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• clustering
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Apache Cassandra Clustering
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When the cluster for Apache Cassandra is
designed, an important point is to select
the right partitioner. Two partitioners exist:
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Apache Cassandra Clustering
RandomPartitioner (RP): This
partitioner randomly distributes the
key-value pairs over the network,
resulting in a good load balancing.
Compared to OPP, more nodes have to
be accessed to get a number of keys.
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Apache Cassandra Clustering
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OrderPreservingPartitioner (OPP): This
partitioner distributes the key-value pairs
in a natural way so that similar keys are
not far away. The advantage is that fewer
nodes have to be accessed. The
drawback is the uneven distribution of the
key-value pairs.
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Cluster analysis Clusters and clusterings
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According to Vladimir Estivill-Castro,
the notion of a "cluster" cannot be
precisely defined, which is one of the
reasons why there are so many
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Cluster analysis Clusters and clusterings
Connectivity models: for example
hierarchical clustering builds models
based on distance connectivity.
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Cluster analysis Clusters and clusterings
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Centroid models: for example the k-means algorithm
represents each cluster by a single mean vector.
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Cluster analysis Clusters and clusterings
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Distribution models: clusters are modeled
using statistical distributions, such as
multivariate normal distributions used by
the Expectation-maximization algorithm.
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Cluster analysis Clusters and clusterings
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Density models: for example DBSCAN
and OPTICS defines clusters as
connected dense regions in the data
space.
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Cluster analysis Clusters and clusterings
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Subspace models: in Biclustering (also
known as Co-clustering or two-modeclustering), clusters are modeled with
both cluster members and relevant
attributes.
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Cluster analysis Clusters and clusterings
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Group models: some algorithms do
not provide a refined model for their
results and just provide the grouping
information.
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Cluster analysis Clusters and clusterings
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Graph-based models: a clique, i.e., a
subset of nodes in a graph such that
every two nodes in the subset are
connected by an edge can be
considered as a prototypical form of
cluster. Relaxations of the complete
connectivity requirement (a fraction of
the edges can be missing) are known
as quasi-cliques.
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Cluster analysis Clusters and clusterings
A "clustering" is essentially a set of
such clusters, usually containing all
objects in the data set. Additionally, it
may specify the relationship of the
clusters to each other, for example a
hierarchy of clusters embedded in each
other. Clusterings can be roughly
distinguished as:
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Cluster analysis Clusters and clusterings
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hard clustering: each
object belongs to a
cluster or not
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Cluster analysis Clusters and clusterings
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soft clustering (also: fuzzy clustering):
each object belongs to each cluster to a
certain degree (e.g. a likelihood of
belonging to the cluster)
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Cluster analysis Clusters and clusterings
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strict partitioning clustering: here each object
belongs to exactly one cluster
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Cluster analysis Clusters and clusterings
strict partitioning clustering with
outliers: objects can also belong to no
cluster, and are considered outliers.
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Cluster analysis Clusters and clusterings
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overlapping clustering (also: alternative
clustering, multi-view clustering): while
usually a hard clustering, objects may
belong to more than one cluster.
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Cluster analysis Clusters and clusterings
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hierarchical clustering: objects that belong to a child
cluster also belong to the parent cluster
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Cluster analysis Clusters and clusterings
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subspace clustering: while an overlapping
clustering, within a uniquely defined
subspace, clusters are not expected to
overlap.
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Cluster analysis Clustering algorithms
Clustering algorithms can be
categorized based on their cluster
model, as listed above. The following
overview will only list the most
prominent examples of clustering
algorithms, as there are possibly over
100 published clustering algorithms. Not
all provide models for their clusters and
can thus not easily be categorized. An
overview of algorithms explained in can
be found in the list of statistics
algorithms.
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Cluster analysis Clustering algorithms
There is no objectively "correct"
clustering algorithm, but as it was
noted, "clustering is in the eye of the
beholder." The most appropriate
clustering algorithm for a particular
problem often needs to be chosen
experimentally, unless there is a
mathematical reason to prefer one
cluster model over another
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Cluster analysis Connectivity based clustering (hierarchical clustering)
At different distances, different
clusters will form, which can be
represented using a dendrogram,
which explains where the common
name "hierarchical clustering"
comes from: these algorithms do not
provide a single partitioning of the
data set, but instead provide an
extensive hierarchy of clusters that
merge with each other at certain
distances
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Cluster analysis Connectivity based clustering (hierarchical clustering)
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Furthermore, hierarchical clustering
can be agglomerative (starting with
single elements and aggregating them
into clusters) or divisive (starting with
the complete data set and dividing it
into partitions).
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Cluster analysis Connectivity based clustering (hierarchical clustering)
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They did however provide inspiration for many
later methods such as density based
clustering.
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Cluster analysis Connectivity based clustering (hierarchical clustering)
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Single-linkage on Gaussian data. At 35
clusters, the biggest cluster starts
fragmenting into smaller parts, while
before it was still connected to the second
largest due to the single-link effect.
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Cluster analysis Connectivity based clustering (hierarchical clustering)
Single-linkage on density-based
clusters. 20 clusters extracted, most of
which contain single elements, since
linkage clustering does not have a
notion of "noise".
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Cluster analysis Centroid-based clustering
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In centroid-based clustering, clusters are
represented by a central vector, which
may not necessarily be a member of the
data set. When the number of clusters is
fixed to k, k-means clustering gives a
formal definition as an optimization
problem: find the cluster centers and
assign the objects to the nearest cluster
center, such that the squared distances
from the cluster are minimized.
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Cluster analysis Centroid-based clustering
Variations of k-means often include
such optimizations as choosing the
best of multiple runs, but also
restricting the centroids to members of
the data set (k-medoids), choosing
medians (k-medians clustering),
choosing the initial centers less
randomly (K-means++) or allowing a
fuzzy cluster assignment (Fuzzy cmeans).
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Cluster analysis Centroid-based clustering
Most k-means-type algorithms
require the number of clusters - - to
be specified in advance, which is
considered to be one of the biggest
drawbacks of these algorithms.
Furthermore, the algorithms prefer
clusters of approximately similar size,
as they will always assign an object to
the nearest centroid. This often leads
to incorrectly cut borders in between
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Cluster analysis Centroid-based clustering
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K-means has a number of interesting
theoretical properties. On the one
hand, it partitions the data space into
a structure known as a Voronoi
diagram. On the other hand, it is
conceptually close to nearest neighbor
classification, and as such is popular
in machine learning. Third, it can be
seen as a variation of model based
classification, and Lloyd's algorithm
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Cluster analysis Centroid-based clustering
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K-means separates data into Voronoi-cells, which
assumes equal-sized clusters (not adequate here)
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Cluster analysis Centroid-based clustering
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K-means cannot represent
density-based clusters
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Cluster analysis Distribution-based clustering
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The clustering model most closely related
to statistics is based on distribution
models. Clusters can then easily be
defined as objects belonging most likely to
the same distribution. A nice property of
this approach is that this closely
resembles the way artificial data sets are
generated: by sampling random objects
from a distribution.
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Cluster analysis Distribution-based clustering
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While the theoretical foundation of these
methods is excellent, they suffer from one
key problem known as overfitting, unless
constraints are put on the model
complexity. A more complex model will
usually always be able to explain the data
better, which makes choosing the
appropriate model complexity inherently
difficult.
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Cluster analysis Distribution-based clustering
In order to obtain a hard clustering,
objects are often then assigned to the
Gaussian distribution they most likely
belong to; for soft clusterings, this is
not necessary.
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Cluster analysis Distribution-based clustering
Distribution-based clustering is a
semantically strong method, as it not
only provides you with clusters, but
also produces complex models for the
clusters that can also capture
correlation and dependence of
attributes
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Cluster analysis Distribution-based clustering
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On Gaussian-distributed data, EM works well,
since it uses Gaussians for modelling clusters
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Cluster analysis Distribution-based clustering
Density-based clusters
cannot be modeled using
Gaussian distributions
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Cluster analysis Density-based clustering
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In density-based clustering, clusters
are defined as areas of higher density
than the remainder of the data set.
Objects in these sparse areas - that are
required to separate clusters - are
usually considered to be noise and
border points.
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Cluster analysis Density-based clustering
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DeLi-Clu, Density-Link-Clustering
combines ideas from single-linkage
clustering and OPTICS, eliminating
the parameter entirely and offering
performance improvements over
OPTICS by using an R-tree index.
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Cluster analysis Density-based clustering
On a data set consisting of mixtures of
Gaussians, these algorithms are nearly
always outperformed by methods such as
EM clustering that are able to precisely
model this kind of data.
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Cluster analysis Density-based clustering
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DBSCAN assumes clusters of similar density, and
may have problems separating nearby clusters
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Cluster analysis Density-based clustering
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OPTICS is a DBSCAN variant that
handles different densities much
better
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Cluster analysis Evaluation of clustering results
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Evaluation of clustering results sometimes is
referred to as cluster validation.
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Cluster analysis Evaluation of clustering results
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There have been several suggestions for a
measure of similarity between two
clusterings. Such a measure can be used
to compare how well different data
clustering algorithms perform on a set of
data. These measures are usually tied to
the type of criterion being considered in
assessing the quality of a clustering
method.
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Cluster analysis Clustering Axioms
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Given that there is a myriad of clustering
algorithms and objectives, it is helpful to
reason about clustering independently of
any particular algorithm, objective function,
or generative data model. This can be
achieved by defining a clustering function
as one that satisfies a set of properties.
This is often termed as an Axiomatic
System. Functions that satisfy the basic
axioms are called clustering functions.
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OpenVMS - Clustering
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OpenVMS supports clustering (first called
VAXcluster and later VMScluster), where
multiple systems share disk storage,
processing, job queues and print queues,
and are connected either by specialized
hardware or an industry-standard LAN
(usually Ethernet). A LAN-based cluster is
often called a LAVc, for Local Area
Network VMScluster, and allows, among
other things, bootstrapping a possibly
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OpenVMS - Clustering
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VAXcluster support was first added in
VMS version 4, which was released in
1984. This version only supported
clustering over CI. Later releases of
version 4 supported clustering over
LAN (LAVC), and support for LAVC
was improved in VMS version 5,
released in 1988.
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OpenVMS - Clustering
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Mixtures of cluster interconnects and
technologies are permitted, including
Gigabit (GbE) Ethernet, SCSI, FDDI,
DSSI, CI and Memory Channel
adapters.
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OpenVMS - Clustering
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OpenVMS supports up to 96 nodes in a
single cluster, and allows mixedarchitecture clusters, where VAX and
Alpha systems, or Alpha and Itanium
systems can co-exist in a single cluster
(Various organizations have demonstrated
triple-architecture clusters and cluster
configurations with up to 150 nodes, but
these configurations are not supported by
HP).
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OpenVMS - Clustering
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Unlike many other clustering solutions,
VAXcluster offers transparent and fully
distributed read-write with record-level
locking, which means that the same disk
and even the same file can be accessed
by several cluster nodes at once; the
locking occurs only at the level of a single
record of a file, which would usually be
one line of text or a single record in a
database. This allows the construction of
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OpenVMS - Clustering
Cluster interconnections can span
upwards of 500 miles, allowing member
nodes to be located in different buildings
on an office campus, or in different cities.
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OpenVMS - Clustering
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Host-based volume shadowing allows
volumes (of the same or of different
sizes) to be shadowed (mirrored)
across multiple controllers and multiple
hosts, allowing the construction of
disaster-tolerant environments.
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OpenVMS - Clustering
Full access into the distributed lock
manager (DLM) is available to application
programmers, and this allows applications
to coordinate arbitrary resources and
activities across all cluster nodes. This
obviously includes file-level coordination,
but the resources and activities and
operations that can be coordinated with
the DLM are completely arbitrary.
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OpenVMS - Clustering
OpenVMS V8.4 offers advances in
clustering technology, including the use of
industry-standard TCP/IP networking to
bring efficiencies to cluster interconnect
technology. Cluster over TCP/IP is
supported with OpenVMS version 8.4
released in 2010.
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OpenVMS - Clustering
With the supported capability of rolling
upgrades and with multiple system disks,
cluster configurations can be maintained
on-line and upgraded incrementally. This
allows cluster configurations to continue to
provide application and data access while
a subset of the member nodes are
upgraded to newer software versions.
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Categorization - Conceptual clustering
Conceptual clustering is a modern
variation of the classical approach, and
derives from attempts to explain how
knowledge is represented. In this
approach, classes (clusters or entities) are
generated by first formulating their
conceptual descriptions and then
classifying the entities according to the
descriptions.
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Categorization - Conceptual clustering
Conceptual clustering developed
mainly during the 1980s, as a machine
paradigm for unsupervised learning. It
is distinguished from ordinary data
clustering by generating a concept
description for each generated
category.
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Categorization - Conceptual clustering
The task of clustering involves
recognizing inherent structure in a data
set and grouping objects together by
similarity into classes
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Categorization - Conceptual clustering
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Conceptual clustering is closely related to
fuzzy set theory, in which objects may
belong to one or more groups, in varying
degrees of fitness.
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Machine learning - Clustering
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Clustering is a method of unsupervised
learning, and a common technique for
statistical data analysis.
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Human genetic variation - Genetic clustering
Genetic data can be used to infer
population structure and assign individuals
to groups that often correspond with their
self-identified geographical ancestry.
Recently, Lynn Jorde and Steven Wooding
argued that "Analysis of many loci now
yields reasonably accurate estimates of
genetic similarity among individuals, rather
than populations. Clustering of individuals
is correlated with geographic origin or
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Unisys OS 2200 operating system - Clustering
OS 2200 systems may be Cluster
(computing)|clustered to achieve
greater performance and availability
than a single system
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Unisys OS 2200 operating system - Clustering
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A clustered environment allows each
system to have its own local files,
databases, and application groups
along with shared files and one or
more shared application groups.
Local files and databases are
accessed only by a single system.
Shared files and databases must be on
disks that are simultaneously
accessible from all systems in the
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Unisys OS 2200 operating system - Clustering
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The XPC-L provides a communication
path among the systems for
coordination of actions. It also
provides a very fast lock engine.
Connection to the XPC-L is via a
special I/O processor that operates with
extremely low latencies. The lock
manager in the XPC-L provides all the
functions required for both file and
database locks. This includes
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Unisys OS 2200 operating system - Clustering
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The XPC-L is implemented with two
physical servers to create a fully
redundant configuration.
Maintenance, including loading new
versions of the XPC-L firmware, may
be performed on one of the servers
while the other continues to run.
Failures, including physical damage
to one server, do not stop the cluster,
as all information is kept in both
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Microsoft Exchange Server - Clustering and high availability
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In fact, support for active-active mode clustering has
been discontinued with Exchange Server 2007.
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Microsoft Exchange Server - Clustering and high availability
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This void has however been filled by ISV's
and storage manufacturers, through site
resilience solutions, such as geoclustering and asynchronous data
replication
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Microsoft Exchange Server - Clustering and high availability
The second type of cluster is the
traditional clustering that was
available in previous versions, and is
now being referred to as SCC (Single
Copy Cluster)
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Microsoft Exchange Server - Clustering and high availability
In November 2007, Microsoft released
SP1 for Exchange Server 2007. This
service pack includes an additional highavailability feature called SCR (Standby
Continuous Replication). Unlike CCR
which requires that both servers belong to
a Windows cluster, typically residing in the
same datacenter, SCR can replicate data
to a non-clustered server, located in a
separate datacenter.
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Microsoft Exchange Server - Clustering and high availability
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With Exchange Server 2010, Microsoft
introduced the concept of the Database
Availability Group (DAG). A DAG
contains Mailbox servers that become
members of the DAG. Once a Mailbox
server is a member of a DAG, the
Mailbox Databases on that server can
be copied to other members of the
DAG. When you add a Mailbox server to
a DAG, the Failover Clustering
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Apache Cassandra - Clustering
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# RandomPartitioner (RP): This partitioner
randomly distributes the key-value pairs
over the network, resulting in a good load
balancing. Compared to OPP, more nodes
have to be accessed to get a number of
keys.
https://store.theartofservice.com/the-clustering-toolkit.html
Apache Cassandra - Clustering
1
# OrderPreservingPartitioner (OPP): This
partitioner distributes the key-value pairs
in a natural way so that similar keys are
not far away. The advantage is that fewer
nodes have to be accessed. The
drawback is the uneven distribution of the
key-value pairs.
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Pattern recognition - Cluster analysis|Clustering algorithms (unsupervised
learning|unsupervised algorithms predicting categorical data|categorical labels)
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*Categorical mixture models
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Pattern recognition - Cluster analysis|Clustering algorithms (unsupervised
learning|unsupervised algorithms predicting categorical data|categorical labels)
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*Hierarchical clustering
(agglomerative or
divisive)
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Pattern recognition - Cluster analysis|Clustering algorithms (unsupervised
learning|unsupervised algorithms predicting categorical data|categorical labels)
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*Correlation clustering
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Single-system image - Features of SSI clustering systems
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Different SSI systems may, depending on
their intended usage, provide some subset
of these features.
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Segmentation (image processing) - Clustering methods
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The K-means algorithm is an iterative
technique that is used to Cluster
analysis|partition an image into K
clusters. The basic algorithm is:
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Segmentation (image processing) - Clustering methods
# Pick K cluster
centers, either
randomly or based
on some heuristic
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Segmentation (image processing) - Clustering methods
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# Assign each pixel in the image to the
cluster that minimizes the distance
between the pixel and the cluster center
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Segmentation (image processing) - Clustering methods
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# Re-compute the cluster centers by averaging all of
the pixels in the cluster
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Segmentation (image processing) - Clustering methods
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# Repeat steps 2 and 3 until convergence is
attained (i.e. no pixels change clusters)
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Segmentation (image processing) - Clustering methods
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In this case, distance is the squared or absolute
difference between a pixel and a cluster center
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List of machine learning algorithms - Hierarchical clustering
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* Conceptual clustering
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EEG microstates - Clustering and processing
Since the brain goes through so many
transformations in such short time scales,
microstate analysis is essentially an
analysis of average EEG states
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EEG microstates - Clustering and processing
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Similarity of EEG spatial configuration of
each prototype map with each of the 10
maps is computed using the coefficient of
determination to omit the maps' polarities
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K-means clustering
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'k-means clustering' is a method of vector
quantization, originally from signal
processing, that is popular for cluster
analysis in data mining. k-means
clustering aims to partition of a
set|partition n observations into k clusters
in which each observation belongs to the
cluster with the nearest mean, serving as
a prototype of the cluster. This results in a
partitioning of the data space into Voronoi
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K-means clustering
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Additionally, they both use cluster centers
to model the data; however, k-means
clustering tends to find clusters of
comparable spatial extent, while the
expectation-maximization mechanism
allows clusters to have different shapes.
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K-means clustering - Free
* Apache Mahout
[http://cwiki.apache.org/MAHOUT/k-meansclustering.html k-Means]
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K-means clustering - Free
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* CrimeStat implements two spatial Kmeans algorithms, one of which allows
the user to define the starting
locations.
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K-means clustering - Free
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* ELKI contains k-means (with Lloyd
and MacQueen iteration, along with
different initializations such as kmeans++ initialization) and various
more advanced clustering algorithms
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K-means clustering - Free
* R (programming language)|R
[http://stat.ethz.ch/R-manual/Rpatched/library/stats/html/kmeans.h
tml kmeans] implements a variety of
algorithms
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K-means clustering - Free
* SciPy
[http://docs.scipy.org/doc/scipy/reference/cl
uster.vq.html vector-quantization]
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K-means clustering - Free
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* Scikit-learn implements a popular python
machine-learning library which contains
various clustering algorithms
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K-means clustering - Free
*
[http://graphlab.org/toolkits/clusteri
ng/ CMU's GraphLab Clustering
library] Efficient multicore
implementation for large scale data.
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K-means clustering - Free
* Weka (machine learning)|Weka
contains k-means and a few variants of
it, including k-means++ and x-means.
1
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K-means clustering - Free
*
[http://spectralpython.sourceforge.net/algo
rithms.html#k-means-clustering Spectral
Python] contains methods for
unsupervised classification including a Kmeans clustering method.
1
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K-means clustering - Free
* [http://scikitlearn.org/dev/modules/generated/sklear
n.cluster.KMeans.html scikit learn]
machine learning in Python contains a
K-Means implementation
1
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K-means clustering - Free
* OpenCV contains a
[http://docs.opencv.org/modules/core/doc/
clustering.html?highlight=kmeans#cv2.km
eans K-means] implementation under Bsd
licence|BSD licence.
1
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K-means clustering - Free
1
* [http://gforge.inria.fr/projects/yael/ Yael]
includes an efficient multi-threaded C
implementation of k-means, with C,
Python and Matlab interfaces.
https://store.theartofservice.com/the-clustering-toolkit.html
K-means clustering - Commercial
*
[http://reference.wolfram.com/math
ematica/ref/ClusteringComponents.h
tml Mathematica
ClusteringComponents function]
1
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K-means clustering - Commercial
* SAS System|SAS
[http://support.sas.com/documentati
on/cdl/en/statug/63033/HTML/defau
lt/fastclus_toc.htm FASTCLUS]
1
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K-means clustering - Commercial
* Stata
[http://www.stata.com/help13.cgi?cl
uster+kmeans kmeans]
1
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K-means clustering - Commercial
*
[http://www.visumap.com/index.asp
x?p=Products VisuMap kMeans
Clustering]
1
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K-means clustering - Source code
1
* ELKI and Weka are
written in Java and
include k-means and
variations
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K-means clustering - Source code
* K-means application in
PHP,http://www25.brinkster.com/den
shade/kmeans.php.htm using
VB,[http://people.revoledu.com/kard
i/tutorial/kMean/download.htm KMeans Clustering Tutorial: Download]
using
Perl,[http://www.lwebzem.com/cgibin/k_means/test3.cgi Perl script for
Kmeans clustering] using
1
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K-means clustering - Source code
* A parallel out-of-core implementation in
Chttp://www.cs.princeton.edu/~wdong/kmeans/
1
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K-means clustering - Source code
* An open-source collection of
clustering algorithms, including kmeans, implemented in
Javascript.http://code.google.com/p/fig
ue/ FIGUE Online
demo.http://jydelort.appspot.com/resou
rces/figue/demo.html
1
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K-means clustering - Visualization, animation and examples
1
* ELKI can visualize k-means using
Voronoi diagram|Voronoi cells and
Delaunay triangulation for 2D data. In
higher dimensionality, only cluster
assignments and cluster centers are
visualized
https://store.theartofservice.com/the-clustering-toolkit.html
K-means clustering - Visualization, animation and examples
* Demos of the K-meansalgorithm[http://home.dei.polimi.it/matteuc
c/Clustering/tutorial_html/AppletKM.html
Clustering - K-means
demo][http://siebn.de/other/yakmeans/
siebn.de - YAKMeans][http://informationandvisualization.d
e/blog/kmeans-and-voronoi-tesselationbuilt-processing k-Means and Voronoi
Tesselation: Built with Processing |
1
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K-means clustering - Visualization, animation and examples
* K-means and K-medoids (Applet),
University of LeicesterE.M. Mirkes,
[http://www.math.le.ac.uk/people/ag153/ho
mepage/KmeansKmedoids/Kmeans_Kme
doids.html K-means and K-medoids
applet]. University of Leicester, 2011.
1
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K-means clustering - Visualization, animation and examples
1
* Clustergram - cluster diagnostic plot
- for visual diagnostics of choosing the
number of (k) clusters (R
(programming language)|R
code)[http://www.rstatistics.com/2010/06/clustergramvisualization-and-diagnostics-forcluster-analysis-r-code/ Clustergram:
visualization and diagnostics for
cluster analysis (R code) | R-statistics
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BIRCH (data clustering)
1
In addition, Birch is recognized as the, first
clustering algorithm proposed in the
database area to handle 'noise' (data
points that are not part of the underlying
pattern) effectively.
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BIRCH (data clustering) - Problem with previous methods
Furthermore, most of Birch's
predecessors inspect all data points
(or all currently existing clusters)
equally for each 'clustering decision'
and do not perform heuristic
weighting based on the distance
between these data points.
1
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BIRCH (data clustering) - Advantages with BIRCH
1
It is local in that each clustering decision is
made without scanning all data points and
currently existing clusters.
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BIRCH (data clustering) - Advantages with BIRCH
1
It exploits the observation that data space
is not usually uniformly occupied and not
every data point is equally important.
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BIRCH (data clustering) - Advantages with BIRCH
1
It makes full use of available memory to
derive the finest possible sub-clusters
while minimizing I/O costs.
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BIRCH (data clustering) - Advantages with BIRCH
1
It is also an incremental method that does not
require the whole data set in advance.
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BIRCH (data clustering) - BIRCH Clustering Algorithm
Given a set of N d-dimensional data
points, the 'clustering feature' CF of the set
is defined as the triple CF = (N,LS,SS),
where LS is the linear sum and SS is the
square sum of data points.
1
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BIRCH (data clustering) - BIRCH Clustering Algorithm
1
Each non-leaf node contains at most B
entries of the form [CF_i,child_i], where
child_i is a pointer to its ith Tree (data
structure)|child node and CF_i the
clustering feature representing the
associated subcluster
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BIRCH (data clustering) - BIRCH Clustering Algorithm
1
Here an agglomerative hierarchical
clustering algorithm is applied
directly to the subclusters represented
by their CF vectors
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Hierarchical clustering
1
In data mining, 'hierarchical clustering' is
a method of cluster analysis which seeks
to build a hierarchy of clusters. Strategies
for hierarchical clustering generally fall into
two types:
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Hierarchical clustering
1
*'Agglomerative': This is a bottom up
approach: each observation starts in its
own cluster, and pairs of clusters are
merged as one moves up the hierarchy.
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Hierarchical clustering
*'Divisive': This is a top down
approach: all observations start in one
cluster, and splits are performed
recursively as one moves down the
hierarchy.
1
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Hierarchical clustering
In general, the merges and splits are
determined in a greedy algorithm|greedy
manner. The results of hierarchical
clustering are usually presented in a
dendrogram.
1
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Hierarchical clustering
In the general case, the complexity of
agglomerative clustering is \mathcal(n^3),
which makes them too slow for large data
sets. Divisive clustering with an exhaustive
search is \mathcal(2^n), which is even
worse. However, for some special cases,
optimal efficient agglomerative methods
(of complexity \mathcal(n^2)) are known:
SLINK for single-linkage and CLINK for
complete-linkage clustering.
1
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Hierarchical clustering - Cluster dissimilarity
1
In order to decide which clusters should be
combined (for agglomerative), or where a
cluster should be split (for divisive), a
measure of dissimilarity between sets of
observations is required. In most methods of
hierarchical clustering, this is achieved by
use of an appropriate metric
(mathematics)|metric (a measure of distance
between pairs of observations), and a linkage
criterion which specifies the dissimilarity of
sets as a function of the pairwise distances of
observations in the sets.
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Hierarchical clustering - Metric
1
The choice of an appropriate metric
will influence the shape of the
clusters, as some elements may be
close to one another according to one
distance and farther away according
to another. For example, in a 2dimensional space, the distance
between the point (1,0) and the origin
(0,0) is always 1 according to the usual
norms, but the distance between the
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Hierarchical clustering - Metric
1
Some commonly used metrics
for hierarchical clustering are:
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Hierarchical clustering - Metric
| Uniform
norm|maximum
distance
1
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Hierarchical clustering - Metric
1
* The probability that candidate clusters spawn
from the same distribution function (Vlinkage).
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Hierarchical clustering - Metric
1
Each agglomeration occurs at a greater
distance between clusters than the
previous agglomeration, and one can
decide to stop clustering either when the
clusters are too far apart to be merged
(distance criterion) or when there is a
sufficiently small number of clusters
(number criterion).
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Hierarchical clustering - Open Source Frameworks
*
[http://bonsai.hgc.jp/~mdehoon/soft
ware/cluster/ Cluster 3.0] provides a
nice Graphical User Interface to
access to different clustering routines
and is available for Windows, Mac OS
X, Linux, Unix.
1
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Hierarchical clustering - Open Source Frameworks
1
* Environment for DeveLoping KDDApplications Supported by IndexStructures|ELKI includes multiple
hierarchical clustering algorithms,
various linkage strategies and also
includes the efficient SLINK
algorithm, flexible cluster extraction
from dendrograms and various other
cluster analysis algorithms.
https://store.theartofservice.com/the-clustering-toolkit.html
Hierarchical clustering - Open Source Frameworks
1
* GNU Octave|Octave, the GNU analog to
MATLAB implements hierarchical
clustering in
[http://octave.sourceforge.net/statistics/fun
ction/linkage.html linkage function]
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Hierarchical clustering - Open Source Frameworks
1
* Orange (software)|Orange, a free data
mining software suite, module
[http://www.ailab.si/orange/doc/modules/or
ngClustering.htm orngClustering] for
scripting in Python (programming
language)|Python, or cluster analysis
through visual programming.
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Hierarchical clustering - Open Source Frameworks
1
* R (programming language)|R has
several functions for hierarchical
clustering: see [http://cran.rproject.org/web/views/Cluster.html
CRAN Task View: Cluster Analysis
Finite Mixture Models] for more
information.
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Hierarchical clustering - Open Source Frameworks
* scikit-learn implements a hierarchical
clustering based on the Ward's method|Ward
algorithm only.
1
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Hierarchical clustering - Standalone implementations
1
* CrimeStat implements two hierarchical
clustering routines, a nearest neighbor
(Nnh) and a risk-adjusted(Rnnh).
https://store.theartofservice.com/the-clustering-toolkit.html
Hierarchical clustering - Standalone implementations
1
* [http://code.google.com/p/figue/ figue] is
a JavaScript package that implements
some agglomerative clustering functions
(single-linkage, complete-linkage,
average-linkage) and functions to visualize
clustering output (e.g. dendrograms).
https://store.theartofservice.com/the-clustering-toolkit.html
Hierarchical clustering - Standalone implementations
1
* [http://code.google.com/p/scipy-cluster/
hcluster] is a Python (programming
language)|Python implementation, based
on NumPy, which supports hierarchical
clustering and plotting.
https://store.theartofservice.com/the-clustering-toolkit.html
Hierarchical clustering - Standalone implementations
*
[http://www.semanticsearchart.com/
researchHAC.html Hierarchical
Agglomerative Clustering]
implemented as C# visual studio
project that includes real text files
processing, building of documentterm matrix with stop words filtering
and stemming.
1
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Hierarchical clustering - Standalone implementations
*
[http://deim.urv.cat/~sgomez/multidendrogr
ams.php MultiDendrograms] An open
source Java (programming
language)|Java application for variablegroup agglomerative hierarchical
clustering, with graphical user interface.
1
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Hierarchical clustering - Standalone implementations
*
[http://www.mathworks.com/matlabc
entral/fileexchange/38018-graphagglomerative-clustering-gac-toolbox
Graph Agglomerative Clustering
(GAC) toolbox] implemented several
graph-based agglomerative clustering
algorithms.
1
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Hierarchical clustering - Commercial
* MathWorks|MATLAB
includes hierarchical cluster
analysis.
1
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Windows Server 2008 - Failover Clustering
Windows Server 2008 offers highavailability to services and
applications through Failover
Clustering. Most server features and
roles can be kept running with little to
no downtime.
1
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Windows Server 2008 - Failover Clustering
This cluster validation process tests the
underlying hardware and software directly,
and individually, to obtain an accurate
assessment of how well failover clustering
can be supported on a given configuration.
1
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Windows Server 2008 - Failover Clustering
1
Note: This feature is only available in Enterprise and
Datacenter editions of Windows Server.
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Community structure - Hierarchical clustering
1
There are several common schemes for
performing the grouping, the two simplest
being single-linkage clustering, in which
two groups are considered separate
communities if and only if all pairs of
nodes in different groups have similarity
lower than a given threshold, and
complete linkage clustering, in which all
nodes within every group have similarity
greater than a threshold.
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Categorization - Conceptual clustering
'Conceptual clustering' is a modern
variation of the classical approach, and
derives from attempts to explain how
knowledge is represented. In this
approach, Class (philosophy)|classes
(clusters or entities) are generated by first
formulating their conceptual descriptions
and then classifying the entities according
to the descriptions.
1
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Categorization - Conceptual clustering
1
The task of clustering involves recognizing
inherent structure in a data set and
grouping objects together by similarity into
classes
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Clustering coefficient
1
In graph theory, a 'clustering coefficient' is
a measure of the degree to which nodes in
a graph tend to cluster together. Evidence
suggests that in most real-world networks,
and in particular social networks, nodes
tend to create tightly knit groups
characterised by a relatively high density
of ties; this likelihood tends to be greater
than the average probability of a tie
randomly established between two nodes
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Clustering coefficient
1
Two versions of this measure exist: the
global and the local. The global version
was designed to give an overall indication
of the clustering in the network, whereas
the local gives an indication of the
embeddedness of single nodes.
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Clustering coefficient - Global clustering coefficient
1
This measure gives an indication of the
clustering in the whole network (global),
and can be applied to both undirected and
directed networks (often called transitivity,
see Wasserman and Faust, 1994, page
243Stanley Wasserman, Kathrine Faust,
1994
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Clustering coefficient - Global clustering coefficient
1
Watts and Strogatz defined the clustering
coefficient as follows, Suppose that a
vertex v has k_v neighbours; then at most
k_v(k_v - 1)/2 edges can exist between
them (this occurs when every neighbour of
v is connected to every other neighbor of
v). Let C_v denote the fraction of these
allowable edges that actually exist. Define
C as the average of C_v over all v.
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Storage engine - Data orientation and clustering
1
In contrast to conventional row-orientation,
relational databases can also be Columnoriented DBMS|column-oriented or
Correlational database|correlational in the
way they store data in any particular
structure.
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Storage engine - Data orientation and clustering
1
Even for in-memory databases clustering
provides performance advantage due to
common utilization of large caches for
input-output operations in memory, with
similar resulting behavior.
https://store.theartofservice.com/the-clustering-toolkit.html
Storage engine - Data orientation and clustering
1
For example it may be beneficial to cluster
a record of an item in stock with all its
respective order records. The decision of
whether to cluster certain objects or not
depends on the objects' utilization
statistics, object sizes, caches sizes,
storage types, etc.
https://store.theartofservice.com/the-clustering-toolkit.html
Redis - Clustering
The Redis project has a cluster
specification,[http://redis.io/topics/clust
er-spec Redis Cluster Specification],
[http://redis.io Redis.io], Retrieved
2013-12-25
1
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OneFS - Clustering
Nodes running OneFS must be
connected together with a high
performance, low-latency back-end
network for optimal performance.
OneFS 1.0-3.0 used Gigabit Ethernet
as that back-end network. Starting
with OneFS 3.5, Isilon offered
Infiniband models. Now all nodes sold
utilize an Infiniband back-end.
1
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OneFS - Clustering
1
Data, metadata, locking, transaction,
group management, allocation, and
event traffic go over the back-end RPC
system. All data and metadata
transfers are zero-copy. All
modification operations to on-disk
structures are transactional and
Journaling file system|journaled.
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Tuxedo (software) - Clustering
The heart of the
Tuxedo system is the
Bulletin Board (BB)
1
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Tuxedo (software) - Clustering
1
Another process on each machine called
the Bridge is responsible for passing
requests from one machine to another
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Tuxedo (software) - Clustering
1
On Oracle Exalogic Tuxedo leverages the
RDMA capabilities of InfiniBand to bypass
the bridge. This allows the client of a
service on one machine to directly make a
request of a server on another machine.
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ZFS - Clustering and high availability
1
ZFS is not a clustered filesystem. However,
clustered ZFS is available from third parties.
https://store.theartofservice.com/the-clustering-toolkit.html
Document clustering
'Document clustering' (or 'Text
clustering') is automatic document
organization, topic extraction and fast
information retrieval or filtering. It is
closely related to data clustering.
1
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Document clustering
Clustering methods can be used to
automatically group the retrieved
documents into a list of meaningful
categories, as is achieved by
Enterprise Search engines such as
Northern Light Group|Northern Light
and Vivisimo, consumer search
engines such as
[http://www.polymeta.com/ PolyMeta]
and [http://www.helioid.com Helioid], or
1
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Document clustering
1
* Clustering divides the results of a search
for cell into groups like biology, battery,
and prison.
https://store.theartofservice.com/the-clustering-toolkit.html
Document clustering
1
* [http://FirstGov.gov FirstGov.gov],
the official Web portal for the U.S.
government, uses document
clustering to automatically organize
its search results into categories. For
example, if a user submits
“immigration”, next to their list of
results they will see categories for
“Immigration Reform”, “Citizenship
and Immigration Services”,
https://store.theartofservice.com/the-clustering-toolkit.html
Document clustering
1
PLSA|Probabilistic Latent Semantic
Analysis (PLSA) can also be conducted
to perform document clustering.
https://store.theartofservice.com/the-clustering-toolkit.html
Document clustering
1
Document clustering involves the use
of descriptors and descriptor
extraction. Descriptors are sets of
words that describe the contents within
the cluster. Document clustering is
generally considered to be a
centralized process. Examples of
document clustering include web
document clustering for search users.
https://store.theartofservice.com/the-clustering-toolkit.html
Document clustering
The application of document
clustering can be categorized to two
types, online and offline. Online
applications are usually constrained
by efficiency problems when
compared offline applications.
1
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Document clustering
1
In general, there are two common
algorithms
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Document clustering
1
Other algorithms involve graph based
clustering, ontology supported clustering
and order sensitive clustering.
https://store.theartofservice.com/the-clustering-toolkit.html
Clustering illusion
1
The 'clustering illusion' is the tendency to
erroneously perceive small samples from
random distributions to have significant
streaks or clusters, caused by a human
tendency to underpredict the amount of
Statistical dispersion|variability likely to
appear in a small sample of random or
semi-random data due to chance.
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Clustering illusion - Examples
1
Gilovich, an early author on the subject,
argues the effect occurs for different types
of random dispersions, including 2dimensional data such as seeing clusters
in the locations of impact of V-1 flying
bombs on 2 dimensional maps of London
during World War II or seeing streaks in
stock market price fluctuations over time
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Clustering illusion - Examples
1
The clustering illusion is central to the
hot hand fallacy, the first study of which
was reported by Gilovich, Robert Vallone
and Amos Tversky. They found that the
idea that basketball players shoot
successfully in streaks, sometimes
called by sportcasters as having a hot
hand and widely believed by Gilovich et
al.'s subjects, was false. In the data they
collected, if anything the success of a
previous throw very slightly predicted a
subsequent miss rather than another
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Clustering illusion - Similar biases
Using this cognitive bias in causal
reasoning may result in the Texas
sharpshooter fallacy. More general forms
of erroneous pattern recognition are
pareidolia and apophenia. Related biases
are the illusion of control which the
clustering illusion could contribute to, and
insensitivity to sample size in which people
don't expect greater variation in smaller
samples. A different cognitive bias
1
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Clustering illusion - Possible causes
1
Daniel Kahneman and Amos Tversky
explained this kind of misprediction as
being caused by the representativeness
heuristic (which itself they also first
proposed).
https://store.theartofservice.com/the-clustering-toolkit.html
Virtual Memory System - Clustering
OpenVMS supports Computer
cluster|clustering (first called VAXcluster
and later VMScluster), where multiple
systems share disk storage, processing,
job queues and print queues, and are
connected either by proprietary
specialized hardware (Cluster
Interconnect) or an industry-standard local
area network|LAN (usually Ethernet)
1
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Virtual Memory System - Clustering
1
VAXcluster support was first added in
VMS version 4, which was released in
1984. This version only supported
clustering over CI. Later releases of
version 4 supported clustering over LAN
(LAVC), and support for LAVC was
improved in VMS version 5, released in
1988.
https://store.theartofservice.com/the-clustering-toolkit.html
Virtual Memory System - Clustering
Unlike many other clustering
solutions, VMScluster offers
transparent and fully distributed readwrite with record-level locking, which
means that the same disk and even
the same file can be accessed by
several cluster nodes at once; the
locking occurs only at the level of a
single record of a file, which would
usually be one line of text or a single
1
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Virtual Memory System - Clustering
1
Cluster connections can span upwards of
500 miles, allowing member nodes to be
located in different buildings on an office
campus, or in different cities.
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Virtual Memory System - Clustering
For more specific details, see the
clustering-related manuals in the [
http://www.hp.com/go/openvms/doc
/ OpenVMS documentation set].
1
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Carrot2 - Clustering algorithms
1
Carrot² offers two specialized document
clustering
algorithms[http://project.carrot2.org/algorit
hms.html Carrotsup2; clustering
algorithms] that place emphasis on the
quality of cluster labels:
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Carrot2 - Clustering algorithms
* Lingo:Stanisław Osiński, Dawid
Weiss: A Concept-Driven Algorithm for
Clustering Search Results. IEEE
Intelligent Systems, May/June, 3 (vol.
20), 2005, pp. 48ndash;54. a clustering
algorithm based on the Singular value
decomposition
1
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Carrot2 - Clustering algorithms
* STC:Oren Zamir, Oren Etzioni: Web
Document Clustering: A Feasibility
Demonstration, Proceedings of the 21st
annual international ACM SIGIR
conference on Research and development
in information retrieval (1998), pp.
46ndash;54 Suffix tree|Suffix Tree
Clustering
1
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Chemometrics - Classification, pattern recognition, clustering
Supervised multivariate classification
techniques are closely related to
multivariate calibration techniques in that a
calibration or training set is used to
develop a mathematical model capable of
classifying future samples
1
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Chemometrics - Classification, pattern recognition, clustering
Unsupervised classification (also
termed cluster analysis) is also
commonly used to discover patterns
in complex data sets, and again many
of the core techniques used in
chemometrics are common to other
fields such as machine learning and
statistical learning.
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Rank Order Clustering
In operations management and
industrial engineering, 'production
flow analysis' refers to methods which
share the following characteristics:
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Rank Order Clustering
3.Generating a binary productmachines matrix (1 if a given product
requires processing in a given
machine, 0 otherwise)
1
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Rank Order Clustering
1
Methods differ on how they group
together machines with products.
These play an important role in
designing manufacturing cells.
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Rank Order Clustering - Similarity Coefficients
1
Given a binary product-machines n-by-m
matrix, the algorithm proceedsAdapted
from MCauley, Machine grouping for
efficient production, Production Engineer
1972
http://ieeexplore.ieee.org/stamp/stamp.jsp
?arnumber=04913845 by the following
steps:
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Rank Order Clustering - Similarity Coefficients
1.Compute the similarity coefficient
s_=max(n_/n_,n_/n_) for all with n_ being
the number of products that need to be
processed on both machine i and machine
j
1
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Rank Order Clustering - Similarity Coefficients
1
2.Group together in cell k the tuple (i*,j*)
with higher similarity coefficient, with k
being the algorithm iteration index
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Rank Order Clustering - Similarity Coefficients
1
3.Remove row i* and column j* from the
original binary matrix and substitute for the
row and column of the cell k,
s_=max(s_,s_)
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Rank Order Clustering - Similarity Coefficients
Unless this procedure is stopped the
algorithm eventually will put all machines
in one single group.
1
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Medical Image Computing - Clustering
1
These techniques borrow ideas from highdimensional clustering and highdimensional pattern-regression to cluster a
given population into homogeneous subpopulations
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Chemical plant - Clustering of Commodity Chemical Plants
1
Chemical Plant particularly for Commodity
chemicals|commodity chemical and
petrochemical manufacture,are located in
relatively few manufacturing locations
around the world largely due to
infrastructural needs.This is less important
for Speciality chemicals|speciality or fine
chemical batch plants
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Operon - Operons versus clustering of prokaryotic genes
gene clustering helps a prokaryotic cell to
produce metabolic enzymes in a correct order
1
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K-medians clustering
1
This has the effect of minimizing error
over all clusters with respect to the 1Norm (mathematics)|norm distance
metric, as opposed to the square of the
2-Norm (mathematics)|norm distance
metric (which k-means clustering|kmeans does.)
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K-medians clustering
1
This relates directly to the 'k-median
problem' which is the problem of
finding k centers such that the
clusters formed by them are the most
compact. Formally, given a set of data
points x, the k centers ci are to be
chosen so as to minimize the sum of
the distances from each x to the
nearestci.
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K-medians clustering
The criterion function formulated in this
way is sometimes a better criterion than
that used in the k-means clustering|kmeans clustering algorithm, in which the
sum of the squared distances is used. The
sum of distances is widely used in
applications such as facility location.
1
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K-medians clustering
1
The proposed algorithm uses Lloydstyle iteration which alternates
between an expectation (E) and
maximization (M) step, making this
an Expectation–maximization
algorithm. In the E step, all objects
are assigned to their nearest median.
In the M step, the medians are
recomputed by using the median in
each single dimension.
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K-medians clustering - Medians and medoids
1
The median is computed in each single
dimension in the Manhattan
distance|Manhattan-distance formulation
of the k-medians problem, so the
individual attributes will come from the
dataset
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K-medians clustering - Medians and medoids
1
This algorithm is often confused with the kmedoids|k-medoids algorithm. However, a
medoid has to be an actual instance from the
dataset, while for the multivariate Manhattandistance median this only holds for single
attribute values. The actual median can thus
be a combination of multiple instances. For
example, given the vectors (0,1), (1,0) and
(2,2), the Manhattan-distance median is (1,1),
which does not exist in the original data, and
thus cannot be a medoid.
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K-medians clustering - Software
* ELKI includes
various k-means
variants, including
k-medians.
1
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Metallic bond - Localization and clustering: from bonding to bonds
1
The metallic bonding in complicated
compounds does not necessarily
involve all constituent elements
equally. It is quite possible to have an
element or more that do not partake at
all. One could picture the conduction
electrons flowing around them like a
river around an island or a big rock. It
is possible to observe which elements
do partake, e.g., by looking at the core
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Metallic bond - Localization and clustering: from bonding to bonds
1
Some intermetallic
materials e.g
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Metallic bond - Localization and clustering: from bonding to bonds
As these phenomena involve the
movement of the atoms towards or
away from each other, they can be
interpreted as the coupling between
the electronic and the vibrational
states (i.e
1
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Subspace clustering
Subspace clustering is the task of
detecting all clusters in all subspaces. This
means that a point might be a member of
multiple clusters, each existing in a
different subspace. Subspaces can either
be axis-parallel or affine. The term is often
used synonymous with general clustering
in high-dimensional data.
1
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Subspace clustering
1
The image on the right shows a mere twodimensional space where a number of
clusters can be identified. In the onedimensional subspaces, the clusters c_a
(in subspace \) and c_b, c_c, c_d (in
subspace \) can be found. c_c cannot be
considered a cluster in a two-dimensional
(sub-)space, since it is too sparsely
distributed in the x axis. In two
dimensions, the two clusters c_ and c_
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Subspace clustering
1
Hence, subspace clustering algorithm
utilize some kind of heuristic to
remain computationally feasible, at
the risk of producing inferior results
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Subspace clustering - Problems
According to , four problems need to be
overcome for clustering in high-dimensional
data:
1
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Subspace clustering - Problems
1
* Multiple dimensions are hard to think in,
impossible to visualize, and, due to the
exponential growth of the number of
possible values with each dimension,
complete enumeration of all subspaces
becomes intractable with increasing
dimensionality. This problem is known as
the curse of dimensionality.
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Subspace clustering - Problems
1
* The concept of distance becomes less
precise as the number of dimensions
grows, since the distance between any
two points in a given dataset converges.
The discrimination of the nearest and
farthest point in particular becomes
meaningless:
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Subspace clustering - Problems
* A cluster is intended to group
objects that are related, based on
observations of their attribute's values
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Subspace clustering - Problems
1
* Given a large number of attributes,
it is likely that some attributes are
correlated. Hence, clusters might
exist in arbitrarily oriented affine
subspaces.
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Subspace clustering - Problems
Recent research by indicates that the
discrimination problems only occur when
there is a high number of irrelevant
dimensions, and that shared-nearestneighbor approaches can improve results.
1
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Subspace clustering - Approaches
1
Approaches towards clustering in axisparallel or arbitrarily oriented affine
subspaces differ in how they interpret
the overall goal, which is finding
clusters in data with high
dimensionality. This distinction is
proposed in . An overall different
approach is to find clusters based on
pattern in the data matrix, often
referred to as biclustering, which is a
technique frequently utilized in
bioinformatics.
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Subspace clustering - Projected clustering
1
Projected clustering seeks to assign each
point to a unique cluster, but clusters may
exist in different subspaces. The general
approach is to use a special distance
function together with a regular cluster
analysis|clustering algorithm.
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Subspace clustering - Projected clustering
For example, the PreDeCon algorithm
checks which attributes seem to support a
clustering for each point, and adjusts the
distance function such that dimensions
with low variance are amplified in the
distance function . In the figure above, the
cluster c_c might be found using DBSCAN
with a distance function that places less
emphasis on the x-axis and thus
exaggerates the low difference in the y1
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Subspace clustering - Projected clustering
1
PROCLUS uses a similar approach with
a k-medoid clustering . Initial medoids
are guessed, and for each medoid the
subspace spanned by attributes with
low variance is determined. Points are
assigned to the medoid closest,
considering only the subspace of that
medoid in determining the distance.
The algorithm then proceeds as the
regular Partitioning Around
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Subspace clustering - Projected clustering
If the distance function weights
attributes differently, but never with 0
(and hence never drops irrelevant
attributes), the algorithm is called a
soft-projected clustering algorithm.
1
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Subspace clustering - Hybrid approaches
Not all algorithms try to either find a
unique cluster assignment for each point
or all clusters in all subspaces; many settle
for a result in between, where a number of
possibly overlapping, but not necessarily
exhaustive set of clusters are found. An
example is FIRES, which is from its basic
approach a subspace clustering algorithm,
but uses a heuristic too aggressive to
credibly produce all subspace clusters .
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Subspace clustering - Correlation clustering
Another type of subspaces is
considered in Correlation
clustering|Correlation clustering
(Data Mining).
1
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Data clustering
1
'Cluster analysis' or 'clustering' is the task
of grouping a set of objects in such a way
that objects in the same group (called a
'cluster') are more similar (in some sense
or another) to each other than to those in
other groups (clusters). It is a main task of
exploratory data mining, and a common
technique for statistics|statistical data
analysis, used in many fields, including
machine learning, pattern recognition,
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Data clustering
1
The appropriate clustering algorithm
and parameter settings (including
values such as the Metric
(mathematics)|distance function to
use, a density threshold or the number
of expected clusters) depend on the
individual data set and intended use
of the results
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Data clustering
1
Besides the term clustering, there are
a number of terms with similar
meanings, including automatic
Statistical
classification|classification,
numerical taxonomy, botryology
(from Greek βότρυς grape) and
typological analysis
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Data clustering
Cluster analysis was originated in
anthropology by Driver and Kroeber in
1932 and introduced to psychology by
Zubin in 1938 and Robert Tryon in 1939
and famously used by Cattell beginning in
1943 for trait theory classification in
personality psychology.
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Data clustering - Definition
1
According to Vladimir Estivill-Castro,
the notion of a cluster cannot be
precisely defined, which is one of the
reasons why there are so many
clustering algorithms
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Data clustering - Definition
1
* Connectivity models: for example
hierarchical clustering builds models
based on distance connectivity.
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Data clustering - Definition
* Centroid models: for example the kmeans algorithm represents each cluster
by a single mean vector.
1
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Data clustering - Definition
* Distribution models: clusters are
modeled using statistical
distributions, such as multivariate
normal distributions used by the
Expectation-maximization algorithm.
1
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Data clustering - Definition
* Density models: for example
DBSCAN and OPTICS defines clusters
as connected dense regions in the data
space.
1
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Data clustering - Definition
1
* Subspace models: in Biclustering
(also known as Co-clustering or twomode-clustering), clusters are
modeled with both cluster members
and relevant attributes.
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Data clustering - Definition
1
* Group models: some algorithms do not
provide a refined model for their results
and just provide the grouping information.
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Data clustering - Definition
1
* Graph-based models: a Clique (graph
theory)|clique, i.e., a subset of nodes in a
Graph (mathematics)|graph such that
every two nodes in the subset are
connected by an edge can be considered
as a prototypical form of cluster.
Relaxations of the complete connectivity
requirement (a fraction of the edges can
be missing) are known as quasi-cliques.
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Data clustering - Definition
1
A clustering is essentially a set of such
clusters, usually containing all objects in
the data set. Additionally, it may specify
the relationship of the clusters to each
other, for example a hierarchy of clusters
embedded in each other. Clusterings can
be roughly distinguished as:
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Data clustering - Definition
1
* hard clustering: each object
belongs to a cluster or not
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Data clustering - Definition
* soft clustering (also: fuzzy
clustering): each object belongs to
each cluster to a certain degree (e.g.a
likelihood of belonging to the cluster)
1
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Data clustering - Definition
1
* strict partitioning clustering: here each object
belongs to exactly one cluster
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Data clustering - Definition
* strict partitioning clustering with
outliers: objects can also belong to no
cluster, and are considered Anomaly
detection|outliers.
1
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Data clustering - Definition
* overlapping clustering (also:
alternative clustering, multi-view
clustering): while usually a hard
clustering, objects may belong to
more than one cluster.
1
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Data clustering - Definition
1
* hierarchical clustering: objects that belong to a
child cluster also belong to the parent cluster
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Data clustering - Definition
* subspace clustering: while an
overlapping clustering, within a
uniquely defined subspace, clusters
are not expected to overlap.
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Data clustering - Algorithms
1
The most appropriate clustering algorithm
for a particular problem often needs to be
chosen experimentally, unless there is a
mathematical reason to prefer one cluster
model over another
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Data clustering - Connectivity based clustering (hierarchical clustering)
At different distances, different
clusters will form, which can be
represented using a dendrogram,
which explains where the common
name hierarchical clustering comes
from: these algorithms do not provide
a single partitioning of the data set,
but instead provide an extensive
hierarchy of clusters that merge with
each other at certain distances
1
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Data clustering - Connectivity based clustering (hierarchical clustering)
1
They did however provide inspiration for many
later methods such as density based
clustering.
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Data clustering - Centroid-based clustering
1
In centroid-based clustering, clusters are
represented by a central vector, which
may not necessarily be a member of the
data set. When the number of clusters is
fixed to k, k-means clustering|k-means
clustering gives a formal definition as an
optimization problem: find the k cluster
centers and assign the objects to the
nearest cluster center, such that the
squared distances from the cluster are
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Data clustering - Centroid-based clustering
1
Variations of k-means often include such
optimizations as choosing the best of
multiple runs, but also restricting the
centroids to members of the data set (kmedoids), choosing medians (k-medians
clustering), choosing the initial centers
less randomly (K-means++) or allowing a
fuzzy cluster assignment (Fuzzy
clustering|Fuzzy c-means).
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Data clustering - Centroid-based clustering
1
Most k-means-type algorithms require the
Determining the number of clusters in a
data set|number of clusters - k - to be
specified in advance, which is considered
to be one of the biggest drawbacks of
these algorithms
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Data clustering - Centroid-based clustering
K-means has a number of interesting
theoretical properties. On the one hand, it
partitions the data space into a structure
known as a Voronoi diagram. On the other
hand, it is conceptually close to nearest
neighbor statistical
classification|classification, and as such is
popular in machine learning. Third, it can be
seen as a variation of model based
classification, and Lloyd's algorithm as a
variation of the Expectation-maximization
algorithm for this model discussed below.
1
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Data clustering - Distribution-based clustering
1
The clustering model most closely
related to statistics is based on
Probability distribution|distribution
models. Clusters can then easily be
defined as objects belonging most
likely to the same distribution. A
convenient property of this approach
is that this closely resembles the way
artificial data sets are generated: by
sampling random objects from a
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Data clustering - Density-based clustering
DeLi-Clu, Density-Link-Clustering
combines ideas from single-linkage
clustering and OPTICS, eliminating the
\varepsilon parameter entirely and offering
performance improvements over OPTICS
by using an R-tree index.
1
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Data clustering - Density-based clustering
1
On a data set consisting of mixtures of
Gaussians, these algorithms are
nearly always outperformed by
methods such as EM clustering that
are able to precisely model this kind
of data.
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Data clustering - Density-based clustering
Mean-shift is a clustering approach where
each object is moved to the densest area in
its vicinity, based on kernel density
estimation. Eventually, objects converge to
local maxima of density. Similar to k-means
clustering, these density attractors can serve
as representatives for the data set, but meanshift can detect arbitrary-shaped clusters
similar to DBSCAN. Due to the expensive
iterative procedure and density estimation,
mean-shift is usually slower than DBSCAN or
k-Means.
1
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Data clustering - Recent developments
1
Various other approaches to clustering have
been tried such as seed based clustering.
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Data clustering - Recent developments
Density-Connected
Subspace Clustering for
High-Dimensional Data
1
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Data clustering - Recent developments
1
Ideas from density-based clustering
methods (in particular the
DBSCAN/OPTICS family of
algorithms) have been adopted to
subspace clustering (HiSC,
hierarchical subspace clustering and
DiSH) and correlation clustering
(HiCO, hierarchical correlation
clustering, 4C using correlation
connectivity and ERiC exploring
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Data clustering - Recent developments
1
Several different clustering systems based on
mutual information have been proposed. One
is Marina Meilă's variation of information
metric; another provides hierarchical
clustering. Using genetic algorithms, a wide
range of different fit-functions can be
optimized, including mutual information. Also
message passing algorithms, a recent
development in Computer Science and
Statistical Physics, has led to the creation of
new types of clustering algorithms.
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Data clustering - Other methods
1
* Basic sequential algorithmic
scheme (BSAS)
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Data clustering - Applications
1
;: cluster analysis is used to describe and
to make spatial and temporal comparisons
of communities (assemblages) of
organisms in heterogeneous
environments; it is also used in
Systematics|plant systematics to generate
artificial Phylogeny|phylogenies or clusters
of organisms (individuals) at the species,
genus or higher level that share a number
of attributes
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Data clustering - Applications
;: clustering is used to build groups of
genes with related expression patterns
(also known as coexpressed genes). Often
such groups contain functionally related
proteins, such as enzymes for a specific
metabolic pathway|pathway, or genes that
are co-regulated. High throughput
experiments using expressed sequence
tags (ESTs) or DNA microarrays can be a
powerful tool for genome annotation, a
1
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Data clustering - Applications
1
;: clustering is used to group homologous
sequences into list of gene families|gene
families. This is a very important concept
in bioinformatics, and evolutionary biology
in general. See evolution by gene
duplication.
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Data clustering - Applications
;;High-throughput
genotype|genotyping
platforms
1
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Data clustering - Applications
;: clustering algorithms are
used to automatically assign
genotypes.
1
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Data clustering - Applications
1
;:The similarity of genetic data is used in
clustering to infer population structures.
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Data clustering - Applications
1
;: On PET scans, cluster analysis can be
used to differentiate between different
types of tissue (biology)|tissue and blood
in a three-dimensional image
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Data clustering - Applications
;: Clustering can be used to divide a
fluence map into distinct regions for
conversion into deliverable fields in MLCbased Radiation Therapy.
1
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Data clustering - Applications
;: Cluster analysis is widely used in
market research when working with
multivariate data from Statistical
survey|surveys and test panels.
Market researchers use cluster
analysis to partition the general
population of consumers into market
segments and to better understand the
relationships between different
groups of consumers/potential
1
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Data clustering - Applications
1
;: Clustering can be used to group all the
shopping items available on the web into a
set of unique products. For example, all
the items on eBay can be grouped into
unique products. (eBay doesn't have the
concept of a Stock-keeping unit|SKU)
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Data clustering - Applications
;: In the study of social networks,
clustering may be used to recognize
communities within large groups of
people.
1
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Data clustering - Applications
1
;;Search result grouping
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Data clustering - Applications
1
;: In the process of intelligent grouping of
the files and websites, clustering may be
used to create a more relevant set of
search results compared to normal search
engines like Google. There are currently a
number of web based clustering tools such
as Clusty.
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Data clustering - Applications
1
;: Flickr's map of photos and other map
sites use clustering to reduce the
number of markers on a map. This
makes it both faster and reduces the
amount of visual clutter.
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Data clustering - Applications
1
;;Software evolution
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Data clustering - Applications
1
;: Clustering is useful in software evolution
as it helps to reduce legacy properties in
code by reforming functionality that has
become dispersed. It is a form of
restructuring and hence is a way of directly
preventative maintenance.
https://store.theartofservice.com/the-clustering-toolkit.html
Data clustering - Applications
1
;: Clustering can be used to divide a
Digital data|digital image into distinct
regions for border detection or object
recognition.
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Data clustering - Applications
1
;;Evolutionary algorithms
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Data clustering - Applications
;: Clustering may be used to identify
different niches within the population of an
evolutionary algorithm so that reproductive
opportunity can be distributed more evenly
amongst the evolving species or
subspecies.
1
https://store.theartofservice.com/the-clustering-toolkit.html
Data clustering - Applications
1
;: Recommender systems are designed to
recommend new items based on a user's
tastes. They sometimes use clustering
algorithms to predict a user's preferences
based on the preferences of other users in
the user's cluster.
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Data clustering - Applications
;: Clustering is often utilized to locate and
characterize extrema in the target distribution.
1
https://store.theartofservice.com/the-clustering-toolkit.html
Data clustering - Applications
;: Cluster analysis can be used to
identify areas where there are greater
incidences of particular types of
crime. By identifying these distinct
areas or hot spots where a similar
crime has happened over a period of
time, it is possible to manage law
enforcement resources more
effectively.
1
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Data clustering - Applications
1
;;Educational data
mining
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Data clustering - Applications
1
;:Cluster analysis is for example used
to identify groups of schools or
students with similar properties.
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Data clustering - Applications
1
;: From poll data, projects such as those
undertaken by the Pew Research Center
use cluster analysis to discern typologies
of opinions, habits, and demographics that
may be useful in politics and marketing.
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Data clustering - Applications
;: Clustering algorithms are used for
robotic situational awareness to track
objects and detect outliers in sensor
data.Bewley A. et al. Real-time volume
estimation of a dragline payload. IEEE
International Conference on Robotics and
Automation,2011: 1571-1576.
1
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Data clustering - Applications
1
;: To find structural similarity, etc., for
example, 3000 chemical compounds
were clustered in the space of 90
topological index|topological
indices.Basak S.C., Magnuson V.R.,
Niemi C.J., Regal R.R. Determining
Structural Similarity of Chemicals
Using Graph Theoretic Indices. Discr.
Appl. Math., '19', 1988: 17-44.
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Data clustering - Applications
1
;: To find weather regimes or preferred
sea level pressure atmospheric
patterns.Huth R. et al. Classifications
of Atmospheric Circulation Patterns:
Recent Advances and Applications.
Ann. N.Y. Acad. Sci., '1146', 2008: 105152
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Data clustering - Applications
;: Cluster analysis is used to
reconstruct missing bottom hole core
data or missing log curves in order to
evaluate reservoir properties.
1
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Data clustering - Applications
;: The clustering of
chemical properties in
different sample
locations.
1
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Relevance (information retrieval) - Clustering and relevance
The cluster
hypothesis, proposed
by C
1
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Relevance (information retrieval) - Clustering and relevance
1
* cluster-based information retrievalW.
B. Croft, “A model of cluster searching
based on classification,” Information
Systems, vol. 5, pp. 189–195, 1980.A.
Griffiths, H. C. Luckhurst, and P. Willett,
“Using interdocument similarity
information in document retrieval
systems,” Journal of the American
Society for Information Science, vol. 37,
no. 1, pp. 3–11, 1986.
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Relevance (information retrieval) - Clustering and relevance
1
* cluster-based document expansion such
as latent semantic analysis or its language
modeling equivalents.X. Liu and W. B.
Croft, “Cluster-based retrieval using
language models,” in SIGIR ’04:
Proceedings of the 27th annual
international conference on Research and
development in information retrieval, (New
York, NY, USA), pp. 186–193, ACM Press,
2004. It is important to ensure that
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Relevance (information retrieval) - Clustering and relevance
1
A second interpretation,
most notably advanced
by Ellen Voorhees,E
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Relevance (information retrieval) - Clustering and relevance
* spreading activationS. Preece, A
spreading activation network model for
information retrieval. PhD thesis,
University of Illinois, Urbana-Champaign,
1981. and relevance propagationT. Qin, T.Y. Liu, X.-D. Zhang, Z. Chen, and W.-Y.
Ma, “A study of relevance propagation for
web search,” in SIGIR ’05: Proceedings of
the 28th annual international ACM SIGIR
conference on Research and development
1
https://store.theartofservice.com/the-clustering-toolkit.html
Relevance (information retrieval) - Clustering and relevance
1
* local document expansionA. Singhal
and F. Pereira, “Document expansion
for speech retrieval,” in SIGIR ’99:
Proceedings of the 22nd annual
international ACM SIGIR conference
on Research and development in
information retrieval, (New York, NY,
USA), pp. 34–41, ACM Press, 1999.
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Relevance (information retrieval) - Clustering and relevance
1
* score regularizationF. Diaz, “Regularizing
query-based retrieval scores,” Information
Retrieval, vol. 10, pp. 531–562, December
2007.
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Relevance (information retrieval) - Clustering and relevance
1
Local methods require an accurate and appropriate
document similarity measure.
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Clustering
1
'Clustering' can refer to the
following:
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Clustering
1
* Clustering (demographics), the
gathering of various populations
based on factors such as ethnicity,
economics or religion.
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Clustering
1
* The formation of clusters of linked nodes in a
network, measured by the clustering
coefficient.
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Clustering
1
* A result of cluster analysis.
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Clustering
* An algorithm for
cluster analysis, a
method for statistical
data analysis.
1
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Clustering
* Cluster (computing), the technique of
linking many computers together to act like
a single computer .
1
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Clustering
1
* Data cluster, an allocation of contiguous storage in
databases and file systems.
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Image segmentation - Clustering methods
1
The K-means algorithm is an iterative
technique that is used to Cluster
analysis|partition an image into K
clusters.Barghout, Lauren, and Jacob
Sheynin. Real-world scene perception
and perceptual organization: Lessons
from Computer Vision. Journal of
Vision 13.9 (2013): 709-709. The basic
algorithm is
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Hierarchical network model - Clustering coefficient
1
In contrast to the other scale-free models
(Erdős–Rényi model|Erdős–Rényi,
Barabási–Albert, Watts–Strogatz) where
the clustering coefficient is independent of
the degree of a specific node, in
hierarchical networks the clustering
coefficient can be expressed as a function
of the degree in the following way:
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Hierarchical network model - Clustering coefficient
1
It has been analytically shown that in
deterministic scale-free networks the
exponent β takes the value of 1.
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Clustering (demographics)
In demographics, 'clustering' is the
gathering of various populations based
on ethnicity, economics, or religion.
1
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Clustering (demographics)
1
Cushing, The Big Sort: Why the Clustering
of Like-Minded America Is Tearing Us
Apart (Houghton Mifflin Harcourt, 2009)
http://books.google.com/books?id=nLNVd
8ZkW0UCsource=gbs_navlinks_s ISBN 0547-23772-3, ISBN 978-0-547-23772-5
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Learning representation - Clustering as feature learning
K-means clustering can be used for
feature learning, by clustering an
unlabeled set to produce k centroids, then
using these centroids to produce k
additional features for a subsequent
supervised learning task
1
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Learning representation - Clustering as feature learning
1
K-means has also been shown to improve
performance in the domain of Natural
language processing|NLP, specifically for
named-entity recognition; there, it
competes with Brown clustering, as well
as with distributed word representations
(also known as neural word embeddings).
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Composer - Clustering
1
Famous composers have a tendency to
cluster in certain cities throughout history.
Based on over 12,000 prominent
composers listed in Grove Music Online
and using word count measurement
techniques the most important cities for
classical music can be quantitatively
identified.
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Composer - Clustering
Paris has been the main hub for
classical music of all times. It was
ranked fifth in the 15th and 16th
centuries but first in the 17th to 20th
centuries inclusive. London was the
second most meaningful city: eight in
the 15th century, seventh in
1
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Composer - Clustering
the 16th, fifth in the 17th,
second in the 18th and 19th
centuries, and
1
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Composer - Clustering
fourth in the 20th century. Rome
topped the rankings in the 15th
century, dropped to second in the 16th
1
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Composer - Clustering
1
and 17th centuries, eight in the 18th
century, ninth in the 19th century but
back at sixth in the 20th century.
Berlin appears in the top ten ranking
only in the 18th century, and was
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Composer - Clustering
1
ranked third most important city in both
the 19th and 20th centuries. New York
entered the rankings in the 19th
century (at fifth place) and stood
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Composer - Clustering
1
at second rank in the 20th
century.
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Human genetic clustering
1
'Human genetic clustering' analysis
uses mathematical cluster analysis of
the degree of similarity of genetic
data between individuals and groups
to infer population structures and
assign individuals to groups that often
correspond with their self-identified
geographical ancestry
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Human genetic clustering - Clusters by Noah Rosenberg|Rosenberg et al.
(2006)
1
In 2004, Lynn Jorde and Steven Wooding
argued that Analysis of many loci now
yields reasonably accurate estimates of
genetic similarity among individuals, rather
than populations. Clustering of individuals
is correlated with geographic origin or
ancestry.Lynn B Jorde Stephen P
Wooding, 2004, Genetic variation,
classification and 'race' in Nature Genetics
36, S28ndash;S33
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Human genetic clustering - Clusters by Noah Rosenberg|Rosenberg et al.
(2006)
1
Studies such as those by Risch and
Noah Rosenberg|Rosenberg use a
computer program called
STRUCTURE to find human
populations (gene clusters)
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Human genetic clustering - Clusters by Noah Rosenberg|Rosenberg et al. (2006)
1
Nevertheless the
Rosenberg et al
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Human genetic clustering - Clusters by Noah Rosenberg|Rosenberg et al.
(2006)
1
Additionally, Edwards (2003) claims in
his essay Lewontin's Fallacy that: It is
not true, as Nature claimed, that 'two
random individuals from any one
group are almost as different as any
two random individuals from the
entire world' and Risch et al
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Human genetic clustering - Clusters by Noah Rosenberg|Rosenberg et al.
(2006)
1
Whereas Edwards claims that it is not
true that the differences between
individuals from different
geographical regions represent only a
small proportion of the variation
within the human population (he
claims that within group differences
between individuals are not almost as
large as between group differences)
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Human genetic clustering - Clusters by Noah Rosenberg|Rosenberg et al. (2006)
1
A study by the The HUGO Pan-Asian
SNP Consortium in 2009 using the
similar principal components
analysis found that East Asian and
South-East Asian populations
clustered together, and suggested a
common origin for these populations
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Human genetic clustering - Criticism
1
The Rosenberg study has been
criticised on several grounds.
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Human genetic clustering - Criticism
1
The existence of allelic clines and the
observation that the bulk of human
variation is continuously distributed,
has led some scientists to conclude
that any categorization schema
attempting to partition that variation
meaningfully will necessarily create
artificial truncations
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Human genetic clustering - Criticism
1
Firstly they maintain that their
clustering analysis is robust
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Human genetic clustering - Criticism
1
Risch et al
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Human genetic clustering - Criticism
In agreement with the observation of
Bamshad et al. (2004), Witherspoon et al.
(2007) have shown that many more than
326 or 377 microsatellite loci are required
in order to show that individuals are
always more similar to individuals in their
own population group than to individuals in
different population groups, even for three
distinct populations.
1
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Human genetic clustering - Criticism
1
Witherspoon et al
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Human genetic clustering - Criticism
Clustering does not particularly
correspond to continental divisions
1
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Human genetic clustering - Criticism
The law professor, Dorothy Roberts
asserts that “the study actually showed
that there are many ways to slice the
expansive range of human genetic
variation. In a 2005 paper, Rosenberg and
his team acknowledged that findings of a
study on human population structure are
highly influenced by the way the study is
designed.
1
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Human genetic clustering - Criticism
1
They reported that the number of loci,
the sample size, the geographic
dispersion of the samples and
assumptions about allele-frequency
correlation all have an effect on the
outcome of the study
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Human genetic clustering - Controversy of genetic clustering and associations with “race”
In the late 1990s Harvard evolutionary
geneticist Richard Lewontin stated that “no
justification can be offered for continuing
the biological concept of Race (human
classification)|race. (...) Genetic data
shows that no matter how racial groups
are defined, two people from the same
racial group are about as different from
each other as two people from any two
different racial groups.
1
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Human genetic clustering - Controversy of genetic clustering and associations
with “race”
Lewontin’s statement came under
attack when new genomic technologies
permitted the analysis of gene clusters.
In 2003, British statistician and
evolutionary biologist A. W. F. Edwards
Human_Genetic_Diversity:_Lewontin's
_Fallacy|faulted Lewontin’s statement
for basing his conclusions on simple
comparison of genes and rather on a
more complex structure of gene
1
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Human genetic clustering - Controversy of genetic clustering and associations
with “race”
1
According to Roberts, “Edwards did
not refute Lewontin’s claim: that there
is more genetic variation within
populations than between them,
especially when it comes to races
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Human genetic clustering - Controversy of genetic clustering and associations
with “race”
1
Genetic clustering was also criticized by
Penn State anthropologists Kenneth
Weiss and Brian Lambert
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Human genetic clustering - Controversy of genetic clustering and associations
with “race”
1
In 2006, Lewontin wrote that any genetic
study requires some priori concept of race
or ethnicity in order to package human
genetic diversity into defined, limited
number of biological groupings. Informed
by geneticist, zoologists have long
discarded the concept of race. Defined on
varying criteria, in the same species widely
varying number of races could be
distinguished. Lewontin notes that genetic
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Human genetic clustering - Controversy of genetic clustering and associations with “race”
Another genetic clustering study used
three sub-Saharan population groups to
represent Africa; Chinese, Japanese, and
Cambodian samples for East Asia;
Northern European and Northern Italian
samples to represent “Caucasians”
1
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Human genetic clustering - Controversy of genetic clustering and associations
with “race”
The model of Big Few fails when including
overlooked geographical regions such as India
1
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Human genetic clustering - Controversy of genetic clustering and associations
with “race”
1
Cavalli-Sforza asserts that classifying
clusters as races would be a “futile
exercise” because “every level of
clustering would determine a
different population and there is no
biological reason to prefer a
particular one.” Bamshad, in 2004
paper published in Nature, asserts
that a more accurate study of human
genetic variation would use an
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Human genetic clustering - Controversy of genetic clustering and associations with “race”
1
When one samples continental groups the
clusters become continental, if one had
chosen other sampling patterns the
clustering would be different
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Human genetic clustering - Commercial ancestry testing and individual
ancestry
1
Limitations of genetic clustering are
intensified when inferred population
structure is applied to individual
ancestry
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Human genetic clustering - Commercial ancestry testing and individual ancestry
Many commercial companies use data
from HapMap’s initial phrase, where
population samples were collected from
four ethnic groups in the world: Han
Chinese, Japanese, Yoruba Nigerian, and
Utah residents of Northern European
ancestry
1
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Human genetic clustering - Geographical and continental groupings
1
Roberts argues against the use of broad
geographical or continental groupings:
“molecular geneticists routinely refer to
African ancestry as if everyone on the
continent is more similar to each other
than they are to people of other
continents, who may be closer both
geographically and genetically
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Human genetic clustering - Usage in scientific journals
1
Some scientific journals have addressed
previous methodological errors by
requiring more rigorous scrutiny of
population variables
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Centre for Innovation and Structural Change - Industry Clustering
1
The goals of this area are to develop a better
understanding of the characteristics and
performance of industry clusters as a newly
identified source of competitive advantage in
the global economy, to construct primary data
sources for research in Ireland using
longitudinal survey and case study
approaches, with the potential for
international comparative analysis, and to
build expertise in collaboration with
international researchers which is
incorporated into undergraduate and
postgraduate teaching programmes.
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Triadic closure - Clustering coefficient
1
One measure for the presence of triadic closure is
clustering coefficient, as follows:
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Triadic closure - Clustering coefficient
Let G = (V,E) be an undirected simple
graph (i.e., a graph having no self-loops or
multiple edges) with V the set of vertices
and E the set of edges. Also, let N = |V|
and M = |E| denote the number of vertices
and edges in G, respectively, and let d_i
be the degree (graph theory)|degree of
vertex i.
1
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Triadic closure - Clustering coefficient
1
Then we can define a triangle among the
triple of vertices i, j, and k to be a set with
the following three edges:
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Triadic closure - Clustering coefficient
1
Now, for a vertex i with d_i \ge 2, the
clustering coefficient c(i) of vertex i is
the fraction of triples for vertex i that
are closed, and can be measured as
\frac. Thus, the clustering coefficient
C(G) of graph G is given by C(G) =
\frac \sum_ c(i), where N_2 is the
number of nodes with degree at least
2.
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Sepp Hochreiter - Biclustering
Sepp Hochreiter developed Factor
Analysis for Bicluster Acquisition (FABIA)
for biclustering that is simultaneously
cluster analysis|clustering rows and
columns of a matrix (mathematics)|matrix
1
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Sepp Hochreiter - Biclustering
1
Bayesian framework. FABIA supplies the information
theory|information content
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Sepp Hochreiter - Biclustering
1
of each bicluster to separate
spurious biclusters from true
biclusters.
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Jem The Bee - Clustering
JEM clustering
[http://www.techterms.com/definition/cluste
r cluster definition] is based on
Hazelcast.[http://www.hazelcast.com
Hazelcast web site] Each Computer
cluster|cluster member (called Node
(computer science)|node) has the same
rights and responsibilities of the others
(with the exception of the oldest member,
that we are going to see in details): this is
1
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Jem The Bee - Clustering
1
When a node starts up, it checks to see if
there's already a cluster in the Computer
network|network. There are two ways to
find this out:
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Jem The Bee - Clustering
1
* Multicast discovery: if multicast discovery
is enabled (this is the default), the node
will send a join request in the form of a
multicast datagram packet.
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Jem The Bee - Clustering
* Unicast discovery: if multicast
discovery is disabled and TCP/IP join is
enabled, the node will try to connect to
the IP address|IPs defined. If it
successfully connects to (at least) one
node, then it will send a join request
through the TCP/IP connection.
1
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Jem The Bee - Clustering
If no cluster (computing)|cluster is
found, the node will be the first
member of the cluster. If multicast is
enabled, it starts a multicast Observer
pattern|listener so that it can respond
to incoming join requests. Otherwise,
it will listen for join request coming
via TCP/IP.
1
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Jem The Bee - Clustering
1
If there is an existing cluster already,
then the 'oldest member' in the cluster
will receive the join request and checks
if the request is for the right group. If
so, the oldest member in the cluster
will start the join process.
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Jem The Bee - Clustering
* tell members to Synchronization
(computer science)|synchronize data in
order to balance the data load
1
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Jem The Bee - Clustering
1
Every member in the cluster has the same
member list in the same order. First
member is the oldest member so if the
oldest member dies, second member in
the list becomes the first member in the list
and the new oldest member. The oldest
member is considered as the JEM cluster
coordinator: it will execute those actions
that must be executed by a single member
(i.e. locks releasing due to a member
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Jem The Bee - Clustering
Aside the normal nodes, there's
another kind of nodes in the cluster,
called supernodes. A supernode is a
lite member of Hazelcast.
1
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Jem The Bee - Clustering
Supernodes are members with no
Computer data storage|storage: they
join the cluster as lite members, not as
data partition (no data on these nodes),
and get super fast access to the cluster
just like any regular member does.
These nodes are used for the Web
Application (running on Apache
Tomcat,[http://tomcat.apache.org/
apache tomcat] as well as on any other
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Concept Mining - Clustering documents by topic
Standard numeric clustering techniques
may be used in concept space as
described above to locate and index
documents by the inferred topic. These
are numerically far more efficient than their
text mining cousins, and tend to behave
more intuitively, in that they map better to
the similarity measures a human would
generate.
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Consensus clustering
'Clustering' is the assignment of objects
into groups (called clusters) so that objects
from the same cluster are more similar to
each other than objects from different
clusters. Often similarity is assessed
according to a metric
(mathematics)|distance measure.
Clustering is a common technique for
statistics|statistical data analysis, which is
used in many fields, including machine
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Consensus clustering
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Consensus clustering for unsupervised
learning is analogous to ensemble
learning in supervised learning.
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Consensus clustering - Issues with existing clustering techniques
1
* Current clustering techniques do not address all
the requirements adequately.
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Consensus clustering - Issues with existing clustering techniques
1
* Dealing with large number of
dimensions and large number of
data items can be problematic
because of time complexity;
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Consensus clustering - Issues with existing clustering techniques
* Effectiveness of the method depends on the
definition of distance (for distance based clustering)
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Consensus clustering - Issues with existing clustering techniques
* If an obvious distance measure
doesn’t exist we must define it, which
is not always easy, especially in
multidimensional spaces.
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Consensus clustering - Issues with existing clustering techniques
1
* The result of the clustering algorithm
(that in many cases can be arbitrary
itself) can be interpreted in different
ways.
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Consensus clustering - Justification for using consensus clustering
1
An extremely important issue in cluster
analysis is the validation of the
clustering results, that is, how to gain
confidence about the significance of
the clusters provided by the clustering
technique (cluster numbers and cluster
assignments)
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Consensus clustering - Justification for using consensus clustering
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However, they lack the intuitive and visual
appeal of Hierarchical clustering
dendrograms, and the number of clusters
must be chosen a priori.
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Consensus clustering - Over-interpretation potential of consensus clustering
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Consensus clustering can be a powerful tool
for identifying clusters, but it needs to be
applied with caution. It has been shown that
consensus clustering is able to claim
apparent stability of chance partitioning of
null datasets drawn from a unimodal
distribution, and thus has the potential to lead
to over-interpretation of cluster stability in a
real study. If clusters are not well separated,
consensus clustering could lead one to
conclude apparent
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Consensus clustering - Over-interpretation potential of consensus clustering
To reduce the false positive potential in
clustering samples (observations),
Şenbabaoğlu et al recommends (1) doing
a formal test of cluster strength using
simulated unimodal data with the same
feature-space correlation structure as in
the empirical data, (2) not relying solely on
the consensus matrix heatmap to declare
the existence of clusters, or to estimate
optimal K, (3) applying the proportion of
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Consensus clustering - Over-interpretation potential of consensus clustering
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A low value of PAC indicates a flat
middle segment, and a low rate of
discordant assignments across
permuted clustering runs
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Consensus clustering - Over-interpretation potential of consensus clustering
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In simulated datasets with known
number of clusters, consensus
clustering+PAC has been shown to
perform better than several other
commonly used methods such as
consensus clustering+Δ(K), CLEST,
GAP, and silhouette width.
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Consensus clustering - Related work
1. 'Clustering ensemble (Strehl and
Ghosh)': They considered various
formulations for the problem, most of
which reduce the problem to a hypergraph partitioning problem. In one of
their formulations they considered the
same graph as in the correlation
clustering problem. The solution they
proposed is to compute the best kpartition of the graph, which does not
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Consensus clustering - Related work
2. 'Clustering aggregation (Fern and
Brodley)': They applied the clustering
aggregation idea to a collection of soft
clusterings they obtained by random
projections. They used an agglomerative
algorithm and did not penalize for merging
dissimilar nodes.
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Consensus clustering - Related work
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3. 'Fred and Jain': They proposed to
use a single linkage algorithm to
combine multiple runs of the k-means
algorithm.
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Consensus clustering - Related work
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4. 'Dana Cristofor and Dan Simovici':
They observed the connection
between clustering aggregation and
clustering of categorical data. They
proposed information theoretic
distance measures, and they propose
genetic algorithms for finding the best
aggregation solution.
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Consensus clustering - Related work
5. 'Topchy et al.': They defined
clustering aggregation as a maximum
likelihood estimation problem, and they
proposed an EM algorithm for finding
the consensus clustering.
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Consensus clustering - Related work
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Therefore, they proposed their work
as a new paradigm of clustering
rather than merely a new ensemble
clustering method.
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Consensus clustering - Hard ensemble clustering
This approach by Strehl and Ghosh
introduces the problem of combining
multiple partitionings of a set of objects
into a single consolidated clustering
without accessing the features or
algorithms that determined these
partitionings
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Consensus clustering - Efficient consensus functions
'1. Cluster-based
similarity partitioning
algorithm (CSPA)'
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Consensus clustering - Efficient consensus functions
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In CSPA the similarity between two datapoints is defined to be directly proportional
to number of constituent clusterings of the
ensemble in which they are clustered
together. The intuition is that the more
similar two data-points are the higher is
the chance that constituent clusterings will
place them in the same cluster. CSPA is
the simplest heuristic, but its
computational and storage complexity are
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Consensus clustering - Efficient consensus functions
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The HGPA algorithm takes a very
different approach to finding the
consensus clustering than the
previous method.
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Consensus clustering - Efficient consensus functions
The cluster ensemble problem is
formulated as partitioning the
hypergraph by cutting a minimal
number of hyperedges. They make use
of
[http://glaros.dtc.umn.edu/gkhome/
metis/hmetis/overview hMETIS]
which is a hypergraph partitioning
package system.
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Consensus clustering - Efficient consensus functions
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'3. Meta-clustering algorithm
(MCLA)'
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Consensus clustering - Efficient consensus functions
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The meta-cLustering algorithm
(MCLA) is based on clustering
clusters.
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Consensus clustering - Efficient consensus functions
First, it tries to solve the cluster
correspondence problem and then
uses voting to place data-points into
the final consensus clusters. The
cluster correspondence problem is
solved by grouping the clusters
identified in the individual clusterings
of the ensemble.
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Consensus clustering - Efficient consensus functions
The clustering is performed using
[http://glaros.dtc.umn.edu/gkhome/views/
metis METIS] and Spectral clustering.
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Consensus clustering - Soft clustering ensembles
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Punera and Ghosh extended the idea of
hard clustering ensembles to the soft
clustering scenario. Each instance in a soft
ensemble is represented by a
concatenation of r posterior membership
probability distributions obtained from the
constituent clustering algorithms. We can
define a distance measure between two
instances using the Kullback–Leibler
divergence|Kullback–Leibler (KL)
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Consensus clustering - Soft clustering ensembles
sCSPA extends CSPA by calculating a
similarity matrix. Each object is visualized
as a point in dimensional space, with each
dimension corresponding to probability of
its belonging to a cluster. This technique
first transforms the objects into a labelspace and then interprets the dot product
between the vectors representing the
objects as their similarity.
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Consensus clustering - Soft clustering ensembles
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sMCLA extends MCLA by accepting soft
clusterings as input. sMCLA’s working can
be divided into the following steps:
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Consensus clustering - Soft clustering ensembles
1
* Group the Clusters into MetaClusters
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Consensus clustering - Soft clustering ensembles
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* Collapse Meta-Clusters using
Weighting
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Consensus clustering - Soft clustering ensembles
HBGF represents the ensemble as a
bipartite graph with clusters and instances
as nodes, and edges between the
instances and the clusters they belong
to.Solving cluster ensemble problems by
bipartite graph partitioning, Xiaoli Zhang
Fern and Carla E
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Consensus clustering - Tunable-tightness partitions
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In some applications like gene clustering,
this matches the biological reality that
many of the genes considered for
clustering in a particular gene discovery
study might be irrelevant to the case of
study in hand and should be ideally not
assigned to any of the output clusters,
moreover, any single gene can be
participating in multiple processes and
would be useful to be included in multiple
clusters simultaneously
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Genetic cluster - Diametrical Clustering
1
Diametrical clustering repeatedly repartitions genes while repeatedly
calculated the dominant singular
vector of each cluster
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Text clustering
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'Document clustering' (or 'text clustering')
is the application of cluster analysis to
textual documents. It has applications in
automatic document organization, topic
(linguistics)|topic extraction and fast
information retrieval or filtering.
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Text clustering - Overview
1
Dimensionality reduction methods can be
considered a subtype of soft clustering; for
documents, these include latent semantic
indexing (truncated singular value
decomposition on term
histograms)http://nlp.stanford.edu/IRbook/pdf/16flat.pdf and topic models.
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Text clustering - Overview
Given a clustering, it can be
beneficial to automatically derive
human-readable labels for the
clusters. Cluster labeling|Various
methods exist for this purpose.
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European windstorm - Clustering
1
Kyrill in 2007 following only four days after Hanno,
and 2008 with Johanna, Kirsten and
Emma.[http://www.airworldwide.com/PublicationsItem.aspx?id=19693
AIR Worldwide: European Windstorms:
Implications of Storm Clustering on Definitions of
Occurrence
Losses][http://www.willisresearchnetwork.com/List
s/Publications/Attachments/63/WRN%20European
%20Windstorm%20Clustering%20%20Briefing%20Paper.pdf European Windstorm
Clustering Briefing Paper] In 2011, Cyclone
Berit|Xaver (Berit) moved across Northern Europe
and just a day later another storm, named Yoda,
hit the https://store.theartofservice.com/the-clustering-toolkit.html
same area
Distributed lock manager - Linux clustering
1
Both Red Hat and Oracle Corporation|Oracle
have developed clustering software for Linux.
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Distributed lock manager - Linux clustering
1
OCFS2, the Oracle Cluster File System
was
added[http://www.kernel.org/git/?p=linux/k
ernel/git/torvalds/linux2.6.git;a=commit;h=29552b1462799afbe0
2af035b243e97579d63350
kernel/git/torvalds/linux.git - Linux kernel
source tree]. Kernel.org. Retrieved on
2013-09-18. to the official Linux kernel
with version 2.6.16, in January 2006. The
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Distributed lock manager - Linux clustering
1
Red Hat's cluster software, including their
DLM and GFS2 was officially added to the
Linux kernel
[http://git.kernel.org/?p=linux/kernel/git/torv
alds/linux2.6.git;a=commit;h=1c1afa3c053d4ccdf44
e5a4e159005cdfd48bfc6
kernel/git/torvalds/linux.git - Linux kernel
source tree]. Git.kernel.org (2006-12-07).
Retrieved on 2013-09-18. with version
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Distributed lock manager - Linux clustering
1
Both systems use a DLM modeled on the
venerable VMS
DLM.[http://lwn.net/Articles/137278/ The
OCFS2 filesystem]. Lwn.net (2005-05-24).
Retrieved on 2013-09-18. Oracle's DLM
has a simpler API. (the core function,
dlmlock(), has eight parameters, whereas
the VMS SYS$ENQ service and Red Hat's
dlm_lock both have 11.)
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Dendrogram - Clustering example
1
For a clustering example, suppose this
data is to be clustered using Euclidean
distance as the Metric
(mathematics)|distance metric.
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Dendrogram - Clustering example
The hierarchical
clustering dendrogram
would be as such:
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Dendrogram - Clustering example
The top row of nodes represents data
(individual observations), and the
remaining nodes represent the clusters to
which the data belong, with the arrows
representing the distance (dissimilarity).
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Dendrogram - Clustering example
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The distance between merged clusters
is monotone increasing with the level
of the merger: the height of each node
in the plot is proportional to the value
of the intergroup dissimilarity
between its two daughters (the top
nodes representing individual
observations are all plotted at zero
height).
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Tally marks - Clustering
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Tally marks are typically clustered in
groups of five for legibility. The cluster
size 5 has the advantages of (a) easy
conversion into decimal for higher
arithmetic operations and (b) avoiding
error, as humans can far more easily
correctly identify a cluster of 5 than
one of 10.
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Sound symbolism - Clustering
Words that share a sound
sometimes have something in
common
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Sound symbolism - Clustering
Another hypothesis states that if a
word begins with a particular
phoneme, then there is likely to be a
number of other words starting with
that phoneme that refer to the same
thing. An example given by Magnus is
if the basic word for 'house' in a given
language starts with a /h/, then by
clustering, disproportionately many
words containing /h/ can be expected
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Sound symbolism - Clustering
Clustering is language dependent,
although closely related languages will
have similar clustering relationships.
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Operons - Operons versus clustering of prokaryotic genes
gene clustering helps a prokaryotic cell to
produce metabolic enzymes in a correct order.
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