Advanced SearchTechniques for Large Scale Data Analytics Pavel Zezula and Jan Sedmidubsky Masaryk University http://disa.fi.muni.cz  Given a cloud of data points we want to understand its structure Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 2 3  Given a set of points, with a notion of distance between points, group the points into some number of clusters, so that ▪ Members of a cluster are close/similar to each other ▪ Members of different clusters are dissimilar  Usually: ▪ Points are in a high-dimensional space ▪ Similarity is defined using a distance measure ▪ Euclidean, Cosine, Jaccard, edit distance, … Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 4 x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x Outlier Cluster Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 5 6  Clustering in two dimensions looks easy  Clustering small amounts of data looks easy  And in most cases, looks are not deceiving  Many applications involve not 2, but 10 or 10,000 dimensions  High-dimensional spaces look different: Almost all pairs of points are at about the same distance Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Intuitively: Music divides into categories, and customers prefer a few categories ▪ But what are categories really?  Represent a CD by a set of customers who bought it: ▪ Similar CDs have similar sets of customers, and vice-versa 7Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Space of all CDs:  Think of a space with one dim. for each customer ▪ Values in a dimension may be 0 or 1 only ▪ A CD is a point in this space (x1, x2,…, xk), where xi = 1 iff the i th customer bought the CD  Task: Find clusters of similar CDs  For Amazon, the dimension is tens of millions Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 8 Finding topics:  Represent a document by a vector (x1, x2,…, xk), where xi = 1 iff the i th word (in some order) appears in the document ▪ It actually doesn’t matter if k is infinite; i.e., we don’t limit the set of words  Documents with similar sets of words may be about the same topic 9Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  As with CDs we have a choice when we think of documents as sets of words or shingles: ▪ Sets as vectors: Measure similarity by the cosine distance ▪ Sets as sets: Measure similarity by the Jaccard distance ▪ Sets as points: Measure similarity by Euclidean distance Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 10 11  Hierarchical: ▪ Agglomerative (bottom up): ▪ Initially, each point is a cluster ▪ Repeatedly combine the two “nearest” clusters into one ▪ Divisive (top down): ▪ Start with one cluster and recursively split it  Point assignment: ▪ Maintain a set of clusters ▪ Points belong to “nearest” cluster Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Key operation: Repeatedly combine two nearest clusters  Three important questions: ▪ 1) How do you represent a cluster of more than one point? ▪ 2) How do you determine the “nearness” of clusters? ▪ 3) When to stop combining clusters? Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 12  Key operation: Repeatedly combine two nearest clusters  (1) How to represent a cluster of many points? ▪ Key problem: As you merge clusters, how do you represent the “location” of each cluster, to tell which pair of clusters is closest?  Euclidean case: each cluster has a centroid = average of its (data)points  (2) How to determine “nearness” of clusters? ▪ Measure cluster distances by distances of centroids Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 13 14 (5,3) o (1,2) o o (2,1) o (4,1) o (0,0) o (5,0) x (1.5,1.5) x (4.5,0.5) x (1,1) x (4.7,1.3) Data: o … data point x … centroid Dendrogram Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) What about the Non-Euclidean case?  The only “locations” we can talk about are the points themselves ▪ i.e., there is no “average” of two points  Approach 1: ▪ (1) How to represent a cluster of many points? clustroid = (data)point “closest” to other points ▪ (2) How do you determine the “nearness” of clusters? Treat clustroid as if it were centroid, when computing inter-cluster distances 15Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  (1) How to represent a cluster of many points? clustroid = point “closest” to other points  Possible meanings of “closest”: ▪ Smallest maximum distance to other points ▪ Smallest average distance to other points ▪ Smallest sum of squares of distances to other points ▪ For distance metric d clustroid c of cluster C is: Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 16 Cx c cxd 2 ),(min Centroid is the avg. of all (data)points in the cluster. This means centroid is an “artificial” point. Clustroid is an existing (data)point that is “closest” to all other points in the cluster. X Cluster on 3 datapoints Centroid Clustroid Datapoint  (2) How do you determine the “nearness” of clusters? ▪ Approach 2: Intercluster distance = minimum of the distances between any two points, one from each cluster ▪ Approach 3: Pick a notion of “cohesion” of clusters, e.g., maximum distance from the clustroid ▪ Merge clusters whose union is most cohesive 17Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Approach 3.1: Use the diameter of the merged cluster = maximum distance between points in the cluster  Approach 3.2: Use the average distance between points in the cluster  Approach 3.3: Use a density-based approach ▪ Take the diameter or avg. distance, e.g., and divide by the number of points in the cluster Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 18  Naïve implementation of hierarchical clustering: ▪ At each step, compute pairwise distances between all pairs of clusters, then merge ▪ O(N3)  Careful implementation using priority queue can reduce time to O(N2 log N) ▪ Still too expensive for really big datasets that do not fit in memory Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 19  Assumes Euclidean space/distance  Start by picking k, the number of clusters  Initialize clusters by picking one point per cluster ▪ Example: Pick one point at random, then k-1 other points, each as far away as possible from the previous points 21Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  1) For each point, place it in the cluster whose current centroid it is nearest  2) After all points are assigned, update the locations of centroids of the k clusters  3) Reassign all points to their closest centroid ▪ Sometimes moves points between clusters  Repeat 2 and 3 until convergence ▪ Convergence: Points don’t move between clusters and centroids stabilize Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 22 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 23 x x x x x x x x x … data point … centroid x x x Clusters after round 1 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 24 x x x x x x x x x … data point … centroid x x x Clusters after round 2 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 25 x x x x x x x x x … data point … centroid x x x Clusters at the end How to select k?  Try different k, looking at the change in the average distance to centroid as k increases  Average falls rapidly until right k, then changes little 26 k Average distance to centroid Best value of k Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 27 x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too few; many long distances to centroid. Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 28 x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Just right; distances rather short. Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 29 x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too many; little improvement in average distance. Extension of k-means to large data  BFR [Bradley-Fayyad-Reina] is a variant of k-means designed to handle very large (disk-resident) data sets  Assumes that clusters are normally distributed around a centroid in a Euclidean space ▪ Standard deviations in different dimensions may vary ▪ Clusters are axis-aligned ellipses  Efficient way to summarize clusters (want memory required O(clusters) and not O(data)) 31Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Points are read from disk one main-memoryfull at a time  Most points from previous memory loads are summarized by simple statistics  To begin, from the initial load we select the initial k centroids by some sensible approach: ▪ Take k random points ▪ Take a small random sample and cluster optimally ▪ Take a sample; pick a random point, and then k–1 more points, each as far from the previously selected points as possible 32Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 3 sets of points which we keep track of:  Discard set (DS): ▪ Points close enough to a centroid to be summarized  Compression set (CS): ▪ Groups of points that are close together but not close to any existing centroid ▪ These points are summarized, but not assigned to a cluster  Retained set (RS): ▪ Isolated points waiting to be assigned to a compression set Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 33 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 34 A cluster. Its points are in the DS. The centroid Compressed sets. Their points are in the CS. Points in the RS Discard set (DS): Close enough to a centroid to be summarized Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points For each cluster, the discard set (DS) is summarized by:  The number of points, N  The vector SUM, whose ith component is the sum of the coordinates of the points in the ith dimension  The vector SUMSQ: ith component = sum of squares of coordinates in ith dimension Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 35 A cluster. All its points are in the DS. The centroid  2d + 1 values represent any size cluster ▪ d = number of dimensions  Average in each dimension (the centroid) can be calculated as SUMi / N ▪ SUMi = ith component of SUM  Variance of a cluster’s discard set in dimension i is: (SUMSQi / N) – (SUMi / N)2 ▪ And standard deviation is the square root of that  Next step: Actual clustering 36Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Note: Dropping the “axis-aligned” clusters assumption would require storing full covariance matrix to summarize the cluster. So, instead of SUMSQ being a d-dim vector, it would be a d x d matrix, which is too big! Processing the “Memory-Load” of points (1):  1) Find those points that are “sufficiently close” to a cluster centroid and add those points to that cluster and the DS ▪ These points are so close to the centroid that they can be summarized and then discarded  2) Use any main-memory clustering algorithm to cluster the remaining points and the old RS ▪ Clusters go to the CS; outlying points to the RS Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 37 Discard set (DS): Close enough to a centroid to be summarized. Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points Processing the “Memory-Load” of points (2):  3) DS set: Adjust statistics of the clusters to account for the new points ▪ Add Ns, SUMs, SUMSQs  4) Consider merging compressed sets in the CS  5) If this is the last round, add all compressed sets in the CS and all RS points into their nearest cluster Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 38 Discard set (DS): Close enough to a centroid to be summarized. Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points  Q1) How do we decide if a point is “close enough” to a cluster that we will add the point to that cluster?  Q2) How do we decide whether two compressed sets (CS) deserve to be combined into one? 39Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Q1) We need a way to decide whether to put a new point into a cluster (and discard) ▪ Using the Mahalanobis distance (MD) – accept a point for a cluster if its MD is < some threshold (e.g., one standard dev. 𝑑) ▪ If clusters are normally distributed in d dimensions, then after normalization, the threshold of one standard deviation 𝑑 means that 68% of the points of the cluster will have a Mahalanobis distance < 𝑑 ▪ For point (x1, …, xd) and centroid (c1, …, cd) 1. Normalize in each dimension: yi = (xi - ci) / i 2. Take sum of the squares of the yi 3. Take the square root 𝑀𝐷 𝑥, 𝑐 = ෍ 𝑖=1 𝑑 𝑦𝑖 2 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 40 σi … standard deviation of points in the cluster in the ith dimension Q2) Should 2 CS subclusters be combined?  Compute the variance of the combined subcluster ▪ N, SUM, and SUMSQ allow us to make that calculation quickly  Combine if the combined variance is below some threshold Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 41 Extension of k-means to clusters of arbitrary shapes  Problem with BFR/k-means: ▪ Assumes clusters are normally distributed in each dimension ▪ And axes are fixed – ellipses at an angle are not OK  CURE (Clustering Using REpresentatives): ▪ Assumes a Euclidean distance ▪ Allows clusters to assume any shape ▪ Uses a collection of representative points to represent clusters 43 Vs. Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 44 e e e e e e e e e e e h h h h h h h h h h h h h salary age Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 2 Pass algorithm. Pass 1:  0) Pick a random sample of points that fit in main memory  1) Initial clusters: ▪ Cluster these points hierarchically – group nearest points/clusters  2) Pick representative points: ▪ For each cluster, pick a sample of points, as dispersed as possible ▪ From the sample, pick representatives by moving them (say) 20% toward the centroid of the cluster Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 45 46 e e e e e e e e e e e h h h h h h h h h h h h h salary age Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 47 e e e e e e e e e e e h h h h h h h h h h h h h salary age Pick (say) 4 remote points for each cluster. Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 48 e e e e e e e e e e e h h h h h h h h h h h h h salary age Move points (say) 20% toward the centroid. Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Pass 2:  Now, rescan the whole dataset and visit each point p in the data set  Place it in the “closest cluster” ▪ Normal definition of “closest”: Find the closest representative to p and assign it to representative’s cluster 49Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) p  Clustering: Given a set of points, with a notion of distance between points, group the points into some number of clusters  Algorithms: ▪ Agglomerative hierarchical clustering: ▪ Centroid and clustroid ▪ k-means: ▪ Initialization, picking k ▪ BFR ▪ CURE Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 50