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R. Zafarani, M. A. Abbasi, and H. Liu, Social Media Mining: An Introduction, Cambridge University Press, 2014. Free book and slides at http://socialmediamining.info/

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Social Media Mining - Chapter 6 (Community Analysis)SocialMediaMining

CS6010 Social Network Analysis Unit IIpkaviya

Web mining TeklayBirhane

Social Media Mining and AnalyticsWeb Science Research Group at Institute of Business Administration, Karachi, Pakistan

Social Data MiningMahesh Meniya

Social network analysis & Big Data - Telecommunications and moreWael Elrifai

- Data Mining Essentials SOCIAL MEDIA MINING
- 2Social Media Mining Measures and Metrics 2Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Dear instructors/users of these slides: Please feel free to include these slides in your own material, or modify them as you see fit. If you decide to incorporate these slides into your presentations, please include the following note: R. Zafarani, M. A. Abbasi, and H. Liu, Social Media Mining: An Introduction, Cambridge University Press, 2014. Free book and slides at http://socialmediamining.info/ or include a link to the website: http://socialmediamining.info/
- 3Social Media Mining Measures and Metrics 3Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Introduction Data production rate has increased dramatically (Big Data) and we are able store much more data β E.g., purchase data, social media data, cellphone data Businesses and customers need useful or actionable knowledge to gain insight from raw data for various purposes β Itβs not just searching data or databases The process of extracting useful patterns from raw data is known as Knowledge Discovery in Databases (KDD)
- 4Social Media Mining Measures and Metrics 4Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ KDD Process
- 5Social Media Mining Measures and Metrics 5Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Collecting Data on Social Media β’ Collect Raw Data β Use site provided APIs β’ Flickrβs: https://www.flickr.com/services/api/ β Scrape information directly β’ Use Provided Repositories β http://socialcomputing.asu.edu β http://snap.Stanford.edu β https://github.com/caesar0301/awesome-public- datasets
- 6Social Media Mining Measures and Metrics 6Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data Mining β’ Extracting/βminingβ knowledge from large-scale data (big data) β’ Data-driven discovery and modeling of hidden patterns in big data β’ Extracting information/knowledge from data that is β implicit, β previously unknown, β unexpected, and β potentially useful Data Mining: the process of discovering hidden and actionable patterns from data It utilizes methods at the intersection of artificial intelligence, machine learning, statistics, and database systems
- 7Social Media Mining Measures and Metrics 7Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data Mining vs. Databases β’ Data mining is the process of extracting hidden and actionable patterns from data β’ Database systems store and manage data β Queries return part of stored data β Queries do not extract hidden patterns β’ Examples of querying databases β Find all employees with income more than $250K β Find top spending customers in last month β Find all students from engineering college with GPA more than average
- 8Social Media Mining Measures and Metrics 8Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Examples of Data Mining Applications β’ Fraud/Spam Detections: Identifying fraudulent transactions of a credit card or spam emails β You are given a userβs purchase history and a new transaction, identify whether the transaction is fraud or not; β Determine whether a given email is spam or not β’ Frequent Patterns: Extracting purchase patterns from existing records β beer β dippers (80%) β’ Forecasting: Forecasting future sales and needs according to some given samples β’ Finding Like-Minded Individuals: Extracting groups of like-minded people in a given network
- 9Social Media Mining Measures and Metrics 9Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data
- 10Social Media Mining Measures and Metrics 10Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data Instances β’ In the KDD process, β Data is in a tabular format (a set of instances) β’ Each instance is a collection of properties and features related to an object or person β A patientβs medical record β A userβs profile β A geneβs information β’ Instances are also called points, data points, or observations Data Instance: Features ( Attributes or measurements) Class Label Feature Value Class Attribute
- 11Social Media Mining Measures and Metrics 11Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data Instances β’ Predicting whether an individual who visits an online book seller is going to buy a specific book β’ Features can be β Continues: values are numeric values β’ Money spent: $25 β Discrete: can take a number of values β’ Money spent: {high, normal, low} Labeled Example Unlabeled Example
- 12Social Media Mining Measures and Metrics 12Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data Types + Permissible Operations (statistics) β’ Nominal β Operations: β’ Mode (most common feature value), Equality Comparison β E.g., {male, female} β’ Ordinal β Feature values have an intrinsic order to them, but the difference is not defined β Operations: β’ same as nominal, feature value rank β E.g., {Low, medium, high} β’ Interval β Operations: β’ Addition and subtractions are allowed whereas divisions and multiplications are not β E.g., 3:08 PM, calendar dates β’ Ratio β Operations: β’ divisions and multiplications are allowed β E.g., Height, weight, money quantities
- 13Social Media Mining Measures and Metrics 13Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Sample Dataset β Twitter Users Activity Date Joined Number of Followers Verified Account? Has Profile Picture? High 2015 50 FALSE no High 2013 300 TRUE no Average 2011 860000 FALSE yes Low 2012 96 FALSE yes High 2008 8,000 FALSE yes Average 2009 5 TRUE no Very High 2010 650,000 TRUE yes Low 2010 95 FALSE no Average 2011 70 FALSE yes Very High 2013 80,000 FALSE yes Low 2014 70 TRUE yes Average 2013 900 TRUE yes High 2011 7500 FALSE yes Low 2010 910 TRUE no NominalOrdinal Interval Ratio Nominal
- 14Social Media Mining Measures and Metrics 14Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Text Representation β’ The most common way to represent documents is to transform them into vectors β Process them with linear algebraic operations β’ This representation is called βBag of Wordsβ β Vector Space Model β’ Weights for words can be assigned by TF-IDF
- 15Social Media Mining Measures and Metrics 15Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Vector Space Model β’ Consider a set of documents π· β’ Each document is a set of words β’ Goal: convert these documents to vectors How to set ππ, π β’ Set π€π, π to 1 when the word π exists in document π and 0 when it does not. β’ We can also set π€π, π to the number, of times the word π is observed in document π (frequency) β’ ππ : document π β’ π€π, π : the weight for word π in document π
- 16Social Media Mining Measures and Metrics 16Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Vector Space Model: An Example β’ Documents: β π1: social media mining β π2: social media data β π3: financial market data β’ Reference vector (Dictionary): β (social, media, mining, data, financial, market) β’ Vector representation:
- 17Social Media Mining Measures and Metrics 17Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ TF-IDF (Term Frequency-Inverse Document Frequency) TF-IDF of term (word) π‘, document π, and document corpus π· is calculated as follows: is the frequency of word π in document π The total number of documents in the corpus The number of documents where the term π appears
- 18Social Media Mining Measures and Metrics 18Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ TF-IDF: An Example Document π1 contains 100 words β Word βappleβ appears 10 times in π1 β Word βorangeβ appears 20 times in π1 We have |π·| = 20 documents β Word βappleβ only appears in document π1 β Word βorangeβ appears in all 20 documents
- 19Social Media Mining Measures and Metrics 19Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ TF-IDF: An Example β’ Documents: β π1: social media mining β π2: social media data β π3: financial market data β’ TF values: β’ TF-IDF
- 20Social Media Mining Measures and Metrics 20Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data Quality When making data ready for mining, data quality needs to be assured β’ Noise β Noise is the distortion of the data β’ Outliers β Outliers are data points that are considerably different from other data points in the dataset β’ Missing Values β Missing feature values in data instances β Solution: β’ Remove instances that have missing values β’ Estimate missing values, and β’ Ignore missing values when running data mining algorithm β’ Duplicate data
- 21Social Media Mining Measures and Metrics 21Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data Preprocessing β’ Aggregation β It is performed when multiple features need to be combined into a single one or when the scale of the features change β Example: image width , image height -> image area (width x height) β’ Discretization β From continues values to discrete values β Example: money spent -> {low, normal, high} β’ Feature Selection β Choose relevant features β’ Feature Extraction β Creating new features from original features β Often, more complicated than aggregation β’ Sampling β Random Sampling β Sampling with or without replacement β Stratified Sampling: useful when having class imbalance β Social Network Sampling
- 22Social Media Mining Measures and Metrics 22Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data Preprocessing Sampling social networks: β’ Start with a small set of nodes (seed nodes) β’ Sample β (a) the connected components they belong to; β (b) the set of nodes (and edges) connected to them directly; or β (c) the set of nodes and edges that are within n-hop distance from them.
- 23Social Media Mining Measures and Metrics 23Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data Mining Algorithms
- 24Social Media Mining Measures and Metrics 24Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Data Mining Algorithms β’ Supervised Learning Algorithm β Classification (class attribute is discrete) β’ Assign data into predefined classes β Spam Detection β Regression (class attribute takes real values) β’ Predict a real value for a given data instance β Predict the price for a given house β’ Unsupervised Learning Algorithm β Group similar items together into some clusters β’ Detect communities in a given social network
- 25Social Media Mining Measures and Metrics 25Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Supervised Learning
- 26Social Media Mining Measures and Metrics 26Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Supervised Learning: The Process β’ We are given a set of labeled records/instances β In the format (πΏ, π¦) β πΏ is a vector of features β π¦ is the class attribute (commonly a scalar) β’ [Training] The supervised learning task is to build model that maps πΏ to π¦ (find a mapping π such that π(πΏ) = π¦) β’ [Testing] Given an unlabeled instance (πΏβ, ? ), we compute π(πΏβ) β E.g., spam/non-spam prediction
- 27Social Media Mining Measures and Metrics 27Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Classification: An Email Example β’ A set of emails is given β Users have manually labeled them as spam / non-spam β’ Use a set of features (x) to identify spam/non-spam status of the email (y) β We can use words in the email β’ In this case, classes are π¦ = {π πππ, πππ β π πππ}
- 28Social Media Mining Measures and Metrics 28Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ A Twitter Example
- 29Social Media Mining Measures and Metrics 29Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Supervised Learning Algorithms β’ Classification β Decision tree learning β Naive Bayes Classifier β π-nearest neighbor classifier β Classification with Network information β’ Regression β Linear Regression β Logistic Regression
- 30Social Media Mining Measures and Metrics 30Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Decision Tree Learning
- 31Social Media Mining Measures and Metrics 31Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Decision Tree β’ A decision tree is learned from the dataset β (training data with known classes) β’ The learned tree is later applied to predict the class attribute value of new data β (test data with unknown classes) β Only the feature values are known
- 32Social Media Mining Measures and Metrics 32Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Decision Tree: Example Class Labels Multiple decision trees can be learned from the same dataset
- 33Social Media Mining Measures and Metrics 33Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Decision Tree Construction β’ Decision trees are constructed recursively β A top-down greedy approach in which features are sequentially selected. β’ After selecting a feature for each node, β based on its attribute values, different branches are created. β’ The training set is then partitioned into subsets based on the feature values, β each of which fall under the respective feature value branch; β The process is continued for these subsets and other nodes β’ When selecting features, we prefer features that partition the set of instances into subsets that are more pure. β’ A pure subset has instances that all have the same class attribute value.
- 34Social Media Mining Measures and Metrics 34Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Decision Tree Construction β’ When reaching pure (or highly pure) subsets under a branch, β the decision tree construction process no longer partitions the subset, β creates a leaf under the branch, and β assigns the class attribute value (or the majority class attribute value) for subset instances as the leafβs predicted class attribute value β’ To measure purity we can use/minimize entropy. β’ Over a subset of training instances, π, with a binary class attribute (values in {+,-}), the entropy of π is defined as: β π+ is the proportion of positive examples in π β πβ is the proportion of negative examples in π
- 35Social Media Mining Measures and Metrics 35Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Entropy Example Assume there is a subset π that has 10 instances: β’ Seven instances have a positive class attribute value β’ Three have a negative class attribute value β’ Denote π as [7+, 3-] β’ The entropy for subset π is In a pure subset, all instances have the same class attribute value (entropy is 0) If the subset contains an unequal number of positive and negative instances β’ The entropy is between 0 and 1.
- 36Social Media Mining Measures and Metrics 36Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ NaΓ―ve Bayes Learning
- 37Social Media Mining Measures and Metrics 37Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Naive Bayes Classifier For two random variables πΏ and π, Bayes theorem states that, class variable the instance features Then class attribute value for instance πΏ We assume that features are independent given the class attribute
- 38Social Media Mining Measures and Metrics 38Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ NBC: An Example
- 39Social Media Mining Measures and Metrics 39Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Nearest Neighbor Classifier
- 40Social Media Mining Measures and Metrics 40Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Nearest Neighbor Classifier β’ π-nearest neighbor or π-NN, β Utilizes the neighbors of an instance to perform classification. β’ It uses the π nearest instances, called neighbors, to perform classification. β’ The instance being classified is assigned the label (class attribute value) that the majority of its π neighbors are assigned β’ When π = 1, the closest neighborβs label is used as the predicted label for the instance being classified β’ To determine the neighbors of an instance, we need to measure its distance to all other instances based on some distance metric. β Often Euclidean distance is employed
- 41Social Media Mining Measures and Metrics 41Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ π-NN: Algorithm
- 42Social Media Mining Measures and Metrics 42Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ π-NN example β’ When π = 5, the predicted label is: β’ When π = 9, the predicted label is: β β‘
- 43Social Media Mining Measures and Metrics 43Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ π-NN: Example 2 Data instance Outlook Temperature Humidity Similarity Label K Prediction 2 1 1 1 3 N 1 N 1 1 0 1 2 N 2 N 4 0 1 1 2 Y 3 N 3 0 0 1 1 Y 4 ? 5 1 0 0 1 Y 5 Y 6 0 0 0 0 N 6 ? 7 0 0 0 0 Y 7 Y Similarity between row 8 and other data instances; (Similarity = 1 if attributes have the same value, otherwise similarity = 0)
- 44Social Media Mining Measures and Metrics 44Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Classification with Network Information
- 45Social Media Mining Measures and Metrics 45Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Classification with Network Information β’ Consider a friendship network on social media and a product being marketed to this network. β’ The product seller wants to know who the potential buyers are for this product. β’ Assume we are given the network with the list of individuals that decided to buy or not buy the product. Our goal is to predict the decision for the undecided individuals. β’ This problem can be formulated as a classification problem based on features gathered from individuals. β’ However, in this case, we have additional friendship information that may be helpful in building better classification models
- 46Social Media Mining Measures and Metrics 46Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Classification with Network Information β’ Let π¦π denote the label for node π. β’ We can assume that β’ How can we estimate P(π¦π = 1|π π£π )?
- 47Social Media Mining Measures and Metrics 47Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Weighted-vote Relational-Neighbor (wvRN) wvRN provides an approach to estimate P(π¦π = 1|π π£π ) β’ In wvRN, to find the label of a node, we compute a weighted vote among its neighbors β’ P(π¦π = 1|π π£π ) is only calculated for unlabeled π£πβs β’ We need to compute these probabilities using some order until convergence [i.e., they donβt change] β What happens for different orders?
- 48Social Media Mining Measures and Metrics 48Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ wvRN example
- 49Social Media Mining Measures and Metrics 49Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ wvRN example
- 50Social Media Mining Measures and Metrics 50Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Regression
- 51Social Media Mining Measures and Metrics 51Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Regression In regression, β’ Class values are real numbers as class values β In classification, class values are categorical π¦ β π(π) Features (regressors) π = (π₯1, π₯2, β¦ , π₯ π) Class attribute (dependent variable) π¦οπ Goal: find the relation between π¦ and vector π = (π₯1, π₯2, β¦ , π₯ π)
- 52Social Media Mining Measures and Metrics 52Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Linear Regression β’ Linear regression: we assume the relation between the class attribute π and feature set πΏ is linear β’ π represents the vector of regression coefficients β’ Regression can be solved by estimating π and ο₯ using the provided dataset and the labels π β βLeast squaresβ is a popular method to solve regression β The goal is to minimize
- 53Social Media Mining Measures and Metrics 53Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Least Squares Find π such that minimizing Ηπ β ππΗ2 for regressors πΏ and labels π
- 54Social Media Mining Measures and Metrics 54Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Simplifying W with SVD SVD of πΏ
- 55Social Media Mining Measures and Metrics 55Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Logistic Regression β’ A probabilistic view of regression β’ Assuming the class attribute is binary, logistic regression finds the probability π β’ We can assume that π can be obtained from π β’ Unbounded: π can take any values and π½ is also unconstrained π: Feature vector
- 56Social Media Mining Measures and Metrics 56Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Logistic Regression β’ Solution: transform π using π(π), such that π(π) is unbounded β Fit π(π) using π½X β’ Known an as the logit function β’ For any π between [0,1] β πΊ(π) is in range [ββ, +β]
- 57Social Media Mining Measures and Metrics 57Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Logistic Regression Logistic Function β Acts as a probability β’ Goal: Find π½ such that π(π|π) is maximized β No closed form solution β Iterative maximum likelihood β’ Prediction: once π½ is determined compute π(π|π) β For a binary class problem if it is more than 0.5, predict 1
- 58Social Media Mining Measures and Metrics 58Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Supervised Learning Evaluation
- 59Social Media Mining Measures and Metrics 59Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Evaluating Supervised Learning β’ Training/Testing Framework: β A training dataset (i.e., the labels are known) is used to train a model β the model is evaluated on a test dataset. β’ The correct labels of the test dataset are unknown, β In practice, the training set is divided into two parts, β’ One used for training and β’ The other used for testing. β’ When testing, the labels from this test set are removed. β After these labels are predicted using the model, the predicted labels are compared with the masked labels (ground truth).
- 60Social Media Mining Measures and Metrics 60Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Evaluating Supervised Learning β’ Dividing the training set into train/test sets β Leave-one-out training β’ Divide the training set into π equally sized partitions β Often called folds β’ Use all folds but one to train and the one left out for testing β π-fold cross validation training β’ Divide the training set into π equally sized sets β’ Run the algorithm π times β’ In round π, we use all folds but fold π for training and fold π for testing. β’ The average performance of the algorithm over π rounds measures the performance of the algorithm.
- 61Social Media Mining Measures and Metrics 61Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Evaluating Supervised Learning β’ As the class labels are discrete, we can measure the accuracy by dividing number of correctly predicted labels (πΆ) by the total number of instances (π) β’ More sophisticated approaches of evaluation β’ AUC β’ F-Measure
- 62Social Media Mining Measures and Metrics 62Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Evaluating Regression Performance β’ The labels cannot be predicted precisely β’ We need to set a margin to accept or reject the predictions β Example. When the observed temperature is 71, any prediction in the range of 71Β±0.5 can be considered as a correct prediction β’ Or, we can use correlation between predicted labels and the ground truth.
- 63Social Media Mining Measures and Metrics 63Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Unsupervised Learning
- 64Social Media Mining Measures and Metrics 64Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Unsupervised Learning β’ Clustering is a form of unsupervised learning β’ Clustering algorithms group together similar items β’ The algorithm does not have examples showing how the samples should be grouped together (unlabeled data) Unsupervised division of instances into groups of similar objects Kernel Density Estimation
- 65Social Media Mining Measures and Metrics 65Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Measuring Distance/Similarity in Clustering β’ Clustering Goal: Group together similar items β’ Instances are put into different clusters based on the distance to other instances β’ Any clustering algorithm requires a distance measure The most popular (dis)similarity measure for continuous features are Euclidean Distance and Pearson Linear Correlation Euclidean Distance
- 66Social Media Mining Measures and Metrics 66Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Similarity Measures β’ π and π are π-dimensional vectors Once a distance measure is selected, instances are grouped using it.
- 67Social Media Mining Measures and Metrics 67Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Clustering β’ Clusters are usually represented by compact and abstract notations. β’ βCluster centroidsβ are one common example of this abstract notation. β’ Partitional Algorithms (most common type) β Partition the dataset into a set of clusters β Each instance is assigned to a cluster exactly once β No instance remains unassigned to clusters. β’ Example: π-means
- 68Social Media Mining Measures and Metrics 68Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ π-means for π = 6
- 69Social Media Mining Measures and Metrics 69Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ π-means The most commonly used clustering algorithm β Related to Expectation Maximization (EM) in statistics.
- 70Social Media Mining Measures and Metrics 70Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ π-means: Algorithm Given data points π₯π and an initial set of π centroids π1 1 , π2 1 ,β¦, π π 1 the algorithm proceeds as follows: β’ Assignment step: Assign each data point to the cluster ππ π‘ with the closest centroid β Each data point goes into exactly one cluster β’ Update step: Calculate the new means to be the centroid of the data points in the cluster β After all points are assigned
- 71Social Media Mining Measures and Metrics 71Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ When do we stop? The procedure is repeated until convergence β’ Convergence: β Whether centroids are no longer changing β Equivalent to clustering assignments not changing β The algorithm can be stopped when the Euclidean distance between the centroids in two consecutive steps is less than some small positive value
- 72Social Media Mining Measures and Metrics 72Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ π-means (alternative!) β’ As an alternative, π-means implementations try to minimize an objective function. β Example: the squared distance error: β π₯π π is the πth instance of cluster π β π(π) is the number of instances in cluster π β ππ is the centroid of cluster π. β’ Stopping Criterion: β when the difference between the objective function values of two consecutive iterations of the π-means algorithm is less than some small value .
- 73Social Media Mining Measures and Metrics 73Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ π-means β’ Finding the global optimum of the π partitions is computationally expensive (NP-hard). β’ This is equivalent to finding the optimal centroids that minimize the objective function β’ Solution: efficient heuristics β Outcome: converge quickly to a local optimum that might not be global β Example: running π-means multiple times β’ Select the clustering assignment that is observed most often or β’ Select the clustering that is more desirable based on an objective function, such as the squared error.
- 74Social Media Mining Measures and Metrics 74Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Unsupervised Learning Evaluation
- 75Social Media Mining Measures and Metrics 75Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Evaluating the Clusterings β’ Evaluation with ground truth β’ Evaluation without ground truth We are given two types of objects β’ In perfect clustering, objects of the same type are clustered together.
- 76Social Media Mining Measures and Metrics 76Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Evaluation with Ground Truth When ground truth is available, β We have prior knowledge on what the clustering should be (the correct clustering assignments) β’ We will discuss these methods in community analysis chapter
- 77Social Media Mining Measures and Metrics 77Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Evaluation without Ground Truth β’ Cohesiveness β In clustering, we are interested in clusters that exhibit cohesiveness β In cohesive clusters, instances inside the clusters are close to each other β’ Separateness β We are also interested in clusterings of the data that generates clusters that are well separated from one another
- 78Social Media Mining Measures and Metrics 78Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Cohesiveness β’ Cohesiveness β In statistics: having a small standard deviation, i.e., being close to the mean value β In clustering: being close to the centroid of the cluster
- 79Social Media Mining Measures and Metrics 79Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Separateness β’ Separateness β In statistics: separateness can be measured by standard deviation β’ Standard deviation is maximized when instances are far from the mean β In clustering: cluster centroids being far from the mean of the entire dataset
- 80Social Media Mining Measures and Metrics 80Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Silhouette Index β’ We are interested in clusters that are both cohesive and separate β Silhouette index β’ It compares β the average distance value between instances in the same cluster To β the average distance value between instances in different clusters β’ In a well-clustered dataset, β the average distance between instances in the same cluster is small (cohesiveness) and β the average distance between instances in different clusters is large (separateness).
- 81Social Media Mining Measures and Metrics 81Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Silhouette Index β’ For any instance π₯ that is a member of cluster πΆ β’ Compute the within-cluster average distance β’ Compute the average distance between π₯ and instances in cluster πΊ β πΊ is closest to π₯ in terms of the average distance between π₯ and members of πΊ
- 82Social Media Mining Measures and Metrics 82Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Silhouette Index β’ Our interest: clusterings where π(π₯) < π(π₯) β’ Silhouette can take values between [β1,1] β’ The best case happens when for all π₯, β π(π₯) = 0, π(π₯) > π(π₯)
- 83Social Media Mining Measures and Metrics 83Social Media Mining Data Mining Essentialshttp://socialmediamining.info/ Silhouette Index - Example

- Beer and chips are not interesting patterns, for example.
- Weather dataset
- number of documents where the term t appearsΒ
- Using the inverse of similarity as a distance measure
- \frac{\partial}{\partial W} {(W^T X^T Y )} = X^T Y \frac{\partial}{\partial W} { (W^T X^T X W) } = 2 X^T X W
- SVD form of matrix X: X = U \Sigma V^T U and V are real valued, they each are an orthogonal matrix U^T = U^{-1} V is orthogonal so: V^T^{-1} = V (V \Sigma^2 V^T)^{-1} = V^T^{-1} \Sigma^{-2} V^T = V \Sigma^{-2} V^T

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