# Classification Using Vector Spaces

###### Classification Using Vector Spaces
Introduction to Information Retrieval Introduction to Information Retrieval Text Classification; Vector space classification www.ThesisScientist.com 1Introduction to Information Retrieval Recap: Naïve Bayes classifiers  Classify based on prior weight of class and conditional parameter for what each word says:   cargmax log P(c ) log P(x c )   NB j i j cC   j  i positions   Training is done by counting and dividing: T N c x c j k j P(x c ) P(c ) k j j T N  c x  j i xV i  Don’t forget to smooth www.ThesisScientist.com 2  Introduction to Information Retrieval The rest of text classification  Today:  Vector space methods for Text Classification  Vector space classification using centroids (Rocchio)  K Nearest Neighbors  Decision boundaries, linear and nonlinear classifiers  Dealing with more than 2 classes  Later in the course  More text classification  Support Vector Machines  Textspecific issues in classification www.ThesisScientist.com 3Introduction to Information Retrieval Sec.14.1 Recall: Vector Space Representation  Each document is a vector, one component for each term (= word).  Normally normalize vectors to unit length.  Highdimensional vector space:  Terms are axes  10,000+ dimensions, or even 100,000+  Docs are vectors in this space  How can we do classification in this space www.ThesisScientist.com 4Introduction to Information Retrieval Sec.14.1 Classification Using Vector Spaces  As before, the training set is a set of documents, each labeled with its class (e.g., topic)  In vector space classification, this set corresponds to a labeled set of points (or, equivalently, vectors) in the vector space  Premise 1: Documents in the same class form a contiguous region of space  Premise 2: Documents from different classes don’t overlap (much)  We define surfaces to delineate classes in the space www.ThesisScientist.com 5Introduction to Information Retrieval Sec.14.1 Documents in a Vector Space Government Science Arts www.ThesisScientist.com 6Introduction to Information Retrieval Sec.14.1 Test Document of what class Government Science Arts www.ThesisScientist.com 7Introduction to Information Retrieval Sec.14.1 Test Document = Government Is this similarity hypothesis true in general Government Science Arts Our main topic today is how to find good separators www.ThesisScientist.com 8Introduction to Information Retrieval Sec.14.1 Aside: 2D/3D graphs can be misleading www.ThesisScientist.com 9Introduction to Information Retrieval Sec.14.2 Using Rocchio for text classification  Relevance feedback methods can be adapted for text categorization  As noted before, relevance feedback can be viewed as 2class classification  Relevant vs. nonrelevant documents  Use standard tfidf weighted vectors to represent text documents  For training documents in each category, compute a prototype vector by summing the vectors of the training documents in the category.  Prototype = centroid of members of class  Assign test documents to the category with the closest prototype vector based on cosine similarity. www.ThesisScientist.com 10Introduction to Information Retrieval Sec.14.2 Illustration of Rocchio Text Categorization www.ThesisScientist.com 11Introduction to Information Retrieval Sec.14.2 Definition of centroid  1  (c) v (d)  D c dD c   Where D is the set of all documents that belong to c class c and v(d) is the vector space representation of d.   Note that centroid will in general not be a unit vector even when the inputs are unit vectors. www.ThesisScientist.com 12Introduction to Information Retrieval Sec.14.2 Rocchio Properties  Forms a simple generalization of the examples in each class (a prototype).  Prototype vector does not need to be averaged or otherwise normalized for length since cosine similarity is insensitive to vector length.  Classification is based on similarity to class prototypes.  Does not guarantee classifications are consistent with the given training data. Why not www.ThesisScientist.com 13Introduction to Information Retrieval Sec.14.2 Rocchio Anomaly  Prototype models have problems with polymorphic (disjunctive) categories. www.ThesisScientist.com 14Introduction to Information Retrieval Sec.14.2 Rocchio classification  Rocchio forms a simple representation for each class: the centroid/prototype  Classification is based on similarity to / distance from the prototype/centroid  It does not guarantee that classifications are consistent with the given training data  It is little used outside text classification  It has been used quite effectively for text classification  But in general worse than Naïve Bayes  Again, cheap to train and test documents www.ThesisScientist.com 15Introduction to Information Retrieval Sec.14.3 k Nearest Neighbor Classification  kNN = k Nearest Neighbor  To classify a document d into class c:  Define kneighborhood N as k nearest neighbors of d  Count number of documents i in N that belong to c  Estimate P(cd) as i/k  Choose as class argmax P(cd) = majority class c www.ThesisScientist.com 16Introduction to Information Retrieval Sec.14.3 Example: k=6 (6NN) P(science ) Government Science Arts www.ThesisScientist.com 17Introduction to Information Retrieval Sec.14.3 NearestNeighbor Learning Algorithm  Learning is just storing the representations of the training examples in D.  Testing instance x (under 1NN):  Compute similarity between x and all examples in D.  Assign x the category of the most similar example in D.  Does not explicitly compute a generalization or category prototypes.  Also called:  Casebased learning  Memorybased learning  Lazy learning  Rationale of kNN: contiguity hypothesis www.ThesisScientist.com 18Introduction to Information Retrieval Sec.14.3 kNN Is Close to Optimal  Cover and Hart (1967)  Asymptotically, the error rate of 1nearestneighbor classification is less than twice the Bayes rate error rate of classifier knowing model that generated data  In particular, asymptotic error rate is 0 if Bayes rate is 0.  Assume: query point coincides with a training point.  Both query point and training point contribute error → 2 times Bayes rate www.ThesisScientist.com 19Introduction to Information Retrieval Sec.14.3 k Nearest Neighbor  Using only the closest example (1NN) to determine the class is subject to errors due to:  A single atypical example.  Noise (i.e., an error) in the category label of a single training example.  More robust alternative is to find the k mostsimilar examples and return the majority category of these k examples.  Value of k is typically odd to avoid ties; 3 and 5 are most common. www.ThesisScientist.com 20Introduction to Information Retrieval Sec.14.3 kNN decision boundaries Boundaries are in principle arbitrary surfaces – but usually polyhedra Government Science Arts kNN gives locally defined decision boundaries between classes – far away points do not influence each classification www.ThesisScientist.com 21 decision (unlike in Naïve Bayes, Rocchio, etc.)Introduction to Information Retrieval Sec.14.3 Similarity Metrics  Nearest neighbor method depends on a similarity (or distance) metric.  Simplest for continuous mdimensional instance space is Euclidean distance.  Simplest for mdimensional binary instance space is Hamming distance (number of feature values that differ).  For text, cosine similarity of tf.idf weighted vectors is typically most effective. www.ThesisScientist.com 22Introduction to Information Retrieval Sec.14.3 Illustration of 3 Nearest Neighbor for Text Vector Space www.ThesisScientist.com 23Introduction to Information Retrieval 3 Nearest Neighbor vs. Rocchio  Nearest Neighbor tends to handle polymorphic categories better than Rocchio/NB. www.ThesisScientist.com 24Introduction to Information Retrieval Sec.14.3 Nearest Neighbor with Inverted Index  Naively, finding nearest neighbors requires a linear search through D documents in collection  But determining k nearest neighbors is the same as determining the k best retrievals using the test document as a query to a database of training documents.  Use standard vector space inverted index methods to find the k nearest neighbors.  Testing Time: O(BV ) where B is the average t number of training documents in which a testdocument word appears.  Typically B D www.ThesisScientist.com 25Introduction to Information Retrieval Sec.14.3 kNN: Discussion  No feature selection necessary  Scales well with large number of classes  Don’t need to train n classifiers for n classes  Classes can influence each other  Small changes to one class can have ripple effect  Scores can be hard to convert to probabilities  No training necessary  Actually: perhaps not true. (Data editing, etc.)  May be expensive at test time  In most cases it’s more accurate than NB or Rocchio www.ThesisScientist.com 26Introduction to Information Retrieval Sec.14.6 kNN vs. Naive Bayes  Bias/Variance tradeoff  Variance ≈ Capacity  kNN has high variance and low bias.  Infinite memory  NB has low variance and high bias.  Decision surface has to be linear (hyperplane – see later)  Consider asking a botanist: Is an object a tree  Too much capacity/variance, low bias  Botanist who memorizes  Will always say “no” to new object (e.g., different of leaves)  Not enough capacity/variance, high bias  Lazy botanist  Says “yes” if the object is green  You want the middle ground (Example due to C. Burges) www.ThesisScientist.com 27Introduction to Information Retrieval Sec.14.6 Bias vs. variance: Choosing the correct model capacity www.ThesisScientist.com 28Introduction to Information Retrieval Sec.14.4 Linear classifiers and binary and multiclass classification  Consider 2 class problems  Deciding between two classes, perhaps, government and nongovernment  Oneversusrest classification  How do we define (and find) the separating surface  How do we decide which region a test doc is in www.ThesisScientist.com 29Introduction to Information Retrieval Sec.14.4 Separation by Hyperplanes  A strong highbias assumption is linear separability:  in 2 dimensions, can separate classes by a line  in higher dimensions, need hyperplanes  Can find separating hyperplane by linear programming (or can iteratively fit solution via perceptron):  separator can be expressed as ax + by = c www.ThesisScientist.com 30Introduction to Information Retrieval Sec.14.4 Linear programming / Perceptron Find a,b,c, such that ax + by c for red points ax + by c for blue points. www.ThesisScientist.com 31Introduction to Information Retrieval Sec.14.4 Which Hyperplane In general, lots of possible solutions for a,b,c. www.ThesisScientist.com 32Introduction to Information Retrieval Sec.14.4 Which Hyperplane  Lots of possible solutions for a,b,c.  Some methods find a separating hyperplane, but not the optimal one according to some criterion of expected goodness  E.g., perceptron  Most methods find an optimal separating hyperplane  Which points should influence optimality  All points  Linear/logistic regression  Naïve Bayes  Only “difficult points” close to decision boundary  Support vector machines www.ThesisScientist.com 33Introduction to Information Retrieval Sec.14.4 Linear classifier: Example  Class: “interest” (as in interest rate)  Example features of a linear classifier  w t w t i i i i • 0.70 prime •−0.71 dlrs • 0.67 rate •−0.35 world • 0.63 interest •−0.33 sees • 0.60 rates •−0.25 year • 0.46 discount •−0.24 group • 0.43 bundesbank •−0.24 dlr  To classify, find dot product of feature vector and weights www.ThesisScientist.com 34Introduction to Information Retrieval Sec.14.4 Linear Classifiers  Many common text classifiers are linear classifiers  Naïve Bayes  Perceptron  Rocchio  Logistic regression  Support vector machines (with linear kernel)  Linear regression with threshold  Despite this similarity, noticeable performance differences  For separable problems, there is an infinite number of separating hyperplanes. Which one do you choose  What to do for nonseparable problems  Different training methods pick different hyperplanes  Classifiers more powerful than linear often don’t perform better on text problems. Why www.ThesisScientist.com 35Introduction to Information Retrieval Sec.14.2 Rocchio is a linear classifier www.ThesisScientist.com 36Introduction to Information Retrieval Sec.14.2 Twoclass Rocchio as a linear classifier  Line or hyperplane defined by: M w d  i i i1  For Rocchio, set:    w  (c ) (c ) 1 2    2 2 0.5( (c ) (c ) ) 1 2  Aside for ML/stats people: Rocchio classification is a simplification of the classic Fisher Linear Discriminant where you don’t model the variance (or assume it is www.ThesisScientist.com 37 spherical). Introduction to Information Retrieval Sec.14.4 Naive Bayes is a linear classifier  Twoclass Naive Bayes. We compute: P(C d) P(C) P(wC) log  log log  P(C d) P(C) P(wC) wd  Decide class C if the odds is greater than 1, i.e., if the log odds is greater than 0.  So decision boundary is hyperplane: P(C) n 0 where  log ;  w w wV P(C ) P(w C)  log ; n of occurrences of w in d w w P(w C ) www.ThesisScientist.com 38Introduction to Information Retrieval Sec.14.4 A nonlinear problem  A linear classifier like Naïve Bayes does badly on this task  kNN will do very well (assuming enough training data) www.ThesisScientist.com 39Introduction to Information Retrieval Sec.14.4 High Dimensional Data  Pictures like the one at right are absolutely misleading  Documents are zero along almost all axes  Most document pairs are very far apart (i.e., not strictly orthogonal, but only share very common words and a few scattered others)  In classification terms: often document sets are separable, for most any classification  This is part of why linear classifiers are quite successful in this domain www.ThesisScientist.com 40Introduction to Information Retrieval Sec.14.5 More Than Two Classes  Anyof or multivalue classification  Classes are independent of each other.  A document can belong to 0, 1, or 1 classes.  Decompose into n binary problems  Quite common for documents  Oneof or multinomial or polytomous classification  Classes are mutually exclusive.  Each document belongs to exactly one class  E.g., digit recognition is polytomous classification  Digits are mutually exclusive www.ThesisScientist.com 41Introduction to Information Retrieval Sec.14.5 Set of Binary Classifiers: Any of  Build a separator between each class and its complementary set (docs from all other classes).  Given test doc, evaluate it for membership in each class.  Apply decision criterion of classifiers independently  Done  Though maybe you could do better by considering dependencies between categories www.ThesisScientist.com 42Introduction to Information Retrieval Sec.14.5 Set of Binary Classifiers: One of  Build a separator between each class and its complementary set (docs from all other classes).  Given test doc, evaluate it for membership in each class.  Assign document to class with:  maximum score  maximum confidence  maximum probability www.ThesisScientist.com 43Introduction to Information Retrieval Summary: Representation of Text Categorization Attributes  Representations of text are usually very high dimensional (one feature for each word)  Highbias algorithms that prevent overfitting in high dimensional space should generally work best  For most text categorization tasks, there are many relevant features and many irrelevant ones  Methods that combine evidence from many or all features (e.g. naive Bayes, kNN) often tend to work better than ones that try to isolate just a few relevant features Although the results are a bit more mixed than often thought www.ThesisScientist.com 44Introduction to Information Retrieval Which classifier do I use for a given text classification problem  Is there a learning method that is optimal for all text classification problems  No, because there is a tradeoff between bias and variance.  Factors to take into account:  How much training data is available  How simple/complex is the problem (linear vs. nonlinear decision boundary)  How noisy is the data  How stable is the problem over time  For an unstable problem, it’s better to use a simple and robust www.ThesisScientist.com 45 classifier.Introduction to Information Retrieval Ch. 14 Resources for today’s lecture  IIR 14  Fabrizio Sebastiani. Machine Learning in Automated Text Categorization. ACM Computing Surveys, 34(1):147, 2002.  Yiming Yang Xin Liu, A reexamination of text categorization methods. Proceedings of SIGIR, 1999.  Trevor Hastie, Robert Tibshirani and Jerome Friedman, Elements of Statistical Learning: Data Mining, Inference and Prediction. SpringerVerlag, New York.  Open Calais: Automatic Semantic Tagging  Free (but they can keep your data), provided by Thompson/Reuters  Weka: A data mining software package that includes an implementation of many ML algorithms www.ThesisScientist.com 46
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