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<!DOCTYPE html>
<html lang="en"><head>
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<title>Introduction to Machine Learning</title>
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<body class="quarto-light">
<div class="reveal">
<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Introduction to Machine Learning</h1>
<p class="subtitle">Fundamentals of Machine Learning for NHS using R</p>
<div class="quarto-title-authors">
</div>
</section>
<section>
<section id="what-is-machine-learning" class="title-slide slide level1 center">
<h1>What is Machine Learning?</h1>
</section>
<section id="machine-learning-is" class="slide level2">
<h2>Machine Learning is…</h2>
<blockquote>
<p>Field of study that gives computers the ability to learn without being explicitly programmed</p>
</blockquote>
<p>Arthur Samuel is credited for this definition, but there is no source of it!</p>
<aside><div>
<p>See <a href="https://datascience.stackexchange.com/questions/37078/source-of-arthur-samuels-definition-of-machine-learning">Arthur Samuel’s definition of machine learning</a>.</p>
</div></aside></section></section>
<section>
<section id="opinionated-quick-history" class="title-slide slide level1 center">
<h1>Opinionated Quick History</h1>
</section>
<section id="arthur-samuel" class="slide level2 scrollable smaller">
<h2>Arthur Samuel</h2>
<div class="columns">
<div class="column" style="width:80%;">
<ul>
<li><p>Was a pioneer of artificial-intelligence research</p></li>
<li><p>In 1959, while working at IBM, he published a paper that popularized the term <em>machine learning</em></p></li>
</ul>
<blockquote>
<p>Programming computers to learn from experience should eventually eliminate the need for much of this detailed programming effort.</p>
</blockquote>
<blockquote>
<p>A computer can be programmed so that it will learn to play a better game of checkers than can be played by the person who wrote the program.</p>
</blockquote>
<p><img data-src="Img/Checkers.jpg"></p>
</div><div class="column" style="width:20%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\arthur_samuel.jpg"></p>
</figure>
</div>
</div>
</div>
</div>
</div>
<aside><div>
<p>References:</p>
<ul>
<li><p>A. L. Samuel, <a href="https://ieeexplore.ieee.org/document/5392560">Some Studies in Machine Learning Using the Game of Checkers,</a> in <em>IBM Journal of Research and Development</em>, vol. 3, no. 3, pp. 210-229, July 1959, doi: 10.1147/rd.33.0210.</p></li>
<li><p>A. L. Samuel, <a href="https://ieeexplore.ieee.org/document/5391906">Some Studies in Machine Learning Using the Game of Checkers. II—Recent Progress,</a> in <em>IBM Journal of Research and Development</em>, vol. 11, no. 6, pp. 601-617, Nov. 1967, doi: 10.1147/rd.116.0601.</p></li>
<li><p><a href="https://en.wikipedia.org/wiki/Arthur_Samuel_(computer_scientist)">Arthur Samuel</a> on Wikipedia</p></li>
</ul>
</div></aside></section>
<section id="deep-blue-1996-1997" class="slide level2 scrollable smaller">
<h2>Deep Blue (1996-1997)</h2>
<div class="quarto-layout-panel">
<div class="quarto-layout-row quarto-layout-valign-top">
<div class="quarto-layout-cell" style="flex-basis: 50.0%;justify-content: center;">
<div class="quarto-figure quarto-figure-right">
<figure>
<p><img data-src="Img/deep_blue.jpg"></p>
</figure>
</div>
</div>
<div class="quarto-layout-cell" style="flex-basis: 50.0%;justify-content: center;">
<div class="quarto-figure quarto-figure-left">
<figure>
<p><img data-src="Img/garry_kasparov.jpg"></p>
</figure>
</div>
</div>
</div>
</div>
<ul>
<li><p>February 1996, first computer program to defeat a world champion in a classical game under tournament regulations (Kasparov–Deep Blue 4–2)</p></li>
<li><p>March 1997, first computer program to defeat a world champion in a match under tournament regulations (Deep Blue–Kasparov 3½–2½)</p></li>
</ul>
<aside><div>
<p>References:</p>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov">Deep Blue versus Garry Kasparov</a> on Wikipedia</li>
</ul>
</div></aside></section>
<section id="alphago-2016" class="slide level2 scrollable smaller">
<h2>AlphaGo (2016)</h2>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img/alpha_go.jpg"></p>
</figure>
</div>
<aside><div>
<p>References:</p>
<ul>
<li><p>BBC News, <a href="https://www.bbc.co.uk/news/technology-35785875">Artificial intelligence: Google’s AlphaGo beats Go master Lee Se-dol</a></p></li>
<li><p>DeepMind, <a href="https://www.youtube.com/watch?v=WXuK6gekU1Y&t=4167s">AlphaGo - The Movie</a> award winning documentary</p></li>
<li><p><a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol">AlphaGo versus Lee Sedol</a> on Wikipedia</p></li>
</ul>
</div></aside></section></section>
<section>
<section id="back-to-our-question-what-is-machine-learning" class="title-slide slide level1 center">
<h1>Back to our Question! What is Machine Learning?</h1>
</section>
<section id="another-definition" class="slide level2">
<h2>Another Definition</h2>
<div class="columns">
<div class="column" style="width:75%;">
<blockquote>
<p>A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P if its performance at tasks in T, as measured by P, improves with experience E</p>
</blockquote>
</div><div class="column" style="width:25%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\mlbook_mitchell.gif" height="300"></p>
</figure>
</div>
</div>
</div>
</div>
</div>
<aside><div>
<p>References:</p>
<ul>
<li>Tom Mitchell, <a href="http://www.cs.cmu.edu/~tom/mlbook.html">Machine Learning</a>, McGraw Hill, 1997</li>
</ul>
</div></aside></section>
<section id="section" class="slide level2">
<h2></h2>
<p><img data-src="Img/algorithms_data_features.jpg"></p>
<aside><div>
<p>Image Credit: Vasily Zubarev, <a href="https://vas3k.com/blog/machine_learning/">Machine Learning for Everyone</a></p>
</div></aside></section>
<section id="section-1" class="slide level2">
<h2></h2>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\ai_ml_nn_dl.jpg" height="600"></p>
</figure>
</div>
</div>
</div>
<aside><div>
<p>Image Credit: Vasily Zubarev, <a href="https://vas3k.com/blog/machine_learning/">Machine Learning for Everyone</a></p>
</div></aside></section>
<section id="so-what-is-machine-learning" class="slide level2">
<h2>So What is Machine Learning?</h2>
<blockquote>
<p>The term <em>machine learning</em> refers to the automated detection of meaningful patterns in data.</p>
</blockquote>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\big_data_wave.jpg" height="300"></p>
</figure>
</div>
</div>
</div>
<aside><div>
<p>References:</p>
<ul>
<li><p>Shai Shalev-Shwartz and Shai Ben-David, <em>Understanding Machine Learning</em>, 2014</p></li>
<li><p>Image Credit <a href="https://www.meetcortex.com/blog/3-easy-steps-to-finding-patterns-in-your-big-data">Cortex</a></p></li>
</ul>
</div></aside></section>
<section id="machine-learning-applications-we-may-use-everyday" class="slide level2 smaller">
<h2>Machine Learning Applications We May Use Everyday</h2>
<ul>
<li><p>Email spam and malware filters (Gmail, Outlook, …)</p></li>
<li><p>Commute estimation (Google Maps, Waze, …)</p></li>
<li><p>Fraud detection (banking)</p></li>
<li><p>Online customer support (chatbots)</p></li>
<li><p>Image recognition (Facebook, Pinterest, …)</p></li>
<li><p>Search engine results refining (Google)</p></li>
<li><p>Product reccomendation (Amazon)</p></li>
<li><p>Healthcare</p></li>
<li><p>Virtual personal smart assistants (Alexa, Cortana, Google Assistant, Siri)</p></li>
</ul>
<p>…</p>
</section></section>
<section>
<section id="supervised-and-unsupervised-learning" class="title-slide slide level1 center">
<h1>Supervised and Unsupervised Learning</h1>
</section>
<section id="learning-settings" class="slide level2 smaller">
<h2>Learning Settings</h2>
<div class="columns">
<div class="column" style="width:50%;">
<p><strong>Learning Problems</strong></p>
<ul>
<li>Supervised Learning</li>
<li>Unsupervised Learning</li>
<li>Reinforcement Learning</li>
</ul>
<p><strong>Statistical Inference</strong></p>
<ul>
<li>Inductive Learning</li>
<li>Deductive Learning</li>
<li>Transductive Learning</li>
</ul>
</div><div class="column" style="width:50%;">
<p><strong>Hybrid Learning Problems</strong></p>
<ul>
<li>Semi Supervised Learning</li>
<li>Self Supervised Learning</li>
<li>Multi Instance Learning</li>
</ul>
<p><strong>Learning Techniques</strong></p>
<ul>
<li>Multi Task Learning</li>
<li>Active Learning</li>
<li>Online Learning</li>
<li>Transfer Learning</li>
<li>Ensemble Learning</li>
</ul>
</div>
</div>
<aside><div>
<p>Jason Brownlee, <a href="https://machinelearningmastery.com/types-of-learning-in-machine-learning/">Different Types of Learning in Machine Learning</a></p>
</div></aside></section>
<section id="section-2" class="slide level2 scrollable">
<h2></h2>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img/types_of_machine_learning.jpg"></p>
</figure>
</div>
<aside><div>
<p>Image Credit: Vasily Zubarev, <a href="https://vas3k.com/blog/machine_learning/">Machine Learning for Everyone</a></p>
</div></aside></section>
<section id="supervised-vs-unsupervised-learning" class="slide level2">
<h2>Supervised vs Unsupervised Learning</h2>
<ul>
<li><p>In <strong>supervised learning</strong> we are given a dataset and already know what our correct output should look like, assuming that there is a relationship between the input and the output.</p></li>
<li><p>In <strong>unsupervised learning</strong> we have no idea what our results should look like. Unsupervised methods try to capture patterns without knowing how these patterns look like.</p></li>
</ul>
</section>
<section id="supervised_unsupervised" class="slide level2 scrollable">
<h2></h2>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\supervised_unsupervised.jpg" height="900"></p>
</figure>
</div>
</div>
</div>
<aside><div>
<p>Image Credit: Vasily Zubarev, <a href="https://vas3k.com/blog/machine_learning/">Machine Learning for Everyone</a></p>
</div></aside></section>
<section id="supervised-vs-unsupervised-learning-1" class="slide level2">
<h2>Supervised vs Unsupervised Learning</h2>
<div class="columns">
<div class="column" style="width:50%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\supervised.png" width="370" height="400"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:50%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\unsupervised.png" width="370" height="400"></p>
</figure>
</div>
</div>
</div>
</div>
</div>
</section>
<section id="in-essence" class="slide level2">
<h2>In Essence</h2>
<ul>
<li><p><strong>Supervised learning</strong> wants to build predictive models</p></li>
<li><p><strong>Unsupervised learning</strong> wants to build descriptive models</p></li>
</ul>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\supervised_and_unsupervised.jpg" height="250"></p>
</figure>
</div>
</div>
</div>
<aside><div>
<p>Bradley Boehmke and Brandon M. Greenwell, <em>Hands-On Machine Learning with R</em></p>
<p>Image credit <a href="https://www.javatpoint.com/difference-between-supervised-and-unsupervised-learning">java T point</a></p>
</div></aside></section></section>
<section id="in-this-course" class="title-slide slide level1 center">
<h1>In This Course</h1>
<p>We will talk ONLY about supervised learning techniques</p>
</section>
<section>
<section id="some-maths" class="title-slide slide level1 smaller center">
<h1>Some Maths</h1>
<p><br></p>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\math_fear.jpg" height="300"></p>
</figure>
</div>
</div>
</div>
<p><a href="https://www.etutorworld.com/blog/online-tutoring-to-overcome-fear-of-math/">Image Credit</a></p>
</section>
<section id="what-are-we-actually-trying-to-do" class="slide level2">
<h2>What Are We Actually Trying to Do?</h2>
<p>In supervised learning we are assuming that there is a relationship between <em>input</em> (usually denoted as <span class="math inline">\(X\)</span>) and <em>output</em> (usually denoted as <span class="math inline">\(Y\)</span>), and we want to express it in terms of a <strong>mathematical function</strong>.</p>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\maths_dance.png" width="825" height="275"></p>
</figure>
</div>
</div>
</div>
<aside><div>
<p>Maths Dance Moves from <a href="https://www.imaginary.org/gallery/maths-dance-moves">Imaginary</a></p>
</div></aside></section>
<section id="what-is-a-mathematical-function" class="slide level2">
<h2>What is a (Mathematical) Function?</h2>
<p>Given two sets, a <strong>function</strong> is a relationship that associates to each element of the first set, <em>one and only one</em> element of the second set.</p>
<p>If <span class="math inline">\(f\)</span> is a function from the set <span class="math inline">\(X\)</span> to the set <span class="math inline">\(Y\)</span> we write</p>
<p><span class="math display">\[f: X \to Y\]</span></p>
<p>The element <span class="math inline">\(y\)</span> of <span class="math inline">\(Y\)</span> associated by <span class="math inline">\(f\)</span> to the element <span class="math inline">\(x\)</span> of <span class="math inline">\(X\)</span> is denoted by <span class="math inline">\(f(x)\)</span>. Therefore we write</p>
<p><span class="math display">\[y = f(x)\]</span></p>
</section>
<section id="terminology" class="slide level2">
<h2>Terminology</h2>
<p>We said that <span class="math inline">\(X\)</span> and <span class="math inline">\(Y\)</span> are sets, more precisely they are the <em>input</em> and the <em>output</em> set respectively.</p>
<p>But we will also use <span class="math inline">\(X\)</span> and <span class="math inline">\(Y\)</span> also to name variables.</p>
<p>Various terms are used interchangeably:</p>
<ul>
<li><p><span class="math inline">\(X\)</span>: “independent variable”, “input variable”, “predictor”, “feature”, “attribute”.</p></li>
<li><p><span class="math inline">\(Y\)</span>: “dependent variable”, “output variable”, “target”, “response”, “outcome”.</p></li>
</ul>
</section>
<section id="hence-we-want-to-find-a-function" class="slide level2">
<h2>Hence We Want to Find a Function!</h2>
<p>Start with <span class="math inline">\(N\)</span> input-output pairs of observed data</p>
<p><span class="math display">\[
\begin{multline}
(x^{(1)},y^{(1)}), (x^{(2)},y^{(2)}), \dots, (x^{(N)},y^{(N)}) = \\
= \left\{(x^{(i)},y^{(i)})\right\}_{i = 1}^N
\end{multline}
\]</span></p>
<p>Problem: find the function <span class="math inline">\(f: X \to Y\)</span> that has generated the observed outputs given the corresponding inputs.</p>
</section>
<section id="a-visual-example" class="slide level2 smaller">
<h2>A Visual Example</h2>
<div class="panel-tabset">
<ul id="tabset-1" class="panel-tabset-tabby"><li><a data-tabby-default="" href="#tabset-1-1">Given the Data…</a></li><li><a href="#tabset-1-2">A Possible Solution</a></li><li><a href="#tabset-1-3">Another Option</a></li></ul>
<div class="tab-content">
<div id="tabset-1-1">
<div class="cell">
<div class="cell-output-display">
<p><img data-src="01_Intro_files/figure-revealjs/unnamed-chunk-12-1.png" width="960"></p>
</div>
</div>
</div>
<div id="tabset-1-2">
<div class="cell">
<div class="cell-output-display">
<p><img data-src="01_Intro_files/figure-revealjs/unnamed-chunk-13-1.png" width="960"></p>
</div>
</div>
</div>
<div id="tabset-1-3">
<div class="cell">
<div class="cell-output-display">
<p><img data-src="01_Intro_files/figure-revealjs/unnamed-chunk-14-1.png" width="960"></p>
</div>
</div>
</div>
</div>
</div>
</section>
<section id="another-visual-example" class="slide level2 smaller">
<h2>Another Visual Example</h2>
<div class="panel-tabset">
<ul id="tabset-2" class="panel-tabset-tabby"><li><a data-tabby-default="" href="#tabset-2-1">Given the Data…</a></li><li><a href="#tabset-2-2">A Possible Solution</a></li><li><a href="#tabset-2-3">Another Option</a></li></ul>
<div class="tab-content">
<div id="tabset-2-1">
<div class="cell">
<div class="cell-output-display">
<p><img data-src="01_Intro_files/figure-revealjs/unnamed-chunk-15-1.png" width="960"></p>
</div>
</div>
</div>
<div id="tabset-2-2">
<div class="cell">
<div class="cell-output-display">
<p><img data-src="01_Intro_files/figure-revealjs/unnamed-chunk-16-1.png" width="960"></p>
</div>
</div>
</div>
<div id="tabset-2-3">
<div class="cell">
<div class="cell-output-display">
<p><img data-src="01_Intro_files/figure-revealjs/unnamed-chunk-17-1.png" width="960"></p>
</div>
</div>
</div>
</div>
</div>
</section>
<section id="so-what-is-the-situation" class="slide level2">
<h2>So, What is the Situation?</h2>
<p>We observe <span class="math inline">\(N\)</span> input-output pairs <span class="math inline">\(\left\{(x^{(i)},y^{(i)})\right\}_{i = 1}^N\)</span>.</p>
<p>We <em>assume</em> that there is some relationship between the predictor <span class="math inline">\(X\)</span> and the response <span class="math inline">\(Y\)</span>, which can be written in the form</p>
<p><span class="math display">\[Y = f(X) + \varepsilon\]</span></p>
<p>where <span class="math inline">\(\varepsilon\)</span> is a <em>random error term</em> that is independent of <span class="math inline">\(X\)</span> and “on average” is equal to zero.</p>
</section>
<section id="it-can-be-a-little-bit-more-complicated" class="slide level2">
<h2>It Can Be a Little Bit More Complicated…</h2>
<p>In general the number of predictors is more than one.</p>
<p>So instead of having just <span class="math inline">\(X\)</span>, we have <span class="math inline">\(X_1, X_2, \dots, X_m\)</span>.</p>
<p>Hence we observe tuples rather than pairs</p>
<p><span class="math display">\[\left\{(x^{(i)}_1, x^{(i)}_2, \dots, x^{(i)}_m,y^{(i)})\right\}_{i = 1}^N\]</span></p>
<p>Therefore the relationship between input and output is written as</p>
<p><span class="math display">\[Y = f(X_1, X_2, \dots, X_m) + \varepsilon\]</span></p>
</section>
<section id="but-we-can-still-write-it-in-a-simple-form" class="slide level2">
<h2>But We Can Still Write It in a Simple Form</h2>
<p>We can write</p>
<p><span class="math display">\[X = (X_1, X_2, \dots, X_m) \]</span></p>
<p>Therefore the relationship between input and output is written as</p>
<p><span class="math display">\[Y = f(X) + \varepsilon\]</span></p>
<p>and we silently remember that the input can have <span class="math inline">\(m\)</span> dimensions.</p>
</section>
<section id="the-problem" class="slide level2">
<h2>The Problem</h2>
<p>The problem is simple to state: find <span class="math inline">\(f\)</span>.</p>
<blockquote>
<p>In essence, statistical learning refers to a set of approaches for estimating <span class="math inline">\(f\)</span>.</p>
</blockquote>
<p>The issue is that we will never find the exact <span class="math inline">\(f\)</span>. Therefore we lower our expectations and we set our goal to find an estimate of <span class="math inline">\(f\)</span>, that we are going to write as <span class="math inline">\(\hat{f}\)</span>.</p>
<aside><div>
<p>Gareth James, Daniela Witten, Trevor Hastie, and Robert Tibshirani, <em>An Introduction to Statistical Learning: with Applications in R</em> (2nd Edition)</p>
</div></aside></section>
<section id="why-estimate-f" class="slide level2 scrollable">
<h2>Why Estimate <span class="math inline">\(f\)</span>?</h2>
<ol type="1">
<li>Make predictions: given new (input) data <span class="math inline">\(X_{\text{new}}\)</span> you want to find</li>
</ol>
<p><span class="math display">\[\hat{Y} = \hat{f}(X_{\text{new}})\]</span></p>
<div class="fragment">
<ol start="2" type="1">
<li>Inference
<ul>
<li>Which predictors are associated with the response?</li>
<li>What is the relationship between the response and each predictor?</li>
<li>Can the relationship between Y and each predictor be adequately summarized using a linear equation, or is the relationship more complicated?</li>
</ul></li>
</ol>
</div>
<aside><div>
<p>Gareth James, Daniela Witten, Trevor Hastie, and Robert Tibshirani, <em>An Introduction to Statistical Learning: with Applications in R</em> (2nd Edition)</p>
</div></aside></section></section>
<section>
<section id="classification-and-regression" class="title-slide slide level1 center">
<h1>Classification and Regression</h1>
</section>
<section id="types-of-variables" class="slide level2">
<h2>Types of Variables</h2>
<ul>
<li><p><strong>Categorical variables</strong> can take a limited number of possible values assigning each observation to a group</p></li>
<li><p><strong>Numerical variables</strong> can take values within a (possibly unbounded) numerical interval</p></li>
</ul>
</section>
<section id="classification-vs-regression-in-words" class="slide level2">
<h2>Classification vs Regression (in words)</h2>
<p>What kind of output we are trying to predict?</p>
<ul>
<li><p>In <strong>classification</strong> the objective is to predict a categorical outcome</p></li>
<li><p>In <strong>regression</strong> the objective is to predict a numerical outcome</p></li>
</ul>
<p>However often in classification problems rather than predict a particular class, we often want predict the probability of a particular class.</p>
</section>
<section id="classification-vs-regression-in-maths" class="slide level2">
<h2>Classification vs Regression (in maths)</h2>
<ul>
<li>In <strong>classification</strong> the output set is of the form
<ul>
<li><span class="math inline">\(Y = \{0, 1\}\)</span> binary classification (2 categories) or</li>
<li><span class="math inline">\(Y = \{0, 1, 2, ..., K\}\)</span> multinomial classification (more that 2 categories)</li>
</ul></li>
<li>In <strong>regression</strong> the output set is of the form
<ul>
<li><span class="math inline">\(Y = \mathbb{R}\)</span> all possible (real) numbers</li>
<li><span class="math inline">\(Y = [a, b]\)</span> all numbers within a certain range (possibly unbounded)</li>
</ul></li>
</ul>
</section>
<section id="classification-vs-regression-in-pictures" class="slide level2">
<h2>Classification vs Regression (in pictures)</h2>
<div class="columns">
<div class="column" style="width:50%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\classification.png" width="370" height="400"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:50%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="Img\regression.png" width="370" height="400"></p>
</figure>
</div>
</div>
</div>
</div>
</div>
</section></section>
<section>
<section id="how-can-we-find-f" class="title-slide slide level1 center">
<h1>How Can We Find <span class="math inline">\(f\)</span>?</h1>
</section>
<section id="two-possible-methods" class="slide level2 smaller">
<h2>Two Possible Methods</h2>
<div class="columns">
<div class="column" style="width:50%;">
<h3 id="parametric">Parametric</h3>
<ol type="1">
<li><p>Assume that <span class="math inline">\(f\)</span> has a particular functional form depending on a set of parameters.</p></li>
<li><p>Estimate parameters using observed data.</p></li>
</ol>
<div>
<ul>
<li class="fragment"><p>Advantages: easy to estimate.</p></li>
<li class="fragment"><p>Disavantages: assumptions may be very wrong!</p></li>
</ul>
</div>
</div><div class="column" style="width:50%;">
<h3 id="non-parametric">Non-Parametric</h3>
<ol type="1">
<li><p>Don’t make particular assumptions on the functional form of <span class="math inline">\(f\)</span>.</p></li>
<li><p>Estimate a function that adapts smoothly to the observed data “avoiding excessive oscillations and edgy behaviours”.</p></li>
</ol>
<div>
<ul>
<li class="fragment"><p>Advantages: can estimate effectively many functions.</p></li>
<li class="fragment"><p>Disadvantages: need a lot of data!</p></li>
</ul>
</div>
</div>
</div>
</section></section>
<section>
<section id="parametric-models-for-regression" class="title-slide slide level1 center">
<h1>Parametric Models for Regression</h1>
</section>
<section id="hypothesis-function" class="slide level2">
<h2>Hypothesis Function</h2>
<p>In Parametric methods, we will make assumption about the functional form of the function <span class="math inline">\(f\)</span>. In other words we assume that <span class="math inline">\(f\)</span> can be written in a particular form. We call it <strong>hypothesis function</strong>, we denote it as <span class="math inline">\(h\)</span>, and it will depend on one or more parameters.</p>
</section>
<section id="a-first-model-for-regression" class="slide level2">
<h2>A First Model for Regression</h2>
<h3 id="model">Model</h3>
<p>A very simple model for regression is given by</p>