本投影片修改自Introduction to Information Retrieval一書之投影片 Ch 16 & 17 Lecture 4 : Clustering 楊立偉教授 wyang@ntu.edu.tw 本投影片修改自Introduction to Information Retrieval一書之投影片 Ch 16 & 17

Clustering : Introduction

Clustering: Definition (Document) clustering is the process of grouping a set of documents into clusters of similar documents. Documents within a cluster should be similar. Documents from different clusters should be dissimilar. Clustering is the most common form of unsupervised learning. Unsupervised = there are no labeled or annotated data. 3

Data set with clear cluster structure Propose algorithm for finding the cluster structure in this example 4

Classification vs. Clustering Supervised learning Classes are human-defined and part of the input to the learning algorithm. Clustering Unsupervised learning Clusters are inferred from the data without human input. 5

Why cluster documents? Whole corpus analysis/navigation Better user interface 提供文件集合的分析與導覽 For improving recall in search applications Better search results 提供更好的搜尋結果 For better navigation of search results Effective "user recall" will be higher 搜尋結果導覽 For speeding up vector space retrieval Faster search 加快搜尋速度

For visualizing a document collection Wise et al, "Visualizing the non-visual" PNNL ThemeScapes, Cartia [Mountain height = cluster size]

For improving search recall Cluster hypothesis - "closely associated documents tend to be relevant to the same requests". Therefore, to improve search recall: Cluster docs in corpus 先將文件做分群 When a query matches a doc D, also return other docs in the cluster containing D 也建議符合的整群 Hope if we do this: The query “car” will also return docs containing automobile Because clustering grouped together docs containing car with those containing automobile. Why might this happen? 具有類似的文件特徵

For better navigation of search results For grouping search results thematically clusty.com / Vivisimo (Enterprise Search – Velocity)

Issues for clustering (1) General goal: put related docs in the same cluster, put unrelated docs in different clusters. Representation for clustering Document representation 如何表示一篇文件 Need a notion of similarity/distance 如何表示相似度 whittle 削切

Issues for clustering (2) How to decide the number of clusters Fixed a priori : assume the number of clusters K is given. Data driven : semiautomatic methods for determining K Avoid very small and very large clusters Define clusters that are easy to explain to the user whittle 削切

Clustering Algorithms Flat (Partitional) algorithms 無階層的聚類演算法 Usually start with a random (partial) partitioning Refine it iteratively 不斷地修正調整 K means clustering Hierarchical algorithms 有階層的聚類演算法 Create a hierarchy Bottom-up, agglomerative 由下往上聚合 Top-down, divisive 由上往下分裂

Flat (Partitioning) Algorithms Partitioning method: Construct a partition of n documents into a set of K clusters 將 n 篇文件分到 K 群中 Given: a set of documents and the number K Find: a partition of K clusters that optimizes the chosen partitioning criterion Globally optimal: exhaustively enumerate all partitions 找出最佳切割 → 通常很耗時 Effective heuristic methods: K-means and K-medoids algorithms 用經驗法則找出近似解即可

Hard vs. Soft clustering Hard clustering: Each document belongs to exactly one cluster. More common and easier to do Soft clustering: A document can belong to more than one cluster. For applications like creating browsable hierarchies Ex. Put sneakers in two clusters: sports apparel, shoes You can only do that with a soft clustering approach. *only hard clustering is discussed in this class. 15

K-means algorithm

K-means Perhaps the best known clustering algorithm Simple, works well in many cases Use as default / baseline for clustering documents 17

K-means In vector space model, Assumes documents are real-valued vectors. Clusters based on centroids (aka the center of gravity 重心 or mean) of points in a cluster, c: Reassignment of instances to clusters is based on distance to the current cluster centroids.

K-means algorithm 1. Select K random docs {s1, s2,… sK} as seeds. 先挑選種子 2. Until clustering converges or other stopping criterion: 重複下列步驟直到收斂或其它停止條件成立 2.1 For each doc di: 針對每一篇文件 Assign di to the cluster cj such that dist(xi, sj) is minimal. 將該文件加入最近的一群 2.2 For each cluster cj sj = (cj) 以各群的重心為種子，再做一次 (Update the seeds to the centroid of each cluster)

K-means algorithm

K-means example (K=2) Pick seeds Reassign clusters Compute centroids Converged! 通常做3至4回就大致穩定（但仍需視資料與群集多寡而調整）

Termination conditions Several possibilities, e.g., A fixed number of iterations. 只做固定幾回合 Doc partition unchanged. 群集不再改變 Centroid positions don’t change. 重心不再改變

Convergence of K-Means Why should the K-means algorithm ever reach a fixed point? A state in which clusters don’t change. 收斂 K-means is a special case of a general procedure known as the Expectation Maximization (EM) algorithm. EM is known to converge. Number of iterations could be large. 在理論上一定會收斂，只是要做幾回合的問題 (逼近法，且一開始逼近得快)

Convergence of K-Means : 證明 Define goodness measure of cluster k as sum of squared distances from cluster centroid: Gk = Σi (di – ck)2 (sum over all di in cluster k) G = Σk Gk 計算每一群中文件與中心的距離平方，然後加總 Reassignment monotonically decreases G since each vector is assigned to the closest centroid. 每回合的動作只會讓G越來越小

Time Complexity Computing distance between two docs is O(m) where m is the dimensionality of the vectors. Reassigning clusters: O(Kn) distance computations, or O(Knm). Computing centroids: Each doc gets added once to some centroid: O(nm). Assume these two steps are each done once for I iterations: O(IKnm). 執行 I 回合；分 K 群；n 篇文件；m 個詞 → 慢且不scalable 改善方法：用 近似估計, 抽樣, 選擇 等技巧來加速

Issue (1) Seed Choice Results can vary based on random seed selection. Some seeds can result in poor convergence rate, or convergence to sub-optimal clusterings. Select good seeds using a heuristic (e.g., doc least similar to any existing mean) Try out multiple starting points Example showing sensitivity to seeds In the above, if you start with B and E as centroids you converge to {A,B,C} and {D,E,F} If you start with D and F you converge to {A,B,D,E} {C,F}

Issue (2) How Many Clusters? Number of clusters K is given Partition n docs into predetermined number of clusters Finding the “right” number of clusters is part of the problem 假設 連應該分成幾群都不知道 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. 查詢結果分群時通常不會預先知道該分幾群

If K not specified in advance Suggest K automatically using heuristics based on N using K vs. Cluster-size diagram Tradeoff between having less clusters (better focus within each cluster) and having too many clusters 如何取捨

方法: 以「組間變異對應 於整體變異的百分比」來 看 (即F檢驗)，每增加一 群所能帶來的邊際變異開 始下降的前一點。 Ref: "Determining the number of clusters in a data set", Wikipedia.

The Calinski-Harabasz index 群間方差和BGSS越大越好，群內方差和WGSS越小越 好，因此得到的分數越高越好。 Ref: "Clustering Indices", clusterCrit package, R project.

K-means variations Recomputing the centroid after every assignment (rather than after all points are re-assigned) can improve speed of convergence of K-means 每個點調整後就重算重心，可以加快收斂 spherical 球面的

Evaluation of Clustering

What Is A Good Clustering? Internal criterion: A good clustering will produce high quality clusters in which: the intra-class (that is, intra-cluster) similarity is high 群內同質性越高越好 the inter-class similarity is low 群間差異大 The measured quality of a clustering depends on both the document representation and the similarity measure used

External criteria for clustering quality Based on a gold standard data set (ground truth) e.g., the Reuters collection we also used for the evaluation of classification Goal: Clustering should reproduce the classes in the gold standard Quality measured by its ability to discover some or all of the hidden patterns 用挑出中間不符合的份子來評估分群好不好

External criterion: Purity Ω= {ω1, ω2, . . . , ωK} is the set of clusters and C = {c1, c2, . . . , cJ} is the set of classes. For each cluster ωk : find class cj with most members nkj in ωk Sum all nkj and divide by total number of points purity是群中最多一類佔該群總數之比例 35

Example for computing purity To compute purity: 5 = maxj |ω1 ∩ cj | (class x, cluster 1) 4 = maxj |ω2 ∩ cj | (class o, cluster 2) 3 = maxj |ω3 ∩ cj | (class ⋄, cluster 3) Purity is (1/17) × (5 + 4 + 3) ≈ 0.71. 36

Rand Index A B C D Number of points Same Cluster in clustering 分在同一群 Different Clusters in clustering 分在不同群 Same class in ground truth 已知同一類 A C Different classes in ground truth 已知不同類 B D

Rand index: symmetric version Compare with standard Precision and Recall.

20 24 72 Rand Index example: 0.68 Number of points Same Cluster in clustering Different Clusters in clustering Same class in ground truth 20 24 Different classes in ground truth 72

DBSCAN algorithm

DBSCAN Density-based clustering : clusters are dense regions in the data space, separated by regions of lower object density. A cluster is defined as a maximal set of density-connected points May discovers clusters of arbitrary shape c.f. K-mean is spherical Ref: Density-Based Spatial Clustering of Applications with Noise 41

DBSCAN Definition Eps-neighborhood of point p : points within radius eps from p "High density" : Eps-neighborhood of a point contains at least MinPts of points For radius ɛ, MinPts=4. Density of p is "high" Density of q is "low" 42

DBSCAN Core points, Border points, and Noise points A point is a core point if it has more than a specified number of points (MinPts) within Eps—These are points that are at the interior of a cluster A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point A noise point is any point that is not a core point nor a border point. 43

Example See http://www.cse.buffalo.edu/~jing/cse601/fa12/materials/clustering_density.pdf

DBSCAN Directly density-reachable point q is directly density-reachable from point p if p is a core object and q is in eps-neighborhood of p q is directly density-reachable from p p is not directly density-reachable from q density-reachability is asymmetric MinPts=4 45

Visualize the algorithm http://www.naftaliharris.com/blog/visualizing-dbscan-clustering/

After clustering.

Hierarchical Clustering

Hierarchical Clustering Build a tree-based hierarchical taxonomy (dendrogram) from a set of documents. One approach: recursive application of a partitional clustering algorithm. 可由每一層不斷執行分群演算法所組成 animal vertebrate fish reptile amphib. mammal worm insect crustacean invertebrate Vertebrate 脊索動物

Dendrogram: Hierarchical Clustering Clustering obtained by cutting the dendrogram at a desired level: each connected component forms a cluster. 另一種思考： 對階層樹橫向切一刀, 留下有連接在一起的, 就構成一群 Dendrogram 樹狀圖

Hierarchical Clustering algorithms Agglomerative (bottom-up): 由下往上聚合 Start with each document being a single cluster. Eventually all documents belong to the same cluster. Divisive (top-down): 由上往下分裂 Start with all documents belong to the same cluster. Eventually each node forms a cluster on its own. Does not require the number of clusters k in advance 不需要先決定要分成幾群 Needs a termination condition

Hierarchical Agglomerative Clustering (HAC) Algorithm Starts with each doc in a separate cluster 每篇文件剛開始都自成一群 then repeatedly joins the closest pair of clusters, until there is only one cluster. 不斷地將最近的二群做連接 The history of merging forms a binary tree or hierarchy. 連接的過程就構成一個二元階層樹

Dendrogram: Document Example As clusters agglomerate, docs likely to fall into a hierarchy of “topics” or concepts. d3 d5 d1 d3,d4,d5 d4 d2 d1,d2 d4,d5 d3

Closest pair of clusters 如何計算最近的二群 Many variants to defining closest pair of clusters Single-link 挑群中最近的一點來代表 Similarity of the most cosine-similar (single-link) Complete-link 挑群中最遠的一點來代表 Similarity of the “furthest” points, the least cosine-similar Centroid 挑群中的重心來代表 Clusters whose centroids (centers of gravity) are the most cosine-similar Average-link 跟群中的所有點計算距離後取平均值 Average cosine between pairs of elements

Single Link Agglomerative Clustering Use maximum similarity of pairs: Can result in “straggly” (long and thin) clusters due to chaining effect. 長而鬆散的群集 After merging ci and cj, the similarity of the resulting cluster to another cluster, ck, is:

Single Link Example

Complete Link Agglomerative Clustering Use minimum similarity of pairs: Makes “tighter,” spherical clusters that are typically preferable. 緊密一點的群集 After merging ci and cj, the similarity of the resulting cluster to another cluster, ck, is: Ci Cj Ck

Complete Link Example

Computational Complexity In the first iteration, all HAC methods need to compute similarity of all pairs of n individual instances which is O(n2). 兩兩文件計算相似性 In each of the subsequent n2 merging iterations, compute the distance between the most recently created cluster and all other existing clusters. 包含合併過程 O(n2 log n)

Key notion: cluster representative We want a notion of a representative point in a cluster 如何代表該群→可以用中心或其它點代表 Representative should be some sort of “typical” or central point in the cluster, e.g.,

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

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. Say 10. Centroid Outlier 簡單說就是距離太遠的點 (ex. 10倍遠)，直接忽略

Using Medoid As Cluster Representative The centroid does not have to be a document. Medoid: A cluster representative that is one of the documents 用以代表該群的某一份文件 Ex. the document closest to the centroid Why use Medoid ? Consider the representative of a large cluster (>1000 documents) The centroid of this cluster will be a dense vector The medoid of this cluster will be a sparse vector

Clustering : discussion

Feature selection 選擇好的詞再來做分群 Which terms to use as axes for vector space? IDF is a form of feature selection the most discriminating terms 鑑別力好的詞 Ex. use only nouns/noun phrases

Labeling 在分好的群上加標記 After clustering algorithm finds clusters - how can they be useful to the end user? Need pithy label for each cluster 加上簡潔厄要的標記 In search results, say “Animal” or “Car” in the jaguar example. In topic trees (Yahoo), need navigational cues. Often done by hand, a posteriori. 事後以人工編輯

How to Label Clusters Show titles of typical documents 用幾份代表文件的標題做標記 Show words/phrases prominent in cluster 用幾個較具代表性的詞做標記 More likely to fully represent cluster Use distinguishing words/phrases 配合自動產生關鍵詞的技術

Labeling Common heuristics - list 5-10 most frequent terms in the centroid vector. 通常用5~10個詞來代表該群 Differential labeling by frequent terms Within a collection “Computers”, clusters all have the word computer as frequent term. Discriminant analysis of centroids. 要挑選有鑑別力的詞

Topic Model 應用分群在主題建模上 在數量龐大的文件集合中自動地發現某些結構 (主題)，並 將每個主題用某些關鍵字的形式表現 (註: 即Bag-of-Word模型)； 隨後，還可以知道每篇文章中各個主題占得比重如何，並 據此判斷兩篇文章的相關程度。 分群演算法就可以將關鍵字群聚成若干主題。

Topic Modelling (with LSA, pLSA, or LDA) brain computer data dna evolve gene genetic life organism nerve neuron number … Clustering words into topics