There are at least eight leaves in the decision tree because there are eight possible outcomes (because each of the eight coins can be the counterfeit lighter coin), and each possible outcome must be represented by at least one leaf. From Corollary 1 of Section 11.1 it follows that the height of the decision tree … permutations of these elements can be the correct order. Decision Trees a decision tree consists of Nodes: test for the value of a certain attribute Edges: correspond to the outcome of a test connect to the next node or leaf Leaves: terminal nodes that predict the outcome to classifiy an example: 1.start at the root 2.perform the test 3.follow the edge corresponding to outcome Trivially, there is a consistent decision tree for any training set w/ one path to leaf for each example (unless f nondeterministic in x) but it probably won’t generalize to new examples Need some kind of regularization to ensure more compact decision trees CS194-10 Fall 2011 Lecture 8 7 (Figure&from&StuartRussell)& The sorting algorithms studied in this book, and most commonly used sorting algorithms, are based on binary comparisons, that is, the comparison of two elements at a time. Lecture notes, lectures 1 - Intro to Linear Programming Lecture notes, lectures 2 - Linear Programming Examples Lecture notes, lectures 5 - Chapters 3, 6, 15 Assignment Problems Lecture notes, lectures 7 - Goal Programming Lecture notes, lectures 8 - Decision Analysis Part 1 Lecture notes, lectures 9 - Decision Analysis Part 2 Each leaf represents one of the n! The biggest problem is their size. The possible solutions of the problem correspond to the paths to the leaves of this rooted tree. �,� ��l���d``TK���s�}����V3���CX���QV��T�D`��)�2vϢ�cs3JL�4�l6\~UE��t�J� �[���-�X��ar5�3�Ql㌅�6rBYMSW��q;�ye�)h=�֒I3k궥Q,�;UW]PR9&(2gu�_Gg��QL�~NK�s��5��� �n`����%% �р�Qؕ�1-����g� Q0�Z�-� (����A&�� `Pa �a�23V� ]@, v�? Decision trees classify the examples by sorting them down the tree from the root to some leaf node, with the leaf node providing the classification to the example. interpretable/intuitive, popular in medical applications because they mimic the way a doctor thinks model discrete outcomes nicely can be very powerful, can be as complex as you need them Thus, a sorting algorithm based on binary comparisons can be represented by a binary decision tree in which each internal vertex represents a comparison of two elements. ), Logical Operations and Logical Connectivity, Theory of inference for the Predicate Calculas, Precedence of Logical Operators and Logic and Bit Operations, Translating from Nested Quantifiers into English, Rules of Inference for Propositional Logic, Using Rules of Inference to Build Arguments, Rules of Inference for Quantified Statements, The Abstract Definition of a Boolean Algebra, Least Upper Bounds and Latest Lower Bounds in a Lattice, Bounded, Complemented and Distributive Lattices, Digramatic Representation of Partial Order Relations and Posets. 344 0 obj <> endobj 352 0 obj <>/Filter/FlateDecode/ID[]/Index[344 21]/Info 343 0 R/Length 60/Prev 417052/Root 345 0 R/Size 365/Type/XRef/W[1 2 1]>>stream To decide whether a particular sorting algorithm is efficient, its complexity is determined. A rooted tree in which each internal vertex corresponds to a decision, with a subtree at these vertices for each possible outcome of the decision, is called a decision tree. It branches out according to the answers. For instance, a binary search tree can be used to locate items based on a series of comparisons, where each comparison tells us whether we have located the item, or whether we should go right or left in a subtree. Today Decision Trees I entropy I information gain Zemel, Urtasun, Fidler (UofT) CSC 411: 06-Decision Trees 2 / 39. Using decision trees as models, a lower bound for the worst-case complexity of sorting algorithms that are based on binary comparisons can be found. Please note that Youtube takes some time to process videos before they become available. THE COMPLEXITY OF COMPARISON-BASED SORTING ALGORITHMS Many different sorting algorithms have been developed. 2 Learning Decision Trees A decision tree is a binary tree in which the internal nodes are labeled with variables and the leafs are labeled with either −1 or +1. They are used in non-linear decision making with simple linear decision surface. Decision Trees MIT 15.097 Course Notes Cynthia Rudin Credit: Russell & Norvig, Mitchell, Kohavi & Quinlan, Carter, Vanden Berghen Why trees? A Decision Tree • A decision tree has 2 kinds of nodes 1. The decision tree that illustrates how this is done is shown in Figure 3. h�b```�f�� �� \$���U �� H��� Y��@�͊����9H #i���o �X� endstream endobj startxref 0 %%EOF 364 0 obj <>stream Note that given n elements, there are n! What is a Decision Tree? And the left and right edges corresponding to any internal node is labeled −1 and +1 respectively. leaves is at least log n!, at least log n! hޤW�r�6�����XK����fl+���ib���4��HlxQHʱ���(Q��Ɲ. Method for solving linear homogeneous recurrence relations with constant coefficients. The largest number of weighings needed to determine the counterfeit coin is the height of the decision tree. possible orderings of these elements, because each of the n! Solution: There are three possibilities for each weighing on a balance scale. Because the height of a binary tree with n! Risk Aversion (a) Thought experiment on large coin-ﬂip gamble Binary decisions trees have some nice properties, but also some less pleasant ones. A binary decision tree of n variables will have 2n1 decision nodes, plus 2nlinks at the lowest level, pointing to the return values 0 and 1. %PDF-1.5 %���� Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of Hildesheim Course on Machine Learning, winter term 2009/2010 3/68 Machine Learning / 1. ?#����>"��&��5�o3%�,``�!����jƷH�lyw�����2��<8� to��A�F�-xT�0���e G��,� ��.%Q��` -q8 endstream endobj 345 0 obj <> endobj 346 0 obj <>/ProcSet[/PDF/Text]>>/Rotate 0/StructParents 0/Type/Page>> endobj 347 0 obj <>stream comparisons are needed, as stated in Theorem 1. Each node in the tree acts as a test case for some attri… In other words, the largest number of comparisons ever needed is equal to the height of the decision tree. 2. The two pans can have equal weight, the first pan can be heavier, or the second pan can be heavier. permutations of n elements. A decision tree is equivalent to a set of such rules, one for each branch. From Corollary 1 of Section 11.1 it follows that the height of the decision tree is at least log3 8 = 2. A decision tree is a tree-like graph with nodes representing the place where we pick an attribute and ask a question; edges represent the answers the to the question; and the leaves represent the actual output or class label. EXAMPLE 1 Suppose there are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. The largest number of weighings needed to determine the counterfeit coin is the height of the decision tree. Each internal node is a question on features. Trivially, there is a consistent decision tree for any training set w/ one path to leaf for each example (unless f nondeterministic in x) but it probably won’t generalize to new examples Need some kind of regularization to ensure more compact decision trees CS194-10 Fall 2011 Lecture 8 7 (Figure&from&StuartRussell)& Refer corollary 5.5 in lecture notes 8. Additional Lecture Notes Lecture 2: Decision Trees Overview The purposes of this lecture are (i) to introduce risk aversion; (ii) to consider the Freemark Abbey Winery case; (iii) to determine the value of information; and (iv) to introduce real options. Example 1 illustrates an application of decision trees. A nice property is canonicity: if we test variables in … Seating Chart Notes 1. Give an algorithm for finding this counterfeit coin. How many weighings are necessary using a balance scale to determine which of the eight coins is the counterfeit one? EXAMPLE 4 We display in Figure 4 a decision tree that orders the elements of the list a, b, c. The complexity of a sort based on binary comparisons is measured in terms of the number of such comparisons used. Rooted trees can be used to model problems in which a series of decisions leads to a solution. Hence, at least two weighings are needed. Decision Trees There are at least eight leaves in the decision tree because there are eight possible outcomes (because each of the eight coins can be the counterfeit lighter coin), and each possible outcome must be represented by at least one leaf. The result of each such comparison narrows down the set of possible orderings. It is possible to determine the counterfeit coin using two weighings. Tuo Zhao | Lecture 6: Decision Tree, Random Forest, and Boosting 22/42. qPart 6: Pros and Cons of Decision Trees qPart 7: Using R to learn Decision Trees Machine Learning Decision Tree Classification Mustafa Jarrar: Lecture Notes on Decision Trees … CS7641/ISYE/CSE 6740: Machine Learning/Computational Data Analysis Decision Tree for Spam Classi cation Boosting Trevor Hastie, Stanford University 10 600/1536 280/1177 180/1065 80/861 80/652 77/423 20/238 19/236 1/2 57/185 48/113 37/101 1/12 9/72 3/229 0/209 100/204 36/123 16/94 14/89 3/5 9/29 16/81 9/112 6/109 …