Morten Hjorth-Jensen
470 lines
28 KiB
HTML
470 lines
28 KiB
HTML
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Overview of week 45', 2, None, 'overview-of-week-45'),
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('Decision trees, overarching aims',
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2,
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None,
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|
'decision-trees-overarching-aims'),
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('Basics of a tree', 2, None, 'basics-of-a-tree'),
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('A Sketch of a Tree, Regression problem',
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2,
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None,
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'a-sketch-of-a-tree-regression-problem'),
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('A Sketch of a Tree, Classification problem',
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2,
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None,
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'a-sketch-of-a-tree-classification-problem'),
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('A typical Decision Tree with its pertinent Jargon, '
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'Classification Problem',
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2,
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None,
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'a-typical-decision-tree-with-its-pertinent-jargon-classification-problem'),
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('General Features', 2, None, 'general-features'),
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('How do we set it up?', 2, None, 'how-do-we-set-it-up'),
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('Decision trees and Regression',
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2,
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None,
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('Building a tree, regression',
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2,
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None,
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'building-a-tree-regression'),
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('A top-down approach, recursive binary splitting',
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2,
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None,
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'a-top-down-approach-recursive-binary-splitting'),
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('Making a tree', 2, None, 'making-a-tree'),
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('Pruning the tree', 2, None, 'pruning-the-tree'),
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('Cost complexity pruning', 2, None, 'cost-complexity-pruning'),
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('Schematic Regression Procedure',
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2,
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None,
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'schematic-regression-procedure'),
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('A Classification Tree', 2, None, 'a-classification-tree'),
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('Growing a classification tree',
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2,
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None,
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'growing-a-classification-tree'),
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('Classification tree, how to split nodes',
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2,
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None,
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'classification-tree-how-to-split-nodes'),
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('Gini Index (or Coefficient or Impurity)',
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2,
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None,
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'gini-index-or-coefficient-or-impurity'),
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('Why binary splits?', 2, None, 'why-binary-splits'),
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('Computing a Tree using the Gini Index',
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2,
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None,
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'computing-a-tree-using-the-gini-index'),
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('The Table', 2, None, 'the-table'),
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('Computing the various Gini Indices',
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2,
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None,
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'computing-the-various-gini-indices'),
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('Computing the various Gini Indices, Hours slept',
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2,
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None,
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'computing-the-various-gini-indices-hours-slept'),
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('Computing the various Gini Indices, Hours studied',
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2,
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('A possible code using Scikit-Learn',
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2,
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None,
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'a-possible-code-using-scikit-learn'),
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('Visualizing Trees, More examples',
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2,
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None,
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'visualizing-trees-more-examples'),
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('Visualizing the Tree, The Moons',
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2,
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None,
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'visualizing-the-tree-the-moons'),
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('Other ways of visualizing the trees',
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2,
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None,
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'other-ways-of-visualizing-the-trees'),
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('Printing out as text', 2, None, 'printing-out-as-text'),
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('Algorithms for Setting up Decision Trees',
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2,
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None,
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'algorithms-for-setting-up-decision-trees'),
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('The CART algorithm for Classification',
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2,
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None,
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'the-cart-algorithm-for-classification'),
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('The CART algorithm for Regression',
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2,
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None,
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'the-cart-algorithm-for-regression'),
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('Computing the Gini index', 2, None, 'computing-the-gini-index'),
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('Simple Python Code to read in Data and perform Classification',
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2,
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None,
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'simple-python-code-to-read-in-data-and-perform-classification'),
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|
('Computing the Gini Factor',
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2,
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None,
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'computing-the-gini-factor'),
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|
('Entropy and the ID3 algorithm',
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2,
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None,
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'entropy-and-the-id3-algorithm'),
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('Cancer Data again now with Decision Trees and other Methods',
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2,
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None,
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'cancer-data-again-now-with-decision-trees-and-other-methods'),
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('Another example, the moons again',
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2,
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None,
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'another-example-the-moons-again'),
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('Playing around with regions',
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|
2,
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None,
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'playing-around-with-regions'),
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('Regression trees', 2, None, 'regression-trees'),
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('Final regressor code', 2, None, 'final-regressor-code'),
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('Pros and cons of trees, pros',
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2,
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|
None,
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'pros-and-cons-of-trees-pros'),
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('Disadvantages', 2, None, 'disadvantages'),
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('Ensemble Methods: From a Single Tree to Many Trees and Extreme '
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'Boosting, Meet the Jungle of Methods',
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2,
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|
None,
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'ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods'),
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('An Overview of Ensemble Methods',
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2,
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None,
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'an-overview-of-ensemble-methods'),
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('Bagging', 2, None, 'bagging'),
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('More bagging', 2, None, 'more-bagging'),
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('Making your own Bootstrap: Changing the Level of the Decision '
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'Tree',
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2,
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None,
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'making-your-own-bootstrap-changing-the-level-of-the-decision-tree'),
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('Why Voting?', 2, None, 'why-voting'),
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('Tossing coins', 2, None, 'tossing-coins'),
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('Standard imports first', 2, None, 'standard-imports-first'),
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('Simple Voting Example, head or tail',
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2,
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None,
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'simple-voting-example-head-or-tail'),
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('Using the Voting Classifier',
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2,
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None,
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'using-the-voting-classifier'),
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('Voting and Bagging', 2, None, 'voting-and-bagging'),
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('Random forests', 2, None, 'random-forests'),
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('Random Forest Algorithm', 2, None, 'random-forest-algorithm'),
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('Random Forests Compared with other Methods on the Cancer Data',
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2,
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None,
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'random-forests-compared-with-other-methods-on-the-cancer-data'),
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('Compare Bagging on Trees with Random Forests',
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|
2,
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None,
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'compare-bagging-on-trees-with-random-forests'),
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("Boosting, a Bird's Eye View",
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2,
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None,
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'boosting-a-bird-s-eye-view'),
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('What is boosting? Additive Modelling/Iterative Fitting',
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2,
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None,
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'what-is-boosting-additive-modelling-iterative-fitting'),
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('Iterative Fitting, Regression and Squared-error Cost Function',
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2,
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None,
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'iterative-fitting-regression-and-squared-error-cost-function'),
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('Squared-Error Example and Iterative Fitting',
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2,
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None,
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'squared-error-example-and-iterative-fitting'),
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('Iterative Fitting, Classification and AdaBoost',
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2,
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None,
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'iterative-fitting-classification-and-adaboost'),
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('Adaptive Boosting, AdaBoost',
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2,
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None,
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'adaptive-boosting-adaboost'),
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('Building up AdaBoost', 2, None, 'building-up-adaboost'),
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('Adaptive boosting: AdaBoost, Basic Algorithm',
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2,
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|
None,
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|
'adaptive-boosting-adaboost-basic-algorithm'),
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|
('Basic Steps of AdaBoost', 2, None, 'basic-steps-of-adaboost'),
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|
('AdaBoost Examples', 2, None, 'adaboost-examples'),
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|
('Gradient boosting: Basics with Steepest Descent/Functional '
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|
'Gradient Descent',
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|
2,
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|
None,
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|
'gradient-boosting-basics-with-steepest-descent-functional-gradient-descent'),
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|
('The Squared-Error again! Steepest Descent',
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|
2,
|
|
None,
|
|
'the-squared-error-again-steepest-descent'),
|
|
('Steepest Descent Example', 2, None, 'steepest-descent-example'),
|
|
('Gradient Boosting, algorithm',
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|
2,
|
|
None,
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|
'gradient-boosting-algorithm'),
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|
('Gradient Boosting, Examples of Regression',
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2,
|
|
None,
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('Gradient Boosting, Classification Example',
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2,
|
|
None,
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'gradient-boosting-classification-example'),
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('XGBoost: Extreme Gradient Boosting',
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2,
|
|
None,
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'xgboost-extreme-gradient-boosting'),
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('Regression Case', 2, None, 'regression-case'),
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('Xgboost on the Cancer Data',
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2,
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None,
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'xgboost-on-the-cancer-data')]}
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<a class="navbar-brand" href="week45-bs.html">Week 45: Decisions Trees, Random Forests, Bagging and Boosting</a>
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<ul class="nav navbar-nav navbar-right">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week45-bs001.html#overview-of-week-45" style="font-size: 80%;">Overview of week 45</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs002.html#decision-trees-overarching-aims" style="font-size: 80%;">Decision trees, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs003.html#basics-of-a-tree" style="font-size: 80%;">Basics of a tree</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs004.html#a-sketch-of-a-tree-regression-problem" style="font-size: 80%;">A Sketch of a Tree, Regression problem</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs005.html#a-sketch-of-a-tree-classification-problem" style="font-size: 80%;">A Sketch of a Tree, Classification problem</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs006.html#a-typical-decision-tree-with-its-pertinent-jargon-classification-problem" style="font-size: 80%;">A typical Decision Tree with its pertinent Jargon, Classification Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs007.html#general-features" style="font-size: 80%;">General Features</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs008.html#how-do-we-set-it-up" style="font-size: 80%;">How do we set it up?</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs009.html#decision-trees-and-regression" style="font-size: 80%;">Decision trees and Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs010.html#building-a-tree-regression" style="font-size: 80%;">Building a tree, regression</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs011.html#a-top-down-approach-recursive-binary-splitting" style="font-size: 80%;">A top-down approach, recursive binary splitting</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs012.html#making-a-tree" style="font-size: 80%;">Making a tree</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs013.html#pruning-the-tree" style="font-size: 80%;">Pruning the tree</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs014.html#cost-complexity-pruning" style="font-size: 80%;">Cost complexity pruning</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs015.html#schematic-regression-procedure" style="font-size: 80%;">Schematic Regression Procedure</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs016.html#a-classification-tree" style="font-size: 80%;">A Classification Tree</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs017.html#growing-a-classification-tree" style="font-size: 80%;">Growing a classification tree</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs018.html#classification-tree-how-to-split-nodes" style="font-size: 80%;">Classification tree, how to split nodes</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs019.html#gini-index-or-coefficient-or-impurity" style="font-size: 80%;">Gini Index (or Coefficient or Impurity)</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs020.html#why-binary-splits" style="font-size: 80%;">Why binary splits?</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs021.html#computing-a-tree-using-the-gini-index" style="font-size: 80%;">Computing a Tree using the Gini Index</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs022.html#the-table" style="font-size: 80%;">The Table</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs023.html#computing-the-various-gini-indices" style="font-size: 80%;">Computing the various Gini Indices</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs024.html#computing-the-various-gini-indices-hours-slept" style="font-size: 80%;">Computing the various Gini Indices, Hours slept</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs025.html#computing-the-various-gini-indices-hours-studied" style="font-size: 80%;">Computing the various Gini Indices, Hours studied</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs026.html#a-possible-code-using-scikit-learn" style="font-size: 80%;">A possible code using Scikit-Learn</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs027.html#visualizing-trees-more-examples" style="font-size: 80%;">Visualizing Trees, More examples</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs028.html#visualizing-the-tree-the-moons" style="font-size: 80%;">Visualizing the Tree, The Moons</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs029.html#other-ways-of-visualizing-the-trees" style="font-size: 80%;">Other ways of visualizing the trees</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs030.html#printing-out-as-text" style="font-size: 80%;">Printing out as text</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs031.html#algorithms-for-setting-up-decision-trees" style="font-size: 80%;">Algorithms for Setting up Decision Trees</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs032.html#the-cart-algorithm-for-classification" style="font-size: 80%;">The CART algorithm for Classification</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs033.html#the-cart-algorithm-for-regression" style="font-size: 80%;">The CART algorithm for Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs034.html#computing-the-gini-index" style="font-size: 80%;">Computing the Gini index</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs035.html#simple-python-code-to-read-in-data-and-perform-classification" style="font-size: 80%;">Simple Python Code to read in Data and perform Classification</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs036.html#computing-the-gini-factor" style="font-size: 80%;">Computing the Gini Factor</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs037.html#entropy-and-the-id3-algorithm" style="font-size: 80%;">Entropy and the ID3 algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs038.html#cancer-data-again-now-with-decision-trees-and-other-methods" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
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|
<!-- navigation toc: --> <li><a href="._week45-bs039.html#another-example-the-moons-again" style="font-size: 80%;">Another example, the moons again</a></li>
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|
<!-- navigation toc: --> <li><a href="._week45-bs040.html#playing-around-with-regions" style="font-size: 80%;">Playing around with regions</a></li>
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|
<!-- navigation toc: --> <li><a href="._week45-bs041.html#regression-trees" style="font-size: 80%;">Regression trees</a></li>
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|
<!-- navigation toc: --> <li><a href="._week45-bs042.html#final-regressor-code" style="font-size: 80%;">Final regressor code</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs043.html#pros-and-cons-of-trees-pros" style="font-size: 80%;">Pros and cons of trees, pros</a></li>
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|
<!-- navigation toc: --> <li><a href="._week45-bs044.html#disadvantages" style="font-size: 80%;">Disadvantages</a></li>
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|
<!-- navigation toc: --> <li><a href="._week45-bs045.html#ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods" style="font-size: 80%;">Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
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|
<!-- navigation toc: --> <li><a href="._week45-bs046.html#an-overview-of-ensemble-methods" style="font-size: 80%;">An Overview of Ensemble Methods</a></li>
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|
<!-- navigation toc: --> <li><a href="._week45-bs047.html#bagging" style="font-size: 80%;">Bagging</a></li>
|
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<!-- navigation toc: --> <li><a href="._week45-bs048.html#more-bagging" style="font-size: 80%;">More bagging</a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0068"></a>
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<!-- !split -->
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<h2 id="basic-steps-of-adaboost" class="anchor">Basic Steps of AdaBoost </h2>
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<p>With the above definitions we are now ready to set up the algorithm for AdaBoost.
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The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.
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</p>
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<ol>
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<li> We start by initializing all weights to \( w_i = 1/n \), with \( i=0,1,2,\dots n-1 \). It is easy to see that we must have \( \sum_{i=0}^{n-1}w_i = 1 \).</li>
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<li> We rewrite the misclassification error as</li>
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</ol>
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$$
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\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
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$$
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<ol>
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<li> Then we start looping over all attempts at classifying, namely we start an iterative process for \( m=1:M \), where \( M \) is the final number of classifications. Our given classifier could for example be a plain decision tree.
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<ol type="a"></li>
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<li> Fit then a given classifier to the training set using the weights \( w_i \).</li>
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<li> Compute then \( \mathrm{err} \) and figure out which events are classified properly and which are classified wrongly.</li>
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<li> Define a quantity \( \alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m} \)</li>
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<li> Set the new weights to \( w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)} \).</li>
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</ol>
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<li> Compute the new classifier \( G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i) \).</li>
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</ol>
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<p>For the iterations with \( m \le 2 \) the weights are modified
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individually at each steps. The observations which were misclassified
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at iteration \( m-1 \) have a weight which is larger than those which were
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classified properly. As this proceeds, the observations which were
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difficult to classifiy correctly are given a larger influence. Each
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new classification step \( m \) is then forced to concentrate on those
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observations that are missed in the previous iterations.
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