Clustering. Introduction to Data Science. Jordan Boyd-Graber and Michael Paul SLIDES ADAPTED FROM LAUREN HANNAH
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1 Clustering Introduction to Data Science Jordan Boyd-Graber and Michael Paul SLIDES ADAPTED FROM LAUREN HANNAH Slides adapted from Tom Mitchell, Eric Xing, and Lauren Hannah Introduction to Data Science Boyd-Graber & Paul Clustering 1 of 12
2 Roadmap Classification: machines labeling data for us Previously: logistic regression This time: SVMs (another) eample of linear classifier State-of-the-art classification Good theoretical properties Introduction to Data Science Boyd-Graber & Paul Clustering 2 of 12
3 Thinking Geometrically Suppose you have two classes: vacations and sports Suppose you have four documents Sports Doc 1 : {ball, ball, ball, travel} Doc 2 : {ball, ball} Vacations What does this look like in vector space? Doc 3 : {travel, ball, travel} Doc 4 : {travel} Introduction to Data Science Boyd-Graber & Paul Clustering 3 of 12
4 Put the documents in vector space Travel Ball Introduction to Data Science Boyd-Graber & Paul Clustering 4 of 12
5 Vector space representation of documents Each document is a vector, one component for each term. Terms are aes. High dimensionality: 10,000s of dimensions and more How can we do classification in this space? Introduction to Data Science Boyd-Graber & Paul Clustering 5 of 12
6 Vector space classification As before, the training set is a set of documents, each labeled with its class. In vector space classification, this set corresponds to a labeled set of points or vectors in the vector space. Premise 1: Documents in the same class form a contiguous region. Premise 2: Documents from different classes don t overlap. We define lines, surfaces, hypersurfaces to divide regions. Introduction to Data Science Boyd-Graber & Paul Clustering 6 of 12
7 Recap Vector space classification Linear classifiers Support Vector Machines D Classes in the vector space Classes in the vector space UK China Kenya Should the document be assigned to China, UK or Introduction to Data Science Boyd-Graber & Paul Clustering 7 of 12
8 Recap Vector space classification Linear classifiers Support Vector Machines D Classes in the vector space Classes in the vector space UK China Kenya Should the document Should the be assigned document to China, be assigned UK or Kenya? to China, UK or Introduction to Data Science Boyd-Graber & Paul Clustering 7 of 12
9 Recap Vector space classification Linear classifiers Support Vector Machines Dis Classes in the vector space Classes in the vector space UK China Kenya Find separators between the classes Find separators between the classes Introduction to Data Science Boyd-Graber & Paul Clustering 7 of 12
10 Recap Vector space classification Linear classifiers Support Vector Machines Discu Classes in the vector space Classes in the vector space UK China Kenya Find separators between the classes Find separators between the classes Introduction to Data Science Boyd-Graber & Paul Clustering 7 of 12
11 Recap Vector space classification Linear classifiers Support Vector Machines Discu Classes in the vector space Classes in the vector space UK China Kenya Find separators between the classes Based on these separators: should be assigned to China Introduction to Data Science Boyd-Graber & Paul Clustering 7 of 12
12 Recap Vector space classification Linear classifiers Support Vector Machines Discu Classes in the vector space Classes in the vector space UK China Kenya Find separators between the classes How do we find separators that do a good job at classifying new documents like? Main topic of today Introduction to Data Science Boyd-Graber & Paul Clustering 7 of 12
13 Linear classifiers Definition: A linear classifier computes a linear combination or weighted sum i β i i of the feature values. Classification decision: i β i i > β 0? (β 0 is our bias)... where β 0 (the threshold) is a parameter. We call this the separator or decision boundary. We find the separator based on training set. Methods for finding separator: logistic regression, linear SVM Assumption: The classes are linearly separable. Introduction to Data Science Boyd-Graber & Paul Clustering 8 of 12
14 Linear classifiers Definition: A linear classifier computes a linear combination or weighted sum i β i i of the feature values. Classification decision: i β i i > β 0? (β 0 is our bias)... where β 0 (the threshold) is a parameter. We call this the separator or decision boundary. We find the separator based on training set. Methods for finding separator: logistic regression, linear SVM Assumption: The classes are linearly separable. Before, we just talked about equations. What s the geometric intuition? Introduction to Data Science Boyd-Graber & Paul Clustering 8 of 12
15 Alinearclassifierin1D A linear classifier in 1D Alinearclassifierin1Dis apoint described by the equation w 1 d 1 = θ = θ/w 1 A linear classifier in 1D is a point described by the equation β 1 1 = β 0 Schütze: Support vector machines 15 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 9 of 12
16 Alinearclassifierin1D A linear classifier in 1D Alinearclassifierin1Dis apoint described by the equation w 1 d 1 = θ = θ/w 1 A linear classifier in 1D is a point described by the equation β 1 1 = β 0 = β 0 /β 1 Schütze: Support vector machines 15 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 9 of 12
17 Alinearclassifierin1D A linear classifier in 1D Alinearclassifierin1Dis apoint described by the equation w 1 d 1 = θ = θ/w 1 Points (d 1 )withw 1 d 1 θ are in the class c. A linear classifier in 1D is a point described by the equation β 1 1 = β 0 = β 0 /β 1 Points ( 1 ) with β 1 1 β 0 are in the class c. Schütze: Support vector machines 15 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 9 of 12
18 Alinearclassifierin1D A linear classifier in 1D Alinearclassifierin1Dis apoint described by the equation w 1 d 1 = θ = θ/w 1 Points (d 1 )withw 1 d 1 θ are in the class c. A linear classifier in 1D is a point described by the equation β 1 1 = β 0 Points (d 1 )withw 1 d 1 < θ are in the complement class c. = β 0 /β 1 Points ( 1 ) with β 1 1 β 0 are in the class c. Points ( 1 ) with β 1 1 < β 0 are in the complement class c. Schütze: Support vector machines 15 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 9 of 12
19 Recap Vector space classification Linear classifiers Support Vector Machines Discussion AAlinearclassifierin2D Alinearclassifierin2Dis alinedescribedbythe equation w 1 d 1 + w 2 d 2 = θ A linear classifier in 2D is a line described by the equation β β 2 2 = β 0 Eample for a 2D linear classifier Schütze: Support vector machines 16 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 10 of 12
20 Recap Vector space classification Linear classifiers Support Vector Machines Discussion AAlinearclassifierin2D Alinearclassifierin2Dis alinedescribedbythe equation w 1 d 1 + w 2 d 2 = θ A linear classifier in 2D is a line described by the equation β β 2 2 = β 0 Eample for a 2D linear classifier Eample for a 2D linear classifier Schütze: Support vector machines 16 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 10 of 12
21 Recap Vector space classification Linear classifiers Support Vector Machines Discussion A linear classifier in 2D Alinearclassifierin2D A linear classifier in 2D is a line described by the equation β β 2 2 = β 0 Eample for a 2D linear Alinearclassifierin2Dis alinedescribedbythe equation w 1 d 1 + w 2 d 2 = θ Eample for a 2D linear classifier Points (d 1 d 2 )with classifier w 1 d 1 + w 2 d 2 θ are in the class c. Points ( 1 2 ) with β β 2 2 β 0 are in the class c. Schütze: Support vector machines 16 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 10 of 12
22 Recap Vector space classification Linear classifiers Support Vector Machines Discussion AAlinearclassifierin2D Alinearclassifierin2Dis alinedescribedbythe equation w 1 d 1 + w 2 d 2 = θ A linear classifier in 2D is a line described by the equation β β 2 2 = β 0 Eample for a 2D linear Eample for a 2D linear classifier Points (d 1 d 2 )with w 1 d 1 + classifier w 2 d 2 θ are in the class c. Points ( 1 2 ) with β β 2 2 β 0 are in the class c. Points (d 1 d 2 )with w 1 d 1 + w 2 d 2 < θ are in the complement class c. Points ( 1 2 ) with β β 2 2 < β 0 are in the complement class c. Schütze: Support vector machines 16 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 10 of 12
23 Recap Vector space classification Linear classifiers Support Vector Machines Discussion A Alinearclassifierin3D 3D Alinearclassifierin3Dis aplanedescribedbythe equation w 1 d 1 + w 2 d 2 + w 3 d 3 = θ A linear classifier in 3D is a plane described by the equation β β β 3 3 = β 0 Schütze: Support vector machines 17 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 11 of 12
24 Recap Vector space classification Linear classifiers Support Vector Machines Discussion A Alinearclassifierin3D Alinearclassifierin3Dis aplanedescribedbythe equation w 1 d 1 + w 2 d 2 + w 3 d 3 = θ A linear classifier in 3D is a plane described by the equation β β β 3 3 = β 0 Eample for a 3D linear classifier Eample for a 3D linear classifier Schütze: Support vector machines 17 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 11 of 12
25 Recap Vector space classification Linear classifiers A linear classifier in 3D Support Vector Machines Discussion Alinearclassifierin3D A linear classifier in 3D is a plane described by the Alinearclassifierin3Dis aplanedescribedbythe equation w 1 d 1 + w 2 dequation 2 + w 3 d 3 = θ β β β 3 3 = β 0 Eample for a 3D linear Eample for a 3D linear classifier Points (d 1 d 2 d 3 )with w 1 d 1 + w 2 dclassifier 2 + w 3 d 3 θ are in the class c. Points ( ) with β β β 3 3 β 0 are in the class c. Schütze: Support vector machines 17 / 55 Introduction to Data Science Boyd-Graber & Paul Clustering 11 of 12
26 Recap Vector space classification Linear classifiers Support Vector Machines Discussion A Alinearclassifierin3D Alinearclassifierin3Dis aplanedescribedbythe equation w 1 d 1 + w 2 d 2 + w 3 d 3 = θ Eample for a 3D linear classifier Points (d 1 d 2 d 3 )with w 1 d 1 + w 2 d 2 + w 3 d 3 θ are in the class c. Points (d 1 d 2 d 3 )with w 1 d 1 + w 2 d 2 + w 3 d 3 < θ are in the complement class c. Schütze: Support vector machines 17 / 55 A linear classifier in 3D is a plane described by the equation β β β 3 3 = β 0 Eample for a 3D linear classifier Points ( ) with β β β 3 3 β 0 are in the class c. Points ( ) with β β β 3 3 < β 0 are in the complement class c. Introduction to Data Science Boyd-Graber & Paul Clustering 11 of 12
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