mirror of
https://github.com/TeachBooks/manual.git
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245 lines
6.8 KiB
Text
245 lines
6.8 KiB
Text
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"tags": []
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},
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"source": [
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"# Programming assignment\n",
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"\n",
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"This page shows how you can convert the classical Jupyter Notebook assignment to a jupyter book format with live coding. A widget has been implemented to check the answers, of which the data is included in a separate python file. The page automatically loads and a custom download page has been added\n",
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"\n",
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"```{custom_download_link} ./example_gumbel_stripped.ipynb\n",
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":replace_default: \"True\"\n",
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"```"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"tags": []
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},
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"source": [
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"## Gumbel Distribution Exercise\n",
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"\n",
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"Imagine you are concerned with the concentration of an airborne contaminant, $X$, measured in ppm. A previous benchmark study estimates that there is a 10% and 1% chance, respectively, of exceeding 4 and 10 ppm, respectively. You have been asked by the regulatory agency to estimate the contaminant concentration with 0.1% probability of being exceeded. Prior studies suggest that a Gumbel distribution can be used to model contaminant concentration, given by the CDF:\n",
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"\n",
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"$$\n",
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"F_X(x) = \\exp{\\Bigg( -\\exp{\\Big(-\\frac{x-\\mu}{\\beta}\\Big)}\\Bigg)}\n",
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"$$\n",
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"\n",
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"Using the cell blocks below as a guide (and also to check your analysis): Task 1) find the parameters of a Gumbel distribution that matches the information provided above, then, Tasks 2-3) use it to estimate the concentration with exceedance probability 0.1%.\n",
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"\n",
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"To complete this assignment, you can use `numpy`, `matplotlib` and from the `math` library, `log` and `e` (these are imported for you when you inialize the notebook)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"tags": [
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"thebe-remove-input-init"
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]
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},
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"outputs": [],
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"source": [
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"import matplotlib.pyplot as plt\n",
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"from math import log, e\n",
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"import numpy as np\n",
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"from example_gumbel import check_example"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**Task 0:** fill in the appropriate fitting points:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"tags": [
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"auto-execute-page"
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]
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},
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"outputs": [],
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"source": [
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"x_1 = _\n",
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"p_1 = _\n",
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"x_2 = _\n",
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"p_2 = _"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**Task 1:** derive the distribution parameters:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"def gumbel_2_points(x_1, p_1, x_2, p_2): \n",
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" \"\"\"Compute Gumbel distribution parameters from two points of the CDF.\n",
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" Arguments:\n",
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" x_1 (float): point one\n",
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" p_1 (float): cumulative probability for point 1\n",
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" x_2 (float): point two\n",
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" p_2 (float): cumulative probability for point 2\n",
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" \"\"\"\n",
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"\n",
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" # YOUR CODE GOES HERE #\n",
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" beta = _\n",
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" mu = _\n",
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" #######################\n",
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"\n",
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" return mu, beta"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The cells below will print your parameter values and create a plot to help confirm you have the right implementation."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"tags": [
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"thebe-remove-input-init"
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]
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},
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"outputs": [],
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"source": [
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"def plot_distribution(mu, beta,\n",
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" x_1, p_1, x_2, p_2):\n",
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" plt.title(\"Gumbel Distribution, $1-F_X(x)$\")\n",
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" plt.xlabel(\"Contaminant Concentration, $X$ [ppm]\")\n",
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" plt.ylabel(\"Exceedance Probability [--]\")\n",
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" plt.grid(color='black', linestyle='-', linewidth=0.3)\n",
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" x_axis = np.arange(0, 20, 0.1)\n",
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" plt.plot(x_axis, np.vectorize(gumbel_distribution)(x_axis), linewidth=2)\n",
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" plt.plot(x_1, 1 - p_1, 'ro')\n",
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" plt.annotate(\"Point 1\", (x_1 + 0.4, 1 - p_1 + 0.02))\n",
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" plt.plot(x_2, 1 - p_2, 'ro')\n",
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" plt.annotate(\"Point 2\", (x_2 + 0.4, 1 - p_2 + 0.001))\n",
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" plt.yscale(\"log\") \n",
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" plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"mu, beta = gumbel_2_points(x_1, p_1, x_2, p_2)\n",
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"\n",
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"print(f\"Your mu: {mu:0.5f}\\nYour beta: {beta:0.5f}\")\n",
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"\n",
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"gumbel_distribution = lambda x: 1 - e**(-e**(-(x - mu)/beta))\n",
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"\n",
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"plot_distribution(mu, beta, x_1, p_1, x_2, p_2)\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**Task 2:** write a function to solve for the random variable value (it will be tested for you with the Check answer button, along with the distribution parameters)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"def find_x_with_probability_p(p): \n",
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" \"\"\" Compute point in the gumbel distribution for which the CDF is p\n",
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" Use the variables mu and beta defined above!\n",
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" Hint: they have been defined globally, so you don't need to \n",
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" include them as arguments.\n",
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" \"\"\"\n",
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" # YOUR CODE GOES HERE #\n",
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" x = _\n",
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" #######################\n",
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"\n",
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" return x"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**Task 3:** use the function `find_x_with_probability_p` to evaluate the random variable value with exceedance probabiltiy of 0.001."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# YOUR CODE GOES HERE #\n",
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"x = _\n",
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"#######################\n",
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"\n",
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"print(f\"The value of x with probability of exceedance 0.001 os {x:0.5f}\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"tags": [
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"thebe-remove-input-init"
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]
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},
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"outputs": [],
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"source": [
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"check_example(globals())"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.9.18"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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