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<title>Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods and Summary of Course &#8212; Applied Data Analysis and Machine Learning</title>
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Applied Data Analysis and Machine Learning
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About the course
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Textbooks
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Review of Statistics with Resampling Techniques and Linear Algebra
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1. Elements of Probability Theory and Statistical Data Analysis
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2. Linear Algebra, Handling of Arrays and more Python Features
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From Regression to Support Vector Machines
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3. Linear Regression
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4. Ridge and Lasso Regression
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5. Resampling Methods
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6. Logistic Regression
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7. Optimization, the central part of any Machine Learning algortithm
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8. Support Vector Machines, overarching aims
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Decision Trees, Ensemble Methods and Boosting
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9. Decision trees, overarching aims
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10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
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Dimensionality Reduction
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11. Basic ideas of the Principal Component Analysis (PCA)
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12. Clustering and Unsupervised Learning
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Deep Learning Methods
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13. Neural networks
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14. Building a Feed Forward Neural Network
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15. Solving Differential Equations with Deep Learning
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16. Convolutional Neural Networks
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17. Recurrent neural networks: Overarching view
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Weekly material, notes and exercises
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Exercises week 34
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Week 34: Introduction to the course, Logistics and Practicalities
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Exercises week 35
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Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
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Exercises week 36
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Week 36: Statistical interpretation of Linear Regression and Resampling techniques
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Exercises week 37
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Week 37: Statistical interpretations and Resampling Methods
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Exercises week 38
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Week 38: Logistic Regression and Optimization
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Exercises week 39
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Week 39: Optimization and Gradient Methods
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Week 40: Gradient descent methods (continued) and start Neural networks
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Exercises week 41
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Week 41 Neural networks and constructing a neural network code
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Exercises week 42
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Week 42 Constructing a Neural Network code with introduction to Tensor flow
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Exercises weeks 43 and 44
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Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations
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Week 44, Convolutional Neural Networks (CNN)
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Week 45, Recurrent Neural Networks
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Week 46: Decision Trees, Ensemble methods and Random Forests
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Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods and Summary of Course
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Exercise week 47
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Projects
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Project 1 on Machine Learning, deadline October 9 (midnight), 2023
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Project 2 on Machine Learning, deadline November 17 (Midnight)
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<i class="fas fa-list"></i> Contents
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<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#plan-for-week-47">
Plan for week 47
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#bagging">
Bagging
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-bagging">
More bagging
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#making-your-own-bootstrap-changing-the-level-of-the-decision-tree">
Making your own Bootstrap: Changing the Level of the Decision Tree
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#random-forests">
Random forests
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#random-forest-algorithm">
Random Forest Algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#random-forests-compared-with-other-methods-on-the-cancer-data">
Random Forests Compared with other Methods on the Cancer Data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#compare-bagging-on-trees-with-random-forests">
Compare Bagging on Trees with Random Forests
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#boosting-a-bird-s-eye-view">
Boosting, a Bird’s Eye View
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-is-boosting-additive-modelling-iterative-fitting">
What is boosting? Additive Modelling/Iterative Fitting
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#iterative-fitting-regression-and-squared-error-cost-function">
Iterative Fitting, Regression and Squared-error Cost Function
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#squared-error-example-and-iterative-fitting">
Squared-Error Example and Iterative Fitting
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#iterative-fitting-classification-and-adaboost">
Iterative Fitting, Classification and AdaBoost
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adaptive-boosting-adaboost">
Adaptive Boosting, AdaBoost
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#building-up-adaboost">
Building up AdaBoost
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adaptive-boosting-adaboost-basic-algorithm">
Adaptive boosting: AdaBoost, Basic Algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#basic-steps-of-adaboost">
Basic Steps of AdaBoost
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adaboost-examples">
AdaBoost Examples
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-boosting-basics-with-steepest-descent-functional-gradient-descent">
Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-squared-error-again-steepest-descent">
The Squared-Error again! Steepest Descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#steepest-descent-example">
Steepest Descent Example
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-boosting-algorithm">
Gradient Boosting, algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-boosting-examples-of-regression">
Gradient Boosting, Examples of Regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-boosting-classification-example">
Gradient Boosting, Classification Example
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#xgboost-extreme-gradient-boosting">
XGBoost: Extreme Gradient Boosting
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#regression-case">
Regression Case
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#xgboost-on-the-cancer-data">
Xgboost on the Cancer Data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#summary-of-course">
Summary of course
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-me-worry-no-final-exam-in-this-course">
What? Me worry? No final exam in this course!
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-is-the-link-between-artificial-intelligence-and-machine-learning-and-some-general-remarks">
What is the link between Artificial Intelligence and Machine Learning and some general Remarks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#going-back-to-the-beginning-of-the-semester">
Going back to the beginning of the semester
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#not-so-sharp-distinctions">
Not so sharp distinctions
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#topics-we-have-covered-this-year">
Topics we have covered this year
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#statistical-analysis-and-optimization-of-data">
Statistical analysis and optimization of data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#machine-learning">
Machine learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#learning-outcomes-and-overarching-aims-of-this-course">
Learning outcomes and overarching aims of this course
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#perspective-on-machine-learning">
Perspective on Machine Learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#machine-learning-research">
Machine Learning Research
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#starting-your-machine-learning-project">
Starting your Machine Learning Project
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#choose-a-model-and-algorithm">
Choose a Model and Algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#preparing-your-data">
Preparing Your Data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#which-activation-and-weights-to-choose-in-neural-networks">
Which Activation and Weights to Choose in Neural Networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#optimization-methods-and-hyperparameters">
Optimization Methods and Hyperparameters
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#resampling">
Resampling
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#other-courses-on-data-science-and-machine-learning-at-uio">
Other courses on Data science and Machine Learning at UiO
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#additional-courses-of-interest">
Additional courses of interest
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-s-the-future-like">
What’s the future like?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#types-of-machine-learning-a-repetition">
Types of Machine Learning, a repetition
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#why-boltzmann-machines">
Why Boltzmann machines?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#boltzmann-machines">
Boltzmann Machines
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#some-similarities-and-differences-from-dnns">
Some similarities and differences from DNNs
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#boltzmann-machines-bm">
Boltzmann machines (BM)
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#a-standard-bm-setup">
A standard BM setup
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-structure-of-the-rbm-network">
The structure of the RBM network
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-network">
The network
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#goals">
Goals
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#joint-distribution">
Joint distribution
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#network-elements-the-energy-function">
Network Elements, the energy function
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#defining-different-types-of-rbms">
Defining different types of RBMs
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-about-rbms">
More about RBMs
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#autoencoders-overarching-view">
Autoencoders: Overarching view
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#bayesian-machine-learning">
Bayesian Machine Learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#reinforcement-learning">
Reinforcement Learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#transfer-learning">
Transfer learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adversarial-learning">
Adversarial learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#dual-learning">
Dual learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#distributed-machine-learning">
Distributed machine learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#meta-learning">
Meta learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-challenges-facing-machine-learning">
The Challenges Facing Machine Learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#explainable-machine-learning">
Explainable machine learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#scientific-machine-learning">
Scientific Machine Learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#quantum-machine-learning">
Quantum machine learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#quantum-machine-learning-algorithms-based-on-linear-algebra">
Quantum machine learning algorithms based on linear algebra
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#quantum-reinforcement-learning">
Quantum reinforcement learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#quantum-deep-learning">
Quantum deep learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#social-machine-learning">
Social machine learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-last-words">
The last words?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#ai-ml-and-some-statements-you-may-have-heard-and-what-do-they-mean">
AI/ML and some statements you may have heard (and what do they mean?)
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#best-wishes-to-you-all-and-thanks-so-much-for-your-heroic-efforts-this-semester">
Best wishes to you all and thanks so much for your heroic efforts this semester
</a>
</li>
</ul>
</nav>
</div>
</div>
</div>
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<h1>Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods and Summary of Course</h1>
<!-- Table of contents -->
<div id="print-main-content">
<div id="jb-print-toc">
<div>
<h2> Contents </h2>
</div>
<nav aria-label="Page">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#plan-for-week-47">
Plan for week 47
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#bagging">
Bagging
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-bagging">
More bagging
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#making-your-own-bootstrap-changing-the-level-of-the-decision-tree">
Making your own Bootstrap: Changing the Level of the Decision Tree
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#random-forests">
Random forests
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#random-forest-algorithm">
Random Forest Algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#random-forests-compared-with-other-methods-on-the-cancer-data">
Random Forests Compared with other Methods on the Cancer Data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#compare-bagging-on-trees-with-random-forests">
Compare Bagging on Trees with Random Forests
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#boosting-a-bird-s-eye-view">
Boosting, a Bird’s Eye View
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-is-boosting-additive-modelling-iterative-fitting">
What is boosting? Additive Modelling/Iterative Fitting
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#iterative-fitting-regression-and-squared-error-cost-function">
Iterative Fitting, Regression and Squared-error Cost Function
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#squared-error-example-and-iterative-fitting">
Squared-Error Example and Iterative Fitting
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#iterative-fitting-classification-and-adaboost">
Iterative Fitting, Classification and AdaBoost
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adaptive-boosting-adaboost">
Adaptive Boosting, AdaBoost
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#building-up-adaboost">
Building up AdaBoost
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adaptive-boosting-adaboost-basic-algorithm">
Adaptive boosting: AdaBoost, Basic Algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#basic-steps-of-adaboost">
Basic Steps of AdaBoost
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adaboost-examples">
AdaBoost Examples
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-boosting-basics-with-steepest-descent-functional-gradient-descent">
Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-squared-error-again-steepest-descent">
The Squared-Error again! Steepest Descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#steepest-descent-example">
Steepest Descent Example
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-boosting-algorithm">
Gradient Boosting, algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-boosting-examples-of-regression">
Gradient Boosting, Examples of Regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-boosting-classification-example">
Gradient Boosting, Classification Example
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#xgboost-extreme-gradient-boosting">
XGBoost: Extreme Gradient Boosting
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#regression-case">
Regression Case
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#xgboost-on-the-cancer-data">
Xgboost on the Cancer Data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#summary-of-course">
Summary of course
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-me-worry-no-final-exam-in-this-course">
What? Me worry? No final exam in this course!
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-is-the-link-between-artificial-intelligence-and-machine-learning-and-some-general-remarks">
What is the link between Artificial Intelligence and Machine Learning and some general Remarks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#going-back-to-the-beginning-of-the-semester">
Going back to the beginning of the semester
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#not-so-sharp-distinctions">
Not so sharp distinctions
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#topics-we-have-covered-this-year">
Topics we have covered this year
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#statistical-analysis-and-optimization-of-data">
Statistical analysis and optimization of data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#machine-learning">
Machine learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#learning-outcomes-and-overarching-aims-of-this-course">
Learning outcomes and overarching aims of this course
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#perspective-on-machine-learning">
Perspective on Machine Learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#machine-learning-research">
Machine Learning Research
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#starting-your-machine-learning-project">
Starting your Machine Learning Project
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#choose-a-model-and-algorithm">
Choose a Model and Algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#preparing-your-data">
Preparing Your Data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#which-activation-and-weights-to-choose-in-neural-networks">
Which Activation and Weights to Choose in Neural Networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#optimization-methods-and-hyperparameters">
Optimization Methods and Hyperparameters
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#resampling">
Resampling
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#other-courses-on-data-science-and-machine-learning-at-uio">
Other courses on Data science and Machine Learning at UiO
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#additional-courses-of-interest">
Additional courses of interest
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-s-the-future-like">
What’s the future like?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#types-of-machine-learning-a-repetition">
Types of Machine Learning, a repetition
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#why-boltzmann-machines">
Why Boltzmann machines?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#boltzmann-machines">
Boltzmann Machines
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#some-similarities-and-differences-from-dnns">
Some similarities and differences from DNNs
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#boltzmann-machines-bm">
Boltzmann machines (BM)
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#a-standard-bm-setup">
A standard BM setup
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-structure-of-the-rbm-network">
The structure of the RBM network
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-network">
The network
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#goals">
Goals
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#joint-distribution">
Joint distribution
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#network-elements-the-energy-function">
Network Elements, the energy function
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#defining-different-types-of-rbms">
Defining different types of RBMs
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-about-rbms">
More about RBMs
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#autoencoders-overarching-view">
Autoencoders: Overarching view
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#bayesian-machine-learning">
Bayesian Machine Learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#reinforcement-learning">
Reinforcement Learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#transfer-learning">
Transfer learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adversarial-learning">
Adversarial learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#dual-learning">
Dual learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#distributed-machine-learning">
Distributed machine learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#meta-learning">
Meta learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-challenges-facing-machine-learning">
The Challenges Facing Machine Learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#explainable-machine-learning">
Explainable machine learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#scientific-machine-learning">
Scientific Machine Learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#quantum-machine-learning">
Quantum machine learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#quantum-machine-learning-algorithms-based-on-linear-algebra">
Quantum machine learning algorithms based on linear algebra
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#quantum-reinforcement-learning">
Quantum reinforcement learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#quantum-deep-learning">
Quantum deep learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#social-machine-learning">
Social machine learning
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-last-words">
The last words?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#ai-ml-and-some-statements-you-may-have-heard-and-what-do-they-mean">
AI/ML and some statements you may have heard (and what do they mean?)
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#best-wishes-to-you-all-and-thanks-so-much-for-your-heroic-efforts-this-semester">
Best wishes to you all and thanks so much for your heroic efforts this semester
</a>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div>
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<!-- dom:TITLE: Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods and Summary of Course --><div class="tex2jax_ignore mathjax_ignore section" id="week-47-from-decision-trees-to-ensemble-methods-random-forests-and-boosting-methods-and-summary-of-course">
<h1>Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods and Summary of Course<a class="headerlink" href="#week-47-from-decision-trees-to-ensemble-methods-random-forests-and-boosting-methods-and-summary-of-course" title="Permalink to this headline"></a></h1>
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University</p>
<p>Date: <strong>November 20-24, 2023</strong></p>
<div class="section" id="plan-for-week-47">
<h2>Plan for week 47<a class="headerlink" href="#plan-for-week-47" title="Permalink to this headline"></a></h2>
<p><strong>Active learning sessions on Tuesday and Wednesday.</strong></p>
<ul class="simple">
<li><p>Work and Discussion of project 3</p></li>
<li><p>Last weekly exercise, course feedback, to be completed by Sunday November 26</p></li>
</ul>
<p><strong>Material for the lecture on Thursday November 23, 2023.</strong></p>
<ul class="simple">
<li><p>Thursday: Basics of decision trees, classification and regression algorithms and ensemble models</p></li>
<li><p>Readings and Videos:</p>
<ul>
<li><p>These lecture notes</p></li>
<li><p><a class="reference external" href="https://youtu.be/SpWXsvn5I9E">Video of Lecture</a></p></li>
<li><p><a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesNov23.pdf">Whiteboard notes</a></p></li>
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=RmajweUFKvM&amp;ab_channel=Simplilearn">Video on Decision trees</a></p></li>
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=wPqtzj5VZus&amp;ab_channel=H2O.ai">Video on boosting methods by Hastie</a></p></li>
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=LsK-xG1cLYA">Video on AdaBoost</a></p></li>
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=3CC4N4z3GJc">Video on Gradient boost, part 1, parts 2-4 follow thereafter</a></p></li>
<li><p>Decision Trees: Geron’s chapter 6 covers decision trees while ensemble models, voting and bagging are discussed in chapter 7. See also lecture from <a class="reference external" href="https://www.uio.no/studier/emner/matnat/math/STK-IN4300/h20/slides/lecture_7.pdf">STK-IN4300, lecture 7</a>. Chapter 9.2 of Hastie et al contains also a good discussion.</p></li>
</ul>
</li>
</ul>
</div>
<div class="section" id="bagging">
<h2>Bagging<a class="headerlink" href="#bagging" title="Permalink to this headline"></a></h2>
<p>The <strong>plain</strong> decision trees suffer from high
variance. This means that if we split the training data into two parts
at random, and fit a decision tree to both halves, the results that we
get could be quite different. In contrast, a procedure with low
variance will yield similar results if applied repeatedly to distinct
data sets; linear regression tends to have low variance, if the ratio
of <span class="math notranslate nohighlight">\(n\)</span> to <span class="math notranslate nohighlight">\(p\)</span> is moderately large.</p>
<p><strong>Bootstrap aggregation</strong>, or just <strong>bagging</strong>, is a
general-purpose procedure for reducing the variance of a statistical
learning method.</p>
</div>
<div class="section" id="more-bagging">
<h2>More bagging<a class="headerlink" href="#more-bagging" title="Permalink to this headline"></a></h2>
<p>Bagging typically results in improved accuracy
over prediction using a single tree. Unfortunately, however, it can be
difficult to interpret the resulting model. Recall that one of the
advantages of decision trees is the attractive and easily interpreted
diagram that results.</p>
<p>However, when we bag a large number of trees, it is no longer
possible to represent the resulting statistical learning procedure
using a single tree, and it is no longer clear which variables are
most important to the procedure. Thus, bagging improves prediction
accuracy at the expense of interpretability. Although the collection
of bagged trees is much more difficult to interpret than a single
tree, one can obtain an overall summary of the importance of each
predictor using the MSE (for bagging regression trees) or the Gini
index (for bagging classification trees). In the case of bagging
regression trees, we can record the total amount that the MSE is
decreased due to splits over a given predictor, averaged over all <span class="math notranslate nohighlight">\(B\)</span> possible
trees. A large value indicates an important predictor. Similarly, in
the context of bagging classification trees, we can add up the total
amount that the Gini index is decreased by splits over a given
predictor, averaged over all <span class="math notranslate nohighlight">\(B\)</span> trees.</p>
</div>
<div class="section" id="making-your-own-bootstrap-changing-the-level-of-the-decision-tree">
<h2>Making your own Bootstrap: Changing the Level of the Decision Tree<a class="headerlink" href="#making-your-own-bootstrap-changing-the-level-of-the-decision-tree" title="Permalink to this headline"></a></h2>
<p>Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with
a decision tree wth different depths and perform a bootstrap aggregate (in this case we perform as many bootstraps as data points <span class="math notranslate nohighlight">\(n\)</span>).</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="c1"># Common imports</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">make_pipeline</span>
<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <span class="n">resample</span>
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span><span class="p">,</span> <span class="n">OneHotEncoder</span>
<span class="kn">from</span> <span class="nn">sklearn.compose</span> <span class="kn">import</span> <span class="n">ColumnTransformer</span>
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
<span class="kn">import</span> <span class="nn">os</span>
<span class="c1"># Where to save the figures and data files</span>
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">&quot;Results&quot;</span>
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">&quot;Results/FigureFiles&quot;</span>
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">&quot;DataFiles/&quot;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;.png&quot;</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">&#39;png&#39;</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">n_boostraps</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">maxdepth</span> <span class="o">=</span> <span class="mi">8</span>
<span class="c1"># Make data set.</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">error</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdepth</span><span class="p">)</span>
<span class="n">bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdepth</span><span class="p">)</span>
<span class="n">variance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdepth</span><span class="p">)</span>
<span class="n">polydegree</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdepth</span><span class="p">)</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="c1"># we produce a simple tree first as benchmark</span>
<span class="n">simpletree</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">simpletree</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">simpleprediction</span> <span class="o">=</span> <span class="n">simpletree</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
<span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">maxdepth</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="n">y_test</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">n_boostraps</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_boostraps</span><span class="p">):</span>
<span class="n">x_</span><span class="p">,</span> <span class="n">y_</span> <span class="o">=</span> <span class="n">resample</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x_</span><span class="p">,</span> <span class="n">y_</span><span class="p">)</span>
<span class="n">y_pred</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span><span class="c1">#.ravel()</span>
<span class="n">polydegree</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">degree</span>
<span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">y_pred</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="p">)</span>
<span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="p">(</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_pred</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
<span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_pred</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Polynomial degree:&#39;</span><span class="p">,</span> <span class="n">degree</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Error:&#39;</span><span class="p">,</span> <span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Bias^2:&#39;</span><span class="p">,</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Var:&#39;</span><span class="p">,</span> <span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1"> &gt;= </span><span class="si">{}</span><span class="s1"> + </span><span class="si">{}</span><span class="s1"> = </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span><span class="o">+</span><span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">]))</span>
<span class="n">mse_simpletree</span><span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">simpleprediction</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Simple tree:&quot;</span><span class="p">,</span><span class="n">mse_simpletree</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">maxdepth</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">error</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;MSE&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;bias&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">variance</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Variance&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;baggingboot&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 1
Error: 0.05485044745873867
Bias^2: 0.05363014989229746
Var: 0.0012202975664411882
0.05485044745873867 &gt;= 0.05363014989229746 + 0.0012202975664411882 = 0.05485044745873865
Polynomial degree: 2
Error: 0.04754825003279861
Bias^2: 0.0362312015777108
Var: 0.01131704845508782
0.04754825003279861 &gt;= 0.0362312015777108 + 0.01131704845508782 = 0.04754825003279862
Polynomial degree: 3
Error: 0.028256917047283964
Bias^2: 0.019709199043491926
Var: 0.008547718003792035
0.028256917047283964 &gt;= 0.019709199043491926 + 0.008547718003792035 = 0.02825691704728396
Polynomial degree: 4
Error: 0.02417252675174287
Bias^2: 0.016541517177965183
Var: 0.007631009573777696
0.02417252675174287 &gt;= 0.016541517177965183 + 0.007631009573777696 = 0.02417252675174288
Polynomial degree: 5
Error: 0.020350773309798075
Bias^2: 0.013742894355267554
Var: 0.006607878954530523
0.020350773309798075 &gt;= 0.013742894355267554 + 0.006607878954530523 = 0.02035077330979808
Polynomial degree: 6
Error: 0.019509108923639135
Bias^2: 0.01312013610582818
Var: 0.006388972817810939
0.019509108923639135 &gt;= 0.01312013610582818 + 0.006388972817810939 = 0.01950910892363912
Polynomial degree: 7
Error: 0.020056743323946562
Bias^2: 0.012815095479733507
Var: 0.007241647844213045
0.020056743323946562 &gt;= 0.012815095479733507 + 0.007241647844213045 = 0.020056743323946552
Simple tree: 0.5601973572808581
</pre></div>
</div>
<img alt="_images/week47_6_1.png" src="_images/week47_6_1.png" />
</div>
</div>
</div>
<div class="section" id="random-forests">
<h2>Random forests<a class="headerlink" href="#random-forests" title="Permalink to this headline"></a></h2>
<p>Random forests provide an improvement over bagged trees by way of a
small tweak that decorrelates the trees.</p>
<p>As in bagging, we build a
number of decision trees on bootstrapped training samples. But when
building these decision trees, each time a split in a tree is
considered, a random sample of <span class="math notranslate nohighlight">\(m\)</span> predictors is chosen as split
candidates from the full set of <span class="math notranslate nohighlight">\(p\)</span> predictors. The split is allowed to
use only one of those <span class="math notranslate nohighlight">\(m\)</span> predictors.</p>
<p>A fresh sample of <span class="math notranslate nohighlight">\(m\)</span> predictors is
taken at each split, and typically we choose</p>
<div class="math notranslate nohighlight">
\[
m\approx \sqrt{p}.
\]</div>
<p>In building a random forest, at
each split in the tree, the algorithm is not even allowed to consider
a majority of the available predictors.</p>
<p>The reason for this is rather clever. Suppose that there is one very
strong predictor in the data set, along with a number of other
moderately strong predictors. Then in the collection of bagged
variable importance random forest trees, most or all of the trees will
use this strong predictor in the top split. Consequently, all of the
bagged trees will look quite similar to each other. Hence the
predictions from the bagged trees will be highly correlated.
Unfortunately, averaging many highly correlated quantities does not
lead to as large of a reduction in variance as averaging many
uncorrelated quantities. In particular, this means that bagging will
not lead to a substantial reduction in variance over a single tree in
this setting.</p>
</div>
<div class="section" id="random-forest-algorithm">
<h2>Random Forest Algorithm<a class="headerlink" href="#random-forest-algorithm" title="Permalink to this headline"></a></h2>
<p>The algorithm described here can be applied to both classification and regression problems.</p>
<p>We will grow of forest of say <span class="math notranslate nohighlight">\(B\)</span> trees.</p>
<ol class="simple">
<li><p>For <span class="math notranslate nohighlight">\(b=1:B\)</span></p></li>
</ol>
<ul class="simple">
<li><p>Draw a bootstrap sample from the training data organized in our <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> matrix.</p></li>
<li><p>We grow then a random forest tree <span class="math notranslate nohighlight">\(T_b\)</span> based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached</p></li>
</ul>
<ol class="simple">
<li><p>we select <span class="math notranslate nohighlight">\(m \le p\)</span> variables at random from the <span class="math notranslate nohighlight">\(p\)</span> predictors/features</p></li>
<li><p>pick the best split point among the <span class="math notranslate nohighlight">\(m\)</span> features using for example the CART algorithm and create a new node</p></li>
<li><p>split the node into daughter nodes</p></li>
<li><p>Output then the ensemble of trees <span class="math notranslate nohighlight">\(\{T_b\}_1^{B}\)</span> and make predictions for either a regression type of problem or a classification type of problem.</p></li>
</ol>
</div>
<div class="section" id="random-forests-compared-with-other-methods-on-the-cancer-data">
<h2>Random Forests Compared with other Methods on the Cancer Data<a class="headerlink" href="#random-forests-compared-with-other-methods-on-the-cancer-data" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <span class="n">SVC</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">BaggingClassifier</span>
<span class="c1"># Load the data</span>
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="c1">#define methods</span>
<span class="c1"># Logistic Regression</span>
<span class="n">logreg</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s1">&#39;lbfgs&#39;</span><span class="p">)</span>
<span class="c1"># Support vector machine</span>
<span class="n">svm</span> <span class="o">=</span> <span class="n">SVC</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span> <span class="n">C</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
<span class="c1"># Decision Trees</span>
<span class="n">deep_tree_clf</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
<span class="c1">#Scale the data</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="c1"># Logistic Regression</span>
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test set accuracy Logistic Regression with scaled data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
<span class="c1"># Support Vector Machine</span>
<span class="n">svm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test set accuracy SVM with scaled data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
<span class="c1"># Decision Trees</span>
<span class="n">deep_tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test set accuracy with Decision Trees and scaled data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">deep_tree_clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">RandomForestClassifier</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">LabelEncoder</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_validate</span>
<span class="c1"># Data set not specificied</span>
<span class="c1">#Instantiate the model with 500 trees and entropy as splitting criteria</span>
<span class="n">Random_Forest_model</span> <span class="o">=</span> <span class="n">RandomForestClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span><span class="n">criterion</span><span class="o">=</span><span class="s2">&quot;entropy&quot;</span><span class="p">)</span>
<span class="n">Random_Forest_model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="c1">#Cross validation</span>
<span class="n">accuracy</span> <span class="o">=</span> <span class="n">cross_validate</span><span class="p">(</span><span class="n">Random_Forest_model</span><span class="p">,</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">,</span><span class="n">cv</span><span class="o">=</span><span class="mi">10</span><span class="p">)[</span><span class="s1">&#39;test_score&#39;</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">accuracy</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test set accuracy with Random Forests and scaled data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">Random_Forest_model</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">Random_Forest_model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_confusion_matrix</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">y_probas</span> <span class="o">=</span> <span class="n">Random_Forest_model</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_roc</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_cumulative_gain</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(426, 30)
(143, 30)
Test set accuracy Logistic Regression with scaled data: 0.96
Test set accuracy SVM with scaled data: 0.96
Test set accuracy with Decision Trees and scaled data: 0.87
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.93333333 0.73333333 0.93333333 1. 1. 0.92857143
1. 0.92857143 0.92857143 1. ]
Test set accuracy with Random Forests and scaled data: 0.98
</pre></div>
</div>
<img alt="_images/week47_12_2.png" src="_images/week47_12_2.png" />
<img alt="_images/week47_12_3.png" src="_images/week47_12_3.png" />
<img alt="_images/week47_12_4.png" src="_images/week47_12_4.png" />
</div>
</div>
<p>Recall that the cumulative gains curve shows the percentage of the
overall number of cases in a given category <em>gained</em> by targeting a
percentage of the total number of cases.</p>
<p>Similarly, the receiver operating characteristic curve, or ROC curve,
displays the diagnostic ability of a binary classifier system as its
discrimination threshold is varied. It plots the true positive rate against the false positive rate.</p>
</div>
<div class="section" id="compare-bagging-on-trees-with-random-forests">
<h2>Compare Bagging on Trees with Random Forests<a class="headerlink" href="#compare-bagging-on-trees-with-random-forests" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">bag_clf</span> <span class="o">=</span> <span class="n">BaggingClassifier</span><span class="p">(</span>
<span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">splitter</span><span class="o">=</span><span class="s2">&quot;random&quot;</span><span class="p">,</span> <span class="n">max_leaf_nodes</span><span class="o">=</span><span class="mi">16</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">),</span>
<span class="n">n_estimators</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">max_samples</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">bootstrap</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">n_jobs</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">bag_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">bag_clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">RandomForestClassifier</span>
<span class="n">rnd_clf</span> <span class="o">=</span> <span class="n">RandomForestClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">max_leaf_nodes</span><span class="o">=</span><span class="mi">16</span><span class="p">,</span> <span class="n">n_jobs</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
<span class="n">rnd_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred_rf</span> <span class="o">=</span> <span class="n">rnd_clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">y_pred</span> <span class="o">==</span> <span class="n">y_pred_rf</span><span class="p">)</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">y_pred</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9790209790209791
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="boosting-a-bird-s-eye-view">
<h2>Boosting, a Bird’s Eye View<a class="headerlink" href="#boosting-a-bird-s-eye-view" title="Permalink to this headline"></a></h2>
<p>The basic idea is to combine weak classifiers in order to create a good
classifier. With a weak classifier we often intend a classifier which
produces results which are only slightly better than we would get by
random guesses.</p>
<p>This is done by applying in an iterative way a weak (or a standard
classifier like decision trees) to modify the data. In each iteration
we emphasize those observations which are misclassified by weighting
them with a factor.</p>
</div>
<div class="section" id="what-is-boosting-additive-modelling-iterative-fitting">
<h2>What is boosting? Additive Modelling/Iterative Fitting<a class="headerlink" href="#what-is-boosting-additive-modelling-iterative-fitting" title="Permalink to this headline"></a></h2>
<p>Boosting is a way of fitting an additive expansion in a set of
elementary basis functions like for example some simple polynomials.
Assume for example that we have a function</p>
<div class="math notranslate nohighlight">
\[
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
\]</div>
<p>where <span class="math notranslate nohighlight">\(\beta_m\)</span> are the expansion parameters to be determined in a
minimization process and <span class="math notranslate nohighlight">\(b(x;\gamma_m)\)</span> are some simple functions of
the multivariable parameter <span class="math notranslate nohighlight">\(x\)</span> which is characterized by the
parameters <span class="math notranslate nohighlight">\(\gamma_m\)</span>.</p>
<p>As an example, consider the Sigmoid function we used in logistic
regression. In that case, we can translate the function
<span class="math notranslate nohighlight">\(b(x;\gamma_m)\)</span> into the Sigmoid function</p>
<div class="math notranslate nohighlight">
\[
\sigma(t) = \frac{1}{1+\exp{(-t)}},
\]</div>
<p>where <span class="math notranslate nohighlight">\(t=\gamma_0+\gamma_1 x\)</span> and the parameters <span class="math notranslate nohighlight">\(\gamma_0\)</span> and
<span class="math notranslate nohighlight">\(\gamma_1\)</span> were determined by the Logistic Regression fitting
algorithm.</p>
<p>As another example, consider the cost function we defined for linear regression</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
\]</div>
<p>In this case the function <span class="math notranslate nohighlight">\(f(x)\)</span> was replaced by the design matrix
<span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> and the unknown linear regression parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>,
that is <span class="math notranslate nohighlight">\(\boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta}\)</span>. In linear regression we can
simply invert a matrix and obtain the parameters <span class="math notranslate nohighlight">\(\beta\)</span> by</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
\]</div>
<p>In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters <span class="math notranslate nohighlight">\(\beta_m\)</span> and <span class="math notranslate nohighlight">\(\gamma_m\)</span>.</p>
</div>
<div class="section" id="iterative-fitting-regression-and-squared-error-cost-function">
<h2>Iterative Fitting, Regression and Squared-error Cost Function<a class="headerlink" href="#iterative-fitting-regression-and-squared-error-cost-function" title="Permalink to this headline"></a></h2>
<p>The way we proceed is as follows (here we specialize to the squared-error cost function)</p>
<ol class="simple">
<li><p>Establish a cost function, here <span class="math notranslate nohighlight">\({\cal C}(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2\)</span> with <span class="math notranslate nohighlight">\(f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m)\)</span>.</p></li>
<li><p>Initialize with a guess <span class="math notranslate nohighlight">\(f_0(x)\)</span>. It could be one or even zero or some random numbers.</p></li>
<li><p>For <span class="math notranslate nohighlight">\(m=1:M\)</span></p></li>
</ol>
<p>a. minimize <span class="math notranslate nohighlight">\(\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2\)</span> wrt <span class="math notranslate nohighlight">\(\gamma\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span></p>
<p>b. This gives the optimal values <span class="math notranslate nohighlight">\(\beta_m\)</span> and <span class="math notranslate nohighlight">\(\gamma_m\)</span></p>
<p>c. Determine then the new values <span class="math notranslate nohighlight">\(f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)\)</span></p>
<p>We could use any of the algorithms we have discussed till now. If we
use trees, <span class="math notranslate nohighlight">\(\gamma\)</span> parameterizes the split variables and split points
at the internal nodes, and the predictions at the terminal nodes.</p>
</div>
<div class="section" id="squared-error-example-and-iterative-fitting">
<h2>Squared-Error Example and Iterative Fitting<a class="headerlink" href="#squared-error-example-and-iterative-fitting" title="Permalink to this headline"></a></h2>
<p>To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.</p>
<p>For simplicity we assume also that our functions <span class="math notranslate nohighlight">\(b(x;\gamma)=1+\gamma x\)</span>.</p>
<p>This means that for every iteration <span class="math notranslate nohighlight">\(m\)</span>, we need to optimize</p>
<div class="math notranslate nohighlight">
\[
(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.
\]</div>
<p>We start our iteration by simply setting <span class="math notranslate nohighlight">\(f_0(x)=0\)</span>.
Taking the derivatives with respect to <span class="math notranslate nohighlight">\(\beta\)</span> and <span class="math notranslate nohighlight">\(\gamma\)</span> we obtain</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0,
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0.
\]</div>
<p>We can then rewrite these equations as (defining <span class="math notranslate nohighlight">\(\boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x})\)</span> with <span class="math notranslate nohighlight">\(\boldsymbol{e}\)</span> being the unit vector)</p>
<div class="math notranslate nohighlight">
\[
\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0,
\]</div>
<p>which gives us <span class="math notranslate nohighlight">\(\beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w})\)</span>. Similarly we have</p>
<div class="math notranslate nohighlight">
\[
\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0,
\]</div>
<p>which leads to <span class="math notranslate nohighlight">\(\gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x})\)</span>. Inserting
for <span class="math notranslate nohighlight">\(\beta\)</span> gives us an equation for <span class="math notranslate nohighlight">\(\gamma\)</span>. This is a non-linear equation in the unknown <span class="math notranslate nohighlight">\(\gamma\)</span> and has to be solved numerically.</p>
<p>The solution to these two equations gives us in turn <span class="math notranslate nohighlight">\(\beta_1\)</span> and <span class="math notranslate nohighlight">\(\gamma_1\)</span> leading to the new expression for <span class="math notranslate nohighlight">\(f_1(x)\)</span> as
<span class="math notranslate nohighlight">\(f_1(x) = \beta_1(1+\gamma_1x)\)</span>. Doing this <span class="math notranslate nohighlight">\(M\)</span> times results in our final estimate for the function <span class="math notranslate nohighlight">\(f\)</span>.</p>
</div>
<div class="section" id="iterative-fitting-classification-and-adaboost">
<h2>Iterative Fitting, Classification and AdaBoost<a class="headerlink" href="#iterative-fitting-classification-and-adaboost" title="Permalink to this headline"></a></h2>
<p>Let us consider a binary classification problem with two outcomes <span class="math notranslate nohighlight">\(y_i \in \{-1,1\}\)</span> and <span class="math notranslate nohighlight">\(i=0,1,2,\dots,n-1\)</span> as our set of
observations. We define a classification function <span class="math notranslate nohighlight">\(G(x)\)</span> which produces a prediction taking one or the other of the two values
<span class="math notranslate nohighlight">\(\{-1,1\}\)</span>.</p>
<p>The error rate of the training sample is then</p>
<div class="math notranslate nohighlight">
\[
\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)).
\]</div>
<p>The iterative procedure starts with defining a weak classifier whose
error rate is barely better than random guessing. The iterative
procedure in boosting is to sequentially apply a weak
classification algorithm to repeatedly modified versions of the data
producing a sequence of weak classifiers <span class="math notranslate nohighlight">\(G_m(x)\)</span>.</p>
<p>Here we will express our function <span class="math notranslate nohighlight">\(f(x)\)</span> in terms of <span class="math notranslate nohighlight">\(G(x)\)</span>. That is</p>
<div class="math notranslate nohighlight">
\[
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
\]</div>
<p>will be a function of</p>
<div class="math notranslate nohighlight">
\[
G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x).
\]</div>
</div>
<div class="section" id="adaptive-boosting-adaboost">
<h2>Adaptive Boosting, AdaBoost<a class="headerlink" href="#adaptive-boosting-adaboost" title="Permalink to this headline"></a></h2>
<p>In our iterative procedure we define thus</p>
<div class="math notranslate nohighlight">
\[
f_m(x) = f_{m-1}(x)+\beta_mG_m(x).
\]</div>
<p>The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the
exponential cost/loss function defined as</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}.
\]</div>
<p>We optimize <span class="math notranslate nohighlight">\(\beta\)</span> and <span class="math notranslate nohighlight">\(G\)</span> for each value of <span class="math notranslate nohighlight">\(m=1:M\)</span> as we did in the regression case.
This is normally done in two steps. Let us however first rewrite the cost function as</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))},
\]</div>
<p>where we have defined <span class="math notranslate nohighlight">\(w_i^m= \exp{(-y_if_{m-1}(x_i))}\)</span>.</p>
</div>
<div class="section" id="building-up-adaboost">
<h2>Building up AdaBoost<a class="headerlink" href="#building-up-adaboost" title="Permalink to this headline"></a></h2>
<p>First, for any <span class="math notranslate nohighlight">\(\beta &gt; 0\)</span>, we optimize <span class="math notranslate nohighlight">\(G\)</span> by setting</p>
<div class="math notranslate nohighlight">
\[
G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)),
\]</div>
<p>which is the classifier that minimizes the weighted error rate in predicting <span class="math notranslate nohighlight">\(y\)</span>.</p>
<p>We can do this by rewriting</p>
<div class="math notranslate nohighlight">
\[
\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m,
\]</div>
<p>which can be rewritten as</p>
<div class="math notranslate nohighlight">
\[
(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0,
\]</div>
<p>which leads to</p>
<div class="math notranslate nohighlight">
\[
\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}},
\]</div>
<p>where we have redefined the error as</p>
<div class="math notranslate nohighlight">
\[
\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m},
\]</div>
<p>which leads to an update of</p>
<div class="math notranslate nohighlight">
\[
f_m(x) = f_{m-1}(x) +\beta_m G_m(x).
\]</div>
<p>This leads to the new weights</p>
<div class="math notranslate nohighlight">
\[
w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))}
\]</div>
</div>
<div class="section" id="adaptive-boosting-adaboost-basic-algorithm">
<h2>Adaptive boosting: AdaBoost, Basic Algorithm<a class="headerlink" href="#adaptive-boosting-adaboost-basic-algorithm" title="Permalink to this headline"></a></h2>
<p>The algorithm here is rather straightforward. Assume that our weak
classifier is a decision tree and we consider a binary set of outputs
with <span class="math notranslate nohighlight">\(y_i \in \{-1,1\}\)</span> and <span class="math notranslate nohighlight">\(i=0,1,2,\dots,n-1\)</span> as our set of
observations. Our design matrix is given in terms of the
feature/predictor vectors
<span class="math notranslate nohighlight">\(\boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}]\)</span>. Finally, we define also a
classifier determined by our data via a function <span class="math notranslate nohighlight">\(G(x)\)</span>. This function tells us how well we are able to classify our outputs/targets <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>.</p>
<p>We have already defined the misclassification error <span class="math notranslate nohighlight">\(\mathrm{err}\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)),
\]</div>
<p>where the function <span class="math notranslate nohighlight">\(I()\)</span> is one if we misclassify and zero if we classify correctly.</p>
</div>
<div class="section" id="basic-steps-of-adaboost">
<h2>Basic Steps of AdaBoost<a class="headerlink" href="#basic-steps-of-adaboost" title="Permalink to this headline"></a></h2>
<p>With the above definitions we are now ready to set up the algorithm for AdaBoost.
The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.</p>
<ol class="simple">
<li><p>We start by initializing all weights to <span class="math notranslate nohighlight">\(w_i = 1/n\)</span>, with <span class="math notranslate nohighlight">\(i=0,1,2,\dots n-1\)</span>. It is easy to see that we must have <span class="math notranslate nohighlight">\(\sum_{i=0}^{n-1}w_i = 1\)</span>.</p></li>
<li><p>We rewrite the misclassification error as</p></li>
</ol>
<div class="math notranslate nohighlight">
\[
\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
\]</div>
<ol class="simple">
<li><p>Then we start looping over all attempts at classifying, namely we start an iterative process for <span class="math notranslate nohighlight">\(m=1:M\)</span>, where <span class="math notranslate nohighlight">\(M\)</span> is the final number of classifications. Our given classifier could for example be a plain decision tree.</p></li>
</ol>
<p>a. Fit then a given classifier to the training set using the weights <span class="math notranslate nohighlight">\(w_i\)</span>.</p>
<p>b. Compute then <span class="math notranslate nohighlight">\(\mathrm{err}\)</span> and figure out which events are classified properly and which are classified wrongly.</p>
<p>c. Define a quantity <span class="math notranslate nohighlight">\(\alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}\)</span></p>
<p>d. Set the new weights to <span class="math notranslate nohighlight">\(w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}\)</span>.</p>
<ol class="simple">
<li><p>Compute the new classifier <span class="math notranslate nohighlight">\(G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i)\)</span>.</p></li>
</ol>
<p>For the iterations with <span class="math notranslate nohighlight">\(m \le 2\)</span> the weights are modified
individually at each steps. The observations which were misclassified
at iteration <span class="math notranslate nohighlight">\(m-1\)</span> have a weight which is larger than those which were
classified properly. As this proceeds, the observations which were
difficult to classifiy correctly are given a larger influence. Each
new classification step <span class="math notranslate nohighlight">\(m\)</span> is then forced to concentrate on those
observations that are missed in the previous iterations.</p>
</div>
<div class="section" id="adaboost-examples">
<h2>AdaBoost Examples<a class="headerlink" href="#adaboost-examples" title="Permalink to this headline"></a></h2>
<p>Using <strong>Scikit-Learn</strong> it is easy to apply the adaptive boosting algorithm, as done here.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">AdaBoostClassifier</span>
<span class="n">ada_clf</span> <span class="o">=</span> <span class="n">AdaBoostClassifier</span><span class="p">(</span>
<span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">),</span> <span class="n">n_estimators</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span>
<span class="n">algorithm</span><span class="o">=</span><span class="s2">&quot;SAMME.R&quot;</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
<span class="n">ada_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">ada_clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_confusion_matrix</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">y_probas</span> <span class="o">=</span> <span class="n">ada_clf</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_roc</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_cumulative_gain</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
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</div>
</div>
<div class="section" id="gradient-boosting-basics-with-steepest-descent-functional-gradient-descent">
<h2>Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent<a class="headerlink" href="#gradient-boosting-basics-with-steepest-descent-functional-gradient-descent" title="Permalink to this headline"></a></h2>
<p>Gradient boosting is again a similar technique to Adaptive boosting,
it combines so-called weak classifiers or regressors into a strong
method via a series of iterations.</p>
<p>In order to understand the method, let us illustrate its basics by
bringing back the essential steps in linear regression, where our cost
function was the least squares function.</p>
</div>
<div class="section" id="the-squared-error-again-steepest-descent">
<h2>The Squared-Error again! Steepest Descent<a class="headerlink" href="#the-squared-error-again-steepest-descent" title="Permalink to this headline"></a></h2>
<p>We start again with our cost function <span class="math notranslate nohighlight">\({\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i))\)</span> where we want to minimize
This means that for every iteration, we need to optimize</p>
<div class="math notranslate nohighlight">
\[
(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
\]</div>
<p>We define a real function <span class="math notranslate nohighlight">\(h_m(x)\)</span> that defines our final function <span class="math notranslate nohighlight">\(f_M(x)\)</span> as</p>
<div class="math notranslate nohighlight">
\[
f_M(x) = \sum_{m=0}^M h_m(x).
\]</div>
<p>In the steepest decent approach we approximate <span class="math notranslate nohighlight">\(h_m(x) = -\rho_m g_m(x)\)</span>, where <span class="math notranslate nohighlight">\(\rho_m\)</span> is a scalar and <span class="math notranslate nohighlight">\(g_m(x)\)</span> the gradient defined as</p>
<div class="math notranslate nohighlight">
\[
g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}.
\]</div>
<p>With the new gradient we can update <span class="math notranslate nohighlight">\(f_m(x) = f_{m-1}(x) -\rho_m g_m(x)\)</span>. Using the above squared-error function we see that
the gradient is <span class="math notranslate nohighlight">\(g_m(x_i) = -2(y_i-f(x_i))\)</span>.</p>
<p>Choosing <span class="math notranslate nohighlight">\(f_0(x)=0\)</span> we obtain <span class="math notranslate nohighlight">\(g_m(x) = -2y_i\)</span> and inserting this into the minimization problem for the cost function we have</p>
<div class="math notranslate nohighlight">
\[
(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2.
\]</div>
</div>
<div class="section" id="steepest-descent-example">
<h2>Steepest Descent Example<a class="headerlink" href="#steepest-descent-example" title="Permalink to this headline"></a></h2>
<p>Optimizing with respect to <span class="math notranslate nohighlight">\(\rho\)</span> we obtain (taking the derivative) that <span class="math notranslate nohighlight">\(\rho_1 = -1/2\)</span>. We have then that</p>
<div class="math notranslate nohighlight">
\[
f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i.
\]</div>
<p>We can then proceed and compute</p>
<div class="math notranslate nohighlight">
\[
g_2(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,
\]</div>
<p>and find a new value for <span class="math notranslate nohighlight">\(\rho_2=-1/2\)</span> and continue till we have reached <span class="math notranslate nohighlight">\(m=M\)</span>. We can modify the steepest descent method, or steepest boosting, by introducing what is called <strong>gradient boosting</strong>.</p>
</div>
<div class="section" id="gradient-boosting-algorithm">
<h2>Gradient Boosting, algorithm<a class="headerlink" href="#gradient-boosting-algorithm" title="Permalink to this headline"></a></h2>
<p>Steepest descent is however not much used, since it only optimizes <span class="math notranslate nohighlight">\(f\)</span> at a fixed set of <span class="math notranslate nohighlight">\(n\)</span> points,
so we do not learn a function that can generalize. However, we can modify the algorithm by
fitting a weak learner to approximate the negative gradient signal.</p>
<p>Suppose we have a cost function <span class="math notranslate nohighlight">\(C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i))\)</span> where <span class="math notranslate nohighlight">\(y_i\)</span> is our target and <span class="math notranslate nohighlight">\(f(x_i)\)</span> the function which is meant to model <span class="math notranslate nohighlight">\(y_i\)</span>. The above cost function could be our standard squared-error function</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2.
\]</div>
<p>The way we proceed in an iterative fashion is to</p>
<ol class="simple">
<li><p>Initialize our estimate <span class="math notranslate nohighlight">\(f_0(x)\)</span>.</p></li>
<li><p>For <span class="math notranslate nohighlight">\(m=1:M\)</span>, we</p></li>
</ol>
<p>a. compute the negative gradient vector <span class="math notranslate nohighlight">\(\boldsymbol{u}_m = -\partial C(\boldsymbol{y},\boldsymbol{f})/\partial \boldsymbol{f}(x)\)</span> at <span class="math notranslate nohighlight">\(f(x) = f_{m-1}(x)\)</span>;</p>
<p>b. fit the so-called base-learner to the negative gradient <span class="math notranslate nohighlight">\(h_m(u_m,x)\)</span>;</p>
<p>c. update the estimate <span class="math notranslate nohighlight">\(f_m(x) = f_{m-1}(x)+h_m(u_m,x)\)</span>;</p>
<ol class="simple">
<li><p>The final estimate is then <span class="math notranslate nohighlight">\(f_M(x) = \sum_{m=1}^M h_m(u_m,x)\)</span>.</p></li>
</ol>
</div>
<div class="section" id="gradient-boosting-examples-of-regression">
<h2>Gradient Boosting, Examples of Regression<a class="headerlink" href="#gradient-boosting-examples-of-regression" title="Permalink to this headline"></a></h2>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">GradientBoostingRegressor</span>
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">maxdegree</span> <span class="o">=</span> <span class="mi">6</span>
<span class="c1"># Make data set.</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">error</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdegree</span><span class="p">)</span>
<span class="n">bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdegree</span><span class="p">)</span>
<span class="n">variance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdegree</span><span class="p">)</span>
<span class="n">polydegree</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdegree</span><span class="p">)</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">maxdegree</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">GradientBoostingRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="n">degree</span><span class="p">,</span> <span class="n">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">1.0</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">polydegree</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">degree</span>
<span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">y_pred</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="p">)</span>
<span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="p">(</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_pred</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
<span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_pred</span><span class="p">)</span> <span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Max depth:&#39;</span><span class="p">,</span> <span class="n">degree</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Error:&#39;</span><span class="p">,</span> <span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Bias^2:&#39;</span><span class="p">,</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Var:&#39;</span><span class="p">,</span> <span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1"> &gt;= </span><span class="si">{}</span><span class="s1"> + </span><span class="si">{}</span><span class="s1"> = </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span><span class="o">+</span><span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">]))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">maxdegree</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">error</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Error&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;bias&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">variance</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Variance&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;gdregression&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Max depth: 1
Error: 0.5069860830630872
Bias^2: 0.28951491799791806
Var: 0.21747116506516934
0.5069860830630872 &gt;= 0.28951491799791806 + 0.21747116506516934 = 0.5069860830630875
Max depth: 2
Error: 0.5222413718621189
Bias^2: 0.28962869031035515
Var: 0.23261268155176382
0.5222413718621189 &gt;= 0.28962869031035515 + 0.23261268155176382 = 0.5222413718621189
Max depth: 3
Error: 0.522240032475565
Bias^2: 0.2896287710119233
Var: 0.2326112614636416
0.522240032475565 &gt;= 0.2896287710119233 + 0.2326112614636416 = 0.5222400324755649
Max depth: 4
Error: 0.5222400329453616
Bias^2: 0.28962877060331055
Var: 0.2326112623420511
0.5222400329453616 &gt;= 0.28962877060331055 + 0.2326112623420511 = 0.5222400329453616
Max depth: 5
Error: 0.5222400329453616
Bias^2: 0.28962877060331055
Var: 0.2326112623420511
0.5222400329453616 &gt;= 0.28962877060331055 + 0.2326112623420511 = 0.5222400329453616
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:494: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
y = column_or_1d(y, warn=True)
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:494: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
y = column_or_1d(y, warn=True)
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:494: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
y = column_or_1d(y, warn=True)
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:494: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
y = column_or_1d(y, warn=True)
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:494: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
y = column_or_1d(y, warn=True)
</pre></div>
</div>
<img alt="_images/week47_92_2.png" src="_images/week47_92_2.png" />
</div>
</div>
</div>
<div class="section" id="gradient-boosting-classification-example">
<h2>Gradient Boosting, Classification Example<a class="headerlink" href="#gradient-boosting-classification-example" title="Permalink to this headline"></a></h2>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">GradientBoostingClassifier</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_validate</span>
<span class="c1"># Load the data</span>
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="c1">#now scale the data</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">gd_clf</span> <span class="o">=</span> <span class="n">GradientBoostingClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">1.0</span><span class="p">)</span>
<span class="n">gd_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="c1">#Cross validation</span>
<span class="n">accuracy</span> <span class="o">=</span> <span class="n">cross_validate</span><span class="p">(</span><span class="n">gd_clf</span><span class="p">,</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">,</span><span class="n">cv</span><span class="o">=</span><span class="mi">10</span><span class="p">)[</span><span class="s1">&#39;test_score&#39;</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">accuracy</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test set accuracy with Gradient boosting and scaled data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">gd_clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">gd_clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_confusion_matrix</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;gdclassiffierconfusion&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">y_probas</span> <span class="o">=</span> <span class="n">gd_clf</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_roc</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;gdclassiffierroc&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_cumulative_gain</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;gdclassiffiercgain&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(426, 30)
(143, 30)
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.93333333 0.93333333 0.8 0.85714286 1. 0.92857143
1. 0.92857143 0.92857143 0.92857143]
Test set accuracy with Gradient boosting and scaled data: 0.97
</pre></div>
</div>
<img alt="_images/week47_94_2.png" src="_images/week47_94_2.png" />
<img alt="_images/week47_94_3.png" src="_images/week47_94_3.png" />
<img alt="_images/week47_94_4.png" src="_images/week47_94_4.png" />
</div>
</div>
</div>
<div class="section" id="xgboost-extreme-gradient-boosting">
<h2>XGBoost: Extreme Gradient Boosting<a class="headerlink" href="#xgboost-extreme-gradient-boosting" title="Permalink to this headline"></a></h2>
<p><a class="reference external" href="https://github.com/dmlc/xgboost">XGBoost</a> or Extreme Gradient
Boosting, is an optimized distributed gradient boosting library
designed to be highly efficient, flexible and portable. It implements
machine learning algorithms under the Gradient Boosting
framework. XGBoost provides a parallel tree boosting that solve many
data science problems in a fast and accurate way. See the <a class="reference external" href="https://arxiv.org/abs/1603.02754">article by Chen and Guestrin</a>.</p>
<p>The authors design and build a highly scalable end-to-end tree
boosting system. It has a theoretically justified weighted quantile
sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.</p>
<p>It is now the algorithm which wins essentially all ML competitions!!!</p>
</div>
<div class="section" id="regression-case">
<h2>Regression Case<a class="headerlink" href="#regression-case" title="Permalink to this headline"></a></h2>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">import</span> <span class="nn">xgboost</span> <span class="k">as</span> <span class="nn">xgb</span>
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">maxdegree</span> <span class="o">=</span> <span class="mi">6</span>
<span class="c1"># Make data set.</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">error</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdegree</span><span class="p">)</span>
<span class="n">bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdegree</span><span class="p">)</span>
<span class="n">variance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdegree</span><span class="p">)</span>
<span class="n">polydegree</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdegree</span><span class="p">)</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">maxdegree</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">xgb</span><span class="o">.</span><span class="n">XGBRegressor</span><span class="p">(</span><span class="n">objective</span> <span class="o">=</span><span class="s1">&#39;reg:squarederror&#39;</span><span class="p">,</span> <span class="n">colsaobjective</span> <span class="o">=</span><span class="s1">&#39;reg:squarederror&#39;</span><span class="p">,</span> <span class="n">colsample_bytree</span> <span class="o">=</span> <span class="mf">0.3</span><span class="p">,</span> <span class="n">learning_rate</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span><span class="n">max_depth</span> <span class="o">=</span> <span class="n">degree</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="n">n_estimators</span> <span class="o">=</span> <span class="mi">200</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">polydegree</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">degree</span>
<span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">y_pred</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="p">)</span>
<span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="p">(</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_pred</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
<span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_pred</span><span class="p">)</span> <span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Max depth:&#39;</span><span class="p">,</span> <span class="n">degree</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Error:&#39;</span><span class="p">,</span> <span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Bias^2:&#39;</span><span class="p">,</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Var:&#39;</span><span class="p">,</span> <span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1"> &gt;= </span><span class="si">{}</span><span class="s1"> + </span><span class="si">{}</span><span class="s1"> = </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span><span class="o">+</span><span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">]))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">maxdegree</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">error</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Error&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;bias&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">variance</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Variance&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/xgboost/compat.py:36: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead.
from pandas import MultiIndex, Int64Index
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576:
Parameters: { &quot;colsaobjective&quot; } might not be used.
This could be a false alarm, with some parameters getting used by language bindings but
then being mistakenly passed down to XGBoost core, or some parameter actually being used
but getting flagged wrongly here. Please open an issue if you find any such cases.
Max depth: 0
Error: 0.35587778675776993
Bias^2: 0.35587778675776993
Var: 0.0
0.35587778675776993 &gt;= 0.35587778675776993 + 0.0 = 0.35587778675776993
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576:
Parameters: { &quot;colsaobjective&quot; } might not be used.
This could be a false alarm, with some parameters getting used by language bindings but
then being mistakenly passed down to XGBoost core, or some parameter actually being used
but getting flagged wrongly here. Please open an issue if you find any such cases.
Max depth: 1
Error: 0.3001669239476101
Bias^2: 0.2731899981798276
Var: 0.0269769337028265
0.3001669239476101 &gt;= 0.2731899981798276 + 0.0269769337028265 = 0.3001669318826541
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576:
Parameters: { &quot;colsaobjective&quot; } might not be used.
This could be a false alarm, with some parameters getting used by language bindings but
then being mistakenly passed down to XGBoost core, or some parameter actually being used
but getting flagged wrongly here. Please open an issue if you find any such cases.
Max depth: 2
Error: 0.30000279381576256
Bias^2: 0.2711099029541577
Var: 0.02889288030564785
0.30000279381576256 &gt;= 0.2711099029541577 + 0.02889288030564785 = 0.30000278325980556
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576:
Parameters: { &quot;colsaobjective&quot; } might not be used.
This could be a false alarm, with some parameters getting used by language bindings but
then being mistakenly passed down to XGBoost core, or some parameter actually being used
but getting flagged wrongly here. Please open an issue if you find any such cases.
Max depth: 3
Error: 0.2999692169766251
Bias^2: 0.2710765113417533
Var: 0.028892725706100464
0.2999692169766251 &gt;= 0.2710765113417533 + 0.028892725706100464 = 0.2999692370478538
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576:
Parameters: { &quot;colsaobjective&quot; } might not be used.
This could be a false alarm, with some parameters getting used by language bindings but
then being mistakenly passed down to XGBoost core, or some parameter actually being used
but getting flagged wrongly here. Please open an issue if you find any such cases.
Max depth: 4
Error: 0.2999728858924457
Bias^2: 0.2710867359084896
Var: 0.02888614870607853
0.2999728858924457 &gt;= 0.2710867359084896 + 0.02888614870607853 = 0.2999728846145681
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576:
Parameters: { &quot;colsaobjective&quot; } might not be used.
This could be a false alarm, with some parameters getting used by language bindings but
then being mistakenly passed down to XGBoost core, or some parameter actually being used
but getting flagged wrongly here. Please open an issue if you find any such cases.
Max depth: 5
Error: 0.29998782047173667
Bias^2: 0.2711004934923823
Var: 0.02888733707368374
0.29998782047173667 &gt;= 0.2711004934923823 + 0.02888733707368374 = 0.29998783056606604
</pre></div>
</div>
<img alt="_images/week47_97_2.png" src="_images/week47_97_2.png" />
</div>
</div>
</div>
<div class="section" id="xgboost-on-the-cancer-data">
<h2>Xgboost on the Cancer Data<a class="headerlink" href="#xgboost-on-the-cancer-data" title="Permalink to this headline"></a></h2>
<p>As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">LabelEncoder</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_validate</span>
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
<span class="kn">import</span> <span class="nn">xgboost</span> <span class="k">as</span> <span class="nn">xgb</span>
<span class="c1"># Load the data</span>
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="c1">#now scale the data</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">xg_clf</span> <span class="o">=</span> <span class="n">xgb</span><span class="o">.</span><span class="n">XGBClassifier</span><span class="p">()</span>
<span class="n">xg_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
<span class="n">y_test</span> <span class="o">=</span> <span class="n">xg_clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test set accuracy with Gradient Boosting and scaled data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">xg_clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">xg_clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_confusion_matrix</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;xdclassiffierconfusion&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">y_probas</span> <span class="o">=</span> <span class="n">xg_clf</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_roc</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;xdclassiffierroc&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_cumulative_gain</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;gdclassiffiercgain&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">xgb</span><span class="o">.</span><span class="n">plot_tree</span><span class="p">(</span><span class="n">xg_clf</span><span class="p">,</span><span class="n">num_trees</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;figure.figsize&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">50</span><span class="p">,</span> <span class="mi">10</span><span class="p">]</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;xgtree&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">xgb</span><span class="o">.</span><span class="n">plot_importance</span><span class="p">(</span><span class="n">xg_clf</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;figure.figsize&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;xgparams&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(426, 30)
(143, 30)
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:1115: Starting in XGBoost 1.3.0, the default evaluation metric used with the objective &#39;binary:logistic&#39; was changed from &#39;error&#39; to &#39;logloss&#39;. Explicitly set eval_metric if you&#39;d like to restore the old behavior.
Test set accuracy with Gradient Boosting and scaled data: 1.00
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/xgboost/sklearn.py:1224: UserWarning: The use of label encoder in XGBClassifier is deprecated and will be removed in a future release. To remove this warning, do the following: 1) Pass option use_label_encoder=False when constructing XGBClassifier object; and 2) Encode your labels (y) as integers starting with 0, i.e. 0, 1, 2, ..., [num_class - 1].
warnings.warn(label_encoder_deprecation_msg, UserWarning)
</pre></div>
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</div>
<div class="section" id="summary-of-course">
<h2>Summary of course<a class="headerlink" href="#summary-of-course" title="Permalink to this headline"></a></h2>
</div>
<div class="section" id="what-me-worry-no-final-exam-in-this-course">
<h2>What? Me worry? No final exam in this course!<a class="headerlink" href="#what-me-worry-no-final-exam-in-this-course" title="Permalink to this headline"></a></h2>
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<p><img src="figures/exam1.jpeg" width="500"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
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<div class="section" id="what-is-the-link-between-artificial-intelligence-and-machine-learning-and-some-general-remarks">
<h2>What is the link between Artificial Intelligence and Machine Learning and some general Remarks<a class="headerlink" href="#what-is-the-link-between-artificial-intelligence-and-machine-learning-and-some-general-remarks" title="Permalink to this headline"></a></h2>
<p>Artificial intelligence is built upon integrated machine learning
algorithms as discussed in this course, which in turn are fundamentally rooted in optimization and
statistical learning.</p>
<p>Can we have Artificial Intelligence without Machine Learning? See <a class="reference external" href="https://www.linkedin.com/pulse/what-artificial-intelligence-without-machine-learning-claudia-pohlink">this post for inspiration</a>.</p>
</div>
<div class="section" id="going-back-to-the-beginning-of-the-semester">
<h2>Going back to the beginning of the semester<a class="headerlink" href="#going-back-to-the-beginning-of-the-semester" title="Permalink to this headline"></a></h2>
<p>Traditionally the field of machine learning has had its main focus on
predictions and correlations. These concepts outline in some sense
the difference between machine learning and what is normally called
Bayesian statistics or Bayesian inference.</p>
<p>In machine learning and prediction based tasks, we are often
interested in developing algorithms that are capable of learning
patterns from given data in an automated fashion, and then using these
learned patterns to make predictions or assessments of newly given
data. In many cases, our primary concern is the quality of the
predictions or assessments, and we are less concerned with the
underlying patterns that were learned in order to make these
predictions. This leads to what normally has been labeled as a
frequentist approach.</p>
</div>
<div class="section" id="not-so-sharp-distinctions">
<h2>Not so sharp distinctions<a class="headerlink" href="#not-so-sharp-distinctions" title="Permalink to this headline"></a></h2>
<p>You should keep in mind that the division between a traditional
frequentist approach with focus on predictions and correlations only
and a Bayesian approach with an emphasis on estimations and
causations, is not that sharp. Machine learning can be frequentist
with ensemble methods (EMB) as examples and Bayesian with Gaussian
Processes as examples.</p>
<p>If one views ML from a statistical learning
perspective, one is then equally interested in estimating errors as
one is in finding correlations and making predictions. It is important
to keep in mind that the frequentist and Bayesian approaches differ
mainly in their interpretations of probability. In the frequentist
world, we can only assign probabilities to repeated random
phenomena. From the observations of these phenomena, we can infer the
probability of occurrence of a specific event. In Bayesian
statistics, we assign probabilities to specific events and the
probability represents the measure of belief/confidence for that
event. The belief can be updated in the light of new evidence.</p>
</div>
<div class="section" id="topics-we-have-covered-this-year">
<h2>Topics we have covered this year<a class="headerlink" href="#topics-we-have-covered-this-year" title="Permalink to this headline"></a></h2>
<p>The course has two central parts</p>
<ol class="simple">
<li><p>Statistical analysis and optimization of data</p></li>
<li><p>Machine learning</p></li>
</ol>
</div>
<div class="section" id="statistical-analysis-and-optimization-of-data">
<h2>Statistical analysis and optimization of data<a class="headerlink" href="#statistical-analysis-and-optimization-of-data" title="Permalink to this headline"></a></h2>
<p>The following topics have been discussed:</p>
<ol class="simple">
<li><p>Basic concepts, expectation values, variance, covariance, correlation functions and errors;</p></li>
<li><p>Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;</p></li>
<li><p>Central elements from linear algebra, matrix inversion and SVD</p></li>
<li><p>Gradient methods for data optimization</p></li>
<li><p>Estimation of errors using cross-validation, bootstrapping and jackknife methods;</p></li>
<li><p>Practical optimization using Singular-value decomposition and least squares for parameterizing data.</p></li>
<li><p>Principal Component Analysis to reduce the number of features.</p></li>
</ol>
</div>
<div class="section" id="machine-learning">
<h2>Machine learning<a class="headerlink" href="#machine-learning" title="Permalink to this headline"></a></h2>
<p>The following topics will be covered</p>
<ol class="simple">
<li><p>Linear methods for regression and classification:</p></li>
</ol>
<p>a. Ordinary Least Squares</p>
<p>b. Ridge regression</p>
<p>c. Lasso regression</p>
<p>d. Logistic regression</p>
<ol class="simple">
<li><p>Neural networks and deep learning:</p></li>
</ol>
<p>a. Feed Forward Neural Networks</p>
<p>b. Convolutional Neural Networks</p>
<p>c. Recurrent Neural Networks</p>
<ol class="simple">
<li><p>Decisions trees and ensemble methods:</p></li>
</ol>
<p>a. Decision trees</p>
<p>b. Bagging and voting</p>
<p>c. Random forests</p>
<p>d. Boosting and gradient boosting</p>
<ol class="simple">
<li><p>Support vector machines, not covered this year but included in notes</p></li>
</ol>
<p>a. Binary classification and multiclass classification</p>
<p>b. Kernel methods</p>
<p>c. Regression</p>
</div>
<div class="section" id="learning-outcomes-and-overarching-aims-of-this-course">
<h2>Learning outcomes and overarching aims of this course<a class="headerlink" href="#learning-outcomes-and-overarching-aims-of-this-course" title="Permalink to this headline"></a></h2>
<p>The course introduces a variety of central algorithms and methods
essential for studies of data analysis and machine learning. The
course is project based and through the various projects, normally
three, you will be exposed to fundamental research problems
in these fields, with the aim to reproduce state of the art scientific
results. The students will learn to develop and structure large codes
for studying these systems, get acquainted with computing facilities
and learn to handle large scientific projects. A good scientific and
ethical conduct is emphasized throughout the course.</p>
<ul class="simple">
<li><p>Understand linear methods for regression and classification;</p></li>
<li><p>Learn about neural network;</p></li>
<li><p>Learn about bagging, boosting and trees</p></li>
<li><p>Support vector machines, not covered</p></li>
<li><p>Learn about basic data analysis;</p></li>
<li><p>Be capable of extending the acquired knowledge to other systems and cases;</p></li>
<li><p>Have an understanding of central algorithms used in data analysis and machine learning;</p></li>
<li><p>Work on numerical projects to illustrate the theory. The projects play a central role and you are expected to know modern programming languages like Python or C++.</p></li>
</ul>
</div>
<div class="section" id="perspective-on-machine-learning">
<h2>Perspective on Machine Learning<a class="headerlink" href="#perspective-on-machine-learning" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>Rapidly emerging application area</p></li>
<li><p>Experiment AND theory are evolving in many many fields. Still many low-hanging fruits.</p></li>
<li><p>Requires education/retraining for more widespread adoption</p></li>
<li><p>A lot of “word-of-mouth” development methods</p></li>
</ol>
<p>Huge amounts of data sets require automation, classical analysis tools often inadequate.
High energy physics hit this wall in the 90’s.
In 2009 single top quark production was determined via <a class="reference external" href="https://arxiv.org/pdf/0903.0850.pdf">Boosted decision trees, Bayesian
Neural Networks, etc.</a>. Similarly, the search for Higgs was a statistical learning tour de force. See this link on <a class="reference external" href="https://www.kaggle.com/c/higgs-boson">Kaggle.com</a>.</p>
</div>
<div class="section" id="machine-learning-research">
<h2>Machine Learning Research<a class="headerlink" href="#machine-learning-research" title="Permalink to this headline"></a></h2>
<p>Where to find recent results:</p>
<ol class="simple">
<li><p>Conference proceedings, arXiv and blog posts!</p></li>
<li><p><strong>NIPS</strong>: <a class="reference external" href="https://papers.nips.cc">Neural Information Processing Systems</a></p></li>
<li><p><strong>ICLR</strong>: <a class="reference external" href="https://openreview.net/group?id=ICLR.cc/2018/Conference#accepted-oral-papers">International Conference on Learning Representations</a></p></li>
<li><p><strong>ICML</strong>: International Conference on Machine Learning</p></li>
<li><p><a class="reference external" href="http://www.jmlr.org/papers/v19/">Journal of Machine Learning Research</a></p></li>
<li><p><a class="reference external" href="https://arxiv.org/list/cs.LG/recent">Follow ML on ArXiv</a></p></li>
</ol>
</div>
<div class="section" id="starting-your-machine-learning-project">
<h2>Starting your Machine Learning Project<a class="headerlink" href="#starting-your-machine-learning-project" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>Identify problem type: classification, regression</p></li>
<li><p>Consider your data carefully</p></li>
<li><p>Choose a simple model that fits 1. and 2.</p></li>
<li><p>Consider your data carefully again! Think of data representation more carefully.</p></li>
<li><p>Based on your results, feedback loop to earliest possible point</p></li>
</ol>
</div>
<div class="section" id="choose-a-model-and-algorithm">
<h2>Choose a Model and Algorithm<a class="headerlink" href="#choose-a-model-and-algorithm" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>Supervised?</p></li>
<li><p>Start with the simplest model that fits your problem</p></li>
<li><p>Start with minimal processing of data</p></li>
</ol>
</div>
<div class="section" id="preparing-your-data">
<h2>Preparing Your Data<a class="headerlink" href="#preparing-your-data" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>Shuffle your data</p></li>
<li><p>Mean center your data</p></li>
</ol>
<ul class="simple">
<li><p>Why?</p></li>
</ul>
<ol class="simple">
<li><p>Normalize the variance</p></li>
</ol>
<ul class="simple">
<li><p>Why?</p></li>
</ul>
<ol class="simple">
<li><p><a class="reference external" href="https://multivariatestatsjl.readthedocs.io/en/latest/whiten.html">Whitening</a></p></li>
</ol>
<ul class="simple">
<li><p>Decorrelates data</p></li>
<li><p>Can be hit or miss</p></li>
</ul>
<ol class="simple">
<li><p>When to do train/test split?</p></li>
</ol>
<p>Whitening is a decorrelation transformation that transforms a set of
random variables into a set of new random variables with identity
covariance (uncorrelated with unit variances).</p>
</div>
<div class="section" id="which-activation-and-weights-to-choose-in-neural-networks">
<h2>Which Activation and Weights to Choose in Neural Networks<a class="headerlink" href="#which-activation-and-weights-to-choose-in-neural-networks" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>RELU? ELU?</p></li>
<li><p>Sigmoid or Tanh?</p></li>
<li><p>Set all weights to 0?</p></li>
</ol>
<ul class="simple">
<li><p>Terrible idea</p></li>
</ul>
<ol class="simple">
<li><p>Set all weights to random values?</p></li>
</ol>
<ul class="simple">
<li><p>Small random values</p></li>
</ul>
</div>
<div class="section" id="optimization-methods-and-hyperparameters">
<h2>Optimization Methods and Hyperparameters<a class="headerlink" href="#optimization-methods-and-hyperparameters" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>Stochastic gradient descent</p></li>
</ol>
<p>a. Stochastic gradient descent + momentum</p>
<ol class="simple">
<li><p>State-of-the-art approaches:</p></li>
</ol>
<ul class="simple">
<li><p>RMSProp</p></li>
<li><p>Adam</p></li>
<li><p>and more</p></li>
</ul>
<p>Which regularization and hyperparameters? <span class="math notranslate nohighlight">\(L_1\)</span> or <span class="math notranslate nohighlight">\(L_2\)</span>, soft
classifiers, depths of trees and many other. Need to explore a large
set of hyperparameters and regularization methods.</p>
</div>
<div class="section" id="resampling">
<h2>Resampling<a class="headerlink" href="#resampling" title="Permalink to this headline"></a></h2>
<p>When do we resample?</p>
<ol class="simple">
<li><p><a class="reference external" href="https://www.cambridge.org/core/books/bootstrap-methods-and-their-application/ED2FD043579F27952363566DC09CBD6A">Bootstrap</a></p></li>
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=fSytzGwwBVw&amp;ab_channel=StatQuestwithJoshStarmer">Cross-validation</a></p></li>
<li><p>Jackknife and many other</p></li>
</ol>
</div>
<div class="section" id="other-courses-on-data-science-and-machine-learning-at-uio">
<h2>Other courses on Data science and Machine Learning at UiO<a class="headerlink" href="#other-courses-on-data-science-and-machine-learning-at-uio" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/fys/FYS5429/index-eng.html">FYS5429 Advanced Machine Learning and Data Analysis for the Physical Sciences</a>. Discussed deep learning and generative deep learning.</p></li>
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/fys/FYS5419/index-eng.html">FYS5419 Quantum Computing and Quantum Machine Learning</a></p></li>
<li><p><a class="reference external" href="http://www.uio.no/studier/emner/matnat/math/STK2100/index-eng.html">STK2100 Machine learning and statistical methods for prediction and classification</a>.</p></li>
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/ifi/IN3050/index-eng.html">IN3050/IN4050 Introduction to Artificial Intelligence and Machine Learning</a>. Introductory course in machine learning and AI with an algorithmic approach.</p></li>
<li><p><a class="reference external" href="http://www.uio.no/studier/emner/matnat/math/STK-INF3000/index-eng.html">STK-INF3000/4000 Selected Topics in Data Science</a>. The course provides insight into selected contemporary relevant topics within Data Science.</p></li>
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/ifi/IN4080/index.html">IN4080 Natural Language Processing</a>. Probabilistic and machine learning techniques applied to natural language processing. o <a class="reference external" href="https://www.uio.no/studier/emner/matnat/math/STK-IN4300/index-eng.html">STK-IN4300 – Statistical learning methods in Data Science</a>. An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.</p></li>
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/ifi/IN-STK5000/index-eng.html">IN-STK5000 Adaptive Methods for Data-Based Decision Making</a>. Methods for adaptive collection and processing of data based on machine learning techniques.</p></li>
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/ifi/IN5400/">IN5400/INF5860 – Machine Learning for Image Analysis</a>. An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.</p></li>
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/its/TEK5040/">TEK5040 – Dyp læring for autonome systemer</a>. The course addresses advanced algorithms and architectures for deep learning with neural networks. The course provides an introduction to how deep-learning techniques can be used in the construction of key parts of advanced autonomous systems that exist in physical environments and cyber environments.</p></li>
</ol>
</div>
<div class="section" id="additional-courses-of-interest">
<h2>Additional courses of interest<a class="headerlink" href="#additional-courses-of-interest" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html">STK4051 Computational Statistics</a></p></li>
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/math/STK4021/index-eng.html">STK4021 Applied Bayesian Analysis and Numerical Methods</a></p></li>
</ol>
</div>
<div class="section" id="what-s-the-future-like">
<h2>What’s the future like?<a class="headerlink" href="#what-s-the-future-like" title="Permalink to this headline"></a></h2>
<p>Based on multi-layer nonlinear neural networks, deep learning can
learn directly from raw data, automatically extract and abstract
features from layer to layer, and then achieve the goal of regression,
classification, or ranking. Deep learning has made breakthroughs in
computer vision, speech processing and natural language, and reached
or even surpassed human level. The success of deep learning is mainly
due to the three factors: big data, big model, and big computing.</p>
<p>In the past few decades, many different architectures of deep neural
networks have been proposed, such as</p>
<ol class="simple">
<li><p>Convolutional neural networks, which are mostly used in image and video data processing, and have also been applied to sequential data such as text processing;</p></li>
<li><p>Recurrent neural networks, which can process sequential data of variable length and have been widely used in natural language understanding and speech processing;</p></li>
<li><p>Encoder-decoder framework, which is mostly used for image or sequence generation, such as machine translation, text summarization, and image captioning.</p></li>
<li><p><strong>Generative deep learning</strong>! Recent textbook by David Foster (and obviously many other ones) at <a class="reference external" href="https://www.oreilly.com/library/view/generative-deep-learning/9781492041931/">https://www.oreilly.com/library/view/generative-deep-learning/9781492041931/</a>”</p></li>
</ol>
</div>
<div class="section" id="types-of-machine-learning-a-repetition">
<h2>Types of Machine Learning, a repetition<a class="headerlink" href="#types-of-machine-learning-a-repetition" title="Permalink to this headline"></a></h2>
<p>The approaches to machine learning are many, but are often split into two main categories.
In <em>supervised learning</em> we know the answer to a problem,
and let the computer deduce the logic behind it. On the other hand, <em>unsupervised learning</em>
is a method for finding patterns and relationship in data sets without any prior knowledge of the system.
Some authours also operate with a third category, namely <em>reinforcement learning</em>. This is a paradigm
of learning inspired by behavioural psychology, where learning is achieved by trial-and-error,
solely from rewards and punishment.</p>
<p>Another way to categorize machine learning tasks is to consider the desired output of a system.
Some of the most common tasks are:</p>
<ul class="simple">
<li><p>Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.</p></li>
<li><p>Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.</p></li>
<li><p>Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.</p></li>
<li><p>Other unsupervised learning algortihms like <strong>Boltzmann machines</strong></p></li>
</ul>
</div>
<div class="section" id="why-boltzmann-machines">
<h2>Why Boltzmann machines?<a class="headerlink" href="#why-boltzmann-machines" title="Permalink to this headline"></a></h2>
<p>What is known as restricted Boltzmann Machines (RMB) have received a lot of attention lately.
One of the major reasons is that they can be stacked layer-wise to build deep neural networks that capture complicated statistics.</p>
<p>The original RBMs had just one visible layer and a hidden layer, but recently so-called Gaussian-binary RBMs have gained quite some popularity in imaging since they are capable of modeling continuous data that are common to natural images.</p>
<p>Furthermore, they have been used to solve complicated <a class="reference external" href="https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.045002">quantum mechanical many-particle problems or classical statistical physics problems like the Ising and Potts classes of models</a>.</p>
</div>
<div class="section" id="boltzmann-machines">
<h2>Boltzmann Machines<a class="headerlink" href="#boltzmann-machines" title="Permalink to this headline"></a></h2>
<p>Why use a generative model rather than the more well known discriminative deep neural networks (DNN)? <strong>Simplest approach to generative deep learning</strong>.</p>
<ul class="simple">
<li><p>Discriminitave methods have several limitations: They are mainly supervised learning methods, thus requiring labeled data. And there are tasks they cannot accomplish, like drawing new examples from an unknown probability distribution.</p></li>
<li><p>A generative model can learn to represent and sample from a probability distribution. The core idea is to learn a parametric model of the probability distribution from which the training data was drawn. As an example</p></li>
</ul>
<p>a. A model for images could learn to draw new examples of cats and dogs, given a training dataset of images of cats and dogs.</p>
<p>b. Generate a sample of an ordered or disordered phase, having been given samples of such phases.</p>
<p>c. Model the trial function for <a class="reference external" href="https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.045002">Monte Carlo calculations</a>.</p>
</div>
<div class="section" id="some-similarities-and-differences-from-dnns">
<h2>Some similarities and differences from DNNs<a class="headerlink" href="#some-similarities-and-differences-from-dnns" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>Both use gradient-descent based learning procedures for minimizing cost functions</p></li>
<li><p>Energy based models don’t use backpropagation and automatic differentiation for computing gradients, instead turning to Markov Chain Monte Carlo methods.</p></li>
<li><p>DNNs often have several hidden layers. A restricted Boltzmann machine has only one hidden layer, however several RBMs can be stacked to make up Deep Belief Networks, of which they constitute the building blocks.</p></li>
</ol>
<p>History: The RBM was developed by amongst others <a class="reference external" href="https://en.wikipedia.org/wiki/Geoffrey_Hinton">Geoffrey Hinton</a>, called by some the “Godfather of Deep Learning”, working with the University of Toronto and Google.</p>
</div>
<div class="section" id="boltzmann-machines-bm">
<h2>Boltzmann machines (BM)<a class="headerlink" href="#boltzmann-machines-bm" title="Permalink to this headline"></a></h2>
<p>A BM is what we would call an undirected probabilistic graphical model
with stochastic continuous or discrete units.</p>
<p>It is interpreted as a stochastic recurrent neural network where the
state of each unit(neurons/nodes) depends on the units it is connected
to. The weights in the network represent thus the strength of the
interaction between various units/nodes.</p>
<p>It turns into a Hopfield network if we choose deterministic rather
than stochastic units. In contrast to a Hopfield network, a BM is a
so-called generative model. It allows us to generate new samples from
the learned distribution.</p>
</div>
<div class="section" id="a-standard-bm-setup">
<h2>A standard BM setup<a class="headerlink" href="#a-standard-bm-setup" title="Permalink to this headline"></a></h2>
<p>A standard BM network is divided into a set of observable and visible units <span class="math notranslate nohighlight">\(\hat{x}\)</span> and a set of unknown hidden units/nodes <span class="math notranslate nohighlight">\(\hat{h}\)</span>.</p>
<p>Additionally there can be bias nodes for the hidden and visible layers. These biases are normally set to <span class="math notranslate nohighlight">\(1\)</span>.</p>
<p>BMs are stackable, meaning they cwe can train a BM which serves as input to another BM. We can construct deep networks for learning complex PDFs. The layers can be trained one after another, a feature which makes them popular in deep learning</p>
<p>However, they are often hard to train. This leads to the introduction of so-called restricted BMs, or RBMS.
Here we take away all lateral connections between nodes in the visible layer as well as connections between nodes in the hidden layer. The network is illustrated in the figure below.</p>
</div>
<div class="section" id="the-structure-of-the-rbm-network">
<h2>The structure of the RBM network<a class="headerlink" href="#the-structure-of-the-rbm-network" title="Permalink to this headline"></a></h2>
<!-- dom:FIGURE: [figures/RBM.png, width=800 frac=1.0] -->
<!-- begin figure -->
<p><img src="figures/RBM.png" width="800"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
<!-- end figure --></div>
<div class="section" id="the-network">
<h2>The network<a class="headerlink" href="#the-network" title="Permalink to this headline"></a></h2>
<p><strong>The network layers</strong>:</p>
<ol class="simple">
<li><p>A function <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> that represents the visible layer, a vector of <span class="math notranslate nohighlight">\(M\)</span> elements (nodes). This layer represents both what the RBM might be given as training input, and what we want it to be able to reconstruct. This might for example be given by the pixels of an image or coefficients representing speech, or the coordinates of a quantum mechanical state function.</p></li>
<li><p>The function <span class="math notranslate nohighlight">\(\mathbf{h}\)</span> represents the hidden, or latent, layer. A vector of <span class="math notranslate nohighlight">\(N\)</span> elements (nodes). Also called “feature detectors”.</p></li>
</ol>
</div>
<div class="section" id="goals">
<h2>Goals<a class="headerlink" href="#goals" title="Permalink to this headline"></a></h2>
<p>The goal of the hidden layer is to increase the model’s expressive
power. We encode complex interactions between visible variables by
introducing additional, hidden variables that interact with visible
degrees of freedom in a simple manner, yet still reproduce the complex
correlations between visible degrees in the data once marginalized
over (integrated out).</p>
<p><strong>The network parameters, to be optimized/learned</strong>:</p>
<ol class="simple">
<li><p><span class="math notranslate nohighlight">\(\mathbf{a}\)</span> represents the visible bias, a vector of same length as <span class="math notranslate nohighlight">\(\mathbf{x}\)</span>.</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{b}\)</span> represents the hidden bias, a vector of same lenght as <span class="math notranslate nohighlight">\(\mathbf{h}\)</span>.</p></li>
<li><p><span class="math notranslate nohighlight">\(W\)</span> represents the interaction weights, a matrix of size <span class="math notranslate nohighlight">\(M\times N\)</span>.</p></li>
</ol>
</div>
<div class="section" id="joint-distribution">
<h2>Joint distribution<a class="headerlink" href="#joint-distribution" title="Permalink to this headline"></a></h2>
<p>The restricted Boltzmann machine is described by a Boltzmann distribution</p>
<!-- Equation labels as ordinary links -->
<div id="_auto1"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
P_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})},
\label{_auto1} \tag{1}
\end{equation}
\]</div>
<p>where <span class="math notranslate nohighlight">\(Z\)</span> is the normalization constant or partition function, defined as</p>
<!-- Equation labels as ordinary links -->
<div id="_auto2"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
Z = \int \int e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})} d\mathbf{x} d\mathbf{h}.
\label{_auto2} \tag{2}
\end{equation}
\]</div>
<p>It is common to ignore <span class="math notranslate nohighlight">\(T_0\)</span> by setting it to one.</p>
</div>
<div class="section" id="network-elements-the-energy-function">
<h2>Network Elements, the energy function<a class="headerlink" href="#network-elements-the-energy-function" title="Permalink to this headline"></a></h2>
<p>The function <span class="math notranslate nohighlight">\(E(\mathbf{x},\mathbf{h})\)</span> gives the <strong>energy</strong> of a
configuration (pair of vectors) <span class="math notranslate nohighlight">\((\mathbf{x}, \mathbf{h})\)</span>. The lower
the energy of a configuration, the higher the probability of it. This
function also depends on the parameters <span class="math notranslate nohighlight">\(\mathbf{a}\)</span>, <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> and
<span class="math notranslate nohighlight">\(W\)</span>. Thus, when we adjust them during the learning procedure, we are
adjusting the energy function to best fit our problem.</p>
<p>An expression for the energy function is</p>
<div class="math notranslate nohighlight">
\[
E(\hat{x},\hat{h}) = -\sum_{ia}^{NA}b_i^a \alpha_i^a(x_i)-\sum_{jd}^{MD}c_j^d \beta_j^d(h_j)-\sum_{ijad}^{NAMD}b_i^a \alpha_i^a(x_i)c_j^d \beta_j^d(h_j)w_{ij}^{ad}.
\]</div>
<p>Here <span class="math notranslate nohighlight">\(\beta_j^d(h_j)\)</span> and <span class="math notranslate nohighlight">\(\alpha_i^a(x_j)\)</span> are so-called transfer functions that map a given input value to a desired feature value. The labels <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(d\)</span> denote that there can be multiple transfer functions per variable. The first sum depends only on the visible units. The second on the hidden ones. <strong>Note</strong> that there is no connection between nodes in a layer.</p>
<p>The quantities <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(c\)</span> can be interpreted as the visible and hidden biases, respectively.</p>
<p>The connection between the nodes in the two layers is given by the weights <span class="math notranslate nohighlight">\(w_{ij}\)</span>.</p>
</div>
<div class="section" id="defining-different-types-of-rbms">
<h2>Defining different types of RBMs<a class="headerlink" href="#defining-different-types-of-rbms" title="Permalink to this headline"></a></h2>
<p>There are different variants of RBMs, and the differences lie in the types of visible and hidden units we choose as well as in the implementation of the energy function <span class="math notranslate nohighlight">\(E(\mathbf{x},\mathbf{h})\)</span>.</p>
<p><strong>Binary-Binary RBM:</strong></p>
<p>RBMs were first developed using binary units in both the visible and hidden layer. The corresponding energy function is defined as follows:</p>
<!-- Equation labels as ordinary links -->
<div id="_auto3"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
E(\mathbf{x}, \mathbf{h}) = - \sum_i^M x_i a_i- \sum_j^N b_j h_j - \sum_{i,j}^{M,N} x_i w_{ij} h_j,
\label{_auto3} \tag{3}
\end{equation}
\]</div>
<p>where the binary values taken on by the nodes are most commonly 0 and 1.</p>
<p><strong>Gaussian-Binary RBM:</strong></p>
<p>Another varient is the RBM where the visible units are Gaussian while the hidden units remain binary:</p>
<!-- Equation labels as ordinary links -->
<div id="_auto4"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
E(\mathbf{x}, \mathbf{h}) = \sum_i^M \frac{(x_i - a_i)^2}{2\sigma_i^2} - \sum_j^N b_j h_j - \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma_i^2}.
\label{_auto4} \tag{4}
\end{equation}
\]</div>
</div>
<div class="section" id="more-about-rbms">
<h2>More about RBMs<a class="headerlink" href="#more-about-rbms" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>Useful when we model continuous data (i.e., we wish <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> to be continuous)</p></li>
<li><p>Requires a smaller learning rate, since there’s no upper bound to the value a component might take in the reconstruction</p></li>
</ol>
<p>Other types of units include:</p>
<ol class="simple">
<li><p>Softmax and multinomial units</p></li>
<li><p>Gaussian visible and hidden units</p></li>
<li><p>Binomial units</p></li>
<li><p>Rectified linear units</p></li>
</ol>
<p>To read more, see <a class="reference external" href="https://github.com/CompPhysics/ComputationalPhysics2/blob/gh-pages/doc/pub/notebook2/ipynb/notebook2.ipynb">Lectures on Boltzmann machines in Physics</a>.</p>
</div>
<div class="section" id="autoencoders-overarching-view">
<h2>Autoencoders: Overarching view<a class="headerlink" href="#autoencoders-overarching-view" title="Permalink to this headline"></a></h2>
<p>Autoencoders are artificial neural networks capable of learning
efficient representations of the input data (these representations are called codings) without
any supervision (i.e., the training set is unlabeled). These codings
typically have a much lower dimensionality than the input data, making
autoencoders useful for dimensionality reduction.</p>
<p>More importantly, autoencoders act as powerful feature detectors, and
they can be used for unsupervised pretraining of deep neural networks.</p>
<p>Lastly, they are capable of randomly generating new data that looks
very similar to the training data; this is called a generative
model. For example, you could train an autoencoder on pictures of
faces, and it would then be able to generate new faces. Surprisingly,
autoencoders work by simply learning to copy their inputs to their
outputs. This may sound like a trivial task, but we will see that
constraining the network in various ways can make it rather
difficult. For example, you can limit the size of the internal
representation, or you can add noise to the inputs and train the
network to recover the original inputs. These constraints prevent the
autoencoder from trivially copying the inputs directly to the outputs,
which forces it to learn efficient ways of representing the data. In
short, the codings are byproducts of the autoencoder’s attempt to
learn the identity function under some constraints.</p>
<p><a class="reference external" href="https://www.coursera.org/lecture/building-deep-learning-models-with-tensorflow/autoencoders-1U4L3">Video on autoencoders</a></p>
<p>See also A. Geron’s textbook, chapter 15.</p>
</div>
<div class="section" id="bayesian-machine-learning">
<h2>Bayesian Machine Learning<a class="headerlink" href="#bayesian-machine-learning" title="Permalink to this headline"></a></h2>
<p>This is an important topic if we aim at extracting a probability
distribution. This gives us also a confidence interval and error
estimates.</p>
<p>Bayesian machine learning allows us to encode our prior beliefs about
what those models should look like, independent of what the data tells
us. This is especially useful when we don’t have a ton of data to
confidently learn our model.</p>
<p><a class="reference external" href="https://www.youtube.com/watch?v=E1qhGw8QxqY&amp;ab_channel=AndrewGordonWilson">Video on Bayesian deep learning</a></p>
<p>See also the <a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Articles/lec03.pdf">slides here</a>.</p>
</div>
<div class="section" id="reinforcement-learning">
<h2>Reinforcement Learning<a class="headerlink" href="#reinforcement-learning" title="Permalink to this headline"></a></h2>
<p>Reinforcement Learning (RL) is one of the most exciting fields of
Machine Learning today, and also one of the oldest. It has been around
since the 1950s, producing many interesting applications over the
years.</p>
<p>It studies
how agents take actions based on trial and error, so as to maximize
some notion of cumulative reward in a dynamic system or
environment. Due to its generality, the problem has also been studied
in many other disciplines, such as game theory, control theory,
operations research, information theory, multi-agent systems, swarm
intelligence, statistics, and genetic algorithms.</p>
<p>In March 2016, AlphaGo, a computer program that plays the board game
Go, beat Lee Sedol in a five-game match. This was the first time a
computer Go program had beaten a 9-dan (highest rank) professional
without handicaps. AlphaGo is based on deep convolutional neural
networks and reinforcement learning. AlphaGo’s victory was a major
milestone in artificial intelligence and it has also made
reinforcement learning a hot research area in the field of machine
learning.</p>
<p><a class="reference external" href="https://www.youtube.com/watch?v=FgzM3zpZ55o&amp;ab_channel=stanfordonline">Lecture on Reinforcement Learning</a>.</p>
<p>See also A. Geron’s textbook, chapter 16.</p>
</div>
<div class="section" id="transfer-learning">
<h2>Transfer learning<a class="headerlink" href="#transfer-learning" title="Permalink to this headline"></a></h2>
<p>The goal of transfer learning is to transfer the model or knowledge
obtained from a source task to the target task, in order to resolve
the issues of insufficient training data in the target task. The
rationality of doing so lies in that usually the source and target
tasks have inter-correlations, and therefore either the features,
samples, or models in the source task might provide useful information
for us to better solve the target task. Transfer learning is a hot
research topic in recent years, with many problems still waiting to be studied.</p>
<p><a class="reference external" href="https://www.ias.edu/video/machinelearning/2020/0331-SamoryKpotufe">Lecture on transfer learning</a>.</p>
</div>
<div class="section" id="adversarial-learning">
<h2>Adversarial learning<a class="headerlink" href="#adversarial-learning" title="Permalink to this headline"></a></h2>
<p>The conventional deep generative model has a potential problem: the
model tends to generate extreme instances to maximize the
probabilistic likelihood, which will hurt its performance. Adversarial
learning utilizes the adversarial behaviors (e.g., generating
adversarial instances or training an adversarial model) to enhance the
robustness of the model and improve the quality of the generated
data. In recent years, one of the most promising unsupervised learning
technologies, generative adversarial networks (GAN), has already been
successfully applied to image, speech, and text.</p>
<p><a class="reference external" href="https://www.youtube.com/watch?v=CIfsB_EYsVI&amp;ab_channel=StanfordUniversitySchoolofEngineering">Lecture on adversial learning</a>.</p>
</div>
<div class="section" id="dual-learning">
<h2>Dual learning<a class="headerlink" href="#dual-learning" title="Permalink to this headline"></a></h2>
<p>Dual learning is a new learning paradigm, the basic idea of which is
to use the primal-dual structure between machine learning tasks to
obtain effective feedback/regularization, and guide and strengthen the
learning process, thus reducing the requirement of large-scale labeled
data for deep learning. The idea of dual learning has been applied to
many problems in machine learning, including machine translation,
image style conversion, question answering and generation, image
classification and generation, text classification and generation,
image-to-text, and text-to-image.</p>
</div>
<div class="section" id="distributed-machine-learning">
<h2>Distributed machine learning<a class="headerlink" href="#distributed-machine-learning" title="Permalink to this headline"></a></h2>
<p>Distributed computation will speed up machine learning algorithms,
significantly improve their efficiency, and thus enlarge their
application. When distributed meets machine learning, more than just
implementing the machine learning algorithms in parallel is required.</p>
</div>
<div class="section" id="meta-learning">
<h2>Meta learning<a class="headerlink" href="#meta-learning" title="Permalink to this headline"></a></h2>
<p>Meta learning is an emerging research direction in machine
learning. Roughly speaking, meta learning concerns learning how to
learn, and focuses on the understanding and adaptation of the learning
itself, instead of just completing a specific learning task. That is,
a meta learner needs to be able to evaluate its own learning methods
and adjust its own learning methods according to specific learning
tasks.</p>
</div>
<div class="section" id="the-challenges-facing-machine-learning">
<h2>The Challenges Facing Machine Learning<a class="headerlink" href="#the-challenges-facing-machine-learning" title="Permalink to this headline"></a></h2>
<p>While there has been much progress in machine learning, there are also challenges.</p>
<p>For example, the mainstream machine learning technologies are
black-box approaches, making us concerned about their potential
risks. To tackle this challenge, we may want to make machine learning
more explainable and controllable. As another example, the
computational complexity of machine learning algorithms is usually
very high and we may want to invent lightweight algorithms or
implementations. Furthermore, in many domains such as physics,
chemistry, biology, and social sciences, people usually seek elegantly
simple equations (e.g., the Schrödinger equation) to uncover the
underlying laws behind various phenomena. In the field of machine
learning, can we reveal simple laws instead of designing more complex
models for data fitting? Although there are many challenges, we are
still very optimistic about the future of machine learning. As we look
forward to the future, here are what we think the research hotspots in
the next ten years will be.</p>
<p>See the article on <a class="reference external" href="https://www.frontiersin.org/articles/10.3389/frai.2020.00025/full">Discovery of Physics From Data: Universal Laws and Discrepancies</a></p>
</div>
<div class="section" id="explainable-machine-learning">
<h2>Explainable machine learning<a class="headerlink" href="#explainable-machine-learning" title="Permalink to this headline"></a></h2>
<p>Machine learning, especially deep learning, evolves rapidly. The
ability gap between machine and human on many complex cognitive tasks
becomes narrower and narrower. However, we are still in the very early
stage in terms of explaining why those effective models work and how
they work.</p>
<p><strong>What is missing: the gap between correlation and causation</strong>. Standard Machine Learning is based on what e have called a frequentist approach.</p>
<p>Most
machine learning techniques, especially the statistical ones, depend
highly on correlations in data sets to make predictions and analyses. In
contrast, rational humans tend to reply on clear and trustworthy
causality relations obtained via logical reasoning on real and clear
facts. It is one of the core goals of explainable machine learning to
transition from solving problems by data correlation to solving
problems by logical reasoning.</p>
<p><strong>Bayesian Machine Learning is one of the exciting research directions in this field</strong>.</p>
</div>
<div class="section" id="scientific-machine-learning">
<h2>Scientific Machine Learning<a class="headerlink" href="#scientific-machine-learning" title="Permalink to this headline"></a></h2>
<p>An important and emerging field is what has been dubbed as scientific ML, see the article by Deiana et al <a class="reference external" href="https://arxiv.org/abs/2110.13041">Applications and Techniques for Fast Machine Learning in Science, arXiv:2110.13041</a></p>
<p>The authors discuss applications and techniques for fast machine
learning (ML) in science - the concept of integrating power ML
methods into the real-time experimental data processing loop to
accelerate scientific discovery. The report covers three main areas</p>
<ol class="simple">
<li><p>applications for fast ML across a number of scientific domains;</p></li>
<li><p>techniques for training and implementing performant and resource-efficient ML algorithms;</p></li>
<li><p>and computing architectures, platforms, and technologies for deploying these algorithms.</p></li>
</ol>
</div>
<div class="section" id="quantum-machine-learning">
<h2>Quantum machine learning<a class="headerlink" href="#quantum-machine-learning" title="Permalink to this headline"></a></h2>
<p>Quantum machine learning is an emerging interdisciplinary research
area at the intersection of quantum computing and machine learning.</p>
<p>Quantum computers use effects such as quantum coherence and quantum
entanglement to process information, which is fundamentally different
from classical computers. Quantum algorithms have surpassed the best
classical algorithms in several problems (e.g., searching for an
unsorted database, inverting a sparse matrix), which we call quantum
acceleration.</p>
<p>When quantum computing meets machine learning, it can be a mutually
beneficial and reinforcing process, as it allows us to take advantage
of quantum computing to improve the performance of classical machine
learning algorithms. In addition, we can also use the machine learning
algorithms (on classic computers) to analyze and improve quantum
computing systems.</p>
<p><a class="reference external" href="https://www.youtube.com/watch?v=Xh9pUu3-WxM&amp;ab_channel=InstituteforPure%26AppliedMathematics%28IPAM%29">Lecture on Quantum ML</a>.</p>
<p><a class="reference external" href="https://physics.aps.org/articles/v13/179?utm_campaign=weekly&amp;utm_medium=email&amp;utm_source=emailalert">Read interview with Maria Schuld on her work on Quantum Machine Learning</a>. See also <a class="reference external" href="https://www.springer.com/gp/book/9783319964232">her recent textbook</a>.</p>
</div>
<div class="section" id="quantum-machine-learning-algorithms-based-on-linear-algebra">
<h2>Quantum machine learning algorithms based on linear algebra<a class="headerlink" href="#quantum-machine-learning-algorithms-based-on-linear-algebra" title="Permalink to this headline"></a></h2>
<p>Many quantum machine learning algorithms are based on variants of
quantum algorithms for solving linear equations, which can efficiently
solve N-variable linear equations with complexity of O(log2 N) under
certain conditions. The quantum matrix inversion algorithm can
accelerate many machine learning methods, such as least square linear
regression, least square version of support vector machine, Gaussian
process, and more. The training of these algorithms can be simplified
to solve linear equations. The key bottleneck of this type of quantum
machine learning algorithms is data input—that is, how to initialize
the quantum system with the entire data set. Although efficient
data-input algorithms exist for certain situations, how to efficiently
input data into a quantum system is as yet unknown for most cases.</p>
</div>
<div class="section" id="quantum-reinforcement-learning">
<h2>Quantum reinforcement learning<a class="headerlink" href="#quantum-reinforcement-learning" title="Permalink to this headline"></a></h2>
<p>In quantum reinforcement learning, a quantum agent interacts with the
classical environment to obtain rewards from the environment, so as to
adjust and improve its behavioral strategies. In some cases, it
achieves quantum acceleration by the quantum processing capabilities
of the agent or the possibility of exploring the environment through
quantum superposition. Such algorithms have been proposed in
superconducting circuits and systems of trapped ions.</p>
</div>
<div class="section" id="quantum-deep-learning">
<h2>Quantum deep learning<a class="headerlink" href="#quantum-deep-learning" title="Permalink to this headline"></a></h2>
<p>Dedicated quantum information processors, such as quantum annealers
and programmable photonic circuits, are well suited for building deep
quantum networks. The simplest deep quantum network is the Boltzmann
machine. The classical Boltzmann machine consists of bits with tunable
interactions and is trained by adjusting the interaction of these bits
so that the distribution of its expression conforms to the statistics
of the data. To quantize the Boltzmann machine, the neural network can
simply be represented as a set of interacting quantum spins that
correspond to an adjustable Ising model. Then, by initializing the
input neurons in the Boltzmann machine to a fixed state and allowing
the system to heat up, we can read out the output qubits to get the
result.</p>
</div>
<div class="section" id="social-machine-learning">
<h2>Social machine learning<a class="headerlink" href="#social-machine-learning" title="Permalink to this headline"></a></h2>
<p>Machine learning aims to imitate how humans
learn. While we have developed successful machine learning algorithms,
until now we have ignored one important fact: humans are social. Each
of us is one part of the total society and it is difficult for us to
live, learn, and improve ourselves, alone and isolated. Therefore, we
should design machines with social properties. Can we let machines
evolve by imitating human society so as to achieve more effective,
intelligent, interpretable “social machine learning”?</p>
<p>And much more.</p>
</div>
<div class="section" id="the-last-words">
<h2>The last words?<a class="headerlink" href="#the-last-words" title="Permalink to this headline"></a></h2>
<p>Early computer scientist Alan Kay said, <strong>The best way to predict the
future is to create it</strong>. Therefore, all machine learning
practitioners, whether scholars or engineers, professors or students,
need to work together to advance these important research
topics. Together, we will not just predict the future, but create it.</p>
</div>
<div class="section" id="ai-ml-and-some-statements-you-may-have-heard-and-what-do-they-mean">
<h2>AI/ML and some statements you may have heard (and what do they mean?)<a class="headerlink" href="#ai-ml-and-some-statements-you-may-have-heard-and-what-do-they-mean" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>Fei-Fei Li on ImageNet: <strong>map out the entire world of objects</strong> (<a class="reference external" href="https://cacm.acm.org/news/219702-the-data-that-transformed-ai-research-and-possibly-the-world/fulltext">The data that transformed AI research</a>)</p></li>
<li><p>Russell and Norvig in their popular textbook: <strong>relevant to any intellectual task; it is truly a universal field</strong> (<a class="reference external" href="http://aima.cs.berkeley.edu/">Artificial Intelligence, A modern approach</a>)</p></li>
<li><p>Woody Bledsoe puts it more bluntly: <strong>in the long run, AI is the only science</strong> (quoted in Pamilla McCorduck, <a class="reference external" href="https://www.pamelamccorduck.com/machines-who-think">Machines who think</a>)</p></li>
</ol>
<p>If you wish to have a critical read on AI/ML from a societal point of view, see <a class="reference external" href="https://www.katecrawford.net/">Kate Crawford’s recent text Atlas of AI</a></p>
<p><strong>Here: with AI/ML we intend a collection of machine learning methods with an emphasis on statistical learning and data analysis</strong></p>
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<div class="section" id="best-wishes-to-you-all-and-thanks-so-much-for-your-heroic-efforts-this-semester">
<h2>Best wishes to you all and thanks so much for your heroic efforts this semester<a class="headerlink" href="#best-wishes-to-you-all-and-thanks-so-much-for-your-heroic-efforts-this-semester" title="Permalink to this headline"></a></h2>
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