drorata/divide-dy-n-1.ipynb

## divide-dy-n-1.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems to be a reappearing question: *what is the difference between [`pandas.std`](https://pandas.pydata.org/pandas-docs/stable/generated/pandas.DataFrame.std.html) and [`numpy.std`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.std.html)?*.\n",
    "See for example [this](https://stackoverflow.com/q/24984178/671013) SO thread.\n",
    "But what does it mean that Pandas uses the unbiased estimator and Numpy does not?\n",
    "In this post I would like to help you understand what is the difference and when which should be used.\n",
    "I will try to achieve this by following the paper *\"Why divide by $N-1$?\"* by J. L. M. Wilkins [(link)](http://www.soe.vt.edu/tandl/pdf/Wilkins/Publications_Wilkins_Why_divide_by_n_1.pdf).\n",
    "In particular instead of using archaic software, I will redo the examples using Numpy and Pandas.\n",
    "\n",
    "## Background\n",
    "\n",
    "What is the average and the standard deviation of height of humans?\n",
    "To answer this question, let us denote by $X$ the *whole* population of humanity (at the moment); each individual would be denoted by $X_i$.\n",
    "The average (or mean), denoted by $\\mu$ can be computed by $$\\mu = \\frac{\\sum_{i=1}^N X_i}{N}$$ where $N$ is the number of individuals.\n",
    "Next, the standard deviation is given by: $$\\sigma = \\sqrt{\\frac{\\sum_{i=1}^N (X_i - \\mu)^2}{N}}$$\n",
    "Recall that the variance is given and denoted by $\\sigma^2$; we will consider the variance from now on.\n",
    "\n",
    "The task at hand is, obviously, impossible to accomplish.\n",
    "What you are likely to do is pick a sample $X_{i(1)}, X_{i(2)}, \\dots, X_{i(n)}$, where $n < N$ and $i(k)$ is a permutation.\n",
    "For the sake of simplicity, let us denote them by $X_i$ where $i=1,\\dots,n$.\n",
    "Using this sample you will compute the sample mean, denoted by $\\bar{X}$ and the sample variance.\n",
    "\n",
    "The first part, that is, the mean of the sample, is easy: $$\\bar{X} = \\frac{\\sum_{j=1}^n X_j}{n}$$\n",
    "It turns out that $\\bar{X}$ is an *unbiased* estimator of $\\mu$.\n",
    "Intuitively saying that it estimates nicely the real mean $\\mu$.\n",
    "See for example [reference](https://en.wikipedia.org/wiki/Bias_of_an_estimator#cite_note-JohnsonWichern2007-1).\n",
    "\n",
    "Things become trickier when it comes to the sample's variance, denoted by $S^2$.\n",
    "One may try to compute $S^2$ by \"generalizing\" the formula for $\\sigma^2$.\n",
    "\n",
    "This would be denoted by $S_n^2$: $$S_n^2 = \\frac{\\sum_{i=1}^n(X_i - \\bar{X})^2}{n}$$\n",
    "However, this *is* biased estimator of $\\sigma^2$.\n",
    "The unbiased estimator is $$S^2 = \\frac{\\sum_{i=1}^n(X_i - \\bar{X})^2}{n-1}$$\n",
    "\n",
    "All this is explained nicely, and in a deeper way, in various places; even [Wikipedia](https://en.wikipedia.org/wiki/Bias_of_an_estimator) is a good start.\n",
    "An intuitive explanation can be found [here](https://stats.stackexchange.com/q/3931/54320).\n",
    "In this post, I will follow the examples of Wilkins which gives a better motivation that indeed $S_n^2$ is biased and $S^2$ is not."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Let the game begin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we start, let us stress that by default, `pandas.std` is equivalent to $S^2$ and `numpy.std` to $S_n^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('ggplot')\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "#\n",
    "# In general tqdm is great and I strongly suggest to try it out, if you run this notebook\n",
    "# locally. However, it doesn't render well in the post generated from this notebook. So I define\n",
    "# A dummy function which does nothing. \n",
    "#\n",
    "# from tqdm import tqdm_notebook as tqdm\n",
    "def tqdm(X):\n",
    "    return X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start with a population $a$ consisting of the numbers $1$ to $100$ (inclusive)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a = np.arange(1,101)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The mean $\\mu$ is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50.5"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we compute $\\sigma^2$; we use the default behavior of `np.std`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "833.25"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(a)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can double check that indeed Numpy delivers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "833.25"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(np.power(a - np.mean(a), 2)) / len(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default, Pandas will fail:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "841.66666666666663"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.power(pd.Series(a).std(), 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But this can be fixed:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "833.25"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.power(pd.Series(a).std(ddof=0), 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Great, this all aligns well with the documentation and the definitions.\n",
    "In the sequence of values you have covers the *WHOLE* of the population in question, you should either use `np.std` or `pd.std(ddof=2)`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sample the whole population\n",
    "\n",
    "Time to play the game.\n",
    "As we cannot have all values for all the individuals in our population, we have to use a sample.\n",
    "We will now do it for various sample sizes.\n",
    "At each step we will compute $S^2$ and $S_n^2$ and compare the results.\n",
    "We take sample sizes of $n = 5, 10, 25, 50$ and $100$.\n",
    "For each sample size, we compute $15000$ times $S^2$ and $S_n^2$.\n",
    "Lastly, we average the results for each sample size."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "SIZES = [5, 10, 25, 50, 100]\n",
    "N=15000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Sn_s = []\n",
    "for SIZE in tqdm(SIZES):\n",
    "    size_res = []\n",
    "    for i in tqdm(range(N)):\n",
    "        size_res.append(np.std(np.random.choice(a, size=SIZE))**2)\n",
    "    Sn_s.append(np.sum(np.array(size_res)) / N)\n",
    "Sn_s = pd.Series(Sn_s, index=SIZES)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "S_s = []\n",
    "for SIZE in tqdm(SIZES):\n",
    "    size_res = []\n",
    "    for i in tqdm(range(N)):\n",
    "        size_res.append(np.std(np.random.choice(a, size=SIZE), ddof=1)**2)\n",
    "    S_s.append(np.sum(np.array(size_res)) / N)\n",
    "\n",
    "S_s = pd.Series(S_s, index=SIZES)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's collect the results into a table."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>S_n^2</th>\n",
       "      <th>S^2</th>\n",
       "      <th>sigma^2</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>665.030288</td>\n",
       "      <td>837.021747</td>\n",
       "      <td>833.25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>748.445851</td>\n",
       "      <td>836.054854</td>\n",
       "      <td>833.25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>800.563841</td>\n",
       "      <td>832.693469</td>\n",
       "      <td>833.25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>816.226962</td>\n",
       "      <td>830.447212</td>\n",
       "      <td>833.25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>100</th>\n",
       "      <td>825.145818</td>\n",
       "      <td>833.501119</td>\n",
       "      <td>833.25</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          S_n^2         S^2  sigma^2\n",
       "5    665.030288  837.021747   833.25\n",
       "10   748.445851  836.054854   833.25\n",
       "25   800.563841  832.693469   833.25\n",
       "50   816.226962  830.447212   833.25\n",
       "100  825.145818  833.501119   833.25"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res_df = pd.concat([Sn_s, S_s], axis=1, keys=['S_n^2', 'S^2'])\n",
    "res_df['sigma^2'] = pd.Series(np.std(a) ** 2 * np.ones(res_df.shape[0]), index=res_df.index)\n",
    "res_df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ed2a6a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "res_df.plot(figsize=(20,5), marker='o')\n",
    "plt.xticks(res_df.index);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall that $\\sigma^2 = 833.25$.\n",
    "It is clear that the $n-1$ normalization yields better approximation of the variance of the population.\n",
    "This approximation is already fairly accurate when using mere $5\\%$ of the population as a sample.\n",
    "\n",
    "## Summary\n",
    "\n",
    "So, if you know that the series of values that you have represents *ALL* members of your population, you should go ahead and use $\\sigma^2$ (using either `np.std` or `pandas.std(ddof=0)`).\n",
    "If however, your series of values is merely a sample of a bigger population, then you should use the unbiased estimator of $\\sigma^2$, denoted by $S^2$ and computed by either `np.std(ddof=1)` or `pandas.std`.\n",
    "\n",
    "## An experiment\n",
    "\n",
    "As a mathematician, I now ask myself how would estimators with different $n$ would behave.\n",
    "Let $$D_n^2 = \\frac{\\sum_{i=1}^n(X_i - \\bar{X})^2}{n}$$\n",
    "In particular, $D_n^2 = S_n^2$ and $D_{n-1}^2 = S^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "res = []\n",
    "for n in tqdm(np.arange(-4,5)):\n",
    "    Dn_s = []\n",
    "    for SIZE in tqdm(SIZES):\n",
    "        size_res = []\n",
    "        for i in tqdm(range(N)):\n",
    "            size_res.append(np.std(np.random.choice(a, size=SIZE), ddof=n)**2)\n",
    "        Dn_s.append(np.sum(np.array(size_res)) / N)\n",
    "    Dn_s = pd.Series(Dn_s, index=SIZES, name=str(n))\n",
    "    res.append(Dn_s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ieov729Ouu+463H777Ugmk8jn85iensYll1yCL33pSxWPa8S3p7VMaFS65z7k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\nZF+Cda2mzy9EtHAMjVaa0HAnZIlWpSVYYaHQQsM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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11205a2b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "df_res = pd.DataFrame(res).T\n",
    "df_res.plot(logy=True, figsize=(20,5))\n",
    "plt.xticks(df_res.index);"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [conda root]",
   "language": "python",
   "name": "conda-root-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}