Huge amount of Raw DATA is available.The Motivation for the Data Mining is to
Charecterize the Data etc...
The Databases are PreProcessed i.e. Cleaned and Integrated and the Data Warehouse is formed.
The Data Warehouse is Selected and Transformed as per the User Requirement and it is submitted to the Data Mining Engine.
The Data Mining Engine will run for ‘n’ iterations/ tuples.
As a Result, We will get some Patterns as Output.
Then the Patterns are Evaluated and finally we will get an Output which is Knowledge.
RDBMS - A Relational database is a collection of tables, each ofwhich is assigned a unique name.
Each table consists of a set of attributes ( columns or fields) and usually stores a large set of tuples ( records or rows). Each tuple in a relational table represents an object identified by a unique key and described by a set of attribute values.
A semantic data model, such as an entity-relationship (ER) data model, is often constructed for relational databases.
An ER data model represents the database as a set of entities and their relationships.
DataWareHouse - A Data warehouse
is a repository of information collected from multiple sources, stored under a unified schema, and that usually resides at a single site. Data warehouses are constructed via a process of data cleaning, data integration, data transformation, data loading, and periodic data refreshing.
Transactional DataBase - In general, a transactional database consists of a filewhere each record represents a transaction. A transaction typically includes a unique transaction identity number ( trans ID) and a list of the items making up the transaction (such as items purchased in a store).
Object Oriented RDBMS - Conceptually, the object-relational data model inherits the essential concepts of object-oriented databases, where, in general terms, each entity is considered as an object. Data and code relating to an object are encapsulated into a single unit. Each object has associated with it the following.
A set of variables that describe the objects. These correspond to attributes in the entity-relationship and relational models.
A set of messages that the object can use to communicate with other objects, or with the rest of the database system.
A set of methods, where each method holds the code to implement a message. Upon receiving a message, the method returns a value in response. For instance, the method for the message get photo(employee) will retrieve and return a photo of the given employee object.
A Temporal database typically stores relational data that include time-related attributes.These attributes may involve several timestamps, each having different semantics.
A Sequence database stores sequences of ordered events, with or without a concrete notion of time. Examples include customer shopping sequences, Web click streams, and biological sequences.
A Time-series database stores sequences of values or events obtained over repeated measurements of time (e.g., hourly, daily, weekly). Examples include data collected from the stock exchange, inventory control, and the observation of natural phenomena (like temperature and wind).
Spatial databases contain spatial-related information. Examples include geographic (map) databases, very large-scale integration (VLSI) or computed-aided design databases, and medical and satellite image databases.
Text databases are databases that contain word descriptions for objects. These word descriptions are usually not simple keywords but rather long sentences or paragraphs, such as product specifications, error or bug reports, warning messages, summary reports,notes, or other documents.
Multimedia databases store image, audio, and video data. They are used in applications such as picture content-based retrieval, voice-mail systems, video-on-demand systems, the World Wide Web, and speech-based user interfaces that recognize spoken commands.
A Heterogeneous database consists of a set of interconnected, autonomous component databases. The components communicate in order to exchange information and answer queries.
Legacy Database formed as a result of long history of IT Development. A legacy database is a group of heterogeneous databases that combines different kinds of data systems, such as relational or object-oriented databases, hierarchical databases, network databases, spreadsheets, multimedia databases, or file systems.
Data Streams - Many applications involve the generation and analysis of a newkind of data, called stream data, where data flow in and out of an observation platform (or window) dynamically.
The World Wide Web and its associated distributed information services, such as Yahoo!, Google, America Online, and AltaVista, provide rich, worldwide, on-line information services, where data objects are linked together to facilitate interactive access.
Characterization and Discrimination
Mining Frequent Patterns, Associations, and Correlations
Classification and Prediction
Data characterization is a summarization of the general characteristics or features of a target class of data. The data corresponding to the user-specified class are typically collected by a database query. For example, to study the characteristics of software products whose sales increased by 10% in the last year, the data related to such products can be collected by executing an SQL query.
Data discrimination is a comparison of the general features of target class data objects with the general features of objects from one or a set of contrasting classes. The target and contrasting classes can be specified by the user, and the corresponding data objects retrieved through database queries. For example, the user may like to compare the general features of software products whose sales increased by 10% in the last year with those whose sales decreased by at least 30% during the same period.
Frequent patterns , as the name suggests, are patterns that occur frequently in data. There are many kinds of frequent patterns, including itemsets, subsequences, and substructures.A frequent itemset typically refers to a set of items that frequently appear together in a transactional data set, such as milk and bread. A frequently occurring subsequence,such as the pattern that customers tend to purchase first a PC, followed by a digital camera, and then a memory card, is a ( frequent) sequential pattern.
Classification is the process of finding a model (or function) that describes and distinguishes data classes or concepts, for the purpose of being able to use the model to predict the class of objects whose class label is unknown.
“ How is the derived model presented?” The derived model may be represented in various forms, such as classification (IF-THEN) rules, decision trees, mathematical formulae, or neural networks.
Prediction is the amount of revenue that each item will generate during an upcoming sale, stock etc..
What is cluster analysis?”Unlike classification and prediction, which analyze class-labeled data objects, clustering analyzes data objects without consulting a known class label.
The objects are clustered or grouped based on the principle of maximizing the intra class similarity and minimizing the interclass similarity.
A database may contain data objects that do not comply with the general behavior or model of the data. These data objects are outliers.
However, in some applications such as fraud detection, the rare events can be more interesting than the more regularly occurring ones.
Data evolution analysis describes and models regularities or trends for objects whose behavior changes over time. Although this may include characterization, discrimination,
association and correlation analysis, classification, prediction, or clustering of time related data, distinct features of such an analysis include time-series data analysis,
sequence or periodicity pattern matching, and similarity-based data analysis.
A data mining system has the potential to generate thousands or even millions of patterns, or rules. “So,” you may ask, “are all of the patterns interesting?” Typically not—only a small fraction of the patterns potentially generated would actually be of interest to any given user. Those are
Rules that do not satisfy a confidence threshold of, say, 50% can be considered uninteresting. Rules below the threshold likely reflect noise , exceptions, or minority cases and are probably of less value.
patterns are interesting if they are unexpected (contradicting a user’s belief).
Patterns that are expected can be interesting if they confirm a hypothesis that the user wished to validate, or resemble a user’s hunch.
The Data Mining Systems are classified according to
kinds of databases mined- relational , transactional, object-relational, or data warehouse mining system etc..
kinds of knowledge mined- characterization, discrimination, association and correlation analysis, classification, prediction, clustering, outlier analysis, and evolution analysis.
kinds of techniques- autonomous systems, interactive exploratory systems, query-driven systems.
The background knowledge to be used in the discovery process
The interestingness measures and thresholds for pattern evaluation
The expected representation for visualizing the discovered patterns
Mining methodology and user interaction issues
Mining different kinds of knowledge in databases
Interactive mining of knowledge at multiple levels of abstraction
Incorporation of background knowledge
Data mining query languages and ad hoc data mining
Presentation and visualization of data mining results
Handling noisy or incomplete data
Pattern evaluation—the interestingness problem.
Efficiency and scalability of data mining algorithms
Parallel, distributed, and incremental mining algorithms
Issues relating to the diversity of database types
Handling of relational and complex types of data
Mining information from heterogeneous databases and
global information systems
Why Preprocess the Data ?
Imagine that you are a manager at AllElectronics and have been charged with analyzing the company’s data with respect to the sales at your branch.You carefully inspect the company’s database and data warehouse, identifying and selecting the attributes or dimensions to be included in your analysis, such as item, price, and units sold.
Alas! You notice that several of the attributes for various tuples have no recorded value.
For your analysis, you would like to include information as to whether each item purchased was advertised as on sale, yet you discover that this information has not been recorded.
Furthermore, users of your database system have reported errors, unusual values, and inconsistencies in the data recorded for some transactions.
In other words, the data you wish to analyze by data mining techniques are incomplete (lacking attribute values or certain attributes of interest, or containing only aggregate data), noisy (containing errors, or outlier values that deviate from the expected), and inconsistent (e.g., containing discrepancies in the department codes used to categorize items).
Real-world data tend to be incomplete, noisy, and inconsistent. Data cleaning (or data cleansing) routines attempt to fill in missing values, smooth out noise while identifying outliers, and correct inconsistencies in the data.
Missing Values –
Ignore the tuple
Fill in the missing value manually
Use a global constant to fill in the missing value
Use the attribute mean to fill in the missing value
Use the attribute mean for all samples belonging to the same class as the given tuple
Use the most probable value to fill in the missing value.
Noisy Data - Noise is a random error or variance in a measured variable.
Noise is Removed in the following three ways
Binning – see the next page
Regression - Data can be smoothed by fitting the data to a function
Clustering - Outliers may be detected by clustering, where similar values are organized into groups, or “clusters.”
Sorted data for price (in dollars): 4, 8, 15, 21, 21, 24, 25, 28, 34
Partition into (equal-frequency) bins:
Bin 1: 4, 8, 15
Bin 2: 21, 21, 24
Bin 3: 25, 28, 34
Smoothing by bin means:
Bin 1: 9, 9, 9
Bin 2: 22, 22, 22
Bin 3: 29, 29, 29
Smoothing by bin boundaries:
Bin 1: 4, 4, 15
Bin 2: 21, 21, 24
Bin 3: 25, 25, 34
To Detect that Data cleaning is required for a particular Data is called Discrepancy Detection and it can be done by using Knowledge and metadata.
It is likely that your data analysis task will involve data integration, which combines data from multiple sources into a coherent data store, as in data warehousing. These sources may include multiple databases, data cubes, or flat files.
There are a number of issues to consider during data integration –
Schema integration and object matching
customer id in one database and cust _number in another
Hence we will perform Data integration by using
Correlation analysis using ϰ 2
The data are transformed or consolidated into forms appropriate
for mining in the following ways.
Smoothing – can be done by binning, regression, clustering
Aggregation - the daily sales data may be aggregated so as to compute monthly and annual total amounts.
Generalization – low level data are Replaced by high level data i.e. street can be generalized to city or country.
Normalization – the data is normalized.
Attribute construction – the new attributes are constructed and added from the given set of attributes.
The data to be Mined will be Generally very Huge, Hence the Data is Reduced in the following ways
Data cube aggregation
Attribute subset selection
Discretization and concept hierarchy generation
By using the Data cube, all the quarter sales can be aggregated to yearly sales.
Hence the Huge data of quarterly is Reduced to yearly..
Data sets for analysis may contain hundreds of attributes, many of which may be irrelevant to the mining task or redundant. For example, if the task is to classify customers as to whether or not they are likely to purchase a popular new CD at AllElectronics when notified of a sale, attributes such as the customer’s telephone number are likely to be irrelevant, unlike attributes such as age or music taste.
The attribute subset selection is done in the following ways
Stepwise forward selection
Stepwise backward elimination
Combination of forward selection and backward elimination
Decision tree induction
The discrete wavelet transform(DWT) is a linear signal processing technique that, when applied to a data vector X, transforms it to a numerically different vector, X0, of wavelet coefficients. The two vectors are of the same length. When applying this technique to data reduction, we consider each tuple as an n-dimensional data vector, that is, X = (x1;x2; : : : ;xn), depicting n measurements made on the tuple from n database attributes.
“ How can this technique be useful for data reduction if the wavelet transformed data are of the same length as the original data?” The usefulness lies in the fact that the wavelet transformed data can be truncated. A compressed approximation of the data can be retained by storing only a small fraction of the strongest of the wavelet coefficients.
Principal Components Analysis
Suppose that the data to be reduced consist of tuples or data vectors described by n attributes or dimensions. Principal components analysis, or PCA (also called the Karhunen-Loeve, or K-L, method), searches for k n-dimensional orthogonal vectors that can best be used to represent the data, where k <= n. The original data are thus projected onto a much smaller space, resulting in dimensionality reduction.
“ Can we reduce the data volume by choosing alternative, ‘smaller’ forms of data representation?”. This can be done as
Regression and Log-Linear Models - the data are modeled to fit a straight line
Histograms - A histogram for an attribute, A, partitions the data distribution of A into disjoint subsets, or buckets.
Clustering - They partition the objects into groups or clusters, so that objects within a cluster are “similar” to one another and
“ dissimilar” to objects in other clusters.
Sampling - it allows a large data set to be represented by a much smaller random sample (or subset) of the data.
Data Discretization and Concept Hierarchy Generation – pls see the next page..
Data discretization techniques can be used to reduce the number of values for a given continuous attribute by dividing the range of the attribute into intervals. This is done in the following ways
Binning – Here we will take the bins with some intervals
Histogram Analysis – Here, the histogram partitions the values into buckets.
Entropy-Based Discretization - T he method selects the value of A that has the minimum entropy as a split-point, and recursively partitions the resulting intervals to arrive at a hierarchical discretization.
Interval Merging by using ϰ 2 Analysis – This contrasts with ChiMerge, which employs a bottom-up approach by finding the best neighboring intervals and then merging these to form larger intervals, recursively.
Cluster Analysis - A clustering algorithm can be applied to discretize a numerical attribute, A, by partitioning the values of A into clusters
Discretization by Intuitive Partitioning - For example, annual salaries broken into ranges like ($50,000, $60,000] are often more desirable than ranges
like ($51,263.98, $60,872.34], obtained by, say, some sophisticated clustering analysis.
Here the concept hierarchy is generated in the following manner. if there are some attributes like state, street, city, country..then the concept hierarchy is generated as
Traditional Databases uses OLAP, whereas DatawareHouse uses OLTP.
The entity-relationship data model is commonly used in the design of relational databases, where a database schema consists of a set of entities and the relationships between them. Such a data model is appropriate for on-line transaction processing.
A data warehouse, however, requires a concise, subject-oriented schema that facilitates on-line data analysis. The most popular data model for a data warehouse is a multidimensional model. Such a model can exist in the form of a star schema, a snowflake schema, or a fact constellation schema. Let’s look at each of these schema types.
Star schema: The most common modeling paradigm is the star schema, in which the data warehouse contains (1) a large central table (fact table) containing the bulk of the data, with no redundancy, and (2) a set of smaller attendant tables (dimension tables), one for each dimension. The schema graph resembles a starburst, with the dimension tables displayed in a radial pattern around the central fact table.
Snowflake schema - For example, the item dimension table now contains the attributes item key, item name, brand, type, and supplier key, where supplier key is linked to the supplier dimension table, containing supplier key and supplier type information. Similarly, the single dimension table for location in the star schema can be normalized into two new tables: location and city.
Fact constellation. A fact constellation schema is shown in Figure 3.6. This schema specifies two fact tables, sales and shipping. The sales table definition is identical to that of the star schema (Figure 3.4). The shipping table has five dimensions, or keys: item key, time key, shipper key, from location, and to location, and two measures: dollars cost and units shipped. A fact constellation schema allows dimension tables to be shared between
fact tables. For example, the dimensions tables for time, item, and location are shared between both the sales and shipping fact tables.
Roll-up: The roll-up operation (also called the drill-up operation by some vendors) performs aggregation on a data cube, either by climbing up a concept hierarchy for a dimension or by dimension reduction. This hierarchy was defined as the total order “ street < city < province or state < country.”
Drill-down: Drill-down is the reverse of roll-up. It navigates from less detailed data to more detailed data. Drill-down can be realized by either stepping down a concept hierarchy for a dimension or introducing additional dimensions. Drill-down operation performed on the central cube by stepping down a concept hierarchy for time defined as “day < month < quarter < year.”
Slice and dice: The slice operation performs a selection on one dimension of the given cube, resulting in a subcube. The dice operation defines a subcube by performing a selection on two or more dimensions.
Pivot (rotate): Pivot (also called rotate) is a visualization operation that rotates the data axes in view in order to provide an alternative presentation of the data.
1. The bottom tier is a warehouse database server that is almost always a relational database system. Back-end tools and utilities are used to feed data into the bottom tier from operational databases or other external sources (such as customer profile information provided by external consultants). These tools and utilities perform data extraction, cleaning, and transformation (e.g., to merge similar data from different sources into a unified format), as well as load and refresh functions to update the data warehouse.
2. The middle tier is an OLAP server that is typically implemented using either (1) a relational OLAP (ROLAP) model, that is, an extended relational DBMS that maps operations on multidimensional data to standard relational operations; or (2) a multidimensional OLAP (MOLAP) model, that is, a special-purpose server that directly implements multidimensional data and operations.
3. The top tier is a front-end client layer, which contains query and reporting tools, analysis tools, and/or data mining tools (e.g., trend analysis, prediction, and so on).
Enterprise warehouse: An enterprise warehouse collects all of the information about subjects spanning the entire organization. It provides corporate-wide data integration, usually from one or more operational systems or external information providers, and is cross-functional in scope.
Data mart: A data mart contains a subset of corporate-wide data that is of value to a specific group of users. The scope is confined to specific selected subjects. For example, a marketing data mart may confine its subjects to customer, item, and sales.
Virtual warehouse: A virtual warehouse is a set of views over operational databases. For efficient query processing, only some of the possible summary views may be materialized.
Relational OLAP (ROLAP) servers: These are the intermediate servers that stand in between a relational back-end server and client front-end tools.
Multidimensional OLAP (MOLAP) servers: These servers support multidimensional views of data through array-based multidimensional storage engines.
Hybrid OLAP (HOLAP) servers: The hybrid OLAP approach combines ROLAP and MOLAP technology, benefiting from the greater scalability of ROLAP and the faster computation of MOLAP.
Specialized SQL servers: To meet the growing demand of OLAP processing in relational databases, some database systemvendors implement specialized SQL servers that provide advanced query language and query processing support for SQL queries over star
and snowflake schemas in a read-only environment.
Data generalization summarizes data by replacing relatively low-level values (such as numeric values for an attribute age) with higher-level concepts (such as young, middle aged, and senior). Given the large amount of data stored in databases, it is useful to be able to describe concepts in concise and succinct terms at generalized (rather than low) levels of abstraction.
Attribute-Oriented Induction for Data Characterization
Before Attribute Induction -
1) First, data focusing should be performed before attribute-oriented induction. This step corresponds to the specification of the task-relevant data (i.e., data for analysis). The data are collected based on the information provided in the data mining query.
2) Specifying the set of relevant attributes. For example, suppose that the dimension birth place is defined by the attributes city, province or state, and country. Of these attributes, let’s say that the user has only thought to specify city. In order to allow generalization on the birth place dimension, the other attributes defining this dimension
should also be included.
3) A correlation-based (Section 2.4.1) or entropy-based (Section 2.6.1) analysis method can be used to perform attribute relevance analysis and filter out statistically irrelevant or weakly relevant attributes from the descriptive mining process.
Attribute Induction takes place in two phases
1) Attribute Removal - If there is a large set of distinct values for an attribute of the initial working relation, but either (1) there is no generalization
operator on the attribute (e.g., there is no concept hierarchy defined for the attribute), or (2) its higher-level concepts are expressed in terms of other attributes, then the attribute should be removed from the working relation.
2) Attribute Generalization - If there is a large set of distinct values for an attribute in the initial working relation, and there exists a set of generalization operators on the attribute, then a generalization operator should be selected and applied to the attribute.
Attribute Generalization can be controlled in 2 ways
1) Attribute generalization threshold control - sets one threshold for each attribute. If the number of distinct values in an attribute is greater than the attribute threshold, further attribute removal or attribute generalization should be performed.
2) Generalized relation threshold control - sets a threshold
for the generalized relation. If the number of (distinct) tuples in the generalized relation is greater than the threshold, further generalization should be performed.
Presentation of the Derived Generalization
The above table is represented in the form of a Bar chart & Pie chart in the following manner.
It is Represented in the form of a 3-D cube in the following way.
Mining Class Comparisons - In many applications, users may not be interested in having a single class (or concept) described or characterized, but rather would prefer to mine a description that compares or distinguishes one class (or concept) from other comparable classes (or concepts).
The class comparison is done in the following procedure
1) Data collection - The set of relevant data in the database is collected by query processing and is partitioned respectively into a target class and one or a set of contrasting class(es).
2) Dimension relevance analysis - If there are many dimensions, then dimension relevance analysis should be performed on these classes to select only the highly relevant dimensions for further analysis. Correlation or entropy-based measures can be used for this step.
3) Synchronous generalization – Generalization is performed on the target class to the level controlled by a user- or expert-specified dimension threshold, which results in a prime target class relation.
4) Presentation of the derived comparison - The resulting class comparison description can be visualized in the form of tables, graphs, and rules.
Frequent patterns are patterns (such as itemsets, subsequences, or substructures) that appear in a data set frequently. For example, a set of items, such as milk and bread , that appear frequently together in a transaction data set is a frequent itemset. A subsequence, such as buying first a PC, then a digital camera, and then a memory card, if it occurs frequently in a shopping history database, is a ( frequent) sequential pattern.
Finding such frequent patterns plays an essential role in mining associations, correlations, and many other interesting relationships among data.
Moreover, it helps in data classification, clustering, and other data mining tasks as well.
Mainly Association mining deals with
Efficient and Scalable Frequent Item set Mining Methods – using Apriori Algorithm
Mining Multilevel Association Rules
Mining Multidimensional Association Rules
from Relational Databases and Data Warehouses
Association Mining to Correlation Analysis
Constraint-Based Association Mining
Consider a particular Departmental store where it shows some transactions
Tid items 1 Bread, milk 2 Bread, diapers, beer, eggs 3 Milk, diapers, beer, cola 4 Bread, milk, diapers, beer 5 Bread, milk, diapers, cola
The above table can be represented in binary format as below
Then we can form a 2-D Grid and cluster them to get the “HDTV” Purchase Zone.
C orrelation between A and B is measured not only by its support and confidence but also by the correlation between item sets A and B.
It is done by using two parameters.
1)lift(A, B) =P(A∪B)/P(A)P(B).
2) Correlation analysis using ϰ 2
Consider an example: Of the 10,000 transactions analyzed, the data show that 6,000 of the customer transactions included computer games, while 7,500 included videos, and 4,000 included both computer games and videos.
We have not only “age”, “income” predicates, but we will have lot of other predicates..and each predicate will have some Range of values..
So, for which set of predicates and for which set of values, the predicate “buys” will be maximum..
We will consider the best set of predicates, best set of values to have the maximum “buys” predicate.
That is Constraint-Based Association Mining.
classification is where a model or classifier is constructed to analyse “safe” or “risky” for the loan application data, “yes” or “no” for the marketing data etc..as in the following table.
prediction is to predict the “yes” or “no” for a particular new tuple
X = (age = youth, income = medium, student = yes, credit rating = fair)
Mainly Classification is done in 5 ways
1) Decision tree Classification
2) Bayesian Classification
3) Rule Based Classification
4) Back propagation Classification
5) Associative Classification
Decision tree classification - Decision tree induction is the learning of decision trees from class-labeled training tuples. A decision tree is a flowchart-like tree structure, where each internal node (non leaf node) denotes a test on an attribute, each branch represents an out come of the test, and each leaf node (or terminal node) holds a class label. The topmost node in a tree is the root node.
Information Gain /Entropy – from the above Buys_computer table,
Gain(A) = Info(D) – Info a (D)
Info(D) – for the complete table – 9 yes, 5 no, 14 tuples
Youth – total 5 values out of 14 tuples, 2 yes and 3 no
Middle aged - total 4 values out of 14 tuples, 4 yes and ‘0’ no
Senior - total 5 values out of 14 tuples, 3 yes and 2 no
Info age (D)=5/14 x (-2/5 log 2 2/5 – 3/5 log 2 3/5)+
4/14 x (-4/4 log 2 4/4 – 0/4 log 2 0/4)+
5/14 x (-3/5 log 2 3/5 – 2/5 log 2 2/5)
Gain(A) = 0.940 – 0.694 = 0.246 bits
Best Classifier Attribute - Entropy or Gain is calculated for all the Attributes Age, Income, Student, Credit_rating and the Attribute having Maximum Entropy value will be the Best Classifier Attribute.
Entropy(Age) is having the Max value..Hence “Age” is acting as the Best classifier Attribute in the following figure.
They can predict class membership probabilities, such as the probability that a given tuple belongs to a particular class.
Bayes Theorem -
Let X be a data tuple. In Bayesian terms, X is considered “evidence.” As usual, it is described by measurements made on a set of n attributes. Let H be some hypothesis, such as that the data tuple X belongs to a specified class C. For classification problems, we want to determine P(H/X), the probability that the hypothesis H holds given the “evidence” or observed data tuple X. In other words, we are looking for the probability that tuple X belongs to class C, given that we know the attribute description of X.
P(H/X) is the posterior probability..
P(H/x) =P(X/H)P(H) / p(X)
The above Buys_computer Table is classified and if we need to predict for a new tuple X,
X = (age = youth, income = medium, student = yes, credit rating = fair),
By using Bayes Theorem, we can calculate in the following way..
P(buys computer = yes) = 9/14 = 0.643
P(buys computer = no) = 5/14 = 0.357
To compute PX/Ci), for i = 1, 2, we compute the following conditional probabilities:
The process of grouping a set of physical or abstract objects into classes of similar objects is called clustering. A cluster is a collection of data objects that are similar to one another within the same cluster and are dissimilar to the objects in other clusters.
Types of Data in Cluster Analysis
1) Data matrix
2) Dissimilarity matrix
3) Interval scaled variables - examples include weight and height, latitude
and longitude coordinates (e.g., when clustering houses), and weather temperature.
4) Binary variables – 0 & 1
Partitioning Methods –
Given D, a data set of n objects, and k, the number of clusters to form, a partitioning algorithm organizes the objects into k partitions (k n), where each partition represents a cluster.
THE K-MEANS METHOD
k: the number of clusters, D: a data set containing n objects.
Output: A set of k clusters.
(1) arbitrarily choose k objects from D as the initial cluster centers;
(3) (re)assign each object to the cluster to which the object is the most similar, based on the mean value of the objects in the cluster;
(4) update the cluster means, i.e., calculate the mean value of the objects for
(5) until no change;
Clustering by k-means partitioning.
Suppose that there is a set of objects located in space as depicted in the rectangle shown in Figure 7.3(a). Let k = 3; that is, the user would like the objects to be partitioned into three clusters.
According to the algorithm in Figure 7.2, we arbitrarily choose three objects as the three initial cluster centers, where cluster centers are marked by a “+”. Each object is distributed to a cluster based on the cluster center to which it is the nearest. Such a distribution forms silhouettes encircled by dotted curves, as shown in Figure 7.3(a).
Next, the cluster centers are updated. That is, the mean value of each cluster is recalculated based on the current objects in the cluster. Using the new cluster centers, the objects are redistributed to the clusters based on which cluster center is the nearest. Such a redistribution forms new silhouettes encircled by dashed curves, as shown in Figure 7.3(b).
This process iterates, leading to Figure 7.3(c). The process of iteratively reassigning objects to clusters to improve the partitioning is referred to as iterative relocation. Eventually, no redistribution of the objects in any cluster occurs, and so the process terminates.
The resulting clusters are returned by the clustering process.
Clustering of a set of objects based on the k-means method. (The mean of each cluster is marked by a “+”.)