Add MLDLM Lab03
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "FZEco2HK6D57"
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},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import pandas as pd\n",
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"from matplotlib import pyplot as plt"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "a3nCUqopXHwv"
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},
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"outputs": [],
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"source": [
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"RANDOM_SEED = 0x0"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "jjTkUw7BWulH"
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},
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"source": [
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"# Lab 03: Linear Regression"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "gNnZUk36Xz7_"
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},
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"source": [
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"For the first few Tasks, we will work with synthetic univariate data.\n",
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"We generate $100$ features $x_i \\in [-1, 1]$ as `x` and two different\n",
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"regression targets `y1` and `y2`."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "Ojta777H2ulb"
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},
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"outputs": [],
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"source": [
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"data_rng = np.random.default_rng(RANDOM_SEED)\n",
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"n = 100\n",
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"x = 2 * data_rng.random(n) - 1 # create n points between -1 and 1\n",
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"\n",
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"# setup synthetic linear data\n",
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"true_offset = 0.5\n",
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"true_slope = 1.25\n",
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"noise = data_rng.normal(loc=0., scale=0.25, size=(n,))\n",
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"\n",
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"y1 = true_offset + true_slope * x + noise\n",
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"\n",
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"\n",
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"# setup synthetic non-linear data\n",
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"y2 = true_offset + np.sin(np.pi * x) + noise"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "ntdpTWzqZqAU"
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},
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"source": [
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"# Task 1 (1 Point): Pearson Correlation"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "JbNJ7WhzbAtm"
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},
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"source": [
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"### Task 1a\n",
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"\n",
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"Plot `x` against the target variable `y1`.\n",
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"\n",
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"* use `plt.scatter`\n",
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"\n",
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"\n",
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"Do you think there is a linear relationship between `x` and the target?"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "MxYMdhfxyYAd"
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},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "6Ak0nQ0PDGpm"
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},
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"source": [
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"Plot `x` against the target variable `y2`.\n",
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"\n",
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"Do you think there is a linear relationship between `x` and the target?"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "HpzwoBdQDd-d"
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},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "HycYQm3tbvyf"
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},
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"source": [
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"### Task 1b\n",
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"\n",
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"In class you have seen the formula for the Pearson Correlation:\n",
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"$\\rho(a, b) = \\frac{\\sum_{i = 1}^{m} (a_i - \\bar{a})(b_i - \\bar{b})}{\\sqrt{\\sum_{i=1}^{m} (a_i - \\bar{a})^2\\sum_{i = 1}^{m}(b_i - \\bar{b})^2}} $, where $\\bar{a} = \\frac{1}{m}\\sum_{i=1}^{m} a_i$ and $\\bar{b} = \\frac{1}{m}\\sum_{i=1}^{m} b_i$.\n",
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"\n",
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"* Compute the Pearson Correlation $\\rho$ between `x` and the target `y1`.\n",
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"* Compute the Pearson Correlation between `x` and `y2`.\n",
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"* Check that you get the same result as the reference implementation"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "EUoJXIrCy0p6"
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},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "L_NesuDQddHS"
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},
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"outputs": [],
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"source": [
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"# Refer to the output of this cell to check whether your implementation of rho\n",
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"# is correct.\n",
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"\n",
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"from scipy.stats import pearsonr\n",
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"\n",
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"print(f\"rho(x, y1): {pearsonr(x, y1)[0]:.4f}\")\n",
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"print(f\"rho(x, y2): {pearsonr(x, y2)[0]:.4f}\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "Kr9OWmCilrAv"
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},
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"source": [
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"## 📢 **HAND-IN** 📢: Report in Moodle whether you solved this task."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "rbjhdwFceHlL"
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},
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"source": [
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"# Task 2 (2 Points): Univariate Linear Regression"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "ucnYGKbmecz_"
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},
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"source": [
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"### Task 2a\n",
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"\n",
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"You will now implement Linear Regression with a single variable. In class you have seen that the underlying model is: $y = \\theta_0 + \\theta_1x$.\n",
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"You also derived the maximum likelihood estimates for $\\theta_0$ and $\\theta_1$:\n",
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"\n",
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"* $\\hat{\\theta}_1 = \\frac{\\sum_{i=1}^{m} (x_i - \\bar{x})(y_i - \\bar{y})}{\\sum_{i=1}^{m}(x_i - \\bar{x})^2}$ with $\\bar{x} = \\frac{1}{m}\\sum_{i=1}^{m} x_i$ and $\\bar{y} = \\frac{1}{m}\\sum_{i=1}^{m} y_i$.\n",
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"* $\\hat{\\theta}_0 = \\bar{y} - \\hat{\\theta}_1\\bar{x}$\n",
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"\n",
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"In the following cell, implement the `.fit` and `.predict` methods:\n",
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"* In the `.predict` method you will have to apply the model to the input `x`\n",
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"* In the `.fit` method you will have to compute $\\hat{\\theta}_0$ and $\\hat{\\theta}_1$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "qS0Oa5Btgk74"
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},
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"outputs": [],
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"source": [
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"class UnivariateLinearRegression:\n",
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"\n",
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" def __init__(self):\n",
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" self.theta_0: float = 0.\n",
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" self.theta_1: float = 0.\n",
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"\n",
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" def predict(self, x):\n",
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" # y = theta_0 + theta_1 * x\n",
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" return None # TODO\n",
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"\n",
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" def fit(self, x, y):\n",
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"\n",
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" self.theta_1 = ... # TODO\n",
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" self.theta_0 = ... # TODO\n",
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"\n",
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" return self"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "9LzenH1UhLOs"
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},
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"source": [
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"### Task 2b\n",
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"\n",
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"Fit your linear model to `x` and the target `y1`.\n",
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"\n",
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"* Create an instance of the class `UnivariateLinearRegression`\n",
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"* fit the model using its `.fit` method\n",
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"* get the predicted values, using `.predict`\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "UHGuDWAntd8R"
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},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "elE3OfjHjBRO"
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},
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"source": [
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"* implement the function `plot_model`\n",
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"* use `plot_model` to plot your linear regression model given the true datapoints"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "T0eKDuRt1YOF"
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},
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"outputs": [],
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"source": [
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"def plot_model(x, y_pred, y_true, title):\n",
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" # TODO\n",
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" ...\n",
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" plt.show()\n",
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"\n",
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"plot_model(...)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "tt2RnAwAG1n9"
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},
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"source": [
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"* Fit another linear model to `x` and `y2`\n",
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"* get the predicted values\n",
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"* plot the model with `plot_model`"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "Ccq3GI17Ga2x"
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},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "E0i3gWvIl7nY"
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},
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"source": [
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"## 📢 **HAND-IN** 📢: A PDF document containing the following:\n",
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"\n",
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"* both plots containing the linear regression model and true datapoints\n",
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"* a short (2-3 sentences) interpretation of the curves: why do you think they look the way\n",
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"they do? can you draw any conclusions?\n",
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"\n",
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"**Solutions for Tasks 2, 3 and 4 should be in the same document: you will only upload 1 document with your solutions for all 3 tasks!**"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "0TK0Pi4ClphY"
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},
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"source": [
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"# Task 3 (4 Points): Univariate Linear Regression using Stochastic Gradient Descent"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "YL31gChVqLpC"
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},
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"source": [
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"### Task 3a\n",
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"\n",
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"In class you have seen an alternative version to estimate the parameters $\\theta_i$ of the linear regression models by using Gradient Descent.\n",
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"\n",
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"For the univariate linear regression model, the stochastic gradient descent updates look like this:\n",
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"* $\\theta_{0}^{(t+1)} = \\theta_{0}^{(t)} - \\alpha (\\theta_{0}^{(t)} + \\theta_{1}^{(t)} x_t - y_t)$\n",
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"* $\\theta_{1}^{(t+1)} = \\theta_{1}^{(t)} - \\alpha (\\theta_{0}^{(t)} + \\theta_{1}^{(t)} x_t - y_t) x_t$\n",
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"\n",
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"Here $\\alpha$ is the learning rate, and $(x_t, y_t)$ is the data point sampled\n",
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"at time $t$.\n",
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"\n",
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"\n",
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"In the following cell, implement the `.fit` and `.predict` methods:\n",
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"* In the `.predict` method you will have to apply the model to the input `x`.\n",
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"* In the `.fit` method you will have to implement the update equations for\n",
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"$\\theta_0$ and $\\theta_1$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "wJMHvQmXmVKr"
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},
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"outputs": [],
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"source": [
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"class SGDUnivariateLinearRegression:\n",
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"\n",
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" def __init__(self):\n",
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" self.theta_0: float = 0.\n",
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" self.theta_1: float = 0.\n",
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" self.rng = np.random.default_rng(RANDOM_SEED)\n",
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"\n",
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" def predict(self, x):\n",
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" # y = theta_0 + theta_1 * x\n",
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" return None # TODO\n",
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"\n",
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" def fit(self, x, y, n_iter: int = 100, learning_rate: float = 1.0):\n",
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" for t in range(n_iter):\n",
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" sample_ix = self.rng.integers(0, len(x))\n",
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"\n",
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" xt = x[sample_ix]\n",
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" yt = y[sample_ix]\n",
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"\n",
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" # TODO: update self.theta_0 and self.theta_1 SIMULTANEOUSLY (!!!) according to their update equations\n",
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"\n",
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" return self"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "MHLBmTm4vK9p"
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},
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"source": [
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"### Task 3b\n",
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"\n",
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"Run SGD for `x` and the target `y1` and compute the mean squared error (MSE).\n",
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"The MSE is defined as: $\\frac{1}{n}\\sum_{i=1}^{n} (\\hat{y}_i - y_i)^2$, where\n",
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"$\\hat{y}$ are the model predictions.\n",
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"\n",
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"* Create an instance of the class `SGDUnivariateLinearRegression`\n",
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"* fit the model using its `.fit` method\n",
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"* get the predicted values, using `.predict`\n",
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"* implement the `mse` function\n",
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"* compute the MSE of your predictions"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "CZ1szyQhK9so"
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},
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"outputs": [],
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"source": [
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"def mse(y_pred, y_true):\n",
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" # TODO\n",
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" return 0."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "V35vBU5Yti8Z"
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},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "hSsE1o6GwA3K"
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},
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"source": [
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"### Task 3c\n",
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"\n",
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"You will now plot the learning curves for different learning rates $\\alpha$.\n",
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"A learning curves shows how a model's performance changes with increasing number of update steps.\n",
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"In our case we will plot the model's MSE as a function of the number of update\n",
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"steps `n_iter` for different values of `learning_rate`.\n",
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"\n",
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"In the following cell we setup most of the scaffold to create this plot. Follow\n",
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"the instructions in the comments to finish the plots."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "4Rr5ix7LNISB"
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},
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"outputs": [],
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"source": [
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"n_iters = [50, 100, 200, 500, 1000, 2000]\n",
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"learning_rates = [1., .1, .01]\n",
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"\n",
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"# we plot the MSE achieved by the closed form model as a reference\n",
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"closed_form = UnivariateLinearRegression()\n",
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"closed_form.fit(x, y1)\n",
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"mse_base = mse(y_pred=closed_form.predict(x), y_true=y1)\n",
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"plt.plot(n_iters, np.ones_like(n_iters) * mse_base, label=\"closed form\", linestyle='--', c='b')\n",
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"\n",
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"for alpha in learning_rates:\n",
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" mses = []\n",
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" for n_iter in n_iters:\n",
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" # fit a SGDUnivariateLinearRegression model using n_iter=n_iter and\n",
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" # learning_rate=alpha\n",
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" # compute its mse and append the mse value to the mses list\n",
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"\n",
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" mse_ = 1. # replace with mse calculation\n",
|
||||
" mses.append(mse_)\n",
|
||||
" plt.plot(n_iters, mses, label=f\"alpha = {alpha:.2f}\")\n",
|
||||
"\n",
|
||||
"plt.xlabel(\"n_iter\")\n",
|
||||
"plt.ylabel(\"MSE\")\n",
|
||||
"plt.legend()\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "SmCkMMJyEEgV"
|
||||
},
|
||||
"source": [
|
||||
"## 📢 **HAND-IN** 📢: A PDF document containing the following:\n",
|
||||
"\n",
|
||||
"* the final plot containing learning curves\n",
|
||||
"* a short (2-3 sentences) interpretation of the curves: why do you think they look the way\n",
|
||||
"they do? can you draw any conclusions?\n",
|
||||
"\n",
|
||||
"In case you were not able to arrive at the final plot:\n",
|
||||
"\n",
|
||||
"* include screenshots of the code you wrote so we can assign partial credit\n",
|
||||
"\n",
|
||||
"**Solutions for Tasks 2, 3 and 4 should be in the same document: you will only upload 1 document with your solutions for all 3 tasks!**\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "dgrNtwsPyigH"
|
||||
},
|
||||
"source": [
|
||||
"# Task 4 (3 Points): Multivariate Linear Regression"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "_sPWegXCg2y1"
|
||||
},
|
||||
"source": [
|
||||
"In this task we will apply linear regression to non-synthetic data.\n",
|
||||
"The variable `X` is a `pandas` `Dataframe` containing features and `y` contains\n",
|
||||
"the target. Read through the description to get an idea of the different variables."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"id": "djGUQ3kVx9ob"
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import load_diabetes\n",
|
||||
"\n",
|
||||
"data = load_diabetes(as_frame=True)\n",
|
||||
"\n",
|
||||
"X = data['data']\n",
|
||||
"y = data['target']\n",
|
||||
"description = data['DESCR']\n",
|
||||
"\n",
|
||||
"print(description)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "byOVt9t9_2c7"
|
||||
},
|
||||
"source": [
|
||||
"### Task 4a\n",
|
||||
"\n",
|
||||
"Implement linear regression using `sklearn`.\n",
|
||||
"\n",
|
||||
"* create an instance of the class `sklearn.linear_model.LinearRegression`. Refer to the documentation at: https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html\n",
|
||||
"* call its `.fit` method\n",
|
||||
"* get the predicted values with `.predict`"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"id": "eyiU4nCQBovr"
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.linear_model import LinearRegression"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"id": "G4AktC189PAc"
|
||||
},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "qQUdYHOXpeLd"
|
||||
},
|
||||
"source": [
|
||||
"### Task 4b\n",
|
||||
"\n",
|
||||
"The estimated parameters $\\theta$ of the linear model can be found in the `.coef_` member variable. The feature names can be found in the `.feature_names_in_` member variable. They are the same as the names of the columns of `X` and should be in the same order.\n",
|
||||
"\n",
|
||||
"Visualize the estimated parameters and the feature names in a bar plot.\n",
|
||||
"\n",
|
||||
"Using these, answer the following questions:\n",
|
||||
"\n",
|
||||
"* Which are the 3 most influential features?\n",
|
||||
"* How do you interpret the sign of the coefficients?\n",
|
||||
"* If you had to exclude 1 feature, which one would you select and why?"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"id": "odXnubfHqrfc"
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "xa_HDxFeolBj"
|
||||
},
|
||||
"source": [
|
||||
"## 📢 **HAND-IN** 📢: A PDF document containing the following:\n",
|
||||
"\n",
|
||||
"* the bar plot\n",
|
||||
"* your answers to the questions in Task 4b\n",
|
||||
"\n",
|
||||
"**Solutions for Tasks 2, 3 and 4 should be in the same document: you will only upload 1 document with your solutions for all 3 tasks!**\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"private_outputs": true,
|
||||
"provenance": []
|
||||
},
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3",
|
||||
"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": "3.8.16"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 0
|
||||
}
|
||||
Loading…
Reference in a new issue