1. Data Mining Classification: Basic Concepts, Decision Trees, and Model Evaluation
Tan, Steinbach, Kumar
© Tan,Steinbach, Kumar Introduction to Data Mining /18/

2. 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.

3. 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

4. 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

5. 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

6. Measures of Node Impurity
Gini Index
Entropy
Misclassification error

7. 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

8. 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

9. 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

10. 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.

11. 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

12. 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)

13. 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.

14. 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

15. 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

16. 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

17. 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.

18. 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

19. 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

20. 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)

21. 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

22. 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

23. Metrics for Performance Evaluation…
Most widely-used metric:
PREDICTED CLASS
ACTUAL CLASS
Class=Yes
Class=No
a (TP)
b (FN)
c (FP)
d (TN)

24. 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

25. 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

26. 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

27. 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)

28. Data Mining Cluster Analysis: Basic Concepts and Algorithms

29. 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

30. Notion of a Cluster can be Ambiguous
How many clusters?
Six Clusters
Two Clusters
Four Clusters

31. 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

32. Partitional Clustering
A Partitional Clustering
Original Points

33. Hierarchical Clustering
Traditional Hierarchical Clustering
Traditional Dendrogram
Non-traditional Hierarchical Clustering
Non-traditional Dendrogram

34. 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

35. Types of Clusters Well-separated clusters Center-based clusters
Contiguous clusters
Density-based clusters
Property or Conceptual
Described by an Objective Function

36. 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

37. 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

38. 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

39. 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

40. 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

41. Clustering Algorithms
K-means and its variants
Hierarchical clustering
Density-based clustering

42. 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

43. 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

44. Two different K-means Clusterings
Original Points
Optimal Clustering
Sub-optimal Clustering

45. Importance of Choosing Initial Centroids

46. Importance of Choosing Initial Centroids …

47. 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

48. 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.

49. Limitations of K-means: Differing Sizes
Original Points
K-means (3 Clusters)

50. Limitations of K-means: Differing Density
Original Points
K-means (3 Clusters)

51. Limitations of K-means: Non-globular Shapes
Original Points
K-means (2 Clusters)

52. 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.

53. Overcoming K-means Limitations
Original Points K-means Clusters

54. Overcoming K-means Limitations
Original Points K-means Clusters

55. 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

56. 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, …)

57. 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