{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Title :\n", "Exercise: Beta Values for Data from Random Universe\n", "\n", "## Description :\n", "\n", "Given a `RandomUniverse(dataframe)->dataframe` function that gives a new dataset from a \"parallel\" universe, calculate the $\\beta_0$'s and $\\beta_1$'s and plot a histogram like the one below.\n", "\n", "\n", "\n", "## Data Description:\n", "\n", "## Instructions:\n", "\n", "- Get a new dataframe using the RandomUniverse function already provided in the exercise\n", "- Calculate $\\beta_0$, $\\beta_1$ for that particular dataframe\n", "- Add the calculated $\\beta_0$ and $\\beta_1$ values to a python list\n", "- Plot a histogram using the lists calculated above\n", "\n", "## Hints: \n", "\n", "$${\\widehat {\\beta_1 }}={\\frac {\\sum _{i=1}^{n}(x_{i}-{\\bar {x}})(y_{i}-{\\bar {y}})}{\\sum _{i=1}^{n}(x_{i}-{\\bar {x}})^{2}}}$$\n", "\n", "$${\\widehat {\\beta_0 }}={\\bar {y}}-{\\widehat {\\beta_1 }}\\,{\\bar {x}}$$\n", "\n", "plt.subplots()\n", "Create a figure and a set of subplots\n", "\n", "ax.hist()\n", "Plot a histogram from a list or series." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "# Import necessary libraries\n", "import numpy as np\n", "import pandas as pd\n", "import matplotlib.pyplot as plt\n", "from randomuniverse import RandomUniverse\n", "%matplotlib inline\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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tvsales
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" ], "text/plain": [ " tv sales\n", "0 230.1 465.26\n", "1 44.5 218.95\n", "2 17.2 195.79\n", "3 151.5 389.47\n", "4 180.8 271.58" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Read the advertising dataset as a pandas dataframe\n", "df = pd.read_csv('Advertising_adj.csv')\n", "\n", "# Take a quick look at the dataframe\n", "df.head()\n" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "# Create two empty lists that will store the beta values\n", "beta0_list, beta1_list = [],[]\n", "\n", "# Choose the number of \"parallel\" Universes to generate \n", "# that many new versions of the dataset\n", "parallelUniverses = 10000\n", "\n", "# Loop over the maximum number of parallel Universes\n", "for i in range(parallelUniverses):\n", "\n", " # Call the RandomUniverse helper function with the dataframe\n", " # read from the data file\n", " df_new = RandomUniverse(df)\n", "\n", " # Find the mean of the predictor values i.e. tv\n", " xmean = df_new.tv.mean()\n", "\n", " # Find the mean of the response values i.e. sales\n", " ymean = df_new.sales.mean()\n", "\n", " # Compute the analytical values of beta0 and beta1 using the \n", " # equation given in the hints\n", " beta1 = np.sum((df_new.tv-xmean) * (df_new.sales-ymean)) / np.sum((df_new.tv-xmean)**2)\n", " beta0 = ymean-beta1*xmean\n", "\n", " # Append the calculated values of beta1 and beta0 to the appropriate lists\n", " beta0_list.append(beta0)\n", " beta1_list.append(beta1)\n" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "### edTest(test_beta) ###\n", "\n", "# Compute the mean of the beta values\n", "beta0_mean = np.mean(beta0_list)\n", "beta1_mean = np.mean(beta1_list)\n" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, 'Frequency')" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Plot histograms of beta_0 and beta_1 using lists created above \n", "fig, ax = plt.subplots(1,2, figsize=(18,8))\n", "ax[0].hist(beta0_list)\n", "ax[1].hist(beta1_list)\n", "ax[0].set_xlabel('Beta 0')\n", "ax[1].set_xlabel('Beta 1')\n", "ax[0].set_ylabel('Frequency');\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ⏸ Increase the number of `parallelUniverses`. Which of the following do you observe?\n", "\n", "#### A. The spread increases\n", "#### B. The frequency of points decreases\n", "#### C. The spread decreases\n", "#### D. There is no change" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "### edTest(test_chow1) ###\n", "# Submit an answer choice as a string below \n", "# (Eg. if you choose option C, put 'C')\n", "answer1 = 'D'\n" ] }, { "cell_type": "code", "execution_count": 0, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }