      ],
      "execution_count": null,
      "outputs": [
        {
          "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": "markdown",
      "metadata": {
        "id": "zAEmsBGuIwo8"
      },
      "source": [
        "Now we fit the data with a exponential using curve_fit."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "8m1CPtlrI4PA"
      },
      "source": [
        "pars, cov = curve_fit(f=exponential, xdata=x, ydata=y, p0=[0, 0], bounds=(-np.inf, np.inf))"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4X4k7smqJQQ6"
      },
      "source": [
        "In this case, we specified that we wanted to try to fit our previously defined function (exponential) to the data. \r\n",
        "\r\n",
        "**p0** specifies the intial guesses for the a and b parameters (intial guess being a=0 and b=0 in this case). \r\n",
        "\r\n",
        "**bounds** specifies what the parameters can be. \r\n",
        "\r\n",
        "**pars** is the array of parameters generated by curve_fit ([a, b] from this fit).\r\n",
        "\r\n",
        "**cov** is the estimated covariance of pars generated by curve_fit which is used to find the standard deiviation."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "c-OyeEGxJ_u1"
      },
      "source": [
        "Now we can find the standard deviation of the fit parameters."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "8sEbY9sxKGhq",
        "outputId": "84bd222f-491a-4131-f0d3-275d19c0f292"
      },
      "source": [
        "# Standard deviation\r\n",
        "stdev = np.sqrt(np.diag(cov)) # stdev is the sqrt(cov)\r\n",
        "\r\n",
        "# Residuals can be calculated easily as well.\r\n",
        "res = y - exponential(x, *pars) # *pars unrolls/unpacks the [a, b] array\r\n",
        "# into a, b. So exponential(x, *pars) is equivalent to exponential(x, a, b)\r\n",
        "\r\n",
        "print(*pars)\r\n",
        "print(*stdev)\r\n",
        "\r\n",
        "# Using print to display the results meaningfully\r\n",
        "print(\"\\n a is %1.3f +/- %1.3f, b is %1.3f +/- %1.3f\" % (pars[0], stdev[0], pars[1], stdev[1]))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "2.084272493217437 1.436989642116566\n",
            "1.7445440016696714 0.1465599235787979\n",
            "\n",
            " a is 2.084 +/- 1.745, b is 1.437 +/- 0.147\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "l0yDKhXIK4Qp"
      },
      "source": [
        "Note: The values of a and b are close to the initial input values (but are definitely not always right on). However, they are generally inside a standard deviation of the initial values."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "O_RCLA_7Ks_D"
      },
      "source": [
        "Now we can plot the fit over the data, like we've done before. It did a pretty good job!"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "mpXhGctRLO1c",
        "outputId": "dcfd1a0a-4726-483a-dc2e-3343f2d5184a"
      },
      "source": [
        "plt.plot(x, y, 'ro')\r\n",
        "plt.plot(x, exponential(x,*pars), 'b--');"
      ],
      "execution_count": null,
      "outputs": [
        {
          "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": "markdown",
      "metadata": {
        "id": "Tdzwl8nGNByX"
      },
      "source": [
        "\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "**Exercise:** Play with the a,b parameters of the exponential, the randomness of the data, or the size of the data set. Are your fitting parameters always within a standard deviation of the initial parameters you set?\n",
        "\n",
        "\n",
        "---\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TQvWRxMrNR6j"
      },
      "source": [
        "\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "**Extra Exercise:** If you have time, try defining a different function (like x, x^2, x^4, sinx, etc.), creating noisy data to fit, fitting the data, and plotting it.\n",
        "\n",
        "\n",
        "---\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dyHrUA6rgpEQ"
      },
      "source": [
        "A common thing is fit in this course is a Gaussian function $y=ae^\\frac{-(x-b)^2}{2c^2}$. Let's follow our exact same steps as before to fit one from a data set."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 408
        },
        "id": "283us9p0hGFP",
        "outputId": "ab429e56-579f-4ee3-9654-413aec5896e0"
      },
      "source": [
        "# Now define our gaussian\r\n",
        "def gaussian(x, a, b, c):\r\n",
        "  return a*np.exp(-np.power(x - b, 2)/(2*np.power(c,2)))\r\n",
        "\r\n",
        "x = np.linspace(start=-10, stop=10, num=150)\r\n",
        "# To create our y data, we have to use x and specify a, b, c\r\n",
        "y = gaussian(x, 5, -1, 3)\r\n",
        "\r\n",
        "# I also am going to create noise (I could do this manually but this is faster)\r\n",
        "# This will make the data set \"imperfect\"\r\n",
        "noise = 1*np.random.normal(size=y.size)\r\n",
        "y = y + noise\r\n",
        "\r\n",
        "# Now plot the data\r\n",
        "plt.plot(x, y, 'ro')\r\n",
        "\r\n",
        "plt.xlabel('x')\r\n",
        "plt.ylabel('y')\r\n",
        "\r\n",
        "pars, cov = curve_fit(f=gaussian, xdata=x, ydata=y, p0=[1, 1, 1], bounds=(-np.inf, np.inf))\r\n",
        "\r\n",
        "# Standard deviation\r\n",
        "stdev = np.sqrt(np.diag(cov)) # stdev is the sqrt(cov)\r\n",
        "\r\n",
        "# Residuals\r\n",
        "res = y - gaussian(x, *pars) # the residuals can be calculated easily as well.. *pars unrolls/unpacks the [a, b, c] array\r\n",
        "\r\n",
        "print(*pars)\r\n",
        "print(*stdev)\r\n",
        "print(\"\\n\")\r\n",
        "\r\n",
        "# Using print to display the fitting parameters and their error meaningfully\r\n",
        "print(\" a is %1.3f +/- %1.3f \\n b is %1.3f +/- %1.3f \\n c is %1.3f +/- %1.3f\" \r\n",
        "     % (pars[0], stdev[0], pars[1], stdev[1], pars[2], stdev[2]))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "5.025400818043974 -1.039634636695472 2.987198578206616\n",
            "0.19708218692598764 0.13524312452566034 0.13535240859180328\n",
            "\n",
            "\n",
            " a is 5.025 +/- 0.197 \n",
            " b is -1.040 +/- 0.135 \n",
            " c is 2.987 +/- 0.135\n"
          ],
          "name": "stdout"
        },
        {
          "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": "markdown",
      "metadata": {
        "id": "5xs_OhsaqoGO"
      },
      "source": [
        "Notice, this time we can't use p0=[0, 0, 0] as we will get a divide by zero answer, so we must try to guess a little about what the parameters would be. You could look at the equation and use it to guess more closely what you think the parameters would be but I just input p0=[1, 1, 1] without thinking about the equation of a gaussian."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "Sd-UrcXri16e",
        "outputId": "591d9171-d4e6-44f4-de9f-a68edfde29c6"
      },
      "source": [
        "plt.plot(x, y, 'ro')\r\n",
        "plt.plot(x, gaussian(x,*pars), 'b--', linewidth=3.0);"
      ],
      "execution_count": null,
      "outputs": [
        {
          "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"
          }
        }
      ]
    }
  ]
}