1. Performance Metrics for Graph Mining Tasks

2. Outline Introduction to Performance Metrics
Supervised Learning Performance Metrics
Unsupervised Learning Performance Metrics
Optimizing Metrics
Statistical Significance Techniques
Model Comparison

3. Outline Introduction to Performance Metrics
Supervised Learning Performance Metrics
Unsupervised Learning Performance Metrics
Optimizing Metrics
Statistical Significance Techniques
Model Comparison

4. Introduction to Performance Metrics
Performance metric measures how well your data mining algorithm is performing on a given dataset. For example, if we apply a classification algorithm on a dataset, we first check to see how many of the data points were classified correctly. This is a performance metric and the formal name for it is “accuracy.” Performance metrics also help us decide is one algorithm is better or worse than another. For example, one classification algorithm A classifies 80% of data points correctly and another classification algorithm B classifies 90% of data points correctly. We immediately realize that algorithm B is doing better. There are some intricacies that we will discuss in this chapter.

5. Outline Introduction to Performance Metrics
Supervised Learning Performance Metrics
Unsupervised Learning Performance Metrics
Optimizing Metrics
Statistical Significance Techniques
Model Comparison

6. Supervised Learning Performance Metrics
Metrics that are applied when the ground truth is known (E.g., Classification tasks)
Outline:
2 X 2 Confusion Matrix
Multi-level Confusion Matrix
Visual Metrics
Cross-validation

7. 2X2 Confusion Matrix f++ f+- C = f++ + f+- f-+ f-- D = f-+ + f--
An 2X2 matrix, is used to tabulate the results of 2-class supervised learning problem and entry (i,j) represents the number of elements with class label i, but predicted to have class label j.
Predicted
Class + -
Actual
f++
f+-
C = f++ + f+-
f-+
f--
D = f-+ + f--
A = f++ + f-+
B = f+- + f--
T = f+++ f-++ f+-+ f--
False Negative
True Positive
False Positive
True Negative
+ and – are two class labels

8. 2X2 Confusion Metrics Example
Results from a Classification Algorithms
Corresponding
2x2 matrix for the given table
Vertex ID
Actual
Class
Predicted 1 + 2 3 4 5 - 6 7 8
Predicted
Class + -
Actual 4 1
C = 5 2
D = 3
A = 6
B = 2
T = 8
True positive = 4
False positive = 1
True Negative = 1
False Negative =2

9. 2X2 Confusion Metrics Performance Metrics
Walk-through different metrics using the following example
Accuracy is proportion of correct predictions
Error rate is proportion of incorrect predictions
3. Recall is the proportion of “+” data points predicted as “+”
4. Precision is the proportion of data points predicted as “+” that are truly “+”

10. Multi-level Confusion Matrix
An nXn matrix, where n is the number of classes and entry (i,j) represents the number of elements with class label i, but predicted to have class label j

11. Multi-level Confusion Matrix Example
Predicted Class
Marginal Sum of
Actual Values
Class 1
Class 2
Class 3
Actual
Class 2 1 4 3 6
Marginal Sum of Predictions 5
T = 14

12. Multi-level Confusion Matrix Conversion to 2X2
Predicted Class
Class 1
Class 2
Class 3
Actual
Class 2 1 3
f-+
f+-
f--
We can now apply all the 2X2 metrics
Predicted Class
Class 1 (+)
Not Class 1 (-)
Actual
Class 2
C = 4 8
D = 10
A = 4
B = 10
T = 14
2X2 Matrix
Specific to Class 1
Accuracy = 2/14
Error = 8/14
Recall = 2/4
Precision = 2/4

13. Multi-level Confusion Matrix Performance Metrics
Predicted Class
Class 1
Class 2
Class 3
Actual
Class 2 1 3
Critical Success Index or Threat Score is the ratio of correct predictions for class L to the sum of vertices that belong to L and those predicted as L
2. Bias - For each class L, it is the ratio of the total points with class label L to the number of points predicted as L.
Bias helps understand if a model is over or
under-predicting a class

14. Confusion Metrics R-code
library(PerformanceMetrics)
data(M) M
[,1] [,2]
[1,] 4 1
[2,] 2 1
twoCrossConfusionMatrixMetrics(M)
data(MultiLevelM)
MultiLevelM
[,1] [,2] [,3]
[1,] 2 1 1
[2,] 1 2 1
[3,] 1 2 3
multilevelConfusionMatrixMetrics(MultiLevelM)

15. Visual Metrics ROC plot
Metrics that are plotted on a graph to obtain the visual picture of the performance of two class classifiers 1
False Positive Rate
True Positive Rate
(0,1) - Ideal
(0,0)
Predicts the –ve
class all the time
(1,1)
Predicts the +ve
Random Guessing Models
AUC = 0.5
ROC plot
Plot the performance of multiple models to decide which one performs best

16. Understanding Model Performance based on ROC Plot
Models that lie in upper right are liberal.
Will predict “+” with little evidence
High False positives
Models that lie in this upper left have good performance
Note: This is where you aim to get the model 1
False Positive Rate
True Positive Rate
Random Guessing Models
AUC = 0.5
Models that lie in lower left are conservative.
Will not predict “+” unless strong evidence
Low False positives but high False Negatives
Models that lie in this area perform worse than random
Note: Models here can be negated to move them to the upper right corner

17. ROC Plot Example M1 (0.1,0.8) M3 (0.3,0.5) M2 (0.5,0.5)1
False Positive Rate
True Positive Rate
M1 (0.1,0.8)
M2 (0.5,0.5)
M3 (0.3,0.5)
M1’s performance occurs furthest
in the upper-right direction and hence is considered the best model.

18. Cross-validation
Cross-validation also called rotation estimation, is a way to analyze how a predictive data mining model will perform on an unknown dataset, i.e., how well the model generalizes
Strategy:
Divide up the dataset into two non-overlapping subsets
One subset is called the “test” and the other the “training”
Build the model using the “training” dataset
Obtain predictions of the “test” set
Utilize the “test” set predictions to calculate all the performance metrics
Typically cross-validation is performed for multiple iterations,
selecting a different non-overlapping test and training set each time

19. Types of Cross-validation
hold-out: Random 1/3rd of the data is used as test and remaining 2/3rd as training
k-fold: Divide the data into k partitions, use one partition as test and remaining k-1 partitions for training
Leave-one-out: Special case of k-fold, where k=1
Note: Selection of data points is typically done in stratified manner, i.e., the class distribution in the test set is similar to the training set

20. Outline Introduction to Performance Metrics
Supervised Learning Performance Metrics
Unsupervised Learning Performance Metrics
Optimizing Metrics
Statistical Significance Techniques
Model Comparison

21. Unsupervised Learning Performance Metrics
Metrics that are applied when the ground truth is not always available (E.g., Clustering tasks)
Outline:
Evaluation Using Prior Knowledge
Evaluation Using Cluster Properties

22. Evaluation Using Prior Knowledge
To test the effectiveness of unsupervised learning methods is by considering a dataset D with known class labels, stripping the labels and providing the set as input to any unsupervised leaning algorithm, U. The resulting clusters are then compared with the knowledge priors to judge the performance of U
To evaluate performance
Contingency Table
Ideal and Observed Matrices

23. Contingency Table Cluster Same Cluster Different Cluster Class
Same Class
u11
u10
Different Class
u01
u00
(A) To fill the table, initialize u11, u01, u10, u00 to 0
(B) Then, for each pair of points of form (v,w):
if v and w belong to the same class and cluster then increment u11
if v and w belong to the same class but different cluster then increment u10
if v and w belong to the different class but same cluster then increment u01
if v and w belong to the different class and cluster then increment u00

24. Contingency Table Performance Metrics
Example Matrix
Cluster
Same Cluster
Different Cluster
Class
Same Class 9 4
Different Class 3 12
Rand Statistic also called simple matching coefficient is a measure where both placing a pair of points with the same class label in the same cluster and placing a pair of points with different class labels in different clusters are given equal importance, i.e., it accounts for both specificity and sensitivity of the clustering
Jaccard Coefficient can be utilized when placing a pair of points with the same class label in the same cluster is primarily important

25. Ideal and Observed Matrices
Given that the number of points is T, the ideal-matrix is a TxT matrix, where each cell (i,j) has a 1 if the points i and j belong to the same class and a 0 if they belong to different clusters. The observed-matrix is a TxT matrix, where a cell (i,j) has a 1 if the points i and j belong to the same cluster and a 0 if they belong to different cluster
Mantel Test is a statistical test of the correlation between two matrices of the same rank. The two matrices, in this case, are symmetric and, hence, it is sufficient to analyze lower or upper diagonals of each matrix

26. Evaluation Using Prior Knowledge R-code
library(PerformanceMetrics)
data(ContingencyTable)
ContingencyTable
[,1] [,2]
[1,] 9 4
[2,] 3 12
contingencyTableMetrics(ContingencyTable)

27. Evaluation Using Cluster Properties
In the absence of prior knowledge we have to rely on the information from the clusters themselves to evaluate performance.
Cohesion measures how closely objects in the same cluster are related
Separation measures how distinct or separated a cluster is from all the other clusters
Here, gi refers to cluster i, W is total number of clusters, x and y are data points, proximity can be any similarity measure (e.g., cosine similarity)
We want the cohesion to be close to 1 and separation to be close to 0

28. Outline Introduction to Performance Metrics
Supervised Learning Performance Metrics
Unsupervised Learning Performance Metrics
Optimizing Metrics
Statistical Significance Techniques
Model Comparison

29. Optimizing Metrics
Performance metrics that act as optimization functions for a data mining algorithm
Outline:
Sum of Squared Errors
Preserved Variability

30. Sum of Squared Errors
Squared sum error (SSE) is typically used in clustering algorithms to measure the quality of the clusters obtained. This parameter takes into consideration the distance between each point in a cluster to its cluster center (centroid or some other chosen representative). For dj, a point in cluster gi, where mi is the cluster center of gi, and W, the total number of clusters, SSE is defined as follows: This value is small when points are close to their cluster center, indicating a good clustering. Similarly, a large SSE indicates a poor clustering. Thus, clustering algorithms aim to minimize SSE.

31. Preserved Variability
Preserved variability is typically used in eigenvector-based dimension reduction techniques to quantify the variance preserved by the chosen dimensions. The objective of the dimension reduction technique is to maximize this parameter. Given that the point is represented in r dimensions (k << r), the eigenvalues are λ1>=λ2>=….. λr-1>=λr. The preserved variability (PV) is calculated as follows: The value of this parameter depends on the number of dimensions chosen: the more included, the higher the value. Choosing all the dimensions will result in the perfect score of 1.

32. Outline Introduction to Performance Metrics
Supervised Learning Performance Metrics
Unsupervised Learning Performance Metrics
Optimizing Metrics
Statistical Significance Techniques
Model Comparison

33. Statistical Significance Techniques
Methods used to asses a p-value for the different performance metrics
Scenario:
We obtain say cohesion =0.99 for clustering algorithm A. From the first look it feels like 0.99 is a very good score.
However, it is possible that the underlying data is structured in such a way that you would get 0.99 no matter how you cluster the data.
Thus, 0.99 is not very significant. One way to decide that is by using statistical significance estimation.
We will discuss the Monte Carlo Procedure in next slide!

34. Monte Carlo Procedure Empirical p-value Estimation
Monte Carlo procedure uses random sampling to assess the significance of a particular performance metric we obtain could have been attained at random.
For example, if we obtain a cohesion score of a cluster of size 5 is 0.99, we would be inclined to think that it is a very cohesive score. However, this value could have resulted due to the nature of the data and not due to the algorithm. To test the significance of this 0.99 value we
Sample N (usually 1000) random sets of size 5 from the dataset
Recalculate the cohesion for each of the 1000 sets
Count R: number of random sets with value >= 0.99 (original score of cluster)
Empirical p-value for the cluster of size 5 with 0.99 score is given by R/N
We apply a cutoff say 0.05 to decide if 0.99 is significant
Steps 1-4 is the Monte Carlo method for p-value estimation.

35. Outline Introduction to Performance Metrics
Supervised Learning Performance Metrics
Unsupervised Learning Performance Metrics
Optimizing Metrics
Statistical Significance Techniques
Model Comparison

36. Model Comparison
Metrics that compare the performance of different algorithms
Scenario:
Model 1 provides an accuracy of 70% and Model 2 provides an accuracy of 75%
From the first look, Model 2 seems better, however it could be that Model 1 is predicting Class1 better than Class2
However, Class1 is indeed more important than Class2 for our problem
We can use model comparison methods to take this notion of “importance” into consideration when we pick one model over another
Cost-based Analysis is an important model comparison method discussed in the next few slides.

37. Cost-based Analysis
In real-world applications, certain aspects of model performance are considered more important than others. For example: if a person with cancer was diagnosed as cancer-free or vice-versa then the prediction model should be especially penalized. This penalty can be introduced in the form of a cost-matrix.
Cost
Matrix
Predicted
Class + -
Actual
c11
c10
c01
c00
Associated with f10 or u10
Associated with f11 or u11
Associated with f01 or u01
Associated with f00 or u00

38. Cost-based Analysis Cost of a Model
The cost and confusion matrices for Model M are given below Cost of Model M is given as:
Confusion Matrix
Predicted
Class + -
Actual
f11
f10
f01
f00
Cost Matrix
Predicted
Class + -
Actual
c11
c10
c01
c00

39. Cost-based Analysis Comparing Two Models
This analysis is typically used to select one model when we have more than one choice through using different algorithms or different parameters to the learning algorithms.
Cost
Matrix
Predicted
Class + -
Actual
-20
100 45
-10
Confusion
Matrix of My
Predicted
Class + -
Actual 3 2 1
Confusion Matrix of Mx
Predicted
Class + -
Actual 4 1 2
Cost of My : 200
Cost of Mx: 100
CMx < CMy
Purely, based on cost model, Mx is a better model

40. Cost-based Analysis R-code
library(PerformanceMetrics)
data(Mx)
data(My)
data(CostMatrix) Mx
[,1] [,2]
[1,] 4 1
[2,] 2 1 My
[1,] 3 2
costAnalysis(Mx,CostMatrix)
costAnalysis(My,CostMatrix)