stat-laws.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Основы анализа данных в Python\n",
    "\n",
    "*Алла Тамбовцева*\n",
    "\n",
    "## Основные законы статистики: закон больших чисел и центральная предельная теорема"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Импортируем необходимые библиотеки и модули:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.stats as st\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Перейдём к визуализации действия основных статистических законов."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Закон больших чисел\n",
    "\n",
    "> С увеличением размера выборки $n$ (в идеале – при стремлении $n$ к бесконечности), среднее значение выборки $\\bar{x}$ становится ближе к среднему значению генеральной совокупности:\n",
    "\n",
    "$$\n",
    "\\lim_{n\\to\\infty}\\sum_{i=1}^n\\frac{x_i}{n} = \\bar{x}.\n",
    "$$\n",
    "\n",
    "\n",
    "\n",
    "Для визуализации этого закона будем извлекать выборки разного размера из нормального распределения со средним значением 168 и стандартным отклонением 2. Вспомним, как выглядит плотность такого распределения:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "mu = 168\n",
    "sigma = 2\n",
    "\n",
    "# linspace() создает массив значений от start до stop из num = 1000 значений, равноудаленных друг от друга\n",
    "# norm.pdf() вычисляет значения функции плотности в каждой точке х\n",
    "# отрисовываем график по точкам и добавляем вертикальную линию x = 168\n",
    "\n",
    "x = np.linspace(start = mu - 3 * sigma, stop = mu + 3 * sigma, num = 1000)\n",
    "plt.plot(x, st.norm.pdf(x, mu, sigma), color = \"navy\");\n",
    "plt.axvline(x = 168, color = 'firebrick', linestyle = \"dashed\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Теперь будем «дёргать» выборки из нормального распределения $N(168, \\sigma=2)$. Содержательно, можно считать, например, что мы будем извлекать выборки из большой генеральной совокупности, которая представляет собой набор значений роста женщин в сантиметрах. Начнём с небольшой выборки в 10 наблюдений. Для этого нам понадобится функция `norm.rvs()` из модуля `stats` внутри `scipy`, которая используется для генерации случайных (псевдослучайных) выборок из нормального распределения:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[167.59930829 166.55450078 165.98712705 167.12926478 169.86067297\n",
      " 166.61057297 161.41621564 166.72202621 168.22877754 168.40813973]\n"
     ]
    }
   ],
   "source": [
    "# norm.rvs(): название распределения + random variable sample\n",
    "# loc: математическое ожидание\n",
    "# scale: стандартное отклонение\n",
    "# random_state: закрепляем для воспроизводимости – чтобы у всех были одинаковые выборки\n",
    "\n",
    "sample01 = st.norm.rvs(loc = 168, scale = 2, size = 10, random_state = 800)\n",
    "print(sample01)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Посчитаем среднее полученной выборки:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N = 10 Mean = 166.85166059439118\n"
     ]
    }
   ],
   "source": [
    "print(\"N = 10\", \"Mean =\", sample01.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Выборочное среднее отличается от среднего генеральной совокупности на примерно на 1.15. Проверим, что будет происходить с увеличением объёма выборки. Возьмём выборку объёма 100 и выборку объёма 2000 из того же нормального распределения."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N = 100 Mean = 167.60198437638792\n"
     ]
    }
   ],
   "source": [
    "# n = 100\n",
    "\n",
    "np.random.seed(800)\n",
    "sample02 = st.norm.rvs(loc = 168, scale = 2, size = 100, random_state = 800)\n",
    "print(\"N = 100\", \"Mean =\", sample02.mean())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N = 2000 Mean = 167.99987039132859\n"
     ]
    }
   ],
   "source": [
    "# n = 2000\n",
    "\n",
    "np.random.seed(800)\n",
    "sample03 = st.norm.rvs(loc = 168, scale = 2, size = 2000, random_state = 800)\n",
    "print(\"N = 2000\", \"Mean =\", sample03.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Как можно заметить, действительно, с увеличением объёма выборки, выборочное среднее меньше отличается от среднего генеральной совокупности, в данном случае от среднего 168 нормальной случайной величины. Попробуем поэкспериментировать – извлечём выборки разного размера от 10 до 10000 наблюдений из нормального распределения $N(168, \\sigma=2)$, зафиксируем выборочные средние и посмотрим на их изменения при увеличении размера выборки."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# извлекаем выборки размера от 10 до 10000 с шагом 10\n",
    "# по каждой считаем среднее и записываем его в список means\n",
    "\n",
    "means = []\n",
    "\n",
    "for k in range(10, 10001, 10):\n",
    "    samp = st.norm.rvs(loc = 168, scale = 2, size = k)\n",
    "    means.append(samp.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Чтобы понять, что происходит, построим график – по оси X будем отмечать объём выборки, а по оси Y – выборочное среднее:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# axhline – горизонтальная линия\n",
    "\n",
    "plt.plot(range(10, 10001, 10), means, color = \"navy\");\n",
    "plt.axhline(y = 168, color = 'firebrick', linestyle = \"dashed\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Возможно, не самый показательный график, но, тем не менее, по нему видно, что при увеличении объёма выборки выборочные средние колеблются вокруг среднего генеральной совокупности с меньшей амплитудой. Ещё лучше этот эффект можно увидеть на гистограммах. Давайте ещё поэкспериментируем: \n",
    "\n",
    "* извлечём 100 выборок объёма 10\n",
    "* извлечём 100 выборок объёма 100\n",
    "* извлечём 100 выборок объёма 1000\n",
    "\n",
    "и посмотрим на то, насколько выборочные средние отличаются от среднего генеральной совокупности 168 при каждом объёме выборки. \n",
    "\n",
    "Сгенерируем 100 выборок объёма $n = 10$, посчитаем по каждой среднее и сохраним средние в список `means10`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "means10 = []\n",
    "for i in range(100):\n",
    "    samp = st.norm.rvs(loc = 168, scale = 2, size = 10)\n",
    "    means10.append(samp.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Построим гистограмму для средних:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# для удобства последующих сравнений:\n",
    "# range = (166, 170) – границы по оси x\n",
    "# bins = np.arange(166, 170, 0.1) – границы столбцов\n",
    "# массив от 166 до 170 с шагом 0.1\n",
    "\n",
    "plt.hist(means10, color = \"firebrick\", edgecolor = \"white\", \n",
    "         range = (166, 170), bins = np.arange(166, 170, 0.1));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Итак, в случае, если выборки маленького размера $n=10$, средние по таким выборкам получаются довольно разнообразными – примерно от 166.5 до 169.5. То есть, теоретически, выборочные средние могут удаляться от среднего генеральной совокупности 168 примерно на 1.5.\n",
    "\n",
    "Попробуем изучить выборки большего размера. Сгенерируем 100 выборок объёма $n=100$, посчитаем по каждой среднее и сохраним средние в список `means100`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "means100 = []\n",
    "for i in range(100):\n",
    "    samp = st.norm.rvs(loc = 168, scale = 2, size = 100)\n",
    "    means100.append(samp.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Снова построим гистограмму для средних:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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iN30evT+aHOmez52h2v4auJ/elTU3A7umVNtlwL9usJ1Z6Ld+apt6vwFnA3/XfU+/AbxnzXZuBZ7RTT+H3g/hQ93yczNU2z91/Xao287ZU6htN/AgcJjeFSvPnpXxNkBtExtvXftNwO9ssPwf0btK5kHgteMab95+QJIaNEunZSRJY2K4S1KDDHdJapDhLkkNMtwlqUGGuyQ1yHCXpAb9D6qDhXkCjRiJAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(means100, color = \"firebrick\", edgecolor = \"white\", \n",
    "         range = (166, 170), bins = np.arange(166, 170, 0.1));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Тут уже разброс выборочных средних явно меньше (специально оставили те же границы значений для наглядности), выборочные средние более скучены вокруг значения 168, они лежат в интервале примерно от 167.5 до 168.5. То есть, максимальное отклонение выборочного среднего от среднего генеральной совокупности уже не 1.5, а примерно 0.5. Закон больших чисел работает! Чтобы окончательно в этом убедиться, возьмём выборки ещё большего размера. \n",
    "\n",
    "Сгенерируем 100 выборок объёма $n=1000$, посчитаем по каждой среднее и сохраним средние в список `means1000`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "means1000 = []\n",
    "for i in range(100):\n",
    "    samp = st.norm.rvs(loc = 168, scale = 2, size = 1000)\n",
    "    means1000.append(samp.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "И вновь строим гистограмму:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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LXZIK+h93mzcwoJFAgAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(means1000, color = \"firebrick\", edgecolor = \"white\", \n",
    "         range = (166, 170), bins = np.arange(166, 170, 0.1));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Гистограмма получилась довольно узкой – выборочные средние значения очень сильно сконцентрированы вокруг 168, разброс значений маленький, от 167.75 до 168.25. Другими словами, увеличив объём выборок ещё в десять раз, мы пришли к выборкам, средние значения которых будут отличаться от среднего генеральной совокупности 168 совсем на чуть-чуть, не больше, чем на 0.25. Закон больших чисел, действительно, работает, мы убедились в этом посредством небольшой серии компьютерных симуляций."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Центральная предельная теорема\n",
    "\n",
    "> Пусть у нас есть генеральная совокупность со средним значением $\\mu$ и стандартным отклонением $\\sigma$. Если мы извлечем из этой совокупности все возможные выборки достаточно большого размера $n$ ($n \\geqslant30$) и посчитаем по каждой выборке среднее значение, то эти средние значения будут примерно нормально распределены со средним $\\mu$ и стандартным отклонением $\\frac{\\sigma}{\\sqrt{n}}$.\n",
    "\n",
    "\n",
    "Почему эта теорема очень мощная и при этом прикладная? Распределение генеральной совокупности нам в реальной жизни часто неизвестно, оно может быть каким угодно, а вот с выборочными средними мы сталкиваемся постоянно. И эта теорема как раз говорит нам о том, что даже если мы не знаем вид распределения генеральной совокупности, про распределение средних мы всё равно знаем многое – оно всегда нормальное и с фиксированными параметрами.\n",
    "\n",
    "Давайте представим себе генеральную совокупность, распределение которой точно не похоже на нормальное. Возьмём какое-нибудь экзотическое [распределение Парето](https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%9F%D0%B0%D1%80%D0%B5%D1%82%D0%BE) (для социально-экономических наук, конечно, не экзотическое, просто мы его не встречали на других курсах, близко к экспоненциальному). Посмотрим, как выглядит плотность распределения Парето с коэффициентом масштаба равным 1 и $k = 3$ (специфические параметры распределения):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(start = 1, stop = 5, num = 1000)\n",
    "plt.plot(x, st.pareto.pdf(x, b = 3), color = \"navy\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Действительно, судя по плотности, на нормальное распределение Парето точно не похоже. А что будет, если мы из него извлечём много случайных выборок и изучим распределение выборочных средних? Генерировать выборки мы уже умеем – извлечём из распределения Парето (генеральной совокупности, имеющей такое распределение) 3000 выборок объёма $n = 900$ и сохраним среднее по каждой выборке в список `p_means`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# еще один способ – \n",
    "\n",
    "pmeans = []\n",
    "\n",
    "for i in range(3000):\n",
    "    samp = st.pareto.rvs(3, size = 900)\n",
    "    pmeans.append(samp.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Построим гистограмму для выборочных средних `pmeans`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(pmeans, color = \"firebrick\", edgecolor = \"white\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Магия! Может, не идеально нормальное распределение, но очень близко к нему. Центральная предельная теорема действует! Осталось проверить, правда ли, что среднее по набору выборочных средних близко к среднему генеральной совокупности, а стандартное отклонение по набору выборочных средних примерно равно стандартному отклонению генеральной совокупности, поделённому на корень из объёма выборок.\n",
    "\n",
    "Математическое ожидание распределения Парето с заданными параметрами равно 1.5, стандартное отклонение равно 0.87. Эти значения можно получить по свойствам и формулам, а можно извлечь из Python:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean: 1.5 Variance: 0.75 St Dev: 0.87\n"
     ]
    }
   ],
   "source": [
    "# моменты – математическое ожидание m и дисперсия v\n",
    "# для случайной величины с распределением Парето с k = 3\n",
    "\n",
    "mean, var = st.pareto.stats(3, moments = 'mv')\n",
    "print(\"Mean:\", mean, \"Variance:\", var, \"St Dev:\", np.sqrt(var).round(2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Проверим, чему равно среднее по массиву выборочных средних `pmeans`, по которому мы построили гистограмму выше:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.5004160213926232\n"
     ]
    }
   ],
   "source": [
    "print(np.mean(pmeans))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Почти 1.5! Проверим, чему равно стандартное отклонение:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.028291541267660973\n"
     ]
    }
   ],
   "source": [
    "print(np.std(pmeans))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Согласно центральной предельной теореме, это стандарное отклонение должно примерно совпадать со стандартным отклонением генеральной совокупности, делённым на корень из объёма выборки. Посчитаем и сравним (мы извлекали выборки с $n = 900$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.028867513459481287\n"
     ]
    }
   ],
   "source": [
    "print(np.sqrt(var) / np.sqrt(900))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Почти совпало! Центральная предельная теорема тоже работает! "
   ]
  }
 ],
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