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**Data Mining Classification: Basic Concepts, Decision Trees, and Model Evaluation**

Tan, Steinbach, Kumar © Tan,Steinbach, Kumar Introduction to Data Mining /18/

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**Classification: Definition**

Given a collection of records (training set ) Each record contains a set of attributes, one of the attributes is the class. Find a model for class attribute as a function of the values of other attributes. Goal: previously unseen records should be assigned a class as accurately as possible. A test set is used to determine the accuracy of the model. Usually, the given data set is divided into training and test sets, with training set used to build the model and test set used to validate it.

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**Example of a Decision Tree**

categorical continuous class Splitting Attributes Refund Yes No NO MarSt Single, Divorced Married TaxInc NO < 80K > 80K NO YES Training Data Model: Decision Tree

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**Hunt’s Algorithm Refund Don’t Cheat Yes No Don’t Cheat Refund Don’t**

Marital Status Single, Divorced Married Refund Don’t Cheat Yes No Marital Status Single, Divorced Married Taxable Income < 80K >= 80K

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**Tree Induction Greedy strategy. Issues**

Split the records based on an attribute test that optimizes certain criterion. Issues Determine how to split the records How to specify the attribute test condition? How to determine the best split? Determine when to stop splitting

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**Measures of Node Impurity**

Gini Index Entropy Misclassification error

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**How to Find the Best Split**

Before Splitting: M0 A? B? Yes No Yes No Node N1 Node N2 Node N3 Node N4 M1 M2 M3 M4 M12 M34 Gain = M0 – M12 vs M0 – M34

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**Measure of Impurity: GINI**

Gini Index for a given node t : (NOTE: p( j | t) is the relative frequency of class j at node t). Maximum (1 - 1/nc) when records are equally distributed among all classes, implying least interesting information Minimum (0.0) when all records belong to one class, implying most interesting information

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**Examples for computing GINI**

P(C1) = 0/6 = P(C2) = 6/6 = 1 Gini = 1 – P(C1)2 – P(C2)2 = 1 – 0 – 1 = 0 P(C1) = 1/ P(C2) = 5/6 Gini = 1 – (1/6)2 – (5/6)2 = 0.278 P(C1) = 2/ P(C2) = 4/6 Gini = 1 – (2/6)2 – (4/6)2 = 0.444

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**Splitting Based on GINI**

Used in CART, SLIQ, SPRINT. When a node p is split into k partitions (children), the quality of split is computed as, where, ni = number of records at child i, n = number of records at node p.

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**Binary Attributes: Computing GINI Index**

Splits into two partitions Effect of Weighing partitions: Larger and Purer Partitions are sought for. B? Yes No Node N1 Node N2 Gini(N1) = 1 – (5/6)2 – (2/6)2 = 0.194 Gini(N2) = 1 – (1/6)2 – (4/6)2 = 0.528 Gini(Children) = 7/12 * /12 * = 0.333

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**Categorical Attributes: Computing Gini Index**

For each distinct value, gather counts for each class in the dataset Use the count matrix to make decisions Multi-way split Two-way split (find best partition of values)

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**Continuous Attributes: Computing Gini Index**

Use Binary Decisions based on one value Several Choices for the splitting value Number of possible splitting values = Number of distinct values Each splitting value has a count matrix associated with it Class counts in each of the partitions, A < v and A v Simple method to choose best v For each v, scan the database to gather count matrix and compute its Gini index Computationally Inefficient! Repetition of work.

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**Continuous Attributes: Computing Gini Index...**

For efficient computation: for each attribute, Sort the attribute on values Linearly scan these values, each time updating the count matrix and computing gini index Choose the split position that has the least gini index Split Positions Sorted Values

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**Alternative Splitting Criteria based on Entropy**

Entropy at a given node t: (NOTE: p( j | t) is the relative frequency of class j at node t). Measures homogeneity of a node. Maximum (log nc) when records are equally distributed among all classes implying least information Minimum (0.0) when all records belong to one class, implying most information Entropy based computations are similar to the GINI index computations

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**Examples for computing Entropy**

P(C1) = 0/6 = P(C2) = 6/6 = 1 Entropy = – 0 log 0 – 1 log 1 = – 0 – 0 = 0 P(C1) = 1/ P(C2) = 5/6 Entropy = – (1/6) log2 (1/6) – (5/6) log2 (1/6) = 0.65 P(C1) = 2/ P(C2) = 4/6 Entropy = – (2/6) log2 (2/6) – (4/6) log2 (4/6) = 0.92

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**Splitting Based on INFO...**

Information Gain: Parent Node, p is split into k partitions; ni is number of records in partition i Measures Reduction in Entropy achieved because of the split. Choose the split that achieves most reduction (maximizes GAIN) Used in ID3 and C4.5 Disadvantage: Tends to prefer splits that result in large number of partitions, each being small but pure.

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**Splitting Based on INFO...**

Gain Ratio: Parent Node, p is split into k partitions ni is the number of records in partition i Adjusts Information Gain by the entropy of the partitioning (SplitINFO). Higher entropy partitioning (large number of small partitions) is penalized! Used in C4.5 Designed to overcome the disadvantage of Information Gain

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**Tree Induction Greedy strategy. Issues**

Split the records based on an attribute test that optimizes certain criterion. Issues Determine how to split the records How to specify the attribute test condition? How to determine the best split? Determine when to stop splitting

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**Stopping Criteria for Tree Induction**

Stop expanding a node when all the records belong to the same class Stop expanding a node when all the records have similar attribute values Early termination (to be discussed later)

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**Decision Tree Based Classification**

Advantages: Inexpensive to construct Extremely fast at classifying unknown records Easy to interpret for small-sized trees Accuracy is comparable to other classification techniques for many simple data sets

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**Metrics for Model Evaluation**

Focus on the predictive capability of a model Rather than how fast it takes to classify or build models, scalability, etc. Confusion Matrix: a: TP (true positive) b: FN (false negative) c: FP (false positive) d: TN (true negative) PREDICTED CLASS ACTUAL CLASS Class=Yes Class=No a b c d

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**Metrics for Performance Evaluation…**

Most widely-used metric: PREDICTED CLASS ACTUAL CLASS Class=Yes Class=No a (TP) b (FN) c (FP) d (TN)

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**Limitation of Accuracy**

Consider a 2-class problem Number of Class 0 examples = 9990 Number of Class 1 examples = 10 If model predicts everything to be class 0, accuracy is 9990/10000 = 99.9 % Accuracy is misleading because model does not detect any class 1 example

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**Cost Matrix PREDICTED CLASS C(i|j) ACTUAL CLASS**

Class=Yes Class=No C(Yes|Yes) C(No|Yes) C(Yes|No) C(No|No) C(i|j): Cost of misclassifying class j example as class i

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**Computing Cost of Classification**

Cost Matrix PREDICTED CLASS ACTUAL CLASS C(i|j) + - -1 100 1 Model M1 PREDICTED CLASS ACTUAL CLASS + - 150 40 60 250 Model M2 PREDICTED CLASS ACTUAL CLASS + - 250 45 5 200 Accuracy = 80% Cost = 3910 Accuracy = 90% Cost = 4255

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**Cost-Sensitive Measures**

Precision is biased towards C(Yes|Yes) & C(Yes|No) Recall is biased towards C(Yes|Yes) & C(No|Yes) F-measure is biased towards all except C(No|No)

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**Data Mining Cluster Analysis: Basic Concepts and Algorithms**

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**What is Cluster Analysis?**

Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups Inter-cluster distances are maximized Intra-cluster distances are minimized

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**Notion of a Cluster can be Ambiguous**

How many clusters? Six Clusters Two Clusters Four Clusters

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**Types of Clusterings A clustering is a set of clusters**

Important distinction between hierarchical and partitional sets of clusters Partitional Clustering A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset Hierarchical clustering A set of nested clusters organized as a hierarchical tree

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**Partitional Clustering**

A Partitional Clustering Original Points

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**Hierarchical Clustering**

Traditional Hierarchical Clustering Traditional Dendrogram Non-traditional Hierarchical Clustering Non-traditional Dendrogram

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**Other Distinctions Between Sets of Clusters**

Exclusive versus non-exclusive In non-exclusive clusterings, points may belong to multiple clusters. Can represent multiple classes or ‘border’ points Fuzzy versus non-fuzzy In fuzzy clustering, a point belongs to every cluster with some weight between 0 and 1 Weights must sum to 1 Probabilistic clustering has similar characteristics Partial versus complete In some cases, we only want to cluster some of the data Heterogeneous versus homogeneous Cluster of widely different sizes, shapes, and densities

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**Types of Clusters Well-separated clusters Center-based clusters**

Contiguous clusters Density-based clusters Property or Conceptual Described by an Objective Function

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**Types of Clusters: Well-Separated**

Well-Separated Clusters: A cluster is a set of points such that any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster. 3 well-separated clusters

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**Types of Clusters: Center-Based**

A cluster is a set of objects such that an object in a cluster is closer (more similar) to the “center” of a cluster, than to the center of any other cluster The center of a cluster is often a centroid, the average of all the points in the cluster, or a medoid, the most “representative” point of a cluster 4 center-based clusters

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**Types of Clusters: Contiguity-Based**

Contiguous Cluster (Nearest neighbor or Transitive) A cluster is a set of points such that a point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster. 8 contiguous clusters

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**Types of Clusters: Density-Based**

A cluster is a dense region of points, which is separated by low-density regions, from other regions of high density. Used when the clusters are irregular or intertwined, and when noise and outliers are present. 6 density-based clusters

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**Types of Clusters: Conceptual Clusters**

Shared Property or Conceptual Clusters Finds clusters that share some common property or represent a particular concept. . 2 Overlapping Circles

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**Clustering Algorithms**

K-means and its variants Hierarchical clustering Density-based clustering

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**K-means Clustering Partitional clustering approach**

Each cluster is associated with a centroid (center point) Each point is assigned to the cluster with the closest centroid Number of clusters, K, must be specified The basic algorithm is very simple

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**K-means Clustering – Details**

Initial centroids are often chosen randomly. Clusters produced vary from one run to another. The centroid is (typically) the mean of the points in the cluster. ‘Closeness’ is measured by Euclidean distance, cosine similarity, correlation, etc. K-means will converge for common similarity measures mentioned above. Most of the convergence happens in the first few iterations. Often the stopping condition is changed to ‘Until relatively few points change clusters’ Complexity is O( n * K * I * d ) n = number of points, K = number of clusters, I = number of iterations, d = number of attributes

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**Two different K-means Clusterings**

Original Points Optimal Clustering Sub-optimal Clustering

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**Importance of Choosing Initial Centroids**

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**Importance of Choosing Initial Centroids …**

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**Evaluating K-means Clusters**

Most common measure is Sum of Squared Error (SSE) For each point, the error is the distance to the nearest cluster To get SSE, we square these errors and sum them. x is a data point in cluster Ci and mi is the representative point for cluster Ci can show that mi corresponds to the center (mean) of the cluster Given two clusters, we can choose the one with the smallest error One easy way to reduce SSE is to increase K, the number of clusters A good clustering with smaller K can have a lower SSE than a poor clustering with higher K

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**Limitations of K-means**

K-means has problems when clusters are of differing Sizes Densities Non-globular shapes K-means has problems when the data contains outliers.

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**Limitations of K-means: Differing Sizes**

Original Points K-means (3 Clusters)

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**Limitations of K-means: Differing Density**

Original Points K-means (3 Clusters)

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**Limitations of K-means: Non-globular Shapes**

Original Points K-means (2 Clusters)

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**Overcoming K-means Limitations**

Original Points K-means Clusters One solution is to use many clusters. Find parts of clusters, but need to put together.

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**Overcoming K-means Limitations**

Original Points K-means Clusters

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**Overcoming K-means Limitations**

Original Points K-means Clusters

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**Hierarchical Clustering**

Produces a set of nested clusters organized as a hierarchical tree Can be visualized as a dendrogram A tree like diagram that records the sequences of merges or splits

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**Strengths of Hierarchical Clustering**

Do not have to assume any particular number of clusters Any desired number of clusters can be obtained by ‘cutting’ the dendogram at the proper level They may correspond to meaningful taxonomies Example in biological sciences (e.g., animal kingdom, phylogeny reconstruction, …)

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**Hierarchical Clustering**

Two main types of hierarchical clustering Agglomerative: Start with the points as individual clusters At each step, merge the closest pair of clusters until only one cluster (or k clusters) left Divisive: Start with one, all-inclusive cluster At each step, split a cluster until each cluster contains a point (or there are k clusters) Traditional hierarchical algorithms use a similarity or distance matrix Merge or split one cluster at a time

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