Morten Hjorth-Jensen
1 line
174 KiB
JavaScript
1 line
174 KiB
JavaScript
Search.setIndex({"docnames": ["chapter1", "chapter10", "chapter11", "chapter12", "chapter13", "chapter2", "chapter3", "chapter4", "chapter5", "chapter6", "chapter7", "chapter8", "chapter9", "chapteroptimization", "clustering", "exercisesweek34", "exercisesweek35", "exercisesweek36", "exercisesweek37", "exercisesweek38", "exercisesweek39", "intro", "linalg", "project1", "schedule", "statistics", "teachers", "textbooks", "week34", "week35", "week36", "week37", "week38", "week39"], "filenames": ["chapter1.ipynb", "chapter10.ipynb", "chapter11.ipynb", "chapter12.ipynb", "chapter13.ipynb", "chapter2.ipynb", "chapter3.ipynb", "chapter4.ipynb", "chapter5.ipynb", "chapter6.ipynb", "chapter7.ipynb", "chapter8.ipynb", "chapter9.ipynb", "chapteroptimization.ipynb", "clustering.ipynb", "exercisesweek34.ipynb", "exercisesweek35.ipynb", "exercisesweek36.ipynb", "exercisesweek37.ipynb", "exercisesweek38.ipynb", "exercisesweek39.ipynb", "intro.md", "linalg.ipynb", "project1.ipynb", "schedule.md", "statistics.ipynb", "teachers.md", "textbooks.md", "week34.ipynb", "week35.ipynb", "week36.ipynb", "week37.ipynb", "week38.ipynb", "week39.ipynb"], "titles": ["<span class=\"section-number\">3. </span>Linear Regression", "<span class=\"section-number\">14. </span>Building a Feed Forward Neural Network", "<span class=\"section-number\">15. </span>Solving Differential Equations with Deep Learning", "<span class=\"section-number\">16. </span>Convolutional Neural Networks", "<span class=\"section-number\">17. </span>Recurrent neural networks: Overarching view", "<span class=\"section-number\">4. </span>Ridge and Lasso Regression", "<span class=\"section-number\">5. </span>Resampling Methods", "<span class=\"section-number\">6. </span>Logistic Regression", "<span class=\"section-number\">8. </span>Support Vector Machines, overarching aims", "<span class=\"section-number\">9. </span>Decision trees, overarching aims", "<span class=\"section-number\">10. </span>Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods", "<span class=\"section-number\">11. </span>Basic ideas of the Principal Component Analysis (PCA)", "<span class=\"section-number\">13. </span>Neural networks", "<span class=\"section-number\">7. </span>Optimization, the central part of any Machine Learning algortithm", "<span class=\"section-number\">12. </span>Clustering and Unsupervised Learning", "Exercises week 34", "Exercises week 35", "Exercises week 36", "Exercises week 37", "Exercises week 38", "Exercises week 39", "Applied Data Analysis and Machine Learning", "<span class=\"section-number\">2. </span>Linear Algebra, Handling of Arrays and more Python Features", "Project 1 on Machine Learning, deadline October 9 (midnight), 2023", "Teaching schedule with links to material", "<span class=\"section-number\">1. </span>Elements of Probability Theory and Statistical Data Analysis", "Teachers and Grading", "Textbooks", "Week 34: Introduction to the course, Logistics and Practicalities", "Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression", "Week 36: Statistical interpretation of Linear Regression and Resampling techniques", "Week 37: Statistical interpretations and Resampling Methods", "Week 38: Logistic Regression and Optimization", "Week 39: Optimization and Gradient Methods"], "terms": {"emphasi": [0, 21, 27, 28], "throughout": [0, 4, 5, 14, 21, 22, 25, 28], "thi": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33], "seri": [0, 1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 22, 28, 29, 30, 31, 32, 33], "lectur": [0, 1, 5, 10, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 33], "understand": [0, 1, 3, 5, 6, 10, 13, 14, 21, 28, 29, 30, 32, 33], "mathemat": [0, 6, 11, 12, 13, 17, 21, 22, 25, 27, 28, 30, 31, 32], "aspect": [0, 6, 21, 23, 28, 29, 30], "differ": [0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32], "algorithm": [0, 1, 2, 4, 5, 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25, 28, 31, 33], "write": [0, 1, 2, 3, 5, 6, 7, 8, 12, 13, 15, 16, 17, 20, 22, 23, 28, 29, 31, 32, 33], "each": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 21, 22, 24, 25, 26, 28, 29, 30, 31, 33], "problem": [0, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17, 21, 22, 23, 25, 31], "build": [0, 4, 5, 6, 10, 22, 25, 28, 30, 31, 32, 33], "its": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 21, 22, 23, 25, 28, 30, 31, 32, 33], "logic": [0, 1, 9, 28], "should": [0, 2, 3, 5, 6, 8, 9, 11, 12, 15, 16, 22, 23, 25, 28, 29, 30, 31], "alwai": [0, 3, 5, 6, 12, 13, 16, 25, 28, 29, 30, 31, 32, 33], "keep": [0, 1, 4, 5, 6, 11, 13, 14, 22, 23, 28, 29, 30, 31, 32, 33], "mind": [0, 6, 13, 23, 28, 29, 30, 31, 32, 33], "extent": [0, 1, 6, 27, 28, 31], "develop": [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 21, 22, 23, 25, 28, 29, 30, 31, 32, 33], "insight": [0, 1, 5, 21, 28, 29, 30, 31], "system": [0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 21, 22, 23, 28, 32, 33], "plai": [0, 3, 4, 5, 6, 8, 11, 21, 28, 29, 30, 31, 32], "role": [0, 2, 5, 6, 8, 21, 28, 29, 30, 31, 32], "when": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 22, 23, 25, 28, 29, 30, 31, 32], "extrem": [0, 1, 4, 5, 6, 7, 8, 9, 13, 16, 22, 29, 30, 31, 32, 33], "rich": [0, 28], "spite": [0, 28], "young": [0, 28], "ag": [0, 7, 24, 28, 29, 32], "increas": [0, 1, 3, 4, 5, 6, 9, 12, 13, 23, 25, 28, 29, 30, 31, 33], "seen": [0, 1, 3, 5, 10, 12, 17, 25, 28], "dure": [0, 1, 3, 4, 8, 9, 11, 17, 20, 21, 23, 28, 31], "last": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 22, 23, 25, 26, 28, 30, 33], "three": [0, 1, 3, 5, 6, 8, 9, 12, 15, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32], "decad": [0, 3, 28], "capabl": [0, 1, 8, 13, 21, 28, 33], "follow": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], "techniqu": [0, 1, 8, 10, 13, 21, 25, 27, 28, 29, 31, 32, 33], "analyz": [0, 1, 3, 4, 5, 6, 16, 17, 23, 25, 28, 29, 30], "handl": [0, 1, 2, 5, 9, 11, 21, 30], "reli": [0, 6, 8, 28], "heavili": [0, 28], "rather": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 25, 28, 29, 30, 31, 32, 33], "new": [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 22, 28, 29, 32, 33], "rapidli": [0, 28], "popular": [0, 1, 3, 6, 7, 8, 9, 11, 12, 15, 21, 22, 23, 25, 28, 29, 32], "written": [0, 2, 3, 5, 11, 12, 13, 16, 21, 22, 23, 25, 28, 29, 30, 31, 32, 33], "pytorch": [0, 21, 23, 28], "kera": [0, 4, 21, 23, 28], "freeli": [0, 23, 28], "respect": [0, 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 23, 25, 28, 29, 30, 31, 32, 33], "github": [0, 15, 17, 19, 21, 23, 24, 26, 27, 28, 29], "site": [0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33], "encompass": [0, 23, 25, 28], "commun": [0, 12, 23], "thousand": [0, 1, 23, 28, 29, 33], "And": [0, 3, 4, 5, 6, 9, 13, 20, 21, 23, 25], "contributor": [0, 23, 28], "Not": [0, 1, 5, 6, 28, 29, 30, 31], "given": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 23, 25, 28, 29, 30, 31, 32, 33], "rigor": [0, 28], "justif": [0, 28], "open": [0, 1, 4, 6, 7, 9, 15, 21, 23, 24, 26, 28, 31, 32], "therebi": [0, 5, 7, 11, 28, 29, 30, 32], "room": [0, 26, 28, 29], "trial": [0, 2, 4, 6, 13, 25, 28, 31, 32, 33], "command": [0, 1, 28], "algebra": [0, 3, 5, 13, 21, 29, 30, 31, 33], "multivari": [0, 2, 10, 11, 21, 25, 28], "theori": [0, 1, 3, 8, 9, 12, 13, 18, 21, 23, 27, 28, 30, 33], "probabl": [0, 1, 3, 4, 6, 7, 10, 13, 21, 28, 29, 32, 33], "mont": [0, 6, 21, 25, 27, 28, 31], "carlo": [0, 6, 21, 25, 27, 28, 31], "proper": [0, 2, 6, 7, 23, 28, 31], "main": [0, 1, 3, 4, 5, 6, 7, 9, 22, 23, 27, 28, 29, 32], "categori": [0, 1, 3, 7, 10, 12, 14, 28, 32], "In": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33], "supervis": [0, 5, 6, 7, 9, 12, 21, 28, 29, 31, 32], "know": [0, 1, 2, 5, 6, 8, 13, 21, 28, 29, 30, 32, 33], "answer": [0, 1, 3, 5, 6, 22, 23, 26, 28, 30, 31], "let": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 22, 25, 28, 29, 30, 31, 32, 33], "deduc": [0, 28], "behind": [0, 1, 6, 8, 13, 28, 32, 33], "On": [0, 3, 15, 24, 25, 26, 27, 28], "other": [0, 1, 2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 19, 21, 23, 24, 25, 26, 27, 30, 31, 33], "unsupervis": [0, 1, 4, 12, 21, 28], "relationship": [0, 4, 9, 28], "ani": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 18, 25, 28, 29, 30, 31], "prior": [0, 5, 6, 28, 29, 30, 31], "oper": [0, 1, 3, 5, 6, 10, 11, 12, 13, 15, 21, 25, 28, 29, 30, 31, 33], "third": [0, 3, 6, 13, 26, 28, 31, 32, 33], "name": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33], "reinforc": [0, 8, 21, 28], "paradigm": [0, 28], "inspir": [0, 1, 12, 28], "behavior": [0, 1, 13, 28, 32, 33], "psycholog": [0, 28], "achiev": [0, 1, 5, 6, 8, 12, 22, 28, 30, 31], "sole": [0, 6, 28], "reward": [0, 4, 28], "punish": [0, 1, 28], "anoth": [0, 1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 22, 23, 25, 28, 29, 32, 33], "wai": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 22, 25, 28, 29, 30, 32, 33], "categor": [0, 1, 3, 9, 11, 28], "consid": [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 16, 19, 22, 23, 25, 28, 29, 30, 31, 32, 33], "desir": [0, 2, 4, 5, 13, 14, 28, 29, 32, 33], "output": [0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 22, 23, 25, 28, 29, 30, 31, 32, 33], "classif": [0, 3, 5, 6, 7, 8, 11, 12, 21, 23, 27, 28, 29, 30, 31, 33], "divid": [0, 1, 3, 5, 6, 7, 8, 9, 11, 12, 25, 28, 29, 30, 31, 32], "goal": [0, 7, 9, 28, 32], "assign": [0, 7, 8, 9, 12, 13, 14, 24, 26, 27, 28, 29, 32, 33], "identifi": [0, 1, 7, 9, 11, 12, 13, 14, 28, 29, 32, 33], "digit": [0, 1, 3, 4, 6, 23, 24, 26, 28], "pictur": [0, 28], "ones": [0, 2, 5, 6, 8, 9, 10, 11, 13, 22, 23, 28, 29, 30, 31, 32, 33], "typic": [0, 1, 2, 3, 4, 5, 7, 9, 10, 12, 13, 25, 28, 29, 30, 31, 32, 33], "between": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 17, 23, 25, 28, 29, 30, 31, 32, 33], "refer": [0, 1, 2, 3, 5, 6, 11, 12, 13, 14, 20, 22, 23, 27, 28, 29, 31, 32, 33], "construct": [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 22, 25, 28, 29, 30, 31, 32], "map": [0, 1, 2, 6, 7, 8, 11, 12, 14, 23, 25, 28, 32], "continu": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 17, 18, 22, 23, 25, 28, 29, 30, 31, 32, 33], "valu": [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 19, 21, 22, 23, 28, 33], "cluster": [0, 1, 4, 6, 11, 21, 28, 31], "group": [0, 6, 7, 9, 14, 21, 23, 24, 26, 28, 31], "certain": [0, 3, 6, 7, 9, 25, 28, 29, 31, 32], "trait": [0, 28], "beforehand": [0, 25, 28], "cover": [0, 5, 17, 21, 26, 27, 29, 30], "irrespect": [0, 28], "deal": [0, 1, 3, 5, 6, 8, 11, 13, 14, 17, 18, 22, 23, 25, 28, 29, 30, 31, 32, 33], "first": [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 22, 23, 25, 26, 27, 31], "ingredi": [0, 9, 28], "subdivid": [0, 22, 28], "valid": [0, 1, 4, 7, 9, 10, 13, 21, 25, 28, 29, 33], "difficult": [0, 1, 6, 10, 13, 23, 25, 28, 31, 33], "part": [0, 1, 3, 5, 6, 10, 17, 18, 22, 24, 25, 26, 28, 30, 31], "meaning": [0, 4, 7, 28, 32], "second": [0, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16, 21, 22, 23, 25, 26, 28, 29, 31, 32], "item": [0, 13, 28, 33], "paramet": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 17, 18, 19, 23, 25, 30, 31], "reflect": [0, 1, 4, 5, 23, 25, 28], "lack": [0, 28], "thereof": [0, 6, 13, 28, 31, 33], "As": [0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 16, 22, 23, 25, 28, 29, 30, 31, 32, 33], "similar": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 21, 22, 23, 28, 30, 31, 32], "would": [0, 1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 23, 25, 28, 29, 30, 31, 32, 33], "determin": [0, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 16, 22, 25, 28, 29, 30, 31, 32, 33], "so": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33], "call": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33], "cost": [0, 2, 3, 5, 6, 7, 8, 9, 12, 13, 16, 17, 19, 23, 28], "loss": [0, 1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 22, 23, 28, 30, 31], "risk": [0, 13, 15, 16, 28, 32, 33], "present": [0, 5, 6, 7, 9, 12, 13, 22, 23, 25, 28, 29, 30, 33], "estim": [0, 1, 5, 6, 7, 10, 11, 13, 21, 25, 28, 29, 32, 33], "reproduc": [0, 5, 6, 9, 12, 15, 16, 21, 23, 25, 28, 29], "suppos": [0, 5, 6, 7, 8, 10, 11, 12, 13, 22, 28, 29, 30, 31, 32, 33], "At": [0, 4, 6, 13, 23, 28, 33], "heart": [0, 7, 28, 32], "encount": [0, 1, 5, 7, 13, 25, 28, 29, 32, 33], "minim": [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 23, 29, 30, 31, 33], "optim": [0, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 16, 17, 18, 19, 23, 30, 31], "famili": [0, 7, 8, 25, 32], "gradient": [0, 3, 4, 7, 8, 9, 12, 21, 28, 29], "hear": [0, 13, 28, 33], "phrase": [0, 28], "causat": [0, 28], "do": [0, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 22, 23, 29, 32], "think": [0, 1, 3, 4, 6, 9, 12, 13, 14, 25, 28, 31, 32, 33], "mai": [0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 18, 21, 22, 23, 25, 29, 30, 31, 32, 33], "classifi": [0, 1, 4, 7, 9, 10, 11, 28], "point": [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 22, 23, 25, 26, 28, 29, 31, 32, 33], "Or": [0, 1, 6, 23, 28, 32], "perhap": [0, 5, 13, 28, 29, 30, 32, 33], "kind": [0, 2, 3, 4, 8, 12, 13, 14, 28, 29, 33], "symmetr": [0, 5, 8, 11, 12, 13, 22, 28, 29, 33], "statement": [0, 7, 22, 32], "b": [0, 1, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 25, 26, 28, 29, 30, 31, 32], "caus": [0, 5, 6, 25, 28, 29, 30, 31], "doe": [0, 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 16, 22, 23, 25, 28, 31, 32, 33], "necessarili": [0, 4, 11, 25, 28], "sens": [0, 4, 6, 8, 23, 28, 31], "autom": [0, 21, 27, 28], "fashion": [0, 9, 10, 28], "assess": [0, 6, 23, 28, 29, 30, 31], "newli": [0, 28], "primari": [0, 7, 28, 32], "concern": [0, 1, 4, 7, 28, 32], "qualiti": [0, 9, 15, 16, 21, 28, 29], "less": [0, 1, 3, 4, 5, 6, 8, 9, 13, 21, 25, 28, 29, 30, 31, 32, 33], "were": [0, 1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 25, 28, 29, 31, 32], "order": [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 22, 23, 25, 28, 29, 30, 31, 32], "distribut": [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 21, 22, 23, 28, 29, 32, 33], "pdf": [0, 3, 4, 5, 6, 9, 23, 27, 28, 30, 31], "turn": [0, 1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 19, 22, 23, 25, 28, 29, 31, 32, 33], "likelihood": [0, 1, 5, 9, 28, 29, 33], "end": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33], "situat": [0, 4, 5, 7, 13, 25, 28, 29, 32, 33], "want": [0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 21, 23, 25, 28, 29, 30, 31, 32, 33], "infer": [0, 1, 4, 6, 27, 28, 31], "quantit": [0, 6, 9, 23, 28, 31], "m": [0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 12, 13, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], "sampl": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 19, 21, 22, 23, 25, 28, 29, 30, 31, 32, 33], "boldsymbol": [0, 1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 23, 28, 32, 33], "x": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 25, 28, 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4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33], "instanc": [0, 1, 2, 4, 6, 9, 11, 13, 28, 29, 31, 32, 33], "discov": [0, 28], "interpret": [0, 1, 6, 9, 10, 12, 13, 17, 19, 22, 23, 25], "addit": [0, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 15, 17, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33], "simplifi": [0, 6, 9, 15, 21, 23, 28, 29, 30, 31], "represent": [0, 1, 3, 6, 25, 28, 31, 32], "help": [0, 1, 4, 12, 13, 16, 23, 28, 33], "futur": [0, 4, 8, 9, 28], "consider": [0, 1, 5, 13, 28, 29, 30, 31, 32, 33], "correct": [0, 1, 2, 3, 4, 5, 7, 13, 22, 25, 28, 29, 30, 31, 33], "elus": [0, 28], "imposs": [0, 5, 28, 29], "fundament": [0, 6, 21, 28, 31], "difficulti": [0, 1, 13, 28, 32, 33], "equal": [0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 22, 23, 25, 28, 29, 30, 31, 32, 33], "question": [0, 5, 6, 9, 11, 12, 13, 23, 26, 28, 29, 30, 31, 33], "lead": [0, 1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 22, 25, 28, 29, 30, 31, 32, 33], "To": [0, 1, 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