From 4a0bf3838be97f8d2499fa6fbc7484caeadc36a0 Mon Sep 17 00:00:00 2001 From: Sam Lau Date: Mon, 27 May 2024 21:47:05 -0700 Subject: [PATCH] Update 05-27 --- resources/lectures/lec16/data/diabetes.csv | 769 + resources/lectures/lec16/dsc80_utils.py | 87 + .../lectures/lec16/imgs/dtree-basic-plot.svg | 22 + resources/lectures/lec16/imgs/dtree-basic.svg | 17 + resources/lectures/lec16/imgs/k-fold.png | Bin 0 -> 69443 bytes resources/lectures/lec16/imgs/ml-taxonomy.svg | 59 + .../lectures/lec16/imgs/train-test-first.png | Bin 0 -> 18425 bytes .../lectures/lec16/imgs/train-test-val.png | Bin 0 -> 31340 bytes resources/lectures/lec16/imgs/tt-errors.png | Bin 0 -> 207420 bytes resources/lectures/lec16/lec16-live.html | 11197 ++++++++ resources/lectures/lec16/lec16-live.ipynb | 22968 ++++++++++++++++ resources/lectures/lec16/lec16.html | 11197 ++++++++ resources/lectures/lec16/lec16.ipynb | 22968 ++++++++++++++++ resources/lectures/lec16/lec16_util.py | 127 + resources/lectures/lec16/rise.css | 77 + 15 files changed, 69488 insertions(+) create mode 100644 resources/lectures/lec16/data/diabetes.csv create mode 100644 resources/lectures/lec16/dsc80_utils.py create mode 100644 resources/lectures/lec16/imgs/dtree-basic-plot.svg create mode 100644 resources/lectures/lec16/imgs/dtree-basic.svg create mode 100644 resources/lectures/lec16/imgs/k-fold.png create mode 100644 resources/lectures/lec16/imgs/ml-taxonomy.svg create mode 100644 resources/lectures/lec16/imgs/train-test-first.png create mode 100644 resources/lectures/lec16/imgs/train-test-val.png create mode 100644 resources/lectures/lec16/imgs/tt-errors.png create mode 100644 resources/lectures/lec16/lec16-live.html create mode 100644 resources/lectures/lec16/lec16-live.ipynb create mode 100644 resources/lectures/lec16/lec16.html create mode 100644 resources/lectures/lec16/lec16.ipynb create mode 100644 resources/lectures/lec16/lec16_util.py create mode 100644 resources/lectures/lec16/rise.css diff --git 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Use `make +setup-lec` to copy this (and custom-rise-styles.css) to the lecture folders. + +Usage: + +from dsc80_utils import * +""" +import pandas as pd +import numpy as np +import matplotlib.pyplot as plt +import seaborn as sns +from matplotlib_inline.backend_inline import set_matplotlib_formats +from IPython.display import display, IFrame, HTML + +import plotly +import plotly.figure_factory as ff +import plotly.graph_objects as go +import plotly.express as px +from plotly.subplots import make_subplots +import plotly.io as pio +pio.renderers.default = "notebook" + +# DSC 80 preferred styles +pio.templates["dsc80"] = go.layout.Template( + layout=dict( + margin=dict(l=30, r=30, t=30, b=30), + autosize=True, + width=600, + height=400, + xaxis=dict(showgrid=True), + yaxis=dict(showgrid=True), + title=dict(x=0.5, xanchor="center"), + ) +) +pio.templates.default = "simple_white+dsc80" + +set_matplotlib_formats("svg") +sns.set_context("poster") +sns.set_style("whitegrid") +plt.rcParams["figure.figsize"] = (10, 5) + +# display options for numpy and pandas +np.set_printoptions(threshold=20, precision=2, suppress=True) +pd.set_option("display.max_rows", 7) +pd.set_option("display.max_columns", 8) +pd.set_option("display.precision", 2) + +# Use plotly as default plotting engine +pd.options.plotting.backend = "plotly" + + +def display_df( + df, rows=pd.options.display.max_rows, cols=pd.options.display.max_columns +): + """Displays n rows and cols from df""" + with pd.option_context( + "display.max_rows", rows, "display.max_columns", cols + ): + display(df) + + +def dfs_side_by_side(*dfs): + """ + Displays two or more dataframes side by side. + """ + display( + HTML( + f""" +
+ {''.join(df.to_html() for df in dfs)} +
+ """ + ) + ) + +from pathlib import Path + +# Used for plotting examples. +def create_kde_plotly(df, group_col, group1, group2, vals_col, title=''): + fig = ff.create_distplot( + hist_data=[df.loc[df[group_col] == group1, vals_col], df.loc[df[group_col] == group2, vals_col]], + group_labels=[group1, group2], + show_rug=False, show_hist=False, + colors=['#ef553b', '#636efb'], + ) + return fig.update_layout(title=title) \ No newline at end of file diff --git a/resources/lectures/lec16/imgs/dtree-basic-plot.svg b/resources/lectures/lec16/imgs/dtree-basic-plot.svg new file mode 100644 index 0000000..611d1d1 --- /dev/null +++ b/resources/lectures/lec16/imgs/dtree-basic-plot.svg @@ -0,0 +1,22 @@ + + + + + + + + + + + + + + + + + + + + + + diff --git a/resources/lectures/lec16/imgs/dtree-basic.svg b/resources/lectures/lec16/imgs/dtree-basic.svg new file mode 100644 index 0000000..eb06485 --- /dev/null +++ b/resources/lectures/lec16/imgs/dtree-basic.svg @@ -0,0 +1,17 @@ + + + + + + + + + + + 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + diff --git a/resources/lectures/lec16/lec16-live.ipynb b/resources/lectures/lec16/lec16-live.ipynb new file mode 100644 index 0000000..c99f330 --- /dev/null +++ b/resources/lectures/lec16/lec16-live.ipynb @@ -0,0 +1,22968 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "id": "8c8a44b8", + "metadata": { + "slideshow": { + "slide_type": "skip" + } + }, + "outputs": [], + "source": [ + "from dsc80_utils import *\n", + "import lec16_util as util" + ] + }, + { + "cell_type": "markdown", + "id": "6ca51859", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Lecture 16 – Hyperparameters, Cross-Validation, and Decision Trees\n", + "\n", + "## DSC 80, Spring 2024" + ] + }, + { + "cell_type": "markdown", + "id": "22824200", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Announcements 📣\n", + "\n", + "- Lab 8 is due on **Wednesday**.\n", + "- Final Project Checkpoint 2 is due **Friday**.\n", + "- The Final Exam is on **Saturday, June 8 from 8-11AM** (room TBD).\n", + " - You can bring two double-sided notes sheets (you can bring your midterm notes sheet, if you want).\n", + " - More details to come." + ] + }, + { + "cell_type": "markdown", + "id": "bd851d5b", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Agenda 📆\n", + "\n", + "- Hyperparameters.\n", + "- Cross-validation.\n", + "- Decision trees." + ] + }, + { + "cell_type": "markdown", + "id": "ea74dd70", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + " \n", + "Remember, you can always ask questions at [**q.dsc80.com**](https://docs.google.com/forms/d/e/1FAIpQLScWbVZv9hBv-wX-ItKHUVRnkPMMtfJZVfErKE9GS7_8dFcRBQ/viewform)!" + ] + }, + { + "cell_type": "markdown", + "id": "b176323a", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Review: Hyperparameters" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "874cfdb7", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [], + "source": [ + "np.random.seed(23) # For reproducibility.\n", + "\n", + "def sample_from_pop(n=100):\n", + " x = np.linspace(-2, 3, n)\n", + " y = x ** 3 + (np.random.normal(0, 3, size=n))\n", + " return pd.DataFrame({'x': x, 'y': y})\n", + "\n", + "sample_1 = sample_from_pop()\n", + "sample_2 = sample_from_pop()" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "d2538011", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + " \n", + " " + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "px.scatter(sample_1, x='x', y='y', title='Sample 1')" + ] + }, + { + "cell_type": "markdown", + "id": "86b338d4", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Example: Polynomial regression\n", + "\n", + "We recently looked at an example of **polynomial regression**." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "72af7566", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig = util.train_and_plot(train_sample=sample_1, test_sample=sample_2, degs=[1, 3, 25], data_name='Sample 2')\n", + "fig.update_layout(title='Trained on Sample 1, Performance on Sample 2')" + ] + }, + { + "cell_type": "markdown", + "id": "42a671d7", + "metadata": {}, + "source": [ + "When building these models:\n", + "- We **got to choose** the degree of the polynomials (i.e. we chose 1, 3, and 25).\n", + "- We didn't get to choose the exact formulas for the three polynomials – their formulas were **learned from data**." + ] + }, + { + "cell_type": "markdown", + "id": "95f0e314", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Parameters vs. hyperparameters" + ] + }, + { + "cell_type": "markdown", + "id": "33d7d0c9", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- A **parameter** defines the relationship between variables in a model. \n", + " - **We learn parameters from data**.\n", + " - For instance, suppose we fit a degree 3 polynomial to data, and end up with\n", + " \n", + " $$H(x) = 1 - 2x + 13x^2 - 4x^3$$\n", + " \n", + " - 1, -2, 13, and -4 are parameters." + ] + }, + { + "cell_type": "markdown", + "id": "6778dd62", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- A **hyperparameter** is a parameter that we get to choose **before our model is fit to the data**.\n", + " - Think of hyperparameters as knobs 🎛 – **we get to pick and tune them!**\n", + " - **Polynomial degree** was a hyperparameter in the previous example, and we tried three different values: 1, 3, and 25." + ] + }, + { + "cell_type": "markdown", + "id": "1d1dfbc3", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- **Question**: How do we choose the \"right\" hyperparameter(s)?" + ] + }, + { + "cell_type": "markdown", + "id": "e2f97b99", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Training error vs. test error" + ] + }, + { + "cell_type": "markdown", + "id": "2cd2e40f", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We know that a model's performance on a **test set** is a good estimate of its ability to generalize to unseen data." + ] + }, + { + "cell_type": "markdown", + "id": "926bc6c1", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We want to find the hyperparameter that leads to the best **test set performance**." + ] + }, + { + "cell_type": "markdown", + "id": "e6e28be9", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Idea:\n", + " 1. Come up with a **list** of hyperparameters to try.\n", + " 2. For each hyperparameter, train the model on the training set and compute its performance on the test set.\n", + " 3. Pick the hyperparameter with the best performance on the test set." + ] + }, + { + "cell_type": "markdown", + "id": "b31001a9", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Training error vs. test error\n", + "\n", + "- Let's try this strategy on Sample 1 from our earlier example. \n", + "\n", + "- We'll try to fit a polynomial model on the dataset; we'll choose the polynomial's degree from the list [1, 2, ..., 25]." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "587b788e", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "px.scatter(sample_1, x='x', y='y', title='Sample 1')" + ] + }, + { + "cell_type": "markdown", + "id": "324c1eb4", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "First, we perform a train-test split." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "184d686f", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.model_selection import train_test_split" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "10ab3ddd", + "metadata": {}, + "outputs": [], + "source": [ + "X = sample_1[['x']]\n", + "y = sample_1['y']\n", + "\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=100)" + ] + }, + { + "cell_type": "markdown", + "id": "e98ff996", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Polynomial degree vs. train/test error\n", + "\n", + "Now, we'll create models with degree-1 through degree-25 polynomial features and compute their train and test errors." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "19abdbc7", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.metrics import mean_squared_error" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "825c48e8", + "metadata": {}, + "outputs": [], + "source": [ + "train_errs = []\n", + "test_errs = []\n", + "\n", + "for d in range(1, 26):\n", + " pl = make_pipeline(PolynomialFeatures(d), LinearRegression())\n", + " pl.fit(X_train, y_train)\n", + " train_errs.append(mean_squared_error(y_train, pl.predict(X_train), squared=False))\n", + " test_errs.append(mean_squared_error(y_test, pl.predict(X_test), squared=False))\n", + "\n", + "errs = pd.DataFrame({'Train Error': train_errs, 'Test Error': test_errs})" + ] + }, + { + "cell_type": "markdown", + "id": "ed06ac65", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "Let's look at the plots of training error vs. degree and test error vs. degree." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "7bdc8ddf", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig = px.line(errs)\n", + "fig.update_layout(showlegend=True, xaxis_title='Polynomial Degree', yaxis_title='RMSE')" + ] + }, + { + "cell_type": "markdown", + "id": "128dfbf6", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Training error appears to decrease as polynomial degree increases." + ] + }, + { + "cell_type": "markdown", + "id": "b724335d", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Test error appears to decrease until a \"valley\", and then increases again." + ] + }, + { + "cell_type": "markdown", + "id": "d76e0bee", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Here, we'd choose a degree of 3, since that degree has the **lowest test error**." + ] + }, + { + "cell_type": "markdown", + "id": "d0abe896", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Training error vs. test error\n", + "\n", + "The pattern we saw in the previous example is true more generally.\n", + "\n", + "
" + ] + }, + { + "cell_type": "markdown", + "id": "e728db66", + "metadata": {}, + "source": [ + "We pick the hyperparameter(s) at the \"valley\" of test error.\n", + "\n", + "Note that training error **tends** to underestimate test error, but it doesn't have to – i.e., it is possible for test error to be lower than training error (say, if the test set is \"easier\" to predict than the training set)." + ] + }, + { + "cell_type": "markdown", + "id": "0f6bc281", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Conducting train-test splits" + ] + }, + { + "cell_type": "markdown", + "id": "88811055", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Recall, training data is used to fit our model, and test data is used to evaluate our model.\n", + "\n", + "
" + ] + }, + { + "cell_type": "markdown", + "id": "ebc6b2ad", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- **Question**: _How_ should we split?\n", + " - `sklearn`'s `train_test_split` splits **randomly**, which usually works well.\n", + " - However, if there is some element of **time** in the training data (say, when predicting the future price of a stock), a better split is \"past\" and \"future\"." + ] + }, + { + "cell_type": "markdown", + "id": "1ec1df4b", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- **Question**: How _large_ should the split be, e.g. 90%-10% vs. 75%-25%?\n", + " - There's a tradeoff – a larger training set should lead to a \"better\" model, while a larger test set should lead to a better estimate of our model's ability to generalize.\n", + " - There's no \"right\" choice, but we usually choose between 10% to 25% for the test set." + ] + }, + { + "cell_type": "markdown", + "id": "5ddf0394", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### But wait..." + ] + }, + { + "cell_type": "markdown", + "id": "17c19ac8", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- With our current strategy, we are choosing the hyperparameter that creates the model that **performs best on the test set**." + ] + }, + { + "cell_type": "markdown", + "id": "4427cc42", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- As such, we are **overfitting to the test set** – the best hyperparameter for the test set might not be the best hyperparameter for a totally unseen dataset!" + ] + }, + { + "cell_type": "markdown", + "id": "01584eef", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- It seems like we need **another** split." + ] + }, + { + "cell_type": "markdown", + "id": "3b36a7aa", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Cross-validation" + ] + }, + { + "cell_type": "markdown", + "id": "39c7b052", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Idea: A single validation set\n", + "\n", + "
" + ] + }, + { + "cell_type": "markdown", + "id": "d344ac51", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "1. Split the data into three sets: training, validation, and test." + ] + }, + { + "cell_type": "markdown", + "id": "ac8d7d91", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "2. For each hyperparameter choice, train the model only on the training set, and evaluate the model's performance on the validation set." + ] + }, + { + "cell_type": "markdown", + "id": "0b66367d", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "3. Find the hyperparameter with the best validation performance." + ] + }, + { + "cell_type": "markdown", + "id": "fad93210", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "4. Retrain the final model on the training and validation sets, and report its performance on the test set." + ] + }, + { + "cell_type": "markdown", + "id": "f30e49e7", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "**Issue**: This strategy is too dependent on the validation set, which may be small and/or not a representative sample of the data. **We're not going to do this.** ❌" + ] + }, + { + "cell_type": "markdown", + "id": "a9131a04", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### A better idea: $k$-fold cross-validation\n", + "\n", + "Instead of relying on a single validation set, we can create $k$ validation sets, where $k$ is some positive integer (5 in the example below).\n", + "\n", + "
\n", + "\n", + "Since each data point is used for training $k-1$ times and validation once, the (averaged) validation performance should be a good metric of a model's ability to generalize to unseen data.\n", + "\n", + "$k$-fold cross-validation (or simply \"cross-validation\") is **the** technique we will use for finding hyperparameters, or more generally, for choosing between different possible models. **It's what you should use in your Final Project!** ✅" + ] + }, + { + "cell_type": "markdown", + "id": "b7a4b3f7", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Creating folds in `sklearn`\n", + "\n", + "`sklearn` has a `KFold` class that splits data into training and validation folds." + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "4555f6c0", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.model_selection import KFold" + ] + }, + { + "cell_type": "markdown", + "id": "319b67ce", + "metadata": {}, + "source": [ + "Let's use a simple dataset for illustration." + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "id": "04fcf601", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([10, 20, 30, 40, 50, 60])" + ] + }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "data = np.arange(10, 70, 10)\n", + "data" + ] + }, + { + "cell_type": "markdown", + "id": "5a43ac2a", + "metadata": {}, + "source": [ + "Let's instantiate a `KFold` object with $k=3$." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "id": "062e7597", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "KFold(n_splits=3, random_state=1, shuffle=True)" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "kfold = KFold(3, shuffle=True, random_state=1)\n", + "kfold" + ] + }, + { + "cell_type": "markdown", + "id": "fbdf8557", + "metadata": {}, + "source": [ + "Finally, let's use `kfold` to `split` `data`:" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "id": "79701be4", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "train: [10 40 50 60], validation: [20 30]\n", + "train: [20 30 40 60], validation: [10 50]\n", + "train: [10 20 30 50], validation: [40 60]\n" + ] + } + ], + "source": [ + "for train, val in kfold.split(data):\n", + " print(f'train: {data[train]}, validation: {data[val]}')" + ] + }, + { + "cell_type": "markdown", + "id": "7367e39b", + "metadata": {}, + "source": [ + "Note that each value in `data` is used for validation exactly once and for training exactly twice. Also note that because we set `shuffle=True` the folds are not simply `[10, 20]`, `[30, 40]`, and `[50, 60]`." + ] + }, + { + "cell_type": "markdown", + "id": "08de2c3f", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### $k$-fold cross-validation\n", + "\n", + "First, **shuffle** the entire training set randomly and **split** it into $k$ disjoint folds, or \"slices\". Then:" + ] + }, + { + "cell_type": "markdown", + "id": "86b39156", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- For each hyperparameter:\n", + " - For each slice:\n", + " - Let the slice be the \"validation set\", $V$.\n", + " - Let the rest of the data be the \"training set\", $T$.\n", + " - Train a model using the selected hyperparameter on the training set $T$.\n", + " - Evaluate the model on the validation set $V$.\n", + " - Compute the **average** validation score (e.g. RMSE) for the particular hyperparameter." + ] + }, + { + "cell_type": "markdown", + "id": "7abb6540", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Choose the hyperparameter with the best average validation score." + ] + }, + { + "cell_type": "markdown", + "id": "4620c78d", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### $k$-fold cross-validation in `sklearn`\n", + "\n", + "While you could manually use `KFold` to perform cross-validation, the `cross_val_score` function in `sklearn` implements $k$-fold cross-validation for us! \n", + "\n", + "```py\n", + "cross_val_score(estimator, X_train, y_train, cv)\n", + "```\n", + "\n", + "Specifically, it takes in:\n", + "- A `Pipeline` or estimator **that has not already been `fit`**.\n", + "- Training data.\n", + "- A value of $k$ (through the `cv` argument).\n", + "- (Optionally) A `scoring` metric.\n", + "\n", + "and performs $k$-fold cross-validation, returning the values of the scoring metric on each fold." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "id": "ca1244ea", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.model_selection import cross_val_score" + ] + }, + { + "cell_type": "markdown", + "id": "b95ca832", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### $k$-fold cross-validation in `sklearn`" + ] + }, + { + "cell_type": "markdown", + "id": "48eaa524", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Let's perform $k$-fold cross validation in order to help us pick a degree for polynomial regression from the list [1, 2, ..., 25]." + ] + }, + { + "cell_type": "markdown", + "id": "af6df303", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We'll use $k=5$ since it's a common choice (and the default in `sklearn`)." + ] + }, + { + "cell_type": "markdown", + "id": "4c570d08", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- For the sake of this example, we'll suppose `sample_1` is our \"training + validation data\", i.e. that our test data is in some other dataset.\n", + " - If this were not true, we'd first need to split `sample_1` into separate training and test sets." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "id": "d2e26f47", + "metadata": {}, + "outputs": [], + "source": [ + "errs_df = pd.DataFrame()\n", + "\n", + "for d in range(1, 26):\n", + " pl = make_pipeline(PolynomialFeatures(d), LinearRegression())\n", + " \n", + " # The `scoring` argument is used to specify that we want to compute the RMSE; \n", + " # the default is R^2. It's called \"neg\" RMSE because, \n", + " # by default, sklearn likes to \"maximize\" scores, and maximizing -RMSE is the same\n", + " # as minimizing RMSE.\n", + " errs = cross_val_score(pl, sample_1[['x']], sample_1['y'], \n", + " cv=5, scoring='neg_root_mean_squared_error')\n", + " errs_df[f'Deg {d}'] = -errs # Negate to turn positive (sklearn computed negative RMSE).\n", + " \n", + "errs_df.index = [f'Fold {i}' for i in range(1, 6)]\n", + "errs_df.index.name = 'Validation Fold'" + ] + }, + { + "cell_type": "markdown", + "id": "e8c7311d", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Next class, we'll look at how to implement this procedure without needing to `for`-loop over values of `d`." + ] + }, + { + "cell_type": "markdown", + "id": "5b6ed7f0", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### $k$-fold cross-validation in `sklearn`\n", + "\n", + "Note that for each choice of degree (our hyperparameter), we have **five** RMSEs, one for each \"fold\" of the data. This means that in total, $5 \\cdot 25 = 125$ models were trained/fit to data!" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "id": "d01ba737", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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" + ], + "text/plain": [ + " Deg 1 Deg 2 Deg 3 Deg 4 ... Deg 22 Deg 23 \\\n", + "Validation Fold ... \n", + "Fold 1 4.79 12.81 5.04 4.93 ... 8.78e+06 6.57e+07 \n", + "Fold 2 3.97 5.36 3.19 3.22 ... 2.93e+01 7.85e+01 \n", + "Fold 3 4.77 2.56 2.08 2.11 ... 3.03e+01 3.09e+01 \n", + "Fold 4 6.13 4.66 2.93 2.93 ... 6.27e+00 3.33e+01 \n", + "Fold 5 11.70 11.92 3.24 4.37 ... 8.36e+06 2.28e+08 \n", + "\n", + " Deg 24 Deg 25 \n", + "Validation Fold \n", + "Fold 1 1.41e+08 5.90e+08 \n", + "Fold 2 7.53e+01 3.13e+01 \n", + "Fold 3 4.24e+01 3.72e+01 \n", + "Fold 4 5.80e+01 9.69e+00 \n", + "Fold 5 8.17e+08 6.63e+09 \n", + "\n", + "[5 rows x 25 columns]" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "errs_df" + ] + }, + { + "cell_type": "markdown", + "id": "c914d676", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "We should choose the degree with the lowest **average** validation RMSE." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "id": "3dc1361c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'Deg 3'" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "errs_df.mean().idxmin()" + ] + }, + { + "cell_type": "markdown", + "id": "43aeacff", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Note that if we didn't perform $k$-fold cross-validation, but instead just used a single validation set, we may have ended up with a different result:" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "id": "de811f8a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Validation Fold\n", + "Fold 1 Deg 1\n", + "Fold 2 Deg 6\n", + "Fold 3 Deg 8\n", + "Fold 4 Deg 3\n", + "Fold 5 Deg 3\n", + "dtype: object" + ] + }, + "execution_count": 21, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "errs_df.idxmin(axis=1)" + ] + }, + { + "cell_type": "markdown", + "id": "50e5db5f", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + " \n", + "The RMSEs in Folds 1 and 5 are much higher than in other folds. Why? How we might fix this?" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "id": "c95f912b", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "px.scatter(sample_1, x='x', y='y', title='Sample 1')" + ] + }, + { + "cell_type": "markdown", + "id": "aa8239bf", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Another example: Tips\n", + "\n", + "We can also use $k$-fold cross-validation to determine which subset of features to use in a linear model that predicts tips by making one pipeline for each subset of features we want to evaluate." + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "id": "a721435f", + "metadata": {}, + "outputs": [], + "source": [ + "# make_column_transformer is a shortcut for the ColumnTransformer class\n", + "from sklearn.compose import make_column_transformer, make_column_selector\n", + "from sklearn.preprocessing import FunctionTransformer, OneHotEncoder" + ] + }, + { + "cell_type": "markdown", + "id": "77f774ac", + "metadata": {}, + "source": [ + "As we should always do, we'll perform a train-test split on `tips` and will only use the training data for cross-validation." + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "id": "e04a69f3", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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" + ], + "text/plain": [ + " total_bill tip sex smoker day time size\n", + "0 16.99 1.01 Female No Sun Dinner 2\n", + "1 10.34 1.66 Male No Sun Dinner 3\n", + "2 21.01 3.50 Male No Sun Dinner 3\n", + "3 23.68 3.31 Male No Sun Dinner 2\n", + "4 24.59 3.61 Female No Sun Dinner 4" + ] + }, + "execution_count": 24, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "tips = sns.load_dataset('tips')\n", + "tips.head()" + ] + }, + { + "cell_type": "code", + "execution_count": 61, + "id": "078f4566", + "metadata": {}, + "outputs": [], + "source": [ + "X = tips.drop('tip', axis=1)\n", + "y = tips['tip']\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=1)" + ] + }, + { + "cell_type": "code", + "execution_count": 62, + "id": "4e2853dc", + "metadata": {}, + "outputs": [], + "source": [ + "# A dictionary that maps names to Pipeline objects.\n", + "select = FunctionTransformer(lambda x: x)\n", + "pipes = {\n", + " 'total_bill only': make_pipeline(\n", + " make_column_transformer( (select, ['total_bill']) ),\n", + " LinearRegression(),\n", + " ),\n", + " 'total_bill + size': make_pipeline(\n", + " make_column_transformer( (select, ['total_bill', 'size']) ),\n", + " LinearRegression(),\n", + " ),\n", + " 'total_bill + size + OHE smoker': make_pipeline(\n", + " make_column_transformer(\n", + " (select, ['total_bill', 'size']),\n", + " (OneHotEncoder(drop='first'), ['smoker']),\n", + " ),\n", + " LinearRegression(),\n", + " ),\n", + " 'total_bill + size + OHE all': make_pipeline(\n", + " make_column_transformer(\n", + " (select, ['total_bill', 'size']),\n", + " (OneHotEncoder(drop='first'), ['smoker', 'sex', 'time', 'day']),\n", + " ),\n", + " LinearRegression(),\n", + " ),\n", + "}" + ] + }, + { + "cell_type": "code", + "execution_count": 63, + "id": "23b42ccf", + "metadata": {}, + "outputs": [], + "source": [ + "pipe_df = pd.DataFrame()\n", + "\n", + "for pipe in pipes:\n", + " errs = cross_val_score(pipes[pipe], X_train, y_train,\n", + " cv=5, scoring='neg_root_mean_squared_error')\n", + " pipe_df[pipe] = -errs\n", + " \n", + "pipe_df.index = [f'Fold {i}' for i in range(1, 6)]\n", + "pipe_df.index.name = 'Validation Fold'" + ] + }, + { + "cell_type": "code", + "execution_count": 64, + "id": "09420210", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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total_bill onlytotal_bill + sizetotal_bill + size + OHE smokertotal_bill + size + OHE all
Validation Fold
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" + ], + "text/plain": [ + " total_bill only total_bill + size \\\n", + "Validation Fold \n", + "Fold 1 1.32 1.27 \n", + "Fold 2 0.95 0.92 \n", + "Fold 3 0.77 0.86 \n", + "Fold 4 0.85 0.84 \n", + "Fold 5 1.10 1.07 \n", + "\n", + " total_bill + size + OHE smoker total_bill + size + OHE all \n", + "Validation Fold \n", + "Fold 1 1.27 1.29 \n", + "Fold 2 0.93 0.93 \n", + "Fold 3 0.86 0.87 \n", + "Fold 4 0.84 0.86 \n", + "Fold 5 1.07 1.08 " + ] + }, + "execution_count": 64, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pipe_df" + ] + }, + { + "cell_type": "code", + "execution_count": 65, + "id": "433f45de", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "total_bill only 1.00\n", + "total_bill + size 0.99\n", + "total_bill + size + OHE smoker 0.99\n", + "total_bill + size + OHE all 1.01\n", + "dtype: float64" + ] + }, + "execution_count": 65, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pipe_df.mean()" + ] + }, + { + "cell_type": "code", + "execution_count": 66, + "id": "be7a3541", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'total_bill + size + OHE smoker'" + ] + }, + "execution_count": 66, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pipe_df.mean().idxmin()" + ] + }, + { + "cell_type": "markdown", + "id": "7df9ae70", + "metadata": {}, + "source": [ + "Even though the third model has the lowest average validation RMSE, its average validation RMSE is very close to that of the other, simpler models, and as a result we'd likely use the simplest model in practice." + ] + }, + { + "cell_type": "markdown", + "id": "80cf1012", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + "\n", + "- Suppose you have a training dataset with 1000 rows.\n", + "- You want to decide between 20 hyperparameters for a particular model.\n", + "- To do so, you perform 10-fold cross-validation.\n", + "- **How many times is the first row in the training dataset (`X.iloc[0]`) used for training a model?**\n" + ] + }, + { + "cell_type": "markdown", + "id": "1c3b3ad4", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Summary: Generalization" + ] + }, + { + "cell_type": "markdown", + "id": "6a7e9ba8", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "1. Split the data into two sets: training and test." + ] + }, + { + "cell_type": "markdown", + "id": "5c3b7200", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "2. Use only the training data when designing, training, and tuning the model.\n", + " - Use $k$-fold cross-validation to choose hyperparameters and estimate the model's ability to generalize.\n", + " - Do not ❌ look at the test data in this step!" + ] + }, + { + "cell_type": "markdown", + "id": "492e6340", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "3. Commit to your final model and train it using the entire training set." + ] + }, + { + "cell_type": "markdown", + "id": "ad9dd3db", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "4. Test the data using the test data. If the performance (e.g. RMSE) is not acceptable, return to step 2." + ] + }, + { + "cell_type": "markdown", + "id": "bc3dc5a7", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "5. Finally, train on **all available data** and ship the model to production! 🛳" + ] + }, + { + "cell_type": "markdown", + "id": "662df112", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "🚨 This is the process you should **always** use! 🚨 " + ] + }, + { + "cell_type": "markdown", + "id": "dc44bead", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + " \n", + "Remember, you can always ask questions at [**q.dsc80.com**](https://docs.google.com/forms/d/e/1FAIpQLScWbVZv9hBv-wX-ItKHUVRnkPMMtfJZVfErKE9GS7_8dFcRBQ/viewform)!" + ] + }, + { + "cell_type": "markdown", + "id": "7e72e409", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Decision trees 🌲" + ] + }, + { + "cell_type": "markdown", + "id": "e6646d38", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "\n", + "Decision trees can be used for both regression and classification. We'll start by using them for **classification**." + ] + }, + { + "cell_type": "markdown", + "id": "942f1289", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Example: Should I get groceries?\n", + "\n", + "Decision trees make classifications by answering a series of yes/no questions.\n", + "\n", + "**Should I go to Trader Joe's to buy groceries today?**" + ] + }, + { + "cell_type": "markdown", + "id": "09e8a413", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "
" + ] + }, + { + "cell_type": "markdown", + "id": "1d8269fb", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Internal **nodes** of trees involve questions; leaf nodes make predictions $H(x)$." + ] + }, + { + "cell_type": "markdown", + "id": "bc8070e7", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "
" + ] + }, + { + "cell_type": "markdown", + "id": "2569dd49", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Example: Predicting diabetes" + ] + }, + { + "cell_type": "code", + "execution_count": 67, + "id": "f83270fa", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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PregnanciesGlucoseBloodPressureSkinThicknessInsulinBMIDiabetesPedigreeFunctionAgeOutcome
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..............................
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" + ], + "text/plain": [ + " Pregnancies Glucose BloodPressure SkinThickness Insulin BMI \\\n", + "0 6 148 72 35 0 33.6 \n", + "1 1 85 66 29 0 26.6 \n", + "2 8 183 64 0 0 23.3 \n", + ".. ... ... ... ... ... ... \n", + "765 5 121 72 23 112 26.2 \n", + "766 1 126 60 0 0 30.1 \n", + "767 1 93 70 31 0 30.4 \n", + "\n", + " DiabetesPedigreeFunction Age Outcome \n", + "0 0.63 50 1 \n", + "1 0.35 31 0 \n", + "2 0.67 32 1 \n", + ".. ... ... ... \n", + "765 0.24 30 0 \n", + "766 0.35 47 1 \n", + "767 0.32 23 0 \n", + "\n", + "[768 rows x 9 columns]" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "diabetes = pd.read_csv(Path('data') / 'diabetes.csv')\n", + "display_df(diabetes, cols=9)" + ] + }, + { + "cell_type": "code", + "execution_count": 68, + "id": "559c4fd6", + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "0 500\n", + "1 268\n", + "Name: Outcome, dtype: int64" + ] + }, + "execution_count": 68, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# 0 means no diabetes, 1 means yes diabetes.\n", + "diabetes['Outcome'].value_counts()" + ] + }, + { + "cell_type": "markdown", + "id": "fd3d4eb8", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- `'Glucose'` is measured in mg/dL (milligrams per deciliter)." + ] + }, + { + "cell_type": "markdown", + "id": "6c2ea02c", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- `'BMI'` is calculated as $\\text{BMI} = \\frac{\\text{weight (kg)}}{\\left[ \\text{height (m)} \\right]^2}$." + ] + }, + { + "cell_type": "markdown", + "id": "82906445", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Let's use `'Glucose'` and `'BMI'` to predict whether or not a patient has diabetes (`'Outcome'`)." + ] + }, + { + "cell_type": "markdown", + "id": "5f4d2b73", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Exploring the dataset\n", + "\n", + "First, a train-test split:" + ] + }, + { + "cell_type": "code", + "execution_count": 69, + "id": "4c699036", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.model_selection import train_test_split\n", + "X_train, X_test, y_train, y_test = (\n", + " train_test_split(diabetes[['Glucose', 'BMI']], diabetes['Outcome'], random_state=1)\n", + ")" + ] + }, + { + "cell_type": "markdown", + "id": "fb537748", + "metadata": {}, + "source": [ + "Class 0 (orange) is \"no diabetes\" and class 1 (blue) is \"diabetes\"." + ] + }, + { + "cell_type": "code", + "execution_count": 70, + "id": "4b4445bb", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig = (\n", + " X_train.assign(Outcome=y_train.astype(str))\n", + " .plot(kind='scatter', x='Glucose', y='BMI', color='Outcome', \n", + " color_discrete_map={'0': 'orange', '1': 'blue'},\n", + " title='Relationship between Glucose, BMI, and Diabetes')\n", + ")\n", + "fig" + ] + }, + { + "cell_type": "markdown", + "id": "c2f94087", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Building a decision tree\n", + "\n", + "Let's build a decision tree and interpret the results.\n", + "\n", + "The relevant class is `DecisionTreeClassifier`, from `sklearn.tree`." + ] + }, + { + "cell_type": "code", + "execution_count": 71, + "id": "12902788", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.tree import DecisionTreeClassifier" + ] + }, + { + "cell_type": "markdown", + "id": "c4b06503", + "metadata": {}, + "source": [ + "Note that we `fit` it the same way we `fit` earlier estimators.\n", + "\n", + "_You may wonder what `max_depth` and `criterion` do – more on this soon!_" + ] + }, + { + "cell_type": "code", + "execution_count": 72, + "id": "ae5efa16", + "metadata": {}, + "outputs": [], + "source": [ + "dt = DecisionTreeClassifier(max_depth=2, criterion='entropy')" + ] + }, + { + "cell_type": "code", + "execution_count": 73, + "id": "b37a3d08", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "DecisionTreeClassifier(criterion='entropy', max_depth=2)" + ] + }, + "execution_count": 73, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dt.fit(X_train, y_train)" + ] + }, + { + "cell_type": "markdown", + "id": "70c4fc97", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Visualizing decision trees\n", + "\n", + "Our fit decision tree is like a \"flowchart\", made up of a series of questions.\n", + "\n", + "As before, orange is \"no diabetes\" and blue is \"diabetes\"." + ] + }, + { + "cell_type": "code", + "execution_count": 74, + "id": "b8ba715f", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.tree import plot_tree" + ] + }, + { + "cell_type": "code", + "execution_count": 75, + "id": "cbe2313a", + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " 2024-05-27T21:27:31.954299\n", + " image/svg+xml\n", + " \n", + " \n", + " Matplotlib v3.5.1, https://matplotlib.org/\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + 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\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + "\n" + ], + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.figure(figsize=(15, 5))\n", + "plot_tree(dt, feature_names=X_train.columns, class_names=['no db', 'yes db'], \n", + " filled=True, fontsize=15, impurity=False);" + ] + }, + { + "cell_type": "markdown", + "id": "066f6931", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- To **classify a new data point**, we start at the top and answer the first question (i.e. \"Glucose <= 129.5\")." + ] + }, + { + "cell_type": "markdown", + "id": "57c96291", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- If the answer is \"**Yes**\", we move to the **left** branch, otherwise we move to the right branch." + ] + }, + { + "cell_type": "markdown", + "id": "f1432dfa", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We repeat this process until we end up at a leaf node, at which point we predict the most common class in that node.\n", + " - Note that each node has a `value` attribute, which describes the number of **training** individuals of each class that fell in that node." + ] + }, + { + "cell_type": "code", + "execution_count": 76, + "id": "9527edfe", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0 304\n", + "1 78\n", + "Name: Outcome, dtype: int64" + ] + }, + "execution_count": 76, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Note that the left node at depth 2 has a `value` of [304, 78].\n", + "y_train[X_train.query('Glucose <= 129.5').index].value_counts()" + ] + }, + { + "cell_type": "markdown", + "id": "7a73e4cc", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Evaluating classifiers\n", + "\n", + "The most common evaluation metric in classification is **accuracy**:\n", + "\n", + "$$\\text{accuracy} = \\frac{\\text{# data points classified correctly}}{\\text{# data points}}$$" + ] + }, + { + "cell_type": "code", + "execution_count": 77, + "id": "6aef8802", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.765625" + ] + }, + "execution_count": 77, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(dt.predict(X_train) == y_train).mean()" + ] + }, + { + "cell_type": "markdown", + "id": "ea2d61d2", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "The `score` method of a classifier computes accuracy by default (just like the `score` method of a regressor computes $R^2$ by default). We want our classifiers to have **high accuracy**." + ] + }, + { + "cell_type": "code", + "execution_count": 78, + "id": "48bde429", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.765625" + ] + }, + "execution_count": 78, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Training accuracy – same number as above\n", + "dt.score(X_train, y_train)" + ] + }, + { + "cell_type": "code", + "execution_count": 79, + "id": "e4045e13", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.7760416666666666" + ] + }, + "execution_count": 79, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Testing accuracy\n", + "dt.score(X_test, y_test)" + ] + }, + { + "cell_type": "markdown", + "id": "83213dcc", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Reflection" + ] + }, + { + "cell_type": "markdown", + "id": "47e36ccd", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Decision trees are easily interpretable: it is clear _how_ they make their predictions." + ] + }, + { + "cell_type": "markdown", + "id": "9f48d00b", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- They work with categorical data without needing to use one hot encoding." + ] + }, + { + "cell_type": "markdown", + "id": "0c046af0", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- They also can be used in multi-class classification problems, e.g. when there are more than 2 possible outcomes." + ] + }, + { + "cell_type": "markdown", + "id": "6ee7e7c8", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The _decision boundary_ of a decision tree can be arbitrarily complicated." + ] + }, + { + "cell_type": "markdown", + "id": "adb19142", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- **How are decision trees trained?**" + ] + }, + { + "cell_type": "markdown", + "id": "0cda55de", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### How are decision trees trained?\n", + "\n", + "Pseudocode:\n", + "\n", + "```python\n", + "def make_tree(X, y):\n", + " if all points in y have the same label C:\n", + " return Leaf(C)\n", + " f = best splitting feature # e.g. Glucose or BMI\n", + " v = best splitting value # e.g. 129.5\n", + " \n", + " X_left, y_left = X, y where (X[f] <= v)\n", + " X_right, y_right = X, y where (X[f] > v)\n", + " \n", + " left = make_tree(X_left, y_left)\n", + " right = make_tree(X_right, y_right)\n", + " \n", + " return Node(f, v, left, right)\n", + " \n", + "make_tree(X_train, y_train)\n", + "```" + ] + }, + { + "cell_type": "markdown", + "id": "670a105e", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### How do we measure the quality of a split?" + ] + }, + { + "cell_type": "markdown", + "id": "214872f7", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Our pseudocode for training a decision tree relies on finding the best way to \"split\" a node – that is, **the best question to ask** to help us narrow down which class to predict." + ] + }, + { + "cell_type": "markdown", + "id": "9621a012", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "**Intuition**: Suppose the distribution within a node looks like this (colors represent classes):\n", + "\n", + "
🟠🟠🟠🟠🟠🟠🔵🔵🔵🔵🔵🔵🔵
\n", + "\n", + "Split A:\n", + "- \"Yes\": 🟠🟠🟠🔵🔵🔵\n", + "- \"No\": 🟠🟠🟠🔵🔵🔵🔵\n", + "\n", + "Split B:\n", + "- \"Yes\": 🔵🔵🔵🔵🔵🔵\n", + "- \"No\": 🔵🟠🟠🟠🟠🟠🟠\n", + "\n", + "**Which split is \"better\"?**" + ] + }, + { + "cell_type": "markdown", + "id": "1701e7e3", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Split B, because there is \"less uncertainty\" in the resulting nodes in Split B than there is in Split A. Let's try and quantify this!" + ] + }, + { + "cell_type": "markdown", + "id": "c6af8be0", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Entropy" + ] + }, + { + "cell_type": "markdown", + "id": "46462eb4", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- For each class $C$ within a node, define $p_C$ as the proportion of points with that class.\n", + " - For example, the two classes may be \"yes diabetes\" and \"no diabetes\"." + ] + }, + { + "cell_type": "markdown", + "id": "a1414f15", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The **surprise** of drawing a point from the node at random and having it be class $C$ is:\n", + "\n", + "$$\n", + "- \\log_2 p_C\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "73b7095a", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The **entropy** of a node is the average surprise over all classes:\n", + "\n", + "\\begin{align}\n", + "\\text{entropy} &= - \\sum_C p_C \\log_2 p_C\n", + "\\end{align}" + ] + }, + { + "cell_type": "markdown", + "id": "dc0d04fd", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The entropy of 🟠🟠🟠🟠🟠🟠🟠🟠 is $ -1 \\log_2(1) = 0 $." + ] + }, + { + "cell_type": "markdown", + "id": "4d3dc897", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The entropy of 🟠🟠🟠🟠🔵🔵🔵🔵 is $ -0.5 \\log_2(0.5) - 0.5 \\log_2(0.5) = 1$." + ] + }, + { + "cell_type": "markdown", + "id": "98afe088", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The entropy of 🟠🔵🟢🟡🟣 is $ - \\log_2 \\frac{1}{5} = \\log_2(5) $\n", + " - In general, if a node has $n$ points, all with different labels, the entropy of the node is $ \\log_2(n) $." + ] + }, + { + "cell_type": "markdown", + "id": "e28fe85d", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Example entropy calculation\n", + "\n", + "Suppose we have:\n", + "\n", + "
🟠🟠🟠🟠🟠🟠🟠🟠🟠🟠🟠🟠🔵🔵🔵🔵🔵🔵
\n", + "\n", + "Split A:\n", + "- \"Yes\": 🟠🟠🟠🟠🟠🟠🔵\n", + "- \"No\": 🟠🟠🟠🟠🟠🟠🔵🔵🔵🔵🔵\n", + "\n", + "Split B:\n", + "- \"Yes\": 🟠🟠🟠🟠🟠🟠🔵🔵🔵\n", + "- \"No\": 🟠🟠🟠🟠🟠🟠🔵🔵🔵" + ] + }, + { + "cell_type": "markdown", + "id": "34bdc7b9", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "We choose the split that has the lowest **weighted entropy**, that is:\n", + "\n", + "$$\\text{entropy of split} = \\frac{\\# \\text{Yes}}{\\# \\text{Yes} + \\# \\text{No}} \\cdot \\text{entropy(Yes)} + \\frac{\\# \\text{No}}{\\# \\text{Yes} + \\# \\text{No}} \\cdot \\text{entropy(No)}$$" + ] + }, + { + "cell_type": "code", + "execution_count": 80, + "id": "49e63931", + "metadata": {}, + "outputs": [], + "source": [ + "def entropy(node):\n", + " props = pd.Series(list(node)).value_counts(normalize=True)\n", + " return -sum(props * np.log2(props))" + ] + }, + { + "cell_type": "code", + "execution_count": 81, + "id": "31a38b3f", + "metadata": {}, + "outputs": [], + "source": [ + "def weighted_entropy(yes_node, no_node):\n", + " yes_entropy = entropy(yes_node)\n", + " no_entropy = entropy(no_node)\n", + " yes_weight = len(yes_node) / (len(yes_node) + len(no_node))\n", + " return yes_weight * yes_entropy + (1 - yes_weight) * no_entropy" + ] + }, + { + "cell_type": "code", + "execution_count": 82, + "id": "e3e62521", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.8375578764623786" + ] + }, + "execution_count": 82, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Split A:\n", + "weighted_entropy(\"🟠🟠🟠🟠🟠🟠🔵\", \"🟠🟠🟠🟠🟠🟠🔵🔵🔵🔵🔵\")" + ] + }, + { + "cell_type": "code", + "execution_count": 83, + "id": "55cd0ce5", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.9182958340544896" + ] + }, + "execution_count": 83, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Split B:\n", + "weighted_entropy(\"🟠🟠🟠🟠🟠🟠🔵🔵🔵\", \"🟠🟠🟠🟠🟠🟠🔵🔵🔵\")" + ] + }, + { + "cell_type": "markdown", + "id": "f8d910d5", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Split A has the lower weighted entropy, so we'll use it." + ] + }, + { + "cell_type": "markdown", + "id": "61477abd", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Understanding entropy" + ] + }, + { + "cell_type": "code", + "execution_count": 84, + "id": "899b6d5e", + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " 2024-05-27T21:28:46.136616\n", + " image/svg+xml\n", + " \n", + " \n", + " Matplotlib v3.5.1, https://matplotlib.org/\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.figure(figsize=(15, 5))\n", + "plot_tree(dt, feature_names=X_train.columns, class_names=['no db', 'yes db'], \n", + " filled=True, fontsize=15, impurity=True);" + ] + }, + { + "cell_type": "markdown", + "id": "3cf312fe", + "metadata": { + "slideshow": { + "slide_type": "" + } + }, + "source": [ + "We can recreate the entropy calculations above! Note that the default `DecisionTreeClassifier` uncertaintly metric _isn't_ entropy; it used entropy because we set `criterion='entropy'` when defining `dt`. (The default metric, [Gini impurity](https://en.wikipedia.org/wiki/Decision_tree_learning#:~:text=Gini%20impurity%20measures%20how%20often,into%20a%20single%20target%20category.), is perfectly fine too!)" + ] + }, + { + "cell_type": "code", + "execution_count": 85, + "id": "648b1555", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.7302263747422792" + ] + }, + "execution_count": 85, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# The first node at depth 2 has an entropy of 0.73,\n", + "# both told to us above and verified here!\n", + "entropy([0] * 304 + [1] * 78)" + ] + }, + { + "cell_type": "markdown", + "id": "26aa9328", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + " \n", + "Remember, you can always ask questions at [**q.dsc80.com**](https://docs.google.com/forms/d/e/1FAIpQLScWbVZv9hBv-wX-ItKHUVRnkPMMtfJZVfErKE9GS7_8dFcRBQ/viewform)!" + ] + }, + { + "cell_type": "markdown", + "id": "63715f3a", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Tree depth\n", + "\n", + "Decision trees are trained by **recursively** picking the best split until:\n", + "\n", + "- all \"leaf nodes\" only contain training examples from a single class (good), or\n", + "- it is impossible to split leaf nodes any further (not good).\n", + "\n", + "By default, there is no \"maximum depth\" for a decision tree. As such, without restriction, decision trees tend to be very deep." + ] + }, + { + "cell_type": "code", + "execution_count": 86, + "id": "df1a32ea", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "DecisionTreeClassifier()" + ] + }, + "execution_count": 86, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dt_no_max = DecisionTreeClassifier()\n", + "dt_no_max.fit(X_train, y_train)" + ] + }, + { + "cell_type": "markdown", + "id": "c8b7ab31", + "metadata": {}, + "source": [ + "A decision tree fit on our training data has a depth of around 20! (It is so deep that `tree.plot_tree` errors when trying to plot it.)" + ] + }, + { + "cell_type": "code", + "execution_count": 87, + "id": "75c06bef", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "21" + ] + }, + "execution_count": 87, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dt_no_max.tree_.max_depth" + ] + }, + { + "cell_type": "markdown", + "id": "30caeb75", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "At first, this tree seems \"better\" than our tree of depth 2, since its training accuracy is much much higher:" + ] + }, + { + "cell_type": "code", + "execution_count": 88, + "id": "811e431b", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.9913194444444444" + ] + }, + "execution_count": 88, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dt_no_max.score(X_train, y_train)" + ] + }, + { + "cell_type": "code", + "execution_count": 89, + "id": "9413b6fa", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.765625" + ] + }, + "execution_count": 89, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Depth 2 tree.\n", + "dt.score(X_train, y_train)" + ] + }, + { + "cell_type": "markdown", + "id": "eaf99510", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "But recall, we truly care about **test set performance**, and this decision tree has **worse accuracy on the test set than our depth 2 tree**." + ] + }, + { + "cell_type": "code", + "execution_count": 90, + "id": "67f35a94", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.7291666666666666" + ] + }, + "execution_count": 90, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dt_no_max.score(X_test, y_test)" + ] + }, + { + "cell_type": "code", + "execution_count": 91, + "id": "7859ac6c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.7760416666666666" + ] + }, + "execution_count": 91, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Depth 2 tree.\n", + "dt.score(X_test, y_test)" + ] + }, + { + "cell_type": "markdown", + "id": "20e1e183", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Decision trees and overfitting" + ] + }, + { + "cell_type": "markdown", + "id": "4769f0d5", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Decision trees have a tendency to overfit. **Why is that?**" + ] + }, + { + "cell_type": "markdown", + "id": "0466852f", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Unlike linear classification techniques (like logistic regression or SVMs), **decision trees are non-linear**.\n", + " - They are also \"non-parametric\" – there are no $w^*$s to learn." + ] + }, + { + "cell_type": "markdown", + "id": "3d76fd16", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- While being trained, decision trees ask enough questions to effectively **memorize** the correct response values in the training set. However, the relationships they learn are often overfit to the noise in the training set, and don't generalize well." + ] + }, + { + "cell_type": "code", + "execution_count": 56, + "id": "b1e165b3", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig" + ] + }, + { + "cell_type": "markdown", + "id": "1f563e29", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- A decision tree whose depth is not restricted will achieve 100% accuracy on any training set, as long as there are no \"overlapping values\" in the training set.\n", + " - Two values overlap when they have the same features $x$ but different response values $y$ (e.g. if two patients have the same glucose levels and BMI, but one has diabetes and one doesn't)." + ] + }, + { + "cell_type": "markdown", + "id": "8564b552", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- **One solution**: Make the decision tree \"less complex\" by limiting the maximum depth." + ] + }, + { + "cell_type": "markdown", + "id": "87429ef6", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "Since `sklearn.tree`'s `plot_tree` can't visualize extremely large decision trees, let's create and visualize some smaller decision trees." + ] + }, + { + "cell_type": "code", + "execution_count": 57, + "id": "66549935", + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " 2024-05-27T12:02:56.199761\n", + " image/svg+xml\n", + " \n", + " \n", + " Matplotlib v3.5.1, https://matplotlib.org/\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " 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\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + "\n" + ], + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "trees = {}\n", + "for d in [2, 4, 8]:\n", + " trees[d] = DecisionTreeClassifier(max_depth=d, random_state=1)\n", + " trees[d].fit(X_train, y_train)\n", + " \n", + " plt.figure(figsize=(15, 5), dpi=100)\n", + " plot_tree(trees[d], feature_names=X_train.columns, class_names=['no db', 'yes db'], \n", + " filled=True, rounded=True, impurity=False)\n", + " \n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "6b9c504a", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "As tree depth increases, complexity increases, and our trees are more prone to overfitting. This means model bias decreases, but model variance increases. \n", + "\n", + "**Question**: What is the \"right\" maximum depth to choose?" + ] + }, + { + "cell_type": "markdown", + "id": "c81ba76e", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Hyperparameters for decision trees" + ] + }, + { + "cell_type": "markdown", + "id": "ce6fcb22", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- `max_depth` is a hyperparameter for `DecisionTreeClassifier`." + ] + }, + { + "cell_type": "markdown", + "id": "bd81005e", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- There are many more hyperparameters we can tweak; look at [the documentation](https://scikit-learn.org/stable/modules/generated/sklearn.tree.DecisionTreeClassifier.html) for examples.\n", + " - `min_samples_split`: The minimum number of samples required to split an internal node.\n", + " - `criterion`: The function to measure the quality of a split (`'gini'` or `'entropy'`)." + ] + }, + { + "cell_type": "markdown", + "id": "2e2dc4ea", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- To ensure that our model generalizes well to unseen data, we need an efficient technique for trying different combinations of hyperparameters!" + ] + }, + { + "cell_type": "markdown", + "id": "da682d20", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We'll see that technique next time: `GridSearchCV`." + ] + }, + { + "cell_type": "markdown", + "id": "b30587d9", + "metadata": {}, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + "\n", + "(Fa23 Final 10.1)\n", + "\n", + "Suppose we fit decision trees of varying depths to predict 'y' using 'x1' and 'x2'. The entire training set is shown in the table below.\n", + "\n", + "
\n", + "\n", + "What is:\n", + "\n", + "1. The entropy of a node containing all the training points.\n", + "1. The lowest possible entropy of a node in a fitted tree with depth 1 (two leaf nodes).\n", + "1. The lowest possible entropy of a node in a fitted tree with depth 2 (four leaf nodes)." + ] + }, + { + "cell_type": "markdown", + "id": "f191f3ca", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Summary, next time" + ] + }, + { + "cell_type": "markdown", + "id": "e08521e0", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Summary\n", + "\n", + "- A hyperparameter is a configuration that we choose before training a model; an important task in machine learning is selecting \"good\" hyperparameters.\n", + "- To choose hyperparameters, we use $k$-fold cross-validation. **This technique is very common, and you are expected to use it in Project 4!**\n", + " - See [Summary: Generalization](#Summary:-Generalization) for more.\n", + "- Decision trees can be used for both regression and classification; we've used them for classification.\n", + " - Decision trees are trained recursively. Each node corresponds to a \"yes\" or \"no\" question.\n", + " - To decide which \"yes\" or \"no\" question to ask, we choose the question with the lowest weighted entropy." + ] + }, + { + "cell_type": "markdown", + "id": "e62eb104", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Next time\n", + "\n", + "- An efficient technique for trying different combinations of hyperparameters.\n", + " - In other words, performing $k$-fold cross-validation _without_ a `for`-loop.\n", + "- Techniques for evaluating classifiers beyond accuracy.\n", + " - You'll need this for Project 4!" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.18" + }, + "livereveal": { + "scroll": true + }, + "rise": { + "transition": "none" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/resources/lectures/lec16/lec16.html b/resources/lectures/lec16/lec16.html new file mode 100644 index 0000000..a799ca7 --- /dev/null +++ b/resources/lectures/lec16/lec16.html @@ -0,0 +1,11197 @@ + + + + + +Notebook + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + diff --git a/resources/lectures/lec16/lec16.ipynb b/resources/lectures/lec16/lec16.ipynb new file mode 100644 index 0000000..c99f330 --- /dev/null +++ b/resources/lectures/lec16/lec16.ipynb @@ -0,0 +1,22968 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "id": "8c8a44b8", + "metadata": { + "slideshow": { + "slide_type": "skip" + } + }, + "outputs": [], + "source": [ + "from dsc80_utils import *\n", + "import lec16_util as util" + ] + }, + { + "cell_type": "markdown", + "id": "6ca51859", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Lecture 16 – Hyperparameters, Cross-Validation, and Decision Trees\n", + "\n", + "## DSC 80, Spring 2024" + ] + }, + { + "cell_type": "markdown", + "id": "22824200", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Announcements 📣\n", + "\n", + "- Lab 8 is due on **Wednesday**.\n", + "- Final Project Checkpoint 2 is due **Friday**.\n", + "- The Final Exam is on **Saturday, June 8 from 8-11AM** (room TBD).\n", + " - You can bring two double-sided notes sheets (you can bring your midterm notes sheet, if you want).\n", + " - More details to come." + ] + }, + { + "cell_type": "markdown", + "id": "bd851d5b", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Agenda 📆\n", + "\n", + "- Hyperparameters.\n", + "- Cross-validation.\n", + "- Decision trees." + ] + }, + { + "cell_type": "markdown", + "id": "ea74dd70", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + " \n", + "Remember, you can always ask questions at [**q.dsc80.com**](https://docs.google.com/forms/d/e/1FAIpQLScWbVZv9hBv-wX-ItKHUVRnkPMMtfJZVfErKE9GS7_8dFcRBQ/viewform)!" + ] + }, + { + "cell_type": "markdown", + "id": "b176323a", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Review: Hyperparameters" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "874cfdb7", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [], + "source": [ + "np.random.seed(23) # For reproducibility.\n", + "\n", + "def sample_from_pop(n=100):\n", + " x = np.linspace(-2, 3, n)\n", + " y = x ** 3 + (np.random.normal(0, 3, size=n))\n", + " return pd.DataFrame({'x': x, 'y': y})\n", + "\n", + "sample_1 = sample_from_pop()\n", + "sample_2 = sample_from_pop()" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "d2538011", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + " \n", + " " + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "px.scatter(sample_1, x='x', y='y', title='Sample 1')" + ] + }, + { + "cell_type": "markdown", + "id": "86b338d4", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Example: Polynomial regression\n", + "\n", + "We recently looked at an example of **polynomial regression**." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "72af7566", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig = util.train_and_plot(train_sample=sample_1, test_sample=sample_2, degs=[1, 3, 25], data_name='Sample 2')\n", + "fig.update_layout(title='Trained on Sample 1, Performance on Sample 2')" + ] + }, + { + "cell_type": "markdown", + "id": "42a671d7", + "metadata": {}, + "source": [ + "When building these models:\n", + "- We **got to choose** the degree of the polynomials (i.e. we chose 1, 3, and 25).\n", + "- We didn't get to choose the exact formulas for the three polynomials – their formulas were **learned from data**." + ] + }, + { + "cell_type": "markdown", + "id": "95f0e314", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Parameters vs. hyperparameters" + ] + }, + { + "cell_type": "markdown", + "id": "33d7d0c9", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- A **parameter** defines the relationship between variables in a model. \n", + " - **We learn parameters from data**.\n", + " - For instance, suppose we fit a degree 3 polynomial to data, and end up with\n", + " \n", + " $$H(x) = 1 - 2x + 13x^2 - 4x^3$$\n", + " \n", + " - 1, -2, 13, and -4 are parameters." + ] + }, + { + "cell_type": "markdown", + "id": "6778dd62", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- A **hyperparameter** is a parameter that we get to choose **before our model is fit to the data**.\n", + " - Think of hyperparameters as knobs 🎛 – **we get to pick and tune them!**\n", + " - **Polynomial degree** was a hyperparameter in the previous example, and we tried three different values: 1, 3, and 25." + ] + }, + { + "cell_type": "markdown", + "id": "1d1dfbc3", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- **Question**: How do we choose the \"right\" hyperparameter(s)?" + ] + }, + { + "cell_type": "markdown", + "id": "e2f97b99", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Training error vs. test error" + ] + }, + { + "cell_type": "markdown", + "id": "2cd2e40f", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We know that a model's performance on a **test set** is a good estimate of its ability to generalize to unseen data." + ] + }, + { + "cell_type": "markdown", + "id": "926bc6c1", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We want to find the hyperparameter that leads to the best **test set performance**." + ] + }, + { + "cell_type": "markdown", + "id": "e6e28be9", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Idea:\n", + " 1. Come up with a **list** of hyperparameters to try.\n", + " 2. For each hyperparameter, train the model on the training set and compute its performance on the test set.\n", + " 3. Pick the hyperparameter with the best performance on the test set." + ] + }, + { + "cell_type": "markdown", + "id": "b31001a9", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Training error vs. test error\n", + "\n", + "- Let's try this strategy on Sample 1 from our earlier example. \n", + "\n", + "- We'll try to fit a polynomial model on the dataset; we'll choose the polynomial's degree from the list [1, 2, ..., 25]." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "587b788e", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "px.scatter(sample_1, x='x', y='y', title='Sample 1')" + ] + }, + { + "cell_type": "markdown", + "id": "324c1eb4", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "First, we perform a train-test split." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "184d686f", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.model_selection import train_test_split" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "10ab3ddd", + "metadata": {}, + "outputs": [], + "source": [ + "X = sample_1[['x']]\n", + "y = sample_1['y']\n", + "\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=100)" + ] + }, + { + "cell_type": "markdown", + "id": "e98ff996", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Polynomial degree vs. train/test error\n", + "\n", + "Now, we'll create models with degree-1 through degree-25 polynomial features and compute their train and test errors." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "19abdbc7", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.metrics import mean_squared_error" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "825c48e8", + "metadata": {}, + "outputs": [], + "source": [ + "train_errs = []\n", + "test_errs = []\n", + "\n", + "for d in range(1, 26):\n", + " pl = make_pipeline(PolynomialFeatures(d), LinearRegression())\n", + " pl.fit(X_train, y_train)\n", + " train_errs.append(mean_squared_error(y_train, pl.predict(X_train), squared=False))\n", + " test_errs.append(mean_squared_error(y_test, pl.predict(X_test), squared=False))\n", + "\n", + "errs = pd.DataFrame({'Train Error': train_errs, 'Test Error': test_errs})" + ] + }, + { + "cell_type": "markdown", + "id": "ed06ac65", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "Let's look at the plots of training error vs. degree and test error vs. degree." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "7bdc8ddf", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig = px.line(errs)\n", + "fig.update_layout(showlegend=True, xaxis_title='Polynomial Degree', yaxis_title='RMSE')" + ] + }, + { + "cell_type": "markdown", + "id": "128dfbf6", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Training error appears to decrease as polynomial degree increases." + ] + }, + { + "cell_type": "markdown", + "id": "b724335d", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Test error appears to decrease until a \"valley\", and then increases again." + ] + }, + { + "cell_type": "markdown", + "id": "d76e0bee", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Here, we'd choose a degree of 3, since that degree has the **lowest test error**." + ] + }, + { + "cell_type": "markdown", + "id": "d0abe896", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Training error vs. test error\n", + "\n", + "The pattern we saw in the previous example is true more generally.\n", + "\n", + "
" + ] + }, + { + "cell_type": "markdown", + "id": "e728db66", + "metadata": {}, + "source": [ + "We pick the hyperparameter(s) at the \"valley\" of test error.\n", + "\n", + "Note that training error **tends** to underestimate test error, but it doesn't have to – i.e., it is possible for test error to be lower than training error (say, if the test set is \"easier\" to predict than the training set)." + ] + }, + { + "cell_type": "markdown", + "id": "0f6bc281", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Conducting train-test splits" + ] + }, + { + "cell_type": "markdown", + "id": "88811055", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Recall, training data is used to fit our model, and test data is used to evaluate our model.\n", + "\n", + "
" + ] + }, + { + "cell_type": "markdown", + "id": "ebc6b2ad", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- **Question**: _How_ should we split?\n", + " - `sklearn`'s `train_test_split` splits **randomly**, which usually works well.\n", + " - However, if there is some element of **time** in the training data (say, when predicting the future price of a stock), a better split is \"past\" and \"future\"." + ] + }, + { + "cell_type": "markdown", + "id": "1ec1df4b", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- **Question**: How _large_ should the split be, e.g. 90%-10% vs. 75%-25%?\n", + " - There's a tradeoff – a larger training set should lead to a \"better\" model, while a larger test set should lead to a better estimate of our model's ability to generalize.\n", + " - There's no \"right\" choice, but we usually choose between 10% to 25% for the test set." + ] + }, + { + "cell_type": "markdown", + "id": "5ddf0394", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### But wait..." + ] + }, + { + "cell_type": "markdown", + "id": "17c19ac8", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- With our current strategy, we are choosing the hyperparameter that creates the model that **performs best on the test set**." + ] + }, + { + "cell_type": "markdown", + "id": "4427cc42", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- As such, we are **overfitting to the test set** – the best hyperparameter for the test set might not be the best hyperparameter for a totally unseen dataset!" + ] + }, + { + "cell_type": "markdown", + "id": "01584eef", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- It seems like we need **another** split." + ] + }, + { + "cell_type": "markdown", + "id": "3b36a7aa", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Cross-validation" + ] + }, + { + "cell_type": "markdown", + "id": "39c7b052", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Idea: A single validation set\n", + "\n", + "
" + ] + }, + { + "cell_type": "markdown", + "id": "d344ac51", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "1. Split the data into three sets: training, validation, and test." + ] + }, + { + "cell_type": "markdown", + "id": "ac8d7d91", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "2. For each hyperparameter choice, train the model only on the training set, and evaluate the model's performance on the validation set." + ] + }, + { + "cell_type": "markdown", + "id": "0b66367d", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "3. Find the hyperparameter with the best validation performance." + ] + }, + { + "cell_type": "markdown", + "id": "fad93210", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "4. Retrain the final model on the training and validation sets, and report its performance on the test set." + ] + }, + { + "cell_type": "markdown", + "id": "f30e49e7", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "**Issue**: This strategy is too dependent on the validation set, which may be small and/or not a representative sample of the data. **We're not going to do this.** ❌" + ] + }, + { + "cell_type": "markdown", + "id": "a9131a04", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### A better idea: $k$-fold cross-validation\n", + "\n", + "Instead of relying on a single validation set, we can create $k$ validation sets, where $k$ is some positive integer (5 in the example below).\n", + "\n", + "
\n", + "\n", + "Since each data point is used for training $k-1$ times and validation once, the (averaged) validation performance should be a good metric of a model's ability to generalize to unseen data.\n", + "\n", + "$k$-fold cross-validation (or simply \"cross-validation\") is **the** technique we will use for finding hyperparameters, or more generally, for choosing between different possible models. **It's what you should use in your Final Project!** ✅" + ] + }, + { + "cell_type": "markdown", + "id": "b7a4b3f7", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Creating folds in `sklearn`\n", + "\n", + "`sklearn` has a `KFold` class that splits data into training and validation folds." + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "4555f6c0", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.model_selection import KFold" + ] + }, + { + "cell_type": "markdown", + "id": "319b67ce", + "metadata": {}, + "source": [ + "Let's use a simple dataset for illustration." + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "id": "04fcf601", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([10, 20, 30, 40, 50, 60])" + ] + }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "data = np.arange(10, 70, 10)\n", + "data" + ] + }, + { + "cell_type": "markdown", + "id": "5a43ac2a", + "metadata": {}, + "source": [ + "Let's instantiate a `KFold` object with $k=3$." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "id": "062e7597", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "KFold(n_splits=3, random_state=1, shuffle=True)" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "kfold = KFold(3, shuffle=True, random_state=1)\n", + "kfold" + ] + }, + { + "cell_type": "markdown", + "id": "fbdf8557", + "metadata": {}, + "source": [ + "Finally, let's use `kfold` to `split` `data`:" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "id": "79701be4", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "train: [10 40 50 60], validation: [20 30]\n", + "train: [20 30 40 60], validation: [10 50]\n", + "train: [10 20 30 50], validation: [40 60]\n" + ] + } + ], + "source": [ + "for train, val in kfold.split(data):\n", + " print(f'train: {data[train]}, validation: {data[val]}')" + ] + }, + { + "cell_type": "markdown", + "id": "7367e39b", + "metadata": {}, + "source": [ + "Note that each value in `data` is used for validation exactly once and for training exactly twice. Also note that because we set `shuffle=True` the folds are not simply `[10, 20]`, `[30, 40]`, and `[50, 60]`." + ] + }, + { + "cell_type": "markdown", + "id": "08de2c3f", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### $k$-fold cross-validation\n", + "\n", + "First, **shuffle** the entire training set randomly and **split** it into $k$ disjoint folds, or \"slices\". Then:" + ] + }, + { + "cell_type": "markdown", + "id": "86b39156", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- For each hyperparameter:\n", + " - For each slice:\n", + " - Let the slice be the \"validation set\", $V$.\n", + " - Let the rest of the data be the \"training set\", $T$.\n", + " - Train a model using the selected hyperparameter on the training set $T$.\n", + " - Evaluate the model on the validation set $V$.\n", + " - Compute the **average** validation score (e.g. RMSE) for the particular hyperparameter." + ] + }, + { + "cell_type": "markdown", + "id": "7abb6540", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Choose the hyperparameter with the best average validation score." + ] + }, + { + "cell_type": "markdown", + "id": "4620c78d", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### $k$-fold cross-validation in `sklearn`\n", + "\n", + "While you could manually use `KFold` to perform cross-validation, the `cross_val_score` function in `sklearn` implements $k$-fold cross-validation for us! \n", + "\n", + "```py\n", + "cross_val_score(estimator, X_train, y_train, cv)\n", + "```\n", + "\n", + "Specifically, it takes in:\n", + "- A `Pipeline` or estimator **that has not already been `fit`**.\n", + "- Training data.\n", + "- A value of $k$ (through the `cv` argument).\n", + "- (Optionally) A `scoring` metric.\n", + "\n", + "and performs $k$-fold cross-validation, returning the values of the scoring metric on each fold." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "id": "ca1244ea", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.model_selection import cross_val_score" + ] + }, + { + "cell_type": "markdown", + "id": "b95ca832", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### $k$-fold cross-validation in `sklearn`" + ] + }, + { + "cell_type": "markdown", + "id": "48eaa524", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Let's perform $k$-fold cross validation in order to help us pick a degree for polynomial regression from the list [1, 2, ..., 25]." + ] + }, + { + "cell_type": "markdown", + "id": "af6df303", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We'll use $k=5$ since it's a common choice (and the default in `sklearn`)." + ] + }, + { + "cell_type": "markdown", + "id": "4c570d08", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- For the sake of this example, we'll suppose `sample_1` is our \"training + validation data\", i.e. that our test data is in some other dataset.\n", + " - If this were not true, we'd first need to split `sample_1` into separate training and test sets." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "id": "d2e26f47", + "metadata": {}, + "outputs": [], + "source": [ + "errs_df = pd.DataFrame()\n", + "\n", + "for d in range(1, 26):\n", + " pl = make_pipeline(PolynomialFeatures(d), LinearRegression())\n", + " \n", + " # The `scoring` argument is used to specify that we want to compute the RMSE; \n", + " # the default is R^2. It's called \"neg\" RMSE because, \n", + " # by default, sklearn likes to \"maximize\" scores, and maximizing -RMSE is the same\n", + " # as minimizing RMSE.\n", + " errs = cross_val_score(pl, sample_1[['x']], sample_1['y'], \n", + " cv=5, scoring='neg_root_mean_squared_error')\n", + " errs_df[f'Deg {d}'] = -errs # Negate to turn positive (sklearn computed negative RMSE).\n", + " \n", + "errs_df.index = [f'Fold {i}' for i in range(1, 6)]\n", + "errs_df.index.name = 'Validation Fold'" + ] + }, + { + "cell_type": "markdown", + "id": "e8c7311d", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Next class, we'll look at how to implement this procedure without needing to `for`-loop over values of `d`." + ] + }, + { + "cell_type": "markdown", + "id": "5b6ed7f0", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### $k$-fold cross-validation in `sklearn`\n", + "\n", + "Note that for each choice of degree (our hyperparameter), we have **five** RMSEs, one for each \"fold\" of the data. This means that in total, $5 \\cdot 25 = 125$ models were trained/fit to data!" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "id": "d01ba737", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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" + ], + "text/plain": [ + " Deg 1 Deg 2 Deg 3 Deg 4 ... Deg 22 Deg 23 \\\n", + "Validation Fold ... \n", + "Fold 1 4.79 12.81 5.04 4.93 ... 8.78e+06 6.57e+07 \n", + "Fold 2 3.97 5.36 3.19 3.22 ... 2.93e+01 7.85e+01 \n", + "Fold 3 4.77 2.56 2.08 2.11 ... 3.03e+01 3.09e+01 \n", + "Fold 4 6.13 4.66 2.93 2.93 ... 6.27e+00 3.33e+01 \n", + "Fold 5 11.70 11.92 3.24 4.37 ... 8.36e+06 2.28e+08 \n", + "\n", + " Deg 24 Deg 25 \n", + "Validation Fold \n", + "Fold 1 1.41e+08 5.90e+08 \n", + "Fold 2 7.53e+01 3.13e+01 \n", + "Fold 3 4.24e+01 3.72e+01 \n", + "Fold 4 5.80e+01 9.69e+00 \n", + "Fold 5 8.17e+08 6.63e+09 \n", + "\n", + "[5 rows x 25 columns]" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "errs_df" + ] + }, + { + "cell_type": "markdown", + "id": "c914d676", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "We should choose the degree with the lowest **average** validation RMSE." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "id": "3dc1361c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'Deg 3'" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "errs_df.mean().idxmin()" + ] + }, + { + "cell_type": "markdown", + "id": "43aeacff", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Note that if we didn't perform $k$-fold cross-validation, but instead just used a single validation set, we may have ended up with a different result:" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "id": "de811f8a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Validation Fold\n", + "Fold 1 Deg 1\n", + "Fold 2 Deg 6\n", + "Fold 3 Deg 8\n", + "Fold 4 Deg 3\n", + "Fold 5 Deg 3\n", + "dtype: object" + ] + }, + "execution_count": 21, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "errs_df.idxmin(axis=1)" + ] + }, + { + "cell_type": "markdown", + "id": "50e5db5f", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + " \n", + "The RMSEs in Folds 1 and 5 are much higher than in other folds. Why? How we might fix this?" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "id": "c95f912b", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "px.scatter(sample_1, x='x', y='y', title='Sample 1')" + ] + }, + { + "cell_type": "markdown", + "id": "aa8239bf", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Another example: Tips\n", + "\n", + "We can also use $k$-fold cross-validation to determine which subset of features to use in a linear model that predicts tips by making one pipeline for each subset of features we want to evaluate." + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "id": "a721435f", + "metadata": {}, + "outputs": [], + "source": [ + "# make_column_transformer is a shortcut for the ColumnTransformer class\n", + "from sklearn.compose import make_column_transformer, make_column_selector\n", + "from sklearn.preprocessing import FunctionTransformer, OneHotEncoder" + ] + }, + { + "cell_type": "markdown", + "id": "77f774ac", + "metadata": {}, + "source": [ + "As we should always do, we'll perform a train-test split on `tips` and will only use the training data for cross-validation." + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "id": "e04a69f3", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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" + ], + "text/plain": [ + " total_bill tip sex smoker day time size\n", + "0 16.99 1.01 Female No Sun Dinner 2\n", + "1 10.34 1.66 Male No Sun Dinner 3\n", + "2 21.01 3.50 Male No Sun Dinner 3\n", + "3 23.68 3.31 Male No Sun Dinner 2\n", + "4 24.59 3.61 Female No Sun Dinner 4" + ] + }, + "execution_count": 24, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "tips = sns.load_dataset('tips')\n", + "tips.head()" + ] + }, + { + "cell_type": "code", + "execution_count": 61, + "id": "078f4566", + "metadata": {}, + "outputs": [], + "source": [ + "X = tips.drop('tip', axis=1)\n", + "y = tips['tip']\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=1)" + ] + }, + { + "cell_type": "code", + "execution_count": 62, + "id": "4e2853dc", + "metadata": {}, + "outputs": [], + "source": [ + "# A dictionary that maps names to Pipeline objects.\n", + "select = FunctionTransformer(lambda x: x)\n", + "pipes = {\n", + " 'total_bill only': make_pipeline(\n", + " make_column_transformer( (select, ['total_bill']) ),\n", + " LinearRegression(),\n", + " ),\n", + " 'total_bill + size': make_pipeline(\n", + " make_column_transformer( (select, ['total_bill', 'size']) ),\n", + " LinearRegression(),\n", + " ),\n", + " 'total_bill + size + OHE smoker': make_pipeline(\n", + " make_column_transformer(\n", + " (select, ['total_bill', 'size']),\n", + " (OneHotEncoder(drop='first'), ['smoker']),\n", + " ),\n", + " LinearRegression(),\n", + " ),\n", + " 'total_bill + size + OHE all': make_pipeline(\n", + " make_column_transformer(\n", + " (select, ['total_bill', 'size']),\n", + " (OneHotEncoder(drop='first'), ['smoker', 'sex', 'time', 'day']),\n", + " ),\n", + " LinearRegression(),\n", + " ),\n", + "}" + ] + }, + { + "cell_type": "code", + "execution_count": 63, + "id": "23b42ccf", + "metadata": {}, + "outputs": [], + "source": [ + "pipe_df = pd.DataFrame()\n", + "\n", + "for pipe in pipes:\n", + " errs = cross_val_score(pipes[pipe], X_train, y_train,\n", + " cv=5, scoring='neg_root_mean_squared_error')\n", + " pipe_df[pipe] = -errs\n", + " \n", + "pipe_df.index = [f'Fold {i}' for i in range(1, 6)]\n", + "pipe_df.index.name = 'Validation Fold'" + ] + }, + { + "cell_type": "code", + "execution_count": 64, + "id": "09420210", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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total_bill onlytotal_bill + sizetotal_bill + size + OHE smokertotal_bill + size + OHE all
Validation Fold
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" + ], + "text/plain": [ + " total_bill only total_bill + size \\\n", + "Validation Fold \n", + "Fold 1 1.32 1.27 \n", + "Fold 2 0.95 0.92 \n", + "Fold 3 0.77 0.86 \n", + "Fold 4 0.85 0.84 \n", + "Fold 5 1.10 1.07 \n", + "\n", + " total_bill + size + OHE smoker total_bill + size + OHE all \n", + "Validation Fold \n", + "Fold 1 1.27 1.29 \n", + "Fold 2 0.93 0.93 \n", + "Fold 3 0.86 0.87 \n", + "Fold 4 0.84 0.86 \n", + "Fold 5 1.07 1.08 " + ] + }, + "execution_count": 64, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pipe_df" + ] + }, + { + "cell_type": "code", + "execution_count": 65, + "id": "433f45de", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "total_bill only 1.00\n", + "total_bill + size 0.99\n", + "total_bill + size + OHE smoker 0.99\n", + "total_bill + size + OHE all 1.01\n", + "dtype: float64" + ] + }, + "execution_count": 65, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pipe_df.mean()" + ] + }, + { + "cell_type": "code", + "execution_count": 66, + "id": "be7a3541", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'total_bill + size + OHE smoker'" + ] + }, + "execution_count": 66, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pipe_df.mean().idxmin()" + ] + }, + { + "cell_type": "markdown", + "id": "7df9ae70", + "metadata": {}, + "source": [ + "Even though the third model has the lowest average validation RMSE, its average validation RMSE is very close to that of the other, simpler models, and as a result we'd likely use the simplest model in practice." + ] + }, + { + "cell_type": "markdown", + "id": "80cf1012", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + "\n", + "- Suppose you have a training dataset with 1000 rows.\n", + "- You want to decide between 20 hyperparameters for a particular model.\n", + "- To do so, you perform 10-fold cross-validation.\n", + "- **How many times is the first row in the training dataset (`X.iloc[0]`) used for training a model?**\n" + ] + }, + { + "cell_type": "markdown", + "id": "1c3b3ad4", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Summary: Generalization" + ] + }, + { + "cell_type": "markdown", + "id": "6a7e9ba8", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "1. Split the data into two sets: training and test." + ] + }, + { + "cell_type": "markdown", + "id": "5c3b7200", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "2. Use only the training data when designing, training, and tuning the model.\n", + " - Use $k$-fold cross-validation to choose hyperparameters and estimate the model's ability to generalize.\n", + " - Do not ❌ look at the test data in this step!" + ] + }, + { + "cell_type": "markdown", + "id": "492e6340", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "3. Commit to your final model and train it using the entire training set." + ] + }, + { + "cell_type": "markdown", + "id": "ad9dd3db", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "4. Test the data using the test data. If the performance (e.g. RMSE) is not acceptable, return to step 2." + ] + }, + { + "cell_type": "markdown", + "id": "bc3dc5a7", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "5. Finally, train on **all available data** and ship the model to production! 🛳" + ] + }, + { + "cell_type": "markdown", + "id": "662df112", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "🚨 This is the process you should **always** use! 🚨 " + ] + }, + { + "cell_type": "markdown", + "id": "dc44bead", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + " \n", + "Remember, you can always ask questions at [**q.dsc80.com**](https://docs.google.com/forms/d/e/1FAIpQLScWbVZv9hBv-wX-ItKHUVRnkPMMtfJZVfErKE9GS7_8dFcRBQ/viewform)!" + ] + }, + { + "cell_type": "markdown", + "id": "7e72e409", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Decision trees 🌲" + ] + }, + { + "cell_type": "markdown", + "id": "e6646d38", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "\n", + "Decision trees can be used for both regression and classification. We'll start by using them for **classification**." + ] + }, + { + "cell_type": "markdown", + "id": "942f1289", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Example: Should I get groceries?\n", + "\n", + "Decision trees make classifications by answering a series of yes/no questions.\n", + "\n", + "**Should I go to Trader Joe's to buy groceries today?**" + ] + }, + { + "cell_type": "markdown", + "id": "09e8a413", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "
" + ] + }, + { + "cell_type": "markdown", + "id": "1d8269fb", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Internal **nodes** of trees involve questions; leaf nodes make predictions $H(x)$." + ] + }, + { + "cell_type": "markdown", + "id": "bc8070e7", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "
" + ] + }, + { + "cell_type": "markdown", + "id": "2569dd49", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Example: Predicting diabetes" + ] + }, + { + "cell_type": "code", + "execution_count": 67, + "id": "f83270fa", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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PregnanciesGlucoseBloodPressureSkinThicknessInsulinBMIDiabetesPedigreeFunctionAgeOutcome
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..............................
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" + ], + "text/plain": [ + " Pregnancies Glucose BloodPressure SkinThickness Insulin BMI \\\n", + "0 6 148 72 35 0 33.6 \n", + "1 1 85 66 29 0 26.6 \n", + "2 8 183 64 0 0 23.3 \n", + ".. ... ... ... ... ... ... \n", + "765 5 121 72 23 112 26.2 \n", + "766 1 126 60 0 0 30.1 \n", + "767 1 93 70 31 0 30.4 \n", + "\n", + " DiabetesPedigreeFunction Age Outcome \n", + "0 0.63 50 1 \n", + "1 0.35 31 0 \n", + "2 0.67 32 1 \n", + ".. ... ... ... \n", + "765 0.24 30 0 \n", + "766 0.35 47 1 \n", + "767 0.32 23 0 \n", + "\n", + "[768 rows x 9 columns]" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "diabetes = pd.read_csv(Path('data') / 'diabetes.csv')\n", + "display_df(diabetes, cols=9)" + ] + }, + { + "cell_type": "code", + "execution_count": 68, + "id": "559c4fd6", + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "0 500\n", + "1 268\n", + "Name: Outcome, dtype: int64" + ] + }, + "execution_count": 68, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# 0 means no diabetes, 1 means yes diabetes.\n", + "diabetes['Outcome'].value_counts()" + ] + }, + { + "cell_type": "markdown", + "id": "fd3d4eb8", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- `'Glucose'` is measured in mg/dL (milligrams per deciliter)." + ] + }, + { + "cell_type": "markdown", + "id": "6c2ea02c", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- `'BMI'` is calculated as $\\text{BMI} = \\frac{\\text{weight (kg)}}{\\left[ \\text{height (m)} \\right]^2}$." + ] + }, + { + "cell_type": "markdown", + "id": "82906445", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Let's use `'Glucose'` and `'BMI'` to predict whether or not a patient has diabetes (`'Outcome'`)." + ] + }, + { + "cell_type": "markdown", + "id": "5f4d2b73", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Exploring the dataset\n", + "\n", + "First, a train-test split:" + ] + }, + { + "cell_type": "code", + "execution_count": 69, + "id": "4c699036", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.model_selection import train_test_split\n", + "X_train, X_test, y_train, y_test = (\n", + " train_test_split(diabetes[['Glucose', 'BMI']], diabetes['Outcome'], random_state=1)\n", + ")" + ] + }, + { + "cell_type": "markdown", + "id": "fb537748", + "metadata": {}, + "source": [ + "Class 0 (orange) is \"no diabetes\" and class 1 (blue) is \"diabetes\"." + ] + }, + { + "cell_type": "code", + "execution_count": 70, + "id": "4b4445bb", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig = (\n", + " X_train.assign(Outcome=y_train.astype(str))\n", + " .plot(kind='scatter', x='Glucose', y='BMI', color='Outcome', \n", + " color_discrete_map={'0': 'orange', '1': 'blue'},\n", + " title='Relationship between Glucose, BMI, and Diabetes')\n", + ")\n", + "fig" + ] + }, + { + "cell_type": "markdown", + "id": "c2f94087", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Building a decision tree\n", + "\n", + "Let's build a decision tree and interpret the results.\n", + "\n", + "The relevant class is `DecisionTreeClassifier`, from `sklearn.tree`." + ] + }, + { + "cell_type": "code", + "execution_count": 71, + "id": "12902788", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.tree import DecisionTreeClassifier" + ] + }, + { + "cell_type": "markdown", + "id": "c4b06503", + "metadata": {}, + "source": [ + "Note that we `fit` it the same way we `fit` earlier estimators.\n", + "\n", + "_You may wonder what `max_depth` and `criterion` do – more on this soon!_" + ] + }, + { + "cell_type": "code", + "execution_count": 72, + "id": "ae5efa16", + "metadata": {}, + "outputs": [], + "source": [ + "dt = DecisionTreeClassifier(max_depth=2, criterion='entropy')" + ] + }, + { + "cell_type": "code", + "execution_count": 73, + "id": "b37a3d08", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "DecisionTreeClassifier(criterion='entropy', max_depth=2)" + ] + }, + "execution_count": 73, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dt.fit(X_train, y_train)" + ] + }, + { + "cell_type": "markdown", + "id": "70c4fc97", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Visualizing decision trees\n", + "\n", + "Our fit decision tree is like a \"flowchart\", made up of a series of questions.\n", + "\n", + "As before, orange is \"no diabetes\" and blue is \"diabetes\"." + ] + }, + { + "cell_type": "code", + "execution_count": 74, + "id": "b8ba715f", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.tree import plot_tree" + ] + }, + { + "cell_type": "code", + "execution_count": 75, + "id": "cbe2313a", + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " 2024-05-27T21:27:31.954299\n", + " image/svg+xml\n", + " \n", + " \n", + " Matplotlib v3.5.1, https://matplotlib.org/\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + 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\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + "\n" + ], + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.figure(figsize=(15, 5))\n", + "plot_tree(dt, feature_names=X_train.columns, class_names=['no db', 'yes db'], \n", + " filled=True, fontsize=15, impurity=False);" + ] + }, + { + "cell_type": "markdown", + "id": "066f6931", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- To **classify a new data point**, we start at the top and answer the first question (i.e. \"Glucose <= 129.5\")." + ] + }, + { + "cell_type": "markdown", + "id": "57c96291", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- If the answer is \"**Yes**\", we move to the **left** branch, otherwise we move to the right branch." + ] + }, + { + "cell_type": "markdown", + "id": "f1432dfa", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We repeat this process until we end up at a leaf node, at which point we predict the most common class in that node.\n", + " - Note that each node has a `value` attribute, which describes the number of **training** individuals of each class that fell in that node." + ] + }, + { + "cell_type": "code", + "execution_count": 76, + "id": "9527edfe", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0 304\n", + "1 78\n", + "Name: Outcome, dtype: int64" + ] + }, + "execution_count": 76, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Note that the left node at depth 2 has a `value` of [304, 78].\n", + "y_train[X_train.query('Glucose <= 129.5').index].value_counts()" + ] + }, + { + "cell_type": "markdown", + "id": "7a73e4cc", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Evaluating classifiers\n", + "\n", + "The most common evaluation metric in classification is **accuracy**:\n", + "\n", + "$$\\text{accuracy} = \\frac{\\text{# data points classified correctly}}{\\text{# data points}}$$" + ] + }, + { + "cell_type": "code", + "execution_count": 77, + "id": "6aef8802", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.765625" + ] + }, + "execution_count": 77, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(dt.predict(X_train) == y_train).mean()" + ] + }, + { + "cell_type": "markdown", + "id": "ea2d61d2", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "The `score` method of a classifier computes accuracy by default (just like the `score` method of a regressor computes $R^2$ by default). We want our classifiers to have **high accuracy**." + ] + }, + { + "cell_type": "code", + "execution_count": 78, + "id": "48bde429", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.765625" + ] + }, + "execution_count": 78, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Training accuracy – same number as above\n", + "dt.score(X_train, y_train)" + ] + }, + { + "cell_type": "code", + "execution_count": 79, + "id": "e4045e13", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.7760416666666666" + ] + }, + "execution_count": 79, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Testing accuracy\n", + "dt.score(X_test, y_test)" + ] + }, + { + "cell_type": "markdown", + "id": "83213dcc", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Reflection" + ] + }, + { + "cell_type": "markdown", + "id": "47e36ccd", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Decision trees are easily interpretable: it is clear _how_ they make their predictions." + ] + }, + { + "cell_type": "markdown", + "id": "9f48d00b", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- They work with categorical data without needing to use one hot encoding." + ] + }, + { + "cell_type": "markdown", + "id": "0c046af0", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- They also can be used in multi-class classification problems, e.g. when there are more than 2 possible outcomes." + ] + }, + { + "cell_type": "markdown", + "id": "6ee7e7c8", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The _decision boundary_ of a decision tree can be arbitrarily complicated." + ] + }, + { + "cell_type": "markdown", + "id": "adb19142", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- **How are decision trees trained?**" + ] + }, + { + "cell_type": "markdown", + "id": "0cda55de", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### How are decision trees trained?\n", + "\n", + "Pseudocode:\n", + "\n", + "```python\n", + "def make_tree(X, y):\n", + " if all points in y have the same label C:\n", + " return Leaf(C)\n", + " f = best splitting feature # e.g. Glucose or BMI\n", + " v = best splitting value # e.g. 129.5\n", + " \n", + " X_left, y_left = X, y where (X[f] <= v)\n", + " X_right, y_right = X, y where (X[f] > v)\n", + " \n", + " left = make_tree(X_left, y_left)\n", + " right = make_tree(X_right, y_right)\n", + " \n", + " return Node(f, v, left, right)\n", + " \n", + "make_tree(X_train, y_train)\n", + "```" + ] + }, + { + "cell_type": "markdown", + "id": "670a105e", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### How do we measure the quality of a split?" + ] + }, + { + "cell_type": "markdown", + "id": "214872f7", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Our pseudocode for training a decision tree relies on finding the best way to \"split\" a node – that is, **the best question to ask** to help us narrow down which class to predict." + ] + }, + { + "cell_type": "markdown", + "id": "9621a012", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "**Intuition**: Suppose the distribution within a node looks like this (colors represent classes):\n", + "\n", + "
🟠🟠🟠🟠🟠🟠🔵🔵🔵🔵🔵🔵🔵
\n", + "\n", + "Split A:\n", + "- \"Yes\": 🟠🟠🟠🔵🔵🔵\n", + "- \"No\": 🟠🟠🟠🔵🔵🔵🔵\n", + "\n", + "Split B:\n", + "- \"Yes\": 🔵🔵🔵🔵🔵🔵\n", + "- \"No\": 🔵🟠🟠🟠🟠🟠🟠\n", + "\n", + "**Which split is \"better\"?**" + ] + }, + { + "cell_type": "markdown", + "id": "1701e7e3", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Split B, because there is \"less uncertainty\" in the resulting nodes in Split B than there is in Split A. Let's try and quantify this!" + ] + }, + { + "cell_type": "markdown", + "id": "c6af8be0", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Entropy" + ] + }, + { + "cell_type": "markdown", + "id": "46462eb4", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- For each class $C$ within a node, define $p_C$ as the proportion of points with that class.\n", + " - For example, the two classes may be \"yes diabetes\" and \"no diabetes\"." + ] + }, + { + "cell_type": "markdown", + "id": "a1414f15", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The **surprise** of drawing a point from the node at random and having it be class $C$ is:\n", + "\n", + "$$\n", + "- \\log_2 p_C\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "73b7095a", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The **entropy** of a node is the average surprise over all classes:\n", + "\n", + "\\begin{align}\n", + "\\text{entropy} &= - \\sum_C p_C \\log_2 p_C\n", + "\\end{align}" + ] + }, + { + "cell_type": "markdown", + "id": "dc0d04fd", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The entropy of 🟠🟠🟠🟠🟠🟠🟠🟠 is $ -1 \\log_2(1) = 0 $." + ] + }, + { + "cell_type": "markdown", + "id": "4d3dc897", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The entropy of 🟠🟠🟠🟠🔵🔵🔵🔵 is $ -0.5 \\log_2(0.5) - 0.5 \\log_2(0.5) = 1$." + ] + }, + { + "cell_type": "markdown", + "id": "98afe088", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The entropy of 🟠🔵🟢🟡🟣 is $ - \\log_2 \\frac{1}{5} = \\log_2(5) $\n", + " - In general, if a node has $n$ points, all with different labels, the entropy of the node is $ \\log_2(n) $." + ] + }, + { + "cell_type": "markdown", + "id": "e28fe85d", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Example entropy calculation\n", + "\n", + "Suppose we have:\n", + "\n", + "
🟠🟠🟠🟠🟠🟠🟠🟠🟠🟠🟠🟠🔵🔵🔵🔵🔵🔵
\n", + "\n", + "Split A:\n", + "- \"Yes\": 🟠🟠🟠🟠🟠🟠🔵\n", + "- \"No\": 🟠🟠🟠🟠🟠🟠🔵🔵🔵🔵🔵\n", + "\n", + "Split B:\n", + "- \"Yes\": 🟠🟠🟠🟠🟠🟠🔵🔵🔵\n", + "- \"No\": 🟠🟠🟠🟠🟠🟠🔵🔵🔵" + ] + }, + { + "cell_type": "markdown", + "id": "34bdc7b9", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "We choose the split that has the lowest **weighted entropy**, that is:\n", + "\n", + "$$\\text{entropy of split} = \\frac{\\# \\text{Yes}}{\\# \\text{Yes} + \\# \\text{No}} \\cdot \\text{entropy(Yes)} + \\frac{\\# \\text{No}}{\\# \\text{Yes} + \\# \\text{No}} \\cdot \\text{entropy(No)}$$" + ] + }, + { + "cell_type": "code", + "execution_count": 80, + "id": "49e63931", + "metadata": {}, + "outputs": [], + "source": [ + "def entropy(node):\n", + " props = pd.Series(list(node)).value_counts(normalize=True)\n", + " return -sum(props * np.log2(props))" + ] + }, + { + "cell_type": "code", + "execution_count": 81, + "id": "31a38b3f", + "metadata": {}, + "outputs": [], + "source": [ + "def weighted_entropy(yes_node, no_node):\n", + " yes_entropy = entropy(yes_node)\n", + " no_entropy = entropy(no_node)\n", + " yes_weight = len(yes_node) / (len(yes_node) + len(no_node))\n", + " return yes_weight * yes_entropy + (1 - yes_weight) * no_entropy" + ] + }, + { + "cell_type": "code", + "execution_count": 82, + "id": "e3e62521", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.8375578764623786" + ] + }, + "execution_count": 82, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Split A:\n", + "weighted_entropy(\"🟠🟠🟠🟠🟠🟠🔵\", \"🟠🟠🟠🟠🟠🟠🔵🔵🔵🔵🔵\")" + ] + }, + { + "cell_type": "code", + "execution_count": 83, + "id": "55cd0ce5", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.9182958340544896" + ] + }, + "execution_count": 83, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Split B:\n", + "weighted_entropy(\"🟠🟠🟠🟠🟠🟠🔵🔵🔵\", \"🟠🟠🟠🟠🟠🟠🔵🔵🔵\")" + ] + }, + { + "cell_type": "markdown", + "id": "f8d910d5", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Split A has the lower weighted entropy, so we'll use it." + ] + }, + { + "cell_type": "markdown", + "id": "61477abd", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Understanding entropy" + ] + }, + { + "cell_type": "code", + "execution_count": 84, + "id": "899b6d5e", + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " 2024-05-27T21:28:46.136616\n", + " image/svg+xml\n", + " \n", + " \n", + " Matplotlib v3.5.1, https://matplotlib.org/\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.figure(figsize=(15, 5))\n", + "plot_tree(dt, feature_names=X_train.columns, class_names=['no db', 'yes db'], \n", + " filled=True, fontsize=15, impurity=True);" + ] + }, + { + "cell_type": "markdown", + "id": "3cf312fe", + "metadata": { + "slideshow": { + "slide_type": "" + } + }, + "source": [ + "We can recreate the entropy calculations above! Note that the default `DecisionTreeClassifier` uncertaintly metric _isn't_ entropy; it used entropy because we set `criterion='entropy'` when defining `dt`. (The default metric, [Gini impurity](https://en.wikipedia.org/wiki/Decision_tree_learning#:~:text=Gini%20impurity%20measures%20how%20often,into%20a%20single%20target%20category.), is perfectly fine too!)" + ] + }, + { + "cell_type": "code", + "execution_count": 85, + "id": "648b1555", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.7302263747422792" + ] + }, + "execution_count": 85, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# The first node at depth 2 has an entropy of 0.73,\n", + "# both told to us above and verified here!\n", + "entropy([0] * 304 + [1] * 78)" + ] + }, + { + "cell_type": "markdown", + "id": "26aa9328", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + " \n", + "Remember, you can always ask questions at [**q.dsc80.com**](https://docs.google.com/forms/d/e/1FAIpQLScWbVZv9hBv-wX-ItKHUVRnkPMMtfJZVfErKE9GS7_8dFcRBQ/viewform)!" + ] + }, + { + "cell_type": "markdown", + "id": "63715f3a", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Tree depth\n", + "\n", + "Decision trees are trained by **recursively** picking the best split until:\n", + "\n", + "- all \"leaf nodes\" only contain training examples from a single class (good), or\n", + "- it is impossible to split leaf nodes any further (not good).\n", + "\n", + "By default, there is no \"maximum depth\" for a decision tree. As such, without restriction, decision trees tend to be very deep." + ] + }, + { + "cell_type": "code", + "execution_count": 86, + "id": "df1a32ea", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "DecisionTreeClassifier()" + ] + }, + "execution_count": 86, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dt_no_max = DecisionTreeClassifier()\n", + "dt_no_max.fit(X_train, y_train)" + ] + }, + { + "cell_type": "markdown", + "id": "c8b7ab31", + "metadata": {}, + "source": [ + "A decision tree fit on our training data has a depth of around 20! (It is so deep that `tree.plot_tree` errors when trying to plot it.)" + ] + }, + { + "cell_type": "code", + "execution_count": 87, + "id": "75c06bef", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "21" + ] + }, + "execution_count": 87, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dt_no_max.tree_.max_depth" + ] + }, + { + "cell_type": "markdown", + "id": "30caeb75", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "At first, this tree seems \"better\" than our tree of depth 2, since its training accuracy is much much higher:" + ] + }, + { + "cell_type": "code", + "execution_count": 88, + "id": "811e431b", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.9913194444444444" + ] + }, + "execution_count": 88, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dt_no_max.score(X_train, y_train)" + ] + }, + { + "cell_type": "code", + "execution_count": 89, + "id": "9413b6fa", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.765625" + ] + }, + "execution_count": 89, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Depth 2 tree.\n", + "dt.score(X_train, y_train)" + ] + }, + { + "cell_type": "markdown", + "id": "eaf99510", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "But recall, we truly care about **test set performance**, and this decision tree has **worse accuracy on the test set than our depth 2 tree**." + ] + }, + { + "cell_type": "code", + "execution_count": 90, + "id": "67f35a94", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.7291666666666666" + ] + }, + "execution_count": 90, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dt_no_max.score(X_test, y_test)" + ] + }, + { + "cell_type": "code", + "execution_count": 91, + "id": "7859ac6c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.7760416666666666" + ] + }, + "execution_count": 91, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Depth 2 tree.\n", + "dt.score(X_test, y_test)" + ] + }, + { + "cell_type": "markdown", + "id": "20e1e183", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Decision trees and overfitting" + ] + }, + { + "cell_type": "markdown", + "id": "4769f0d5", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Decision trees have a tendency to overfit. **Why is that?**" + ] + }, + { + "cell_type": "markdown", + "id": "0466852f", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Unlike linear classification techniques (like logistic regression or SVMs), **decision trees are non-linear**.\n", + " - They are also \"non-parametric\" – there are no $w^*$s to learn." + ] + }, + { + "cell_type": "markdown", + "id": "3d76fd16", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- While being trained, decision trees ask enough questions to effectively **memorize** the correct response values in the training set. However, the relationships they learn are often overfit to the noise in the training set, and don't generalize well." + ] + }, + { + "cell_type": "code", + "execution_count": 56, + "id": "b1e165b3", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig" + ] + }, + { + "cell_type": "markdown", + "id": "1f563e29", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- A decision tree whose depth is not restricted will achieve 100% accuracy on any training set, as long as there are no \"overlapping values\" in the training set.\n", + " - Two values overlap when they have the same features $x$ but different response values $y$ (e.g. if two patients have the same glucose levels and BMI, but one has diabetes and one doesn't)." + ] + }, + { + "cell_type": "markdown", + "id": "8564b552", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- **One solution**: Make the decision tree \"less complex\" by limiting the maximum depth." + ] + }, + { + "cell_type": "markdown", + "id": "87429ef6", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "Since `sklearn.tree`'s `plot_tree` can't visualize extremely large decision trees, let's create and visualize some smaller decision trees." + ] + }, + { + "cell_type": "code", + "execution_count": 57, + "id": "66549935", + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " 2024-05-27T12:02:56.199761\n", + " image/svg+xml\n", + " \n", + " \n", + " Matplotlib v3.5.1, https://matplotlib.org/\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "trees = {}\n", + "for d in [2, 4, 8]:\n", + " trees[d] = DecisionTreeClassifier(max_depth=d, random_state=1)\n", + " trees[d].fit(X_train, y_train)\n", + " \n", + " plt.figure(figsize=(15, 5), dpi=100)\n", + " plot_tree(trees[d], feature_names=X_train.columns, class_names=['no db', 'yes db'], \n", + " filled=True, rounded=True, impurity=False)\n", + " \n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "6b9c504a", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "As tree depth increases, complexity increases, and our trees are more prone to overfitting. This means model bias decreases, but model variance increases. \n", + "\n", + "**Question**: What is the \"right\" maximum depth to choose?" + ] + }, + { + "cell_type": "markdown", + "id": "c81ba76e", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Hyperparameters for decision trees" + ] + }, + { + "cell_type": "markdown", + "id": "ce6fcb22", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- `max_depth` is a hyperparameter for `DecisionTreeClassifier`." + ] + }, + { + "cell_type": "markdown", + "id": "bd81005e", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- There are many more hyperparameters we can tweak; look at [the documentation](https://scikit-learn.org/stable/modules/generated/sklearn.tree.DecisionTreeClassifier.html) for examples.\n", + " - `min_samples_split`: The minimum number of samples required to split an internal node.\n", + " - `criterion`: The function to measure the quality of a split (`'gini'` or `'entropy'`)." + ] + }, + { + "cell_type": "markdown", + "id": "2e2dc4ea", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- To ensure that our model generalizes well to unseen data, we need an efficient technique for trying different combinations of hyperparameters!" + ] + }, + { + "cell_type": "markdown", + "id": "da682d20", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We'll see that technique next time: `GridSearchCV`." + ] + }, + { + "cell_type": "markdown", + "id": "b30587d9", + "metadata": {}, + "source": [ + "
\n", + "

Question 🤔 (Answer at q.dsc80.com)

\n", + "
\n", + "\n", + "(Fa23 Final 10.1)\n", + "\n", + "Suppose we fit decision trees of varying depths to predict 'y' using 'x1' and 'x2'. The entire training set is shown in the table below.\n", + "\n", + "
\n", + "\n", + "What is:\n", + "\n", + "1. The entropy of a node containing all the training points.\n", + "1. The lowest possible entropy of a node in a fitted tree with depth 1 (two leaf nodes).\n", + "1. The lowest possible entropy of a node in a fitted tree with depth 2 (four leaf nodes)." + ] + }, + { + "cell_type": "markdown", + "id": "f191f3ca", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Summary, next time" + ] + }, + { + "cell_type": "markdown", + "id": "e08521e0", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Summary\n", + "\n", + "- A hyperparameter is a configuration that we choose before training a model; an important task in machine learning is selecting \"good\" hyperparameters.\n", + "- To choose hyperparameters, we use $k$-fold cross-validation. **This technique is very common, and you are expected to use it in Project 4!**\n", + " - See [Summary: Generalization](#Summary:-Generalization) for more.\n", + "- Decision trees can be used for both regression and classification; we've used them for classification.\n", + " - Decision trees are trained recursively. Each node corresponds to a \"yes\" or \"no\" question.\n", + " - To decide which \"yes\" or \"no\" question to ask, we choose the question with the lowest weighted entropy." + ] + }, + { + "cell_type": "markdown", + "id": "e62eb104", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Next time\n", + "\n", + "- An efficient technique for trying different combinations of hyperparameters.\n", + " - In other words, performing $k$-fold cross-validation _without_ a `for`-loop.\n", + "- Techniques for evaluating classifiers beyond accuracy.\n", + " - You'll need this for Project 4!" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.18" + }, + "livereveal": { + "scroll": true + }, + "rise": { + "transition": "none" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/resources/lectures/lec16/lec16_util.py b/resources/lectures/lec16/lec16_util.py new file mode 100644 index 0000000..2c06732 --- /dev/null +++ b/resources/lectures/lec16/lec16_util.py @@ -0,0 +1,127 @@ +import pandas as pd +import numpy as np + +import plotly.graph_objects as go +from plotly.subplots import make_subplots + +from sklearn.preprocessing import PolynomialFeatures +from sklearn.pipeline import Pipeline +from sklearn.linear_model import LinearRegression + +from dsc80_utils import * + + +def fit_polys(train_sample, degs): + # Create all three Pipelines. + pls = [ + Pipeline( + [("poly", PolynomialFeatures(d)), ("lin-reg", LinearRegression())] + ) + for d in degs + ] + + # Fit all three Pipelines. + [pl.fit(train_sample[["x"]], train_sample["y"]) for pl in pls] + + # Return the relevant fit Pipelines. + return pls + + +def train_and_plot(train_sample, test_sample, degs, data_name): + pred_funcs = fit_polys(train_sample, degs) + + rmses = [ + np.sqrt( + np.mean( + (pred_funcs[i].predict(test_sample[["x"]]) - test_sample["y"]) + ** 2 + ) + ) + for i in range(len(pred_funcs)) + ] + titles = [ + f"Degree {degs[i]}, RMSE: {np.round(rmses[i], 3)}" + for i in range(len(pred_funcs)) + ] + + fig = make_subplots(rows=1, cols=len(pred_funcs)) + + for i in range(len(pred_funcs)): + fig.add_trace( + go.Scatter( + x=test_sample["x"], + y=test_sample["y"], + name="Actual Data", + mode="markers", + marker={"color": "#4c72b0"}, + ), + row=1, + col=i + 1, + ) + + fig.add_trace( + go.Scatter( + x=test_sample["x"], + y=pred_funcs[i].predict(test_sample[["x"]]), + name="Predictions", + line={"color": "orange", "width": 4}, + ), + row=1, + col=i + 1, + ) + + fig["data"][0]["name"] = data_name + fig["data"][1]["name"] = "Fit Model" + + for i in range(2, len(fig["data"])): + fig["data"][i]["showlegend"] = False + + for i, val in enumerate(titles): + fig['layout'][f'xaxis{"" if i == 0 else i + 1}']['title'] = val + # fig['data'][i]['xaxis_title'] = titles[i] + + fig.update_layout(width=950, height=500) + + return fig + + +def plot_multiple_models(sample_1, sample_2, degs): + sample_1_funcs = fit_polys(sample_1, degs) + sample_2_funcs = fit_polys(sample_2, degs) + + fig = make_subplots( + rows=1, + cols=len(degs), + subplot_titles=[f"Degree {deg}" for deg in degs], + ) + + for i in range(len(degs)): + fig.add_trace( + go.Scatter( + x=sample_1["x"], + y=sample_1_funcs[i].predict(sample_1[["x"]]), + line={"color": "orange", "width": 4}, + ), + row=1, + col=i + 1, + ) + + fig.add_trace( + go.Scatter( + x=sample_2["x"], + y=sample_2_funcs[i].predict(sample_2[["x"]]), + line={"color": "purple", "width": 4}, + ), + row=1, + col=i + 1, + ) + + for i in range(2): + fig["data"][i]["name"] = f"Trained on Sample {i+1}" + + for i in range(2, len(fig["data"])): + fig["data"][i]["showlegend"] = False + + fig.update_layout(width=950, height=500) + + return fig diff --git a/resources/lectures/lec16/rise.css b/resources/lectures/lec16/rise.css new file mode 100644 index 0000000..00d2aa4 --- /dev/null +++ b/resources/lectures/lec16/rise.css @@ -0,0 +1,77 @@ +/* +We want custom styles for RISE slideshows, so we copy this file as rise.css +into each lecture subfolder and RISE will automatically include it. + +Use `make cp-styles` in the project root to copy this file to all lectures +*/ + +.reveal, +.reveal h1, +.reveal h2, +.reveal h3, +.reveal h4, +.reveal h5, +.reveal h6 { + /* reset slideshow fonts to system default */ + font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Helvetica, + Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; +} + +.reveal h1, +.reveal h2, +.reveal h3, +.reveal h4, +.reveal h5, +.reveal h6 { + font-weight: bold; +} + +/* Make table styles more similar to pandas */ +.reveal table th, +.reveal table td { + border: none; + padding: 0.5em; +} + +/* Table font is too small by default */ +.rise-enabled .jp-RenderedHTMLCommon table { + font-size: 125%; +} + +/* Make code font bigger */ +.cm-editor { + font-size: 18px; +} + +.rise-enabled .reveal pre { + margin: 0; +} + +/* +RISE sets table font-size to 75%, but this makes pandas tutor tables super +small +*/ +table.wstRoot { + font-size: 18px !important; +} + +.wstRoot td.rowLabel, +.wstRoot td.rowLabelLevelName, +.wstRoot .columnLabel, +.wstRoot .columnLabelLevelName { + font-size: 20px !important; +} + +/* using .sbs in a cell lets us put things side by side */ +.sbs { + display: flex; + gap: 1rem; +} + +.sbs > div { + flex-grow: 1; +} + +.sbs > div:last-child { + max-width: 30%; +}