Advanced Search Techniques 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)
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http://www.cs.toronto.edu/~laurens/drtoronto/Dimensionality_Reduction_@_Toronto_files/shapeimage_2.
png

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¡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)

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x        x
x  x      x  x
x   x x  x
x     x  x
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x x   x
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     x   x
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x
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
x        x
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Outlier
Cluster

Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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http://tagc.univ-mrs.fr/tagc/images/dputhier/tb2.jpg

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¡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)

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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)
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Pavel 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)
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http://www.mathworks.com/help/toolbox/stats/dendrogram.gif
http://www.ima.umn.edu/~iwen/REU/2Ddata.jpg
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)
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http://www.mathworks.com/help/toolbox/stats/dendrogram.gif

¡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)
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  (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)

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Pavel 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)
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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
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Pavel 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)
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Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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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)
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Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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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)
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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)
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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
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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)
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x        x
x  x      x  x
x   x x  x
x     x  x
x   x
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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)
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x        x
x  x      x  x
x   x x  x
x     x  x
x   x
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xx    x
x  x
x    x  x
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x x   x
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x
Just right;
distances
rather short.

Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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x        x
x  x      x  x
x   x x  x
x     x  x
x   x
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xx    x
x  x
x    x  x
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x x   x
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     x   x
x  x    x    x
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x
x
Too many;
little improvement
in average
distance.

Extension of k-means to large data



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Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
http://hyperphysics.phy-astr.gsu.edu/hbase/math/immath/gauds.gif

¡Points are read from disk one main-memory-full 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
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Pavel 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)
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Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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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)
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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
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Pavel 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)
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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

Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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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

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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)
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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)
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Extension of k-means to clusters
of arbitrary shapes


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http://www.ima.umn.edu/~iwen/REU/2Ddata.jpg
Vs.
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
http://www.ml.uni-saarland.de/code/pSpectralClustering/images/eigenvector11b.png

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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)
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Pick (say) 4
remote points
for each
cluster.
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)

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Move points
(say) 20%
toward the
centroid.
Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)

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Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)
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¡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)
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