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807 - TEXT ANALYTICS Massimo Poesio Lecture 2: Text (Document) clustering

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Reminder: Clustering Or, discovering similarities between objects – Individuals, Documents, … Applications: – Recommender systems – Document organization

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Recommending: restaurants We have a list of all Wivenhoe restaurants – with and ratings for some – as provided by some Uni Essex students / staff Which restaurant(s) should I recommend to you?

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Input

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Algorithm 0 Recommend to you the most popular restaurants – say # positive votes minus # negative votes Ignores your culinary preferences – And judgements of those with similar preferences How can we exploit the wisdom of “like- minded” people?

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Another look at the input - a matrix

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Now that we have a matrix View all other entries as zeros for now.

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8. PREFERENCE-DEFINED DATA SPACE..............................

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Similarity between two people Similarity between their preference vectors. Inner products are a good start. Dave has similarity 3 with Estie – but -2 with Cindy. Perhaps recommend Black Buoy to Dave – and Bakehouse to Bob, etc.

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Algorithm 1.1 You give me your preferences and I need to give you a recommendation. I find the person “most similar” to you in my database and recommend something he likes. Aspects to consider: – No attempt to discern cuisines, etc. – What if you’ve been to all the restaurants he has? – Do you want to rely on one person’s opinions?

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Algorithm 1.k You give me your preferences and I need to give you a recommendation. I find the k people “most similar” to you in my database and recommend what’s most popular amongst them. Issues: – A priori unclear what k should be – Risks being influenced by “unlike minds”

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Slightly more sophisticated attempt Group similar users together into clusters You give your preferences and seek a recommendation, then – Find the “nearest cluster” (what’s this?) – Recommend the restaurants most popular in this cluster Features: – avoids data sparsity issues – still no attempt to discern why you’re recommended what you’re recommended – how do you cluster?

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CLUSTERS Can cluster Dave, Estie Alice, Bob …. Fred.. Cindy

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How do you cluster? Must keep similar people together in a cluster Separate dissimilar people Factors: – Need a notion of similarity/distance – Vector space? Normalization? – How many clusters? Fixed a priori? Completely data driven? – Avoid “trivial” clusters - too large or small

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Looking beyond Clustering people for restaurant recommendations Clustering other things (documents, web pages) Other approaches to recommendation General unsupervised machine learning. Amazon.com

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Text clustering Search results clustering Document clustering

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Navigating search results Given the results of a search (say jaguar), partition into groups of related docs – sense disambiguation Approach followed by Uni Essex site – Kruschwitz / al Bakouri / Lungley Other examples: IBM InfoSphere Data Explorer – (was: vivisimo.com)

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Results list clustering example Cluster 1: Jaguar Motor Cars’ home page Mike’s XJS resource page Vermont Jaguar owners’ club Cluster 2: Big cats My summer safari trip Pictures of jaguars, leopards and lions Cluster 3: Jacksonville Jaguars’ Home Page AFC East Football Teams

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Search results clustering: example

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Why cluster documents? For improving recall in search applications For speeding up vector space retrieval Corpus analysis/navigation – Sense disambiguation in search results

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Improving search recall Cluster hypothesis - Documents with similar text are related Ergo, to improve search recall: – Cluster docs in corpus a priori – When a query matches a doc D, also return other docs in the cluster containing D Hope: docs containing automobile returned on a query for car because – clustering grouped together docs containing car with those containing automobile. Why might this happen?

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Speeding up vector space retrieval In vector space retrieval, must find nearest doc vectors to query vector This would entail finding the similarity of the query to every doc - slow! By clustering docs in corpus a priori – find nearest docs in cluster(s) close to query – inexact but avoids exhaustive similarity computation Exercise: Make up a simple example with points on a line in 2 clusters where this inexactness shows up.

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Corpus analysis/navigation Given a corpus, partition it into groups of related docs – Recursively, can induce a tree of topics – Allows user to browse through corpus to home in on information – Crucial need: meaningful labels for topic nodes.

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CLUSTERING DOCUMENTS IN A (VERY) LARGE COLLECTION: GOOGLE NEWS

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CLUSTERING DOCUMENTS IN A VERY LARGE COLLECTION: JRC’S NEWS EXPLORER

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What makes docs “related”? Ideal: semantic similarity. Practical: statistical similarity – We will use cosine similarity. – Docs as vectors. – For many algorithms, easier to think in terms of a distance (rather than similarity) between docs. – We will describe algorithms in terms of cosine similarity.

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DOCUMENTS AS BAGS OF WORDS broad tech stock rally may signal trend - traders. technology stocks rallied on tuesday, with gains scored broadly across many sectors, amid what some traders called a recovery from recent doldrums. broad may rally rallied signal stock stocks tech technology traders traders trend DOCUMENT INDEX

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Doc as vector Each doc j is a vector of tf idf values, one component for each term. Can normalize to unit length. So we have a vector space – terms are axes - aka features – n docs live in this space – even with stemming, may have 10000+ dimensions – do we really want to use all terms?

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TERM WEIGHTING IN VECTOR SPACE MODELS: THE TF.IDF MEASURE FREQUENCY of term i in document k Number of documents with term i

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Intuition Postulate: Documents that are “close together” in vector space talk about the same things. t 1 D2D2 D1 D3 D4 t 3 t 2 x y

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Cosine similarity

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Two flavors of clustering Given n docs and a positive integer k, partition docs into k (disjoint) subsets. Given docs, partition into an “appropriate” number of subsets. – E.g., for query results - ideal value of k not known up front - though UI may impose limits. Can usually take an algorithm for one flavor and convert to the other.

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Thought experiment Consider clustering a large set of computer science documents – what do you expect to see in the vector space?

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Thought experiment Consider clustering a large set of computer science documents – what do you expect to see in the vector space? NLP Graphics AI Theory Arch.

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Decision boundaries Could we use these blobs to infer the subject of a new document? NLP Graphics AI Theory Arch.

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Deciding what a new doc is about Check which region the new doc falls into – can output “softer” decisions as well. NLP Graphics AI Theory Arch. = AI

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Setup Given “training” docs for each category – Theory, AI, NLP, etc. Cast them into a decision space – generally a vector space with each doc viewed as a bag of words Build a classifier that will classify new docs – Essentially, partition the decision space Given a new doc, figure out which partition it falls into

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Supervised vs. unsupervised learning This setup is called supervised learning in the terminology of Machine Learning In the domain of text, various names – Text classification, text categorization – Document classification/categorization – “Automatic” categorization – Routing, filtering … In contrast, the earlier setting of clustering is called unsupervised learning – Presumes no availability of training samples – Clusters output may not be thematically unified.

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“Which is better?” Depends – on your setting – on your application Can use in combination – Analyze a corpus using clustering – Hand-tweak the clusters and label them – Use clusters as training input for classification – Subsequent docs get classified Computationally, methods quite different

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What more can these methods do? Assigning a category label to a document is one way of adding structure to it. Can add others, e.g.,extract from the doc – people – places – dates – organizations … This process is known as information extraction – can also be addressed using supervised learning.

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Clustering: a bit of terminology REPRESENTATIVE CENTROID OUTLIER

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Key notion: cluster representative In the algorithms to follow, will generally need a notion of a representative point in a cluster Representative should be some sort of “typical” or central point in the cluster, e.g., – point inducing smallest radii to docs in cluster – smallest squared distances, etc. – point that is the “average” of all docs in the cluster Need not be a document

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Key notion: cluster centroid Centroid of a cluster = component-wise average of vectors in a cluster - is a vector. – Need not be a doc. Centroid of (1,2,3); (4,5,6); (7,2,6) is (4,3,5). Centroid

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(Outliers in centroid computation) Can ignore outliers when computing centroid. What is an outlier? – Lots of statistical definitions, e.g. – moment of point to centroid > M some cluster moment. Centroid Outlier Say 10.

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Clustering algorithms Partitional vs. hierarchical Agglomerative K-means

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Partitional Clustering Original Points A Partitional Clustering

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Hierarchical Clustering Traditional Hierarchical Clustering Non-traditional Hierarchical Clustering Non-traditional Dendrogram Traditional Dendrogram

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Agglomerative clustering Given target number of clusters k. Initially, each doc viewed as a cluster – start with n clusters; Repeat: – while there are > k clusters, find the “closest pair” of clusters and merge them.

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“Closest pair” of clusters Many variants to defining closest pair of clusters – Clusters whose centroids are the most cosine- similar – … whose “closest” points are the most cosine- similar – … whose “furthest” points are the most cosine- similar

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Example: n=6, k=3, closest pair of centroids d1 d2 d3 d4 d5 d6 Centroid after first step. Centroid after second step.

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Issues Have to support finding closest pairs continually – compare all pairs? Potentially n 3 cosine similarity computations To avoid: use approximations. – “points” are changing as centroids change. Changes at each step are not localized – on a large corpus, memory management an issue – sometimes addressed by clustering a sample. Why? What’s this?

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Exercise Consider agglomerative clustering on n points on a line. Explain how you could avoid n 3 distance computations - how many will your scheme use?

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“Using approximations” In standard algorithm, must find closest pair of centroids at each step Approximation: instead, find nearly closest pair – use some data structure that makes this approximation easier to maintain – simplistic example: maintain closest pair based on distances in projection on a random line Random line

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A partitional clustering algorithm: K-MEANS Given k - the number of clusters desired. Each cluster associated with a centroid. Each point assigned to the cluster with the closest centroid. Iterate.

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Different algorithm: k-means Given k - the number of clusters desired. Iterative algorithm. More locality within each iteration. Hard to get good bounds on the number of iterations.

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Basic iteration At the start of the iteration, we have k centroids. Each doc assigned to the nearest centroid. All docs assigned to the same centroid are averaged to compute a new centroid; – thus have k new centroids.

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Iteration example Current centroids Docs

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Iteration example New centroids Docs

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K-means Clustering: the full algorithm

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K-means Clustering – Details Initial centroids are often chosen randomly. – Clusters produced vary from one run to another. The centroid is (typically) the mean of the points in the cluster. ‘Closeness’ is measured by Euclidean distance, cosine similarity, correlation, etc. K-means will converge for common similarity measures mentioned above. Most of the convergence happens in the first few iterations. – Often the stopping condition is changed to ‘Until relatively few points change clusters’ Complexity is O( n * K * I * d ) – n = number of points, K = number of clusters, I = number of iterations, d = number of attributes

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Effect of initial choice of centroids: Two different K-means Clusterings Sub-optimal ClusteringOptimal Clustering Original Points

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Importance of Choosing Initial Centroids

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k-means clustering Begin with k docs as centroids – could be any k docs, but k random docs are better. Repeat Basic Iteration until termination condition satisfied.

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Termination conditions Several possibilities, e.g., – A fixed number of iterations. – Doc partition unchanged. – Centroid positions don’t change. Does this mean that the docs in a cluster are unchanged?

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Convergence Why should the k-means algorithm ever reach a fixed point? – A state in which clusters don’t change. k-means is a special case of a general procedure known as the EM algorithm. – Under reasonable conditions, known to converge. – Number of iterations could be large.

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Exercise Consider running 2-means clustering on a corpus, each doc of which is from one of two different languages. What are the two clusters we would expect to see? Is agglomerative clustering likely to produce different results?

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k not specified in advance Say, the results of a query. Solve an optimization problem: penalize having lots of clusters – application dependant, e.g., compressed summary of search results list. Tradeoff between having more clusters (better focus within each cluster) and having too many clusters

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k not specified in advance Given a clustering, define the Benefit for a doc to be the cosine similarity to its centroid Define the Total Benefit to be the sum of the individual doc Benefits. Why is there always a clustering of Total Benefit n?

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Penalize lots of clusters For each cluster, we have a Cost C. Thus for a clustering with k clusters, the Total Cost is kC. Define the Value of a cluster to be = Total Benefit - Total Cost. Find the clustering of highest Value, over all choices of k.

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Evaluating K-means Clusters Most common measure is Sum of Squared Error (SSE) – For each point, the error is the distance to the nearest cluster – To get SSE, we square these errors and sum them. – x is a data point in cluster C i and m i is the representative point for cluster C i can show that m i corresponds to the center (mean) of the cluster – Given two clusters, we can choose the one with the smallest error – One easy way to reduce SSE is to increase K, the number of clusters A good clustering with smaller K can have a lower SSE than a poor clustering with higher K

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Back to agglomerative clustering In a run of agglomerative clustering, we can try all values of k=n,n-1,n-2, … 1. At each, we can measure our Value, then pick the best choice of k.

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Exercise Suppose a run of agglomerative clustering finds k=7 to have the highest Value amongst all k. Have we found the highest-Value clustering amongst all clusterings with k=7?

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Limitations of K-means K-means has problems when clusters are of differing – Sizes – Densities – Non-globular shapes K-means has problems when the data contains outliers.

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Limitations of K-means: Differing Sizes Original Points K-means (3 Clusters)

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Limitations of K-means: Differing Density Original Points K-means (3 Clusters)

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Limitations of K-means: Non-globular Shapes Original Points K-means (2 Clusters)

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Hierarchical clustering As clusters agglomerate, docs likely to fall into a hierarchy of “topics” or concepts. d1 d2 d3 d4 d5 d1,d2 d4,d5 d3 d3,d4,d 5

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Hierarchical Clustering Produces a set of nested clusters organized as a hierarchical tree Can be visualized as a dendrogram – A tree like diagram that records the sequences of merges or splits

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Strengths of Hierarchical Clustering Do not have to assume any particular number of clusters – Any desired number of clusters can be obtained by ‘cutting’ the dendogram at the proper level They may correspond to meaningful taxonomies – Example in biological sciences (e.g., animal kingdom, phylogeny reconstruction, …)

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Hierarchical Clustering Two main types of hierarchical clustering – Agglomerative: Start with the points as individual clusters At each step, merge the closest pair of clusters until only one cluster (or k clusters) left – Divisive: Start with one, all-inclusive cluster At each step, split a cluster until each cluster contains a point (or there are k clusters) Traditional hierarchical algorithms use a similarity or distance matrix – Merge or split one cluster at a time

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81 Agglomerative vs. Divisive Clustering Agglomerative (bottom-up) methods start with each example in its own cluster and iteratively combine them to form larger and larger clusters. Divisive (partitional, top-down) separate all examples immediately into clusters.

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82 Hierarchical Agglomerative Clustering (HAC) More popular hierarchical clustering technique Assumes a similarity function for determining the similarity of two instances. Starts with all instances in a separate cluster and then repeatedly joins the two clusters that are most similar until there is only one cluster. The history of merging forms a binary tree or hierarchy.

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Agglomerative Clustering Algorithm Basic algorithm is straightforward 1.Compute the proximity matrix 2.Let each data point be a cluster 3.Repeat 4.Merge the two closest clusters 5.Update the proximity matrix 6.Until only a single cluster remains Key operation is the computation of the proximity of two clusters – Different approaches to defining the distance between clusters distinguish the different algorithms

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Starting Situation Start with clusters of individual points and a proximity matrix p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix

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Intermediate Situation After some merging steps, we have some clusters C1 C4 C2 C5 C3 C2C1 C3 C5 C4 C2 C3C4C5 Proximity Matrix

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Intermediate Situation We want to merge the two closest clusters (C2 and C5) and update the proximity matrix. C1 C4 C2 C5 C3 C2C1 C3 C5 C4 C2 C3C4C5 Proximity Matrix

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After Merging The question is “How do we update the proximity matrix?” C1 C4 C2 U C5 C3 ? ? ? ? ? C2 U C5 C1 C3 C4 C2 U C5 C3C4 Proximity Matrix

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How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Similarity? l MIN l MAX l Group Average l Distance Between Centroids l Other methods driven by an objective function – Ward’s Method uses squared error Proximity Matrix

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How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids l Other methods driven by an objective function – Ward’s Method uses squared error

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How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids l Other methods driven by an objective function – Ward’s Method uses squared error

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How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids l Other methods driven by an objective function – Ward’s Method uses squared error

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Text clustering issues and applications

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List of issues/applications covered Term vs. document space clustering Multi-lingual docs Feature selection Speeding up scoring Building navigation structures – “Automatic taxonomy induction” Labeling

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Term vs. document space Thus far, we clustered docs based on their similarities in terms space For some applications, e.g., topic analysis for inducing navigation structures, can “dualize”: – use docs as axes – represent (some) terms as vectors – proximity based on co-occurrence of terms in docs – now clustering terms, not docs

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Term vs. document space If terms carefully chosen (say nouns) – fixed number of pairs for distance computation independent of corpus size – clusters have clean descriptions in terms of noun phrase co-occurrence - easier labeling? – left with problem of binding docs to these clusters

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Multi-lingual docs E.g., News Explorer, Canadian government docs. Every doc in English and equivalent French. – Must cluster by concepts rather than language Simplest: pad docs in one lang with dictionary equivalents in the other – thus each doc has a representation in both languages Axes are terms in both languages

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Feature selection Which terms to use as axes for vector space? Huge body of (ongoing) research IDF is a form of feature selection – can exaggerate noise e.g., mis-spellings Pseudo-linguistic heuristics, e.g., – drop stop-words – stemming/lemmatization – use only nouns/noun phrases Good clustering should “figure out” some of these

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Clustering to speed up scoring From CS276a, recall sampling and pre- grouping – Wanted to find, given a query Q, the nearest docs in the corpus – Wanted to avoid computing cosine similarity of Q to each of n docs in the corpus.

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Sampling and pre-grouping First run a clustering phase – pick a representative leader for each cluster. Process a query as follows: – Given query Q, find its nearest leader L. – Seek nearest docs from L’s followers only avoid cosine similarity to all docs.

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Navigation structure Given a corpus, agglomerate into a hierarchy Throw away lower layers so you don’t have n leaf topics each having a single doc – Many principled methods for this pruning such as MDL. d1 d2 d3 d4 d5 d1,d2d4,d5 d3 d3,d4,d 5

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Navigation structure Can also induce hierarchy top-down - e.g., use k-means, then recur on the clusters. – Need to figure out what k should be at each point. Topics induced by clustering need human ratification – can override mechanical pruning. Need to address issues like partitioning at the top level by language.

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Major issue - labeling After clustering algorithm finds clusters - how can they be useful to the end user? Need pithy label for each cluster – In search results, say “Football” or “Car” in the jaguar example. – In topic trees, need navigational cues. Often done by hand, a posteriori.

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How to Label Clusters Show titles of typical documents – Titles are easy to scan – Authors create them for quick scanning! – But you can only show a few titles which may not fully represent cluster Show words/phrases prominent in cluster – More likely to fully represent cluster – Use distinguishing words/phrases – But harder to scan

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Labeling Common heuristics - list 5-10 most frequent terms in the centroid vector. – Drop stop-words; stem. Differential labeling by frequent terms – Within the cluster “Computers”, child clusters all have the word computer as frequent terms. – Discriminant analysis of sub-tree centroids.

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The biggest issue in clustering? How do you compare two alternatives? Computation (time/space) is only one metric of performance How do you look at the “goodness” of the clustering produced by a method

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READINGS Ingersoll / Morton / Farris – chapter 6

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Resources Scatter/Gather: A Cluster-based Approach to Browsing Large Document Collections (1992) – Cutting/Karger/Pederesen/Tukey – http://citeseer.nj.nec.com/cutting92scattergather.html Data Clustering: A Review (1999) – Jain/Murty/Flynn – http://citeseer.nj.nec.com/jain99data.html

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Acknowledgments Many of these slides were borrowed, from – Chris Manning’s Stanford module – Tan, Steinbach and Kumar, Intro to Data Mining – Ray Mooney’s Austin module

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