RiskEngineering notebooks

basic-statistics.ipynb

coins-dice.ipynb

correlations.ipynb

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   "source": [
    "<img src=\"https://risk-engineering.org/static/img/logo-RE.png\" width=\"100\" alt=\"\" style=\"float:right;margin:15px;\">\n",
    "\n",
    "This notebook is an element of the [risk-engineering.org courseware](https://risk-engineering.org/). It can be distributed under the terms of the [Creative Commons Attribution-ShareAlike licence](https://creativecommons.org/licenses/by-sa/4.0/).\n",
    "\n",
    "Author: Eric Marsden <eric.marsden@risk-engineering.org>.\n",
    "\n",
    "---\n",
    "\n",
    "In this notebook, we illustrate NumPy features for working with correlated data. Check the [associated lecture slides](https://risk-engineering.org/correlation/) for background material and to download this content as a Jupyter/Python notebook. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear correlation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats\n",
    "plt.style.use(\"bmh\")\n",
    "%config InlineBackend.figure_formats=[\"svg\"]"
   ]
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  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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    "X = numpy.random.normal(10, 1, 100)\n",
    "X"
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   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "Y = -X + numpy.random.normal(0, 1, 100)"
   ]
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  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "plt.scatter(X, Y);"
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   "metadata": {},
   "source": [
    "Looking at the scatterplot above, we can see that the random variables $X$ and $Y$ are *correlated*. There are various statistical measures that allow us to quantify the degree of linear correlation. The most commonly used is [Pearson’s product-moment correlation coefficient](https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient). It is available in `scipy.stats`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.7136727388061208, 7.877150437410328e-17)"
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     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "scipy.stats.pearsonr(X, Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first return value is the linear correlation coefficient, a value between -1 and 1 which measures the strength of the linear correlation. A value greater than 0.9 indicates a strong positive linear correlation, and a value lower than -0.9 indicates strong negative linear correlation (when $X$ increases, $Y$ decreases).\n",
    "\n",
    "(The second return value is a *p-value*, which is a measure of the confidence which can be placed in the estimation of the correlation coefficient (smaller = more confidence). It tells you the probability of an uncorrelated system producing datasets that have a Pearson correlation at least as extreme as the one computed from these datasets. Here we have a very very low p-value, so high confidence in the estimated value of the correlation coefficient.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: show that when the error in $Y$ decreases, the correlation coefficient increases."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: produce data and a plot with a negative correlation coefficient."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Anscombe’s quartet"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let’s examine four datasets produced by the statistician [Francis Anscombe](https://en.wikipedia.org/wiki/Frank_Anscombe) to illustrate the importance of exploring your data qualitatively (for example by plotting the data), rather than relying only on summary statistics such as the linear correlation coefficient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "x =  numpy.array([10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5])\n",
    "y1 = numpy.array([8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68])\n",
    "y2 = numpy.array([9.14, 8.14, 8.74, 8.77, 9.26, 8.10, 6.13, 3.10, 9.13, 7.26, 4.74])\n",
    "y3 = numpy.array([7.46, 6.77, 12.74, 7.11, 7.81, 8.84, 6.08, 5.39, 8.15, 6.42, 5.73])\n",
    "x4 = numpy.array([8,8,8,8,8,8,8,19,8,8,8])\n",
    "y4 = numpy.array([6.58,5.76,7.71,8.84,8.47,7.04,5.25,12.50,5.56,7.91,6.89])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
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    "plt.plot(x, y1, \"ks\", marker=\"o\")\n",
    "plt.title(\"Anscombe quartet n°1\")\n",
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   "metadata": {},
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    "plt.plot(x, y2, \"ks\", marker=\"o\")\n",
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   "source": [
    "plt.plot(x, y3, \"ks\", marker=\"o\")\n",
    "plt.title(\"Anscombe quartet n°3\")\n",
    "plt.margins(0.1)"
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   "execution_count": 12,
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "dark"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x4, y4, \"ks\", marker=\"o\")\n",
    "plt.title(\"Anscombe quartet n°4\")\n",
    "plt.margins(0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.8165214368885028, 0.0021646023471972235)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scipy.stats.pearsonr(x4, y4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that the linear correlation coefficient (Pearson's $r$) of the four datasets is identical, though clearly the relationship between $X$ and $Y$ is very different in each case! This illustrates the risks of depending only on quantitative descriptors to understand your datasets: you should also use different types of plots to give you a better overview of the data."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Datasaurus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The [Datasaurus](http://www.thefunctionalart.com/2016/08/download-datasaurus-never-trust-summary.html) provides another illustration of the importance of plotting your data to make sure it doesn't contain any surprises, rather than relying only on summary statistics. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "  <thead>\n",
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       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>142.000000</td>\n",
       "      <td>142.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>54.263273</td>\n",
       "      <td>47.832253</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>16.765142</td>\n",
       "      <td>26.935403</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>22.307700</td>\n",
       "      <td>2.948700</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>44.102600</td>\n",
       "      <td>25.288450</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>53.333300</td>\n",
       "      <td>46.025600</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>64.743600</td>\n",
       "      <td>68.525675</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>98.205100</td>\n",
       "      <td>99.487200</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                0           1\n",
       "count  142.000000  142.000000\n",
       "mean    54.263273   47.832253\n",
       "std     16.765142   26.935403\n",
       "min     22.307700    2.948700\n",
       "25%     44.102600   25.288450\n",
       "50%     53.333300   46.025600\n",
       "75%     64.743600   68.525675\n",
       "max     98.205100   99.487200"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas\n",
    "\n",
    "ds = pandas.read_csv(\"https://risk-engineering.org/static/data/datasaurus.csv\", header=None)\n",
    "ds.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.06447185270095164, 0.44589659802470616)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scipy.stats.pearsonr(ds[0], ds[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These summary statistics don't look too nasty, but check out the data once it's plotted!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
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     },
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   ],
   "source": [
    "plt.scatter(ds[0], ds[1]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[This article](https://www.autodeskresearch.com/publications/samestats) titled *Same Stats, Different Graphs: Generating Datasets with Varied Appearance and Identical Statistics through Simulated Annealing* runs a lot further with this concept."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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