RiskEngineering notebooks

basic-statistics.ipynb

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   "source": [
    "# Basic statistics"
   ]
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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 free [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 <[email protected]>. \n",
    "\n",
    "---\n",
    "\n",
    "This notebook contains an introduction to use of Python and the NumPy library for basic statistical calculations.\n",
    "See the [associated course materials](https://risk-engineering.org/statistical-modelling/) for background information and to download this content as a Jupyter notebook.\n",
    "\n",
    "We start by importing the numpy library, which makes it possible to use functions and variables from the library, prefixed by `numpy`."
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   "source": [
    "import numpy"
   ]
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   "source": [
    "We can use Python as simple interactive calculator:"
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   "source": [
    "Here we call the `sqrt` function from the numpy library."
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       "2.0"
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   "source": [
    "numpy.sqrt(2 + 2)"
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   "source": [
    "Some useful constants are predefined."
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   "source": [
    "numpy.pi"
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       "1.2246467991473532e-16"
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     "execution_count": 5,
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   "source": [
    "numpy.sin(numpy.pi)"
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   "source": [
    "The notation `e-16` above means $10^{-16}$; the number above is very very small (it’s a numerical approximation to the mathematical answer of zero)."
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    "We can generate a random number from a uniform distribution between 20 and 30. If you evaluate this several times (in most Jupyter interfaces, press `Shift-Enter` or press on the `Run` button in the toolbar above), it will generate a different random number each time."
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    "numpy.random.uniform(20, 30)"
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    "We can generate an **array** of random numbers by passing a third argument to the `numpy.random.uniform` function, saying how many random numbers we want. We store the array in a *variable* named `obs`."
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       "array([25.11043932, 25.88812475, 23.23260568, 29.27667097, 28.66875271,\n",
       "       20.76826133, 28.55084839, 23.99177767, 27.64333244, 22.33271169])"
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   "source": [
    "obs = numpy.random.uniform(20, 30, 10)\n",
    "obs"
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   "source": [
    "The builtin function `len` in Python tells us the length of an array or a list."
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   "source": [
    "len(obs)"
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   "source": [
    "We can do arithmetic on arrays, adding them together or subtracting a constant from each element."
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       "array([50.22087865, 51.7762495 , 46.46521136, 58.55334195, 57.33750542,\n",
       "       41.53652267, 57.10169679, 47.98355533, 55.28666488, 44.66542338])"
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    "obs + obs"
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       "array([ 0.11043932,  0.88812475, -1.76739432,  4.27667097,  3.66875271,\n",
       "       -4.23173867,  3.55084839, -1.00822233,  2.64333244, -2.66728831])"
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   "source": [
    "obs - 25"
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   "source": [
    "We can apply a numpy function to all the elements of an array. "
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   "execution_count": 12,
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       "array([5.01103176, 5.08803742, 4.82002134, 5.41079208, 5.35432094,\n",
       "       4.55722079, 5.34329939, 4.89814023, 5.25769269, 4.72574985])"
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   "source": [
    "numpy.sqrt(obs)"
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   "source": [
    "The array has *methods*, a kind of function that acts on the array."
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   "source": [
    "obs.mean()"
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   "execution_count": 14,
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   "source": [
    "obs.sum()"
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   "cell_type": "code",
   "execution_count": 15,
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       "20.76826133300001"
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     "execution_count": 15,
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   "source": [
    "obs.min()"
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   "source": [
    "There are similar functions in the `numpy` library that take an array as argument:"
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   "execution_count": 16,
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   "source": [
    "numpy.mean(obs)"
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   "execution_count": 17,
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    "numpy.sum(obs)"
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   "execution_count": 18,
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    "numpy.min(obs)"
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   "source": [
    "## Simple plotting"
   ]
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   "source": [
    "The matplotlib library allows you to generate many types of plots and statistical graphs in a convenient way. The [online gallery](https://matplotlib.org/gallery.html) shows the variety of plots available, and the [documentation](https://matplotlib.org/contents.html) is also available online. We import the `pyplot` component of matplotlib and give it an alias `plt`. "
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   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.style.use(\"bmh\")  # this affects the style (colors etc.) of plots\n",
    "%config InlineBackend.figure_formats=[\"svg\"]"
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    "X = numpy.random.uniform(20, 30, 10)\n",
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    "We can add two vectors together, assuming that all their dimensions are identical. Our array $x$ has one dimension of size 100. We can add another random vector of size 100 to it, containing numbers drawn from a uniform probability distribution between -0.1 and 0.1 (these represent some random “noise” which is added to our sine curve)."
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    "x = numpy.linspace(0, 10, 100)\n",
    "obs = numpy.sin(x) + numpy.random.uniform(-0.1, 0.1, 100)\n",
    "plt.plot(x, obs);"
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    "The **central limit theorem** states that the sum of a number of independent random variables tends toward a normal distribution even if the original variables themselves are not normally distributed. We illustrate this result by examining the distribution of the sums of 1000 realizations of a uniformly distributed random variable, plotting the distribution as a histogram. "
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   ],
   "source": [
    "N = 10000\n",
    "sim = numpy.zeros(N)\n",
    "for i in range(N):\n",
    "    sim[i] = numpy.random.uniform(30, 40, 100).sum()\n",
    "plt.hist(sim, bins=40, alpha=0.5, density=True);"
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coins-dice.ipynb

correlations.ipynb

Poisson-processes.ipynb

Semmelweis-fever.ipynb

data-analysis-earthquakes.ipynb

data-analysis-speed-light.ipynb

data-analysis-weather.ipynb

dike-case-study.ipynb

distributions.ipynb

elementary-effects.ipynb

exercise-descriptive-statistics.ipynb

exercises-reliability-engineering.ipynb

helmet-data-bootstrapping.ipynb

intervention-analysis-hierarchical-model.ipynb

monte-carlo-hoop-strength.ipynb

monte-carlo-hydrogen-tank-failure-rust.ipynb

monte-carlo-hydrogen-tank-failure.ipynb

monte-carlo-integration.ipynb

monte-carlo-LHS.ipynb

monte-carlo-pi.ipynb

monte-carlo-project-risk.ipynb

monte-carlo-reliability.ipynb

monte-carlo-slope-stability.ipynb

poker.ipynb

regression-CCPP.ipynb

regression-piecewise-linear.ipynb

regression-smoking.ipynb

saint-petersburg.ipynb

sensitivity-analysis.ipynb

simple-descriptive-statistics.ipynb

stock-market.ipynb

uncertainty-propagation.ipynb