statkclee/chap08soln-kor.ipynb

## chap08soln-kor.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "통계적 사고 (2판) 연습문제 ([thinkstats2.com](thinkstats2.com), [think-stat.xwmooc.org](http://think-stat.xwmooc.org))<br>\n",
    "Allen Downey / 이광춘(xwMOOC)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from __future__ import print_function\n",
    "\n",
    "import thinkstats2\n",
    "import thinkplot\n",
    "\n",
    "import math\n",
    "import random\n",
    "import numpy as np\n",
    "\n",
    "from scipy import stats\n",
    "from estimation import RMSE, MeanError"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 연습문제 8.1\n",
    "\n",
    "이번 장에서, $\\mu$를 추정하는데 xbar 와 중위수를 사용했고, xbar가 MSE 하한을 산출함을 알아냈다. 또한, $\\sigma$를 추정하는데  $S^2$ 와  $S_{n-1}^2$ 을 사용했고,  $S^2$ 은 편향되었고, $S_{n-1}^2$은 불편향임을 알아냈다.\n",
    "유사한 실험을 실행해서, xbar와 중위수가 $\\mu$의 편향된 추정값임을 알아내라. 또한, $S^2$ 혹은 $S_{n-1}^2$ 가 MSE 하한을 산출하는지 검사하라."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Experiment 1\n",
      "mean error xbar -0.000501853563251\n",
      "mean error median -5.92353811989e-05\n",
      "Experiment 2\n",
      "RMSE biased 0.515294279494\n",
      "RMSE unbiased 0.577482542019\n"
     ]
    }
   ],
   "source": [
    "\"\"\"Mean error for xbar and median as estimators of population mean.\n",
    "\n",
    "    n: sample size\n",
    "    m: number of iterations\n",
    "\"\"\"\n",
    "n = 7\n",
    "m=100000\n",
    "\n",
    "mu = 0\n",
    "sigma = 1\n",
    "\n",
    "means = []\n",
    "medians = []\n",
    "for _ in range(m):\n",
    "    xs = [random.gauss(mu, sigma) for i in range(n)]\n",
    "    xbar = np.mean(xs)\n",
    "    median = np.median(xs)\n",
    "    means.append(xbar)\n",
    "    medians.append(median)\n",
    "\n",
    "print('Experiment 1')\n",
    "print('mean error xbar', MeanError(means, mu))\n",
    "print('mean error median', MeanError(medians, mu))\n",
    "\n",
    "\"\"\"RMSE for biased and unbiased estimators of population variance.\n",
    "\n",
    "n: sample size\n",
    "m: number of iterations\n",
    "\"\"\"\n",
    "mu = 0\n",
    "sigma = 1\n",
    "\n",
    "estimates1 = []\n",
    "estimates2 = []\n",
    "for _ in range(m):\n",
    "    xs = [random.gauss(mu, sigma) for i in range(n)]\n",
    "    biased = np.var(xs)\n",
    "    unbiased = np.var(xs, ddof=1)\n",
    "    estimates1.append(biased)\n",
    "    estimates2.append(unbiased)\n",
    "\n",
    "print('Experiment 2')\n",
    "print('RMSE biased', RMSE(estimates1, sigma**2))\n",
    "print('RMSE unbiased', RMSE(estimates2, sigma**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. xbar 와 중위수가 m이 증가함에따라 더 낮은 평균오차를 산출한다. 실험으로부터 식별할 수 있는한 어떤 것도 분명히 편향되지 않았다.\n",
    "1. 분산에 대한 불편 추정량이 불편 추정량보다 더 낮은 약 10% 낮은 RMSE를 산출한다. 그리고 m이 증가함에 따라 차이는 유지된다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 연습문제 8.2  \n",
    "모수 $\\lambda=2$를 갖는 지수분포에서 표본 $n=10$개를 추출한다고 가정하자. 실험을 1000번 모의시험하고 추정값 lamhat의 표본 분포를 도식화한다. 추정값의 표준오차와 90% 신뢰구간을 계산하라.\n",
    "\n",
    "다른 $n$ 값을 갖는 실험을 반복하고, $n$ 값과 표준오차를 도식화한다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "standard error 0.784058129255\n",
      "confidence interval (1.2783641952567839, 3.5793120939847975)\n"
     ]
    },
    {
     "data": {
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GYNPG4BlBrzx9Hdq03t1RIhGpiIqAZEzyMQAAGPb8jWipYwEikaXhIMmIcRO+LLfu0gu6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\nwzyHx4XiAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xf21afd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7adeb50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"Sampling distribution of L as an estimator of exponential parameter.\n",
    "\n",
    "lam: parameter of an exponential distribution\n",
    "n: sample size\n",
    "m: number of iterations\n",
    "\"\"\"\n",
    "lam=2\n",
    "n=10\n",
    "m=1000\n",
    "\n",
    "def VertLine(x, y=1):\n",
    "    thinkplot.Plot([x, x], [0, y], color='0.8', linewidth=3)\n",
    "\n",
    "estimates = []\n",
    "for j in range(m):\n",
    "    xs = np.random.exponential(1.0/lam, n)\n",
    "    lamhat = 1.0 / np.mean(xs)\n",
    "    estimates.append(lamhat)\n",
    "\n",
    "stderr = RMSE(estimates, lam)\n",
    "print('standard error', stderr)\n",
    "\n",
    "cdf = thinkstats2.Cdf(estimates)\n",
    "ci = cdf.Percentile(5), cdf.Percentile(95)\n",
    "print('confidence interval', ci)\n",
    "VertLine(ci[0])\n",
    "VertLine(ci[1])\n",
    "\n",
    "# plot the CDF\n",
    "thinkplot.Cdf(cdf)\n",
    "thinkplot.Show(xlabel='estimate',\n",
    "               ylabel='CDF',\n",
    "               title='Sampling distribution')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. 표본크기 10인경우\n",
    "    - 표준오차 : 0.896717911545\n",
    "    - 신뢰구간: (1.2901330772324622, 3.8692334892427911)\n",
    "1. 표본크기가 증가함에 따라, 표준오차와 CI 구간범위가 줄어든다:\n",
    "    - 10      0.90    (1.3, 3.9)\n",
    "    - 100     0.21    (1.7, 2.4)\n",
    "    - 1000    0.06    (1.9, 2.1)\n",
    "상기 신뢰구간 세개 모두 실제 값 2를 포함하고 있다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## 연습문제 8.3  \n",
    "하키와 축구같은 스포츠 게임에서 득점 사이 시간은 대체로 지수를 따른다. 그래서 게임에서 한 팀이 득점한 골을 관측함으로써 득점을 추정할 수 있다. 이 추정 과정은 득점 사이 시간을 표집하는 것과 약간 다르다. 그래서 작동방법을 살펴보다.\n",
    "\n",
    "게임당 골로 득점률 ${\\tt lam}$을 인자로 받고, 전체 시간이 1 게임 경과할 때까지 득점사이 시간을 생성함으로서 게임을 모의시험하고 나서, 득점한 점수를 반환하는 함수를 작성하라.\n",
    "\n",
    "많은 게임을 모의시험하고, ${\\tt lam}$ 추정값을 저장하고 나서 평균 오차와 RMSE를 계산하는 또다른 함수를 작성하라.\n",
    "\n",
    "추정값을 이와 같은 방식으로 만드는 것이 편향됐을까? 추정값에 대한 표본분포와 90% 신뢰구간을 도식화하시오. 표준오차는 얼마인가? ${\\tt lam}$ 값을 크게하면, 표집오차에 무슨일이 생길까?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Experiment 4\n",
      "rmse L 1.41495194265\n",
      "mean error L -0.000607\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0xebbc870>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0xebbc910>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def SimulateGame(lam):\n",
    "    \"\"\"Simulates a game and returns the estimated goal-scoring rate.\n",
    "\n",
    "    lam: actual goal scoring rate in goals per game\n",
    "    \"\"\"\n",
    "    goals = 0\n",
    "    t = 0\n",
    "    while True:\n",
    "        time_between_goals = random.expovariate(lam)\n",
    "        t += time_between_goals\n",
    "        if t > 1:\n",
    "            break\n",
    "        goals += 1\n",
    "\n",
    "    # estimated goal-scoring rate is the actual number of goals scored\n",
    "    L = goals\n",
    "    return L\n",
    "\n",
    "def Estimate4(lam=2, m=1000000):\n",
    "\n",
    "    estimates = []\n",
    "    for i in range(m):\n",
    "        L = SimulateGame(lam)\n",
    "        estimates.append(L)\n",
    "\n",
    "    print('Experiment 4')\n",
    "    print('rmse L', RMSE(estimates, lam))\n",
    "    print('mean error L', MeanError(estimates, lam))\n",
    "    \n",
    "    pmf = thinkstats2.Pmf(estimates)\n",
    "\n",
    "    thinkplot.Hist(pmf)\n",
    "    thinkplot.Show()\n",
    "\n",
    "Estimate4()    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. 이런 방식으로 lambda 추정에 대한 RMSE는 1.4.\n",
    "1. 평균오차는 작고, m이 커짐에 따라 줄어든다. 그래서, 추정량은 편향되지 않은 것으로 보인다.\n",
    "주의: 득점 사이 시간이 지수분포라면, 게임에서 득점된 골의 분포는 포아송이다. [https://en.wikipedia.org/wiki/Poisson_distribution](https://en.wikipedia.org/wiki/Poisson_distribution) 참조한다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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