Data Mining Anomaly Detection Lecture Notes for Chapter 10 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Anomaly/Outlier Detection l What are anomalies/outliers? –The set of data points that are considerably different than the remainder of the data l Variants of Anomaly/Outlier Detection Problems –Given a database D  find all the data points x  D with anomaly scores greater than some threshold t  find all the data points x  D having the top-k largest anomaly scores  containing mostly normal (but unlabeled) data points, and a test point x, compute the anomaly score of x with respect to D l Applications: –Credit card fraud detection, telecommunication fraud detection, network intrusion detection, fault detection

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Causes of anomalies l Data from different classes –Hawkins: differs so much from other observations  Could be generated by a different mechanism l Natural Variation –Some people are very tall/short/… l Data Measurement and Collection Errors

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Importance of Anomaly Detection Ozone Depletion History l In 1985 three researchers (Farman, Gardinar and Shanklin) were puzzled by data gathered by the British Antarctic Survey showing that ozone levels for Antarctica had dropped 10% below normal levels l Why did the Nimbus 7 satellite, which had instruments aboard for recording ozone levels, not record similarly low ozone concentrations? l The ozone concentrations recorded by the satellite were so low they were being treated as outliers by a computer program and discarded! Sources:

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Use of class labels l Class labels: normal, anomaly l Supervised AD –Both class labels are available –Not usually considered as AD — just Classification l Unsupervised AD –No class labels –Discussed in this chapter l Semi-supervised AD –All training instances are known normal –Generate an anomaly score for a test instance

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Issues l Number of attributes used to define an anomaly –2-foot tall is common, 200-pound heavy is common –2-foot and 200-pound together is not common l Global versus local perspective –6.5-foot is unusually tall, but not in basketball teams. l Degree/score of anomaly l Identify one at a time vs multiple at once –One: masking—some anomalies hide the rest –Multiple: swamping—some normal objects got lumped into the anomaly group l Evaluation l Efficiency

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Anomaly Detection l Challenges –How many outliers are there in the data? –Method is unsupervised  Validation can be quite challenging (just like for clustering) –Finding needle in a haystack l Working assumption: –There are considerably more “normal” observations than “abnormal” observations (outliers/anomalies) in the data

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Anomaly Detection Schemes l General Steps –Build a profile of the “normal” behavior  Profile can be patterns or summary statistics for the overall population –Use the “normal” profile to detect anomalies  Anomalies are observations whose characteristics differ significantly from the normal profile l Types of anomaly detection schemes –Graphical & Statistical-based –Distance-based –Density-based –Clustering-based

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Graphical Approaches l Boxplot (1-D), Scatter plot (2-D), Spin plot (3-D) l Limitations –Time consuming –Subjective

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Convex Hull Method l Extreme points are assumed to be outliers l Use convex hull method to detect extreme values l What if the outlier occurs in the middle of the data?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Anomaly/Outlier Detection l Proximity-based (10.3) l Density-based (10.4) l Clustering-based (10.5) l Statistical (10.1)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Nearest-Neighbor Based Approach l Approach: –Compute the distance between every pair of data points –There are various ways to define outliers:  fewer than p neighboring points within a distance d  The top n whose distance to the kth nearest neighbor is greatest  The top n whose average distance to the k nearest neighbors is greatest

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Strengths and Weaknesses l Simple l O(m^2) time [m data points] l Sensitive to the choice of parameters l Cannot handle different densities

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Outliers in Lower Dimensional Projection l In high-dimensional space, data is sparse and notion of proximity becomes meaningless –Every point is an almost equally good outlier from the perspective of proximity-based definitions l Lower-dimensional projection methods –A point is an outlier if in some lower dimensional projection, it is present in a local region of abnormally low density

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Outliers in Lower Dimensional Projection l Divide each attribute into  equal-depth intervals –Each interval contains a fraction f = 1/  of the records l Consider a k-dimensional cube created by picking grid ranges from k different dimensions –If attributes are independent, we expect region to contain a fraction f k of the records –If there are N points, we can measure sparsity of a cube D as: –Negative sparsity indicates cube contains smaller number of points than expected

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Example l N=100,  = 5, f = 1/5 = 0.2, N  f 2 = 4

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Anomaly/Outlier Detection l Proximity-based (10.3) l Density-based (10.4) l Clustering-based (10.5) l Statistical (10.1)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Handling different densities l Example from the original paper p 2  p 1 

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Density-based: LOF approach

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Density-based: LOF approach p 2  p 1  In the NN approach, p 2 is not considered as outlier, while LOF approach find both p 1 and p 2 as outliers

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Strengths and weaknesses l Handles different densities l O(m^2) l Selecting k –Vary k, use the max outlier score

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Anomaly/Outlier Detection l Proximity-based (10.3) l Density-based (10.4) l Clustering-based (10.5) l Statistical (10.1)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Clustering-based l How would you use clustering algorithms for anomaly detection?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Distant small clusters l Cluster the data into groups l Choose points in small cluster as candidate outliers l Compute the distance between candidate points and non-candidate clusters.  If candidate points are far from all other non-candidate points, they are outliers

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Distant Small Clusters l Parameters: –How small is small? –How far is far? l Sensitive to number of clusters (k) –When k is large, more “small” clusters

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Strength of cluster membership l Outlier if not strongly belong to any cluster –Prototype-based clustering  Distance from centroid –Density-based clustering  Low density (noise points in DBSCAN)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Prototype based clustering l Strength of cluster membership –Distance to centroid –What if clusters have different densities

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Prototype based clustering l Strength of cluster membership –Distance to centroid –What if clusters have different densities  Ideas?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Prototype based clustering

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Prototype based clustering

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Prototype based clustering

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Impact of Outliers to Initial Clustering l We cluster first –But outliers affect clustering l Simple approach –Cluster first –Remove outliers –Cluster again

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Impact of Outliers to Initial Clustering l More sophisticated approach –Special group of potential outliers  don’t fit well in any cluster –While clustering  Add to group if an object doesn’t fit well  Remove from group if an object fits well

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Number of clusters to use l No simple answer –Try different numbers –Try more clusters  Clusters are smaller –More cohesive –Detected outliers are more likely to be real outliers –However, outliers can form small clusters

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Strengths and weaknesses l Definition of clustering is complementary to outliers l Sensitive to number of clusters l Sensitive to the clustering algorithm

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Anomaly/Outlier Detection l Proximity-based (10.3) l Density-based (10.4) l Clustering-based (10.5) l Statistical (10.1)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Statistical Approaches l Univariate (10.2.1) l Multivariate (10.2.2) l Mixture Model (10.2.3)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Univariate Distribution l Assume a parametric model describing the distribution of the data (e.g., normal distribution) l Apply a statistical test that depends on –Data distribution –Parameter of distribution (e.g., mean, variance) –Number of expected outliers (confidence limit)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Grubbs’ Test l Detect outliers in univariate data l Assume data comes from normal distribution l Detects one outlier at a time, remove the outlier, and repeat –H 0 : There is no outlier in data –H A : There is at least one outlier l Grubbs’ test statistic: l Reject H 0 if:

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Statistical Approaches l Univariate (10.2.1) l Multivariate (10.2.2) l Mixture Model (10.2.3)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Multivariate distribution

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Note the x and y axes don’t have the same scale

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Mahalanobis Distance

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Statistical Approaches l Univariate (10.2.1) l Multivariate (10.2.2) l Mixture Model (10.2.3)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Mixture Model Approach l Assume the data set D contains samples from a mixture of two probability distributions: –M (majority distribution) –A (anomalous distribution)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Mixture Model Approach l Data distribution, D = (1 – ) M + A l M is a probability distribution estimated from data –Can be based on any modeling method (naïve Bayes, maximum entropy, etc) l A is initially assumed to be uniform distribution l Likelihood at time t:

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Mixture Model Approach l General Algorithm: –Initially, assume all the data points belong to M –Let LL t (D) be the log likelihood of D at time t –For each point x t that belongs to M, move it to A  Let LL t+1 (D) be the new log likelihood.  Compute the difference,  = LL t (D) – LL t+1 (D)  If  > c (some threshold), then x t is declared as an anomaly and moved permanently from M to A

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Strengths and weaknesses l Grounded in statistics l Most of the tests are for a single attribute (univariate) l In many cases, data distribution may not be known l For high dimensional data, it may be difficult to estimate the true distribution

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Evaluation of Anomaly Detection l Accuracy is not appropriate for anomaly detection –why?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Evaluation of Anomaly Detection

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Evaluation l Accuracy is not appropriate for anomaly detection –Easy to get high accuracy--just predict normal l Ideas?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Evaluation l Accuracy is not appropriate for anomaly detection –Easy to get high accuracy--just predict normal l We discussed these in Classification –Cost matrix for different errors –Precision and Recall (F measure) –True-positive and false-positive rates  Receiver Operating Characteristic (ROC) curve –Area under the curve (AUC)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Base Rate Fallacy (Axelsson, 1999) l Diagnosing Disease –Test is 99% accurate  True-positive is 99%  True-negative is 99% l Bad news –Test is positive l Good news –1 in 10,000 has the disease l Are you more likely to have the disease or not?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Base Rate Fallacy l Bayes theorem: l More generally:

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Total Probability

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Base Rate Fallacy l S=Disease; P=Test l Even though the test is 99% certain, your chance of having the disease is 1/100, because the population of healthy people is much larger than sick people

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Base Rate Fallacy in Intrusion Detection l I: intrusive behavior,  I: non-intrusive behavior A: alarm  A: no alarm l Detection rate (true positive rate): P(A|I) l False alarm rate (false positive rate): P(A|  I) l Goal is to maximize both –“Bayesian detection rate,” P(I|A) [precision] –P(  I|  A)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Detection Rate vs False Alarm Rate l Suppose: l Then: l False alarm rate becomes more dominant if P(I) is very low

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Detection Rate vs False Alarm Rate l Axelsson: We need a very low false alarm rate to achieve a reasonable Bayesian detection rate