{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "63f2cb41",
   "metadata": {},
   "source": [
    "# IntroStat Week 5\n",
    "\n",
    "Simulation of the samples and QQ-plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "88fc8695",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats as stats\n",
    "\n",
    "# NEW: import statsmodels.api to do automated q-q-plot\n",
    "import statsmodels.api as sm"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "abe4110a",
   "metadata": {},
   "source": [
    "### Simulation: Sample from Normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "55af8038",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(24)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e6c55bac",
   "metadata": {},
   "outputs": [],
   "source": [
    "# (repeat this cell many times)\n",
    "\n",
    "# population parameters:\n",
    "mu = 100\n",
    "sigma = 12\n",
    "\n",
    "# simulated sample:\n",
    "n = 10\n",
    "data = stats.norm.rvs(mu, sigma, size=n)\n",
    "\n",
    "# significance level (for plotting confidence interval of the mean)\n",
    "a = 0.05\n",
    "\n",
    "# plotting\n",
    "x = np.arange(70,130,0.1)\n",
    "plt.xlim(70,130)\n",
    "plt.plot(x, .2*np.sqrt(2*np.pi*sigma**2)*stats.norm.pdf(x, loc=mu, scale=sigma), color=\"red\",alpha=0.3)\n",
    "plt.plot(x, .2*np.sqrt(2*np.pi*sigma**2/n)*stats.norm.pdf(x, loc=mu, scale=sigma/np.sqrt(n)), color=\"black\", linestyle='--',alpha=0.3)\n",
    "plt.hist(data, density=True, color='deepskyblue', bins=6)\n",
    "plt.axvline(data.mean(), linestyle='-', color=\"blue\", ymin=0, ymax=1)\n",
    "plt.plot([data.mean()-data.std(ddof=1)/np.sqrt(n), data.mean()+data.std(ddof=1)/np.sqrt(n)], [0.06,0.06], linestyle='-', color=\"blue\")\n",
    "plt.axvline(stats.t.ppf(a/2, df=n-1)*data.std(ddof=1)/np.sqrt(n)+data.mean(), linestyle='--', color=\"blue\", ymin=0, ymax=1)\n",
    "plt.axvline(stats.t.ppf(1-a/2, df=n-1)*data.std(ddof=1)/np.sqrt(n)+data.mean(), linestyle='--', color=\"blue\", ymin=0, ymax=1)\n",
    "plt.tick_params(left = False, right = False , labelleft = False) \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "abcd18b7",
   "metadata": {},
   "source": [
    "The histogram of the sample does not always look very \"normal\"\n",
    "\n",
    "(back to slides)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89ce4224",
   "metadata": {},
   "source": [
    "### ECDF and Q-Q plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ef145d9b",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(24)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c162aac2",
   "metadata": {},
   "outputs": [],
   "source": [
    "# (repeat this cell many times)\n",
    "data = stats.norm.rvs(mu, sigma, size=n)\n",
    "\n",
    "# split plot in 2 \n",
    "fig, axs = plt.subplots(1, 2, figsize=(10*2/3,4))\n",
    "\n",
    "# plot histogram\n",
    "axs[0].hist(data, density=True, bins=6)\n",
    "axs[0].set_xlim([70,130])\n",
    "\n",
    "# plot ecdf\n",
    "axs[1].ecdf(data)\n",
    "axs[1].plot(np.arange(70,130,1), stats.norm.cdf(np.arange(70,130,1), loc=mu, scale=sigma))\n",
    "axs[1].set_xlim([70,130])\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "545a754a",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(24)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c13cc996",
   "metadata": {},
   "outputs": [],
   "source": [
    "# (repeat this cell many times)\n",
    "data = stats.norm.rvs(mu, sigma, size=n)\n",
    "\n",
    "# split plot in 3\n",
    "fig, axs = plt.subplots(1, 3, figsize=(10,4))\n",
    "\n",
    "# plot histogram\n",
    "axs[0].hist(data, density=True, bins=6)\n",
    "axs[0].set_xlim([70,130])\n",
    "\n",
    "# plot ecdf\n",
    "axs[1].ecdf(data)\n",
    "axs[1].plot(np.arange(70,130,1), stats.norm.cdf(np.arange(70,130,1), loc=mu, scale=sigma))\n",
    "axs[1].set_xlim([70,130])\n",
    "\n",
    "# plot qq-plot\n",
    "sm.qqplot(data,line=\"q\",a=3/8,ax=axs[2])\n",
    "# OBS: \"a = 3/8\" is preferred for n <= 10 \n",
    "#     (\"a = 1/2\" is preferred for n >  10)  \n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ecaea6b5",
   "metadata": {},
   "source": [
    "Now try larger samples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "03f7723b",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(24)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a16dc51d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# (repeat this cell many times)\n",
    "\n",
    "n = 100 # larger sample size\n",
    "data = stats.norm.rvs(mu, sigma, size=n)\n",
    "\n",
    "# split plot in 3\n",
    "fig, axs = plt.subplots(1, 3, figsize=(10,4))\n",
    "\n",
    "# plot histogram\n",
    "axs[0].hist(data, density=True)\n",
    "axs[0].set_xlim([70,130])\n",
    "\n",
    "# plot ecdf\n",
    "axs[1].ecdf(data)\n",
    "axs[1].plot(np.arange(70,130,1), stats.norm.cdf(np.arange(70,130,1), loc=mu, scale=sigma))\n",
    "axs[1].set_xlim([70,130])\n",
    "\n",
    "# plot qq-plot\n",
    "sm.qqplot(data,line=\"q\",a=1/2,ax=axs[2])\n",
    "# OBS: \"a = 3/8\" is preferred for n <= 10 \n",
    "#     (\"a = 1/2\" is preferred for n >  10)  \n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d2712d4e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# examples with exponentially distributed data:\n",
    "\n",
    "n = 10 # try both n=10 and n=100\n",
    "data = stats.expon.rvs(mu, sigma, size=n)\n",
    "fig, axs = plt.subplots(1, 3, figsize=(10,4))\n",
    "axs[0].hist(data, density=True)\n",
    "axs[1].ecdf(data)\n",
    "axs[1].plot(np.arange(70,150,1), stats.norm.cdf(np.arange(70,150,1), loc=stats.expon.mean(loc=mu, scale=sigma), scale=stats.expon.std(loc=mu, scale=sigma)))\n",
    "axs[1].set_xlim(70,150)\n",
    "sm.qqplot(data,line=\"q\",a=1/2,ax=axs[2])\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9afd5a65",
   "metadata": {},
   "outputs": [],
   "source": [
    "# examples with uniformly distributed data:\n",
    "\n",
    "n = 10 # try both n=10 and n=100\n",
    "data = stats.uniform.rvs(mu, sigma, size=n)\n",
    "fig, axs = plt.subplots(1, 3, figsize=(10,4))\n",
    "axs[0].hist(data, density=True)\n",
    "axs[1].ecdf(data)\n",
    "axs[1].plot(np.arange(90,120,1), stats.norm.cdf(np.arange(90,120,1), loc=stats.uniform.mean(loc=mu, scale=sigma), scale=stats.uniform.std(loc=mu, scale=sigma)))\n",
    "axs[1].set_xlim(90,120)\n",
    "sm.qqplot(data,line=\"q\",a=1/2,ax=axs[2])\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  }
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