From ca0b6db1df931043e78b46f467d9299f8e8042d0 Mon Sep 17 00:00:00 2001 From: m1nce Date: Fri, 17 May 2024 22:48:38 -0700 Subject: [PATCH] added permutation test LaTeX rendering needs to be fixed --- docs/statistical-inference/permutation.md | 118 + docusaurus.config.js | 16 +- package-lock.json | 243 +- package.json | 7 +- .../permutation-test.png | Bin 0 -> 14107 bytes static/katex.tar | Bin 0 -> 3143680 bytes static/katex/README.md | 125 + static/katex/contrib/auto-render.js | 341 + static/katex/contrib/auto-render.min.js | 1 + static/katex/contrib/auto-render.mjs | 244 + static/katex/contrib/copy-tex.css | 14 + static/katex/contrib/copy-tex.js | 127 + static/katex/contrib/copy-tex.min.css | 1 + static/katex/contrib/copy-tex.min.js | 1 + static/katex/contrib/copy-tex.mjs | 105 + static/katex/contrib/mathtex-script-type.js | 112 + .../katex/contrib/mathtex-script-type.min.js | 1 + static/katex/contrib/mathtex-script-type.mjs | 24 + static/katex/contrib/mhchem.js | 3216 +++ 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/dev/null +++ b/docs/statistical-inference/permutation.md @@ -0,0 +1,118 @@ +--- +sidebar_position: 2 +sidebar_label: Permutation Testing +title: Permutation Testing +--- + +import DataFrameComponent from '../../components/DataFrameComponent.jsx'; + +We use a permutation test to test whether two samples were drawn from the same population. + +--- + +## 1. State the question/hypothesis + +Our pair of hypotheses is: +* **Null Hypothesis:** The mean weights of dogs and cats are the *same*. +* **Alternative Hypothesis:** The mean weights of dogs and cats are *different*. + +Since the alternative hypothesis is of the form "A and B are different," the test statistic should measure distance and use an absolute value. + +$\therefore$ Use **absolute difference in group means** as the test statistic. + +--- + +## 2. Query the DataFrame + +Since we want to compare the distributions of only cats and dogs, we need to make sure to only include the relevant pieces of data (e.g., cats and dogs weights). + +```python +# Create a boolean Series that references which rows fulfill either condition. +querying_condition = (full_pets.get('Species') == 'dog') | (full_pets.get('Species') == 'cat') +# Query. +cats_dogs = full_pets[querying_condition] +# Display the first 5 rows only. +cats_dogs.take(np.arange(5)) +``` + + + +--- + +## 3. Create a function to calculate test statistic + +Since our hypotheses depend on the test statistic, create a function to be able to calculate it during every trial of our permutation test. + +```python +def difference_in_means(cats_dogs): + """ + Calculate the absolute difference in the mean weight of dogs and cats. + --- + Input: + cats_dogs: a DataFrame containing the columns 'Species' and 'Weight'. + --- + Output: + The absolute difference in the mean weight of dogs and cats. + """ + means = cats_dogs.groupby('Species').mean() + return np.abs((means.get('Weight').loc['dog'] - means.get('Weight').loc['cat'])) +``` + +*Note: Although we can simply do this in the function, we can practice good coding habits by separating our code into readable bits!* + +--- + +## 4. Simulate the permutation test + +```python +n = 500 # Number of simulations. +statistics = np.array([]) # Array to keep track of the difference in means for each iteration. +for i in np.arange(n): # Run the simulation `n` number of times + # 1. Shuffle the species. + shuffled = cats_dogs.assign(Species=np.random.permutation(cats_dogs.get('Species'))) + + # 2. Compute the test statistic. + statistic = difference_in_means(shuffled) + + # 3. Save the result. + statistics = np.append(statistics, statistic) +``` + +This code will run the permutation test 500 times, but a different reasonable number can be used instead. It is **crucial** to keep track of the difference in means each time our for-loop runs so that the number of simulated values can be displayed. + +--- + +## 5. Conclusion + +```python +observed = difference_in_means(cats_dogs) +p_value = np.count_nonzero(statistics >= observed) / n + +print("The observed value of the test statistic is:", observed) +print("The p-value is:", p_value) +``` +**The observed value of the test statistic is: 30.361111111111107**
+**The p-value is: 0.004** + +Using a significance level of 0.05... + +**Conclusion:** +* Under the null hypothesis, we rarely see a difference greater than the observed value. +* Therefore, we reject the null hypothesis: the evidence implies that the two groups do not come from the same distribution. +* Still, we cannot conclude that species causes a different weight because there may be other factors at play. + +--- + +## 6. Extra + +Let's see how our observed statistic compares to the overall simulated values! + +```python +# Create the histogram. +bpd.DataFrame().assign(statistics=statistics).plot(kind='hist', bins=20, density=True, ec='w') +# Don't worry about this line - you won't need to know it for this course! +plt.axvline(x=observed, c='black', linewidth=4, label='population difference in means') +``` +![Distribution](/img/statistical-inference-plots/permutation-test.png) + +From this graph, we can tell that there is barely any data to the **right** of the black vertical line, meaning we have a very low p-value! \ No newline at end of file diff --git a/docusaurus.config.js b/docusaurus.config.js index 02bcfb12..2e58020a 100644 --- a/docusaurus.config.js +++ b/docusaurus.config.js @@ -5,6 +5,8 @@ // See: https://docusaurus.io/docs/api/docusaurus-config import {themes as prismThemes} from 'prism-react-renderer'; +const math = require('remark-math') +const katex = require('rehype-katex') /** @type {import('@docusaurus/types').Config} */ const config = { @@ -20,7 +22,7 @@ const config = { // GitHub pages deployment config. // If you aren't using GitHub pages, you don't need these. - organizationName: 'dsc-courses', // Usually your GitHub org/user name. + organizationName: 'dsc-courses', // Usually your GitHub org/username. projectName: 'bpd-reference', // Usually your repo name. onBrokenLinks: 'throw', @@ -44,6 +46,8 @@ const config = { ({ docs: { sidebarPath: './sidebars.js', + remarkPlugins: [math], + rehypePlugins: [katex], // Please change this to your repo. // Remove this to remove the "edit this page" links. editUrl: @@ -56,6 +60,16 @@ const config = { ], ], + stylesheets: [ + { + href: '/katex/katex.min.css', + type: 'text/css', + integrity: + 'sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X', + crossorigin: 'anonymous', + }, + ], + themeConfig: /** @type {import('@docusaurus/preset-classic').ThemeConfig} */ ({ diff --git a/package-lock.json b/package-lock.json index b0bae5dc..6d538324 100644 --- a/package-lock.json +++ b/package-lock.json @@ -9,13 +9,16 @@ "version": "0.0.0", "dependencies": { "@docusaurus/core": "3.2.1", - "@docusaurus/preset-classic": "3.2.1", + "@docusaurus/preset-classic": "^3.2.1", "@docusaurus/theme-live-codeblock": "^3.2.1", "@mdx-js/react": "^3.0.0", "clsx": "^2.0.0", + "katex": "^0.16.10", "prism-react-renderer": "^2.3.0", "react": "^18.0.0", - "react-dom": "^18.0.0" + "react-dom": "^18.0.0", + "rehype-katex": "^7.0.0", + "remark-math": "^6.0.0" }, "devDependencies": { "@docusaurus/module-type-aliases": "3.2.1", @@ -3654,6 +3657,11 @@ "resolved": "https://registry.npmjs.org/@types/json-schema/-/json-schema-7.0.15.tgz", "integrity": "sha512-5+fP8P8MFNC+AyZCDxrB2pkZFPGzqQWUzpSeuuVLvm8VMcorNYavBqoFcxK8bQz4Qsbn4oUEEem4wDLfcysGHA==" }, + "node_modules/@types/katex": { + "version": "0.16.7", + "resolved": "https://registry.npmjs.org/@types/katex/-/katex-0.16.7.tgz", + "integrity": 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