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Data Mining
Anomaly Detection

Lecture Notes for Chapter 10

Introduction to Data Mining

Tan, Steinbach, Kumar

Anomaly/Outlier Detection

What are anomalies/outliers?
The set of data points that are considerably different than the remainder of the data
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
Given a database D, find all the data points x  D having the top-n largest anomaly scores f(x)
Given a database D, containing mostly normal (but unlabeled) data points, and a test point x, compute the anomaly score of x with respect to D
Applications:
Credit card fraud detection, telecommunication fraud detection, network intrusion detection, fault detection

Importance of Anomaly Detection

Ozone Depletion History

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

Why did the Nimbus 7 satellite, which had instruments aboard for recording ozone levels, not record similarly low ozone concentrations?

The ozone concentrations recorded by the satellite were so low they were being treated as outliers by a computer program and discarded!

Sources:
http://exploringdata.cqu.edu.au/ozone.html
http://www.epa.gov/ozone/science/hole/size.html

Anomaly Detection

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

Working assumption:
There are considerably more “normal” observations than “abnormal” observations (outliers/anomalies) in the data

Anomaly Detection Schemes

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

Types of anomaly detection
schemes
Graphical & Statistical-based
Distance-based
Model-based

Graphical Approaches

Boxplot (1-D), Scatter plot (2-D), Spin plot (3-D)

Limitations
Time consuming
Subjective

Convex Hull Method

Extreme points are assumed to be outliers
Use convex hull method to detect extreme values

What if the outlier occurs in the middle of the data?

Statistical Approaches

Assume a parametric model describing the distribution of the data (e.g., normal distribution)

Apply a statistical test that depends on
Data distribution
Parameter of distribution (e.g., mean, variance)
Number of expected outliers (confidence limit)

Grubbs’ Test

Detect outliers in univariate data
Assume data comes from normal distribution
Detects one outlier at a time, remove the outlier, and repeat
H0: There is no outlier in data
HA: There is at least one outlier
Grubbs’ test statistic:

Reject H0 if:

Statistical-based – Likelihood Approach

Assume the data set D contains samples from a mixture of two probability distributions:
M (majority distribution)
A (anomalous distribution)
General Approach:
Initially, assume all the data points belong to M
Let Lt(D) be the log likelihood of D at time t
For each point xt that belongs to M, move it to A
Let Lt+1 (D) be the new log likelihood.
Compute the difference,  = Lt(D) – Lt+1 (D)
If  > c (some threshold), then xt is declared as an anomaly and moved permanently from M to A

Statistical-based – Likelihood Approach

Data distribution, D = (1 – ) M +  A
M is a probability distribution estimated from data
Can be based on any modeling method (naïve Bayes, maximum entropy, etc)
A is initially assumed to be uniform distribution
Likelihood at time t:

Limitations of Statistical Approaches

Most of the tests are for a single attribute

In many cases, data distribution may not be known

For high dimensional data, it may be difficult to estimate the true distribution

Distance-based Approaches

Data is represented as a vector of features

Three major approaches
Nearest-neighbor based
Density based
Clustering based

Nearest-Neighbor Based Approach

Approach:
Compute the distance between every pair of data points

There are various ways to define outliers:
Data points for which there are fewer than p neighboring points within a distance D

The top n data points whose distance to the kth nearest neighbor is greatest

The top n data points whose average distance to the k nearest neighbors is greatest

Outliers in Lower Dimensional Projection

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

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

Outliers in Lower Dimensional Projection

Divide each attribute into  equal-depth intervals
Each interval contains a fraction f = 1/ of the records
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 fk 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

Example

N=100,  = 5, f = 1/5 = 0.2, N  f2 = 4

Density-based: LOF approach

For each point, compute the density of its local neighborhood
Compute local outlier factor (LOF) of a sample p as the average of the ratios of the density of sample p and the density of its nearest neighbors
Outliers are points with largest LOF value

In the NN approach, p2 is not considered as outlier, while LOF approach find both p1 and p2 as outliers



Clustering-Based

Basic idea:
Cluster the data into groups of different density
Choose points in small cluster as candidate outliers
Compute the distance between candidate points and non-candidate clusters.
If candidate points are far from all other non-candidate points, they are outliers

Base Rate Fallacy

Bayes theorem:

More generally:

Base Rate Fallacy (Axelsson, 1999)

Base Rate Fallacy

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

Base Rate Fallacy in Intrusion Detection

I: intrusive behavior,
I: non-intrusive behavior
A: alarm
A: no alarm

Detection rate (true positive rate): P(A|I)
False alarm rate: P(A|I)

Goal is to maximize both
Bayesian detection rate, P(I|A)
P(I|A)

Detection Rate vs False Alarm Rate

Suppose:

Then:

False alarm rate becomes more dominant if P(I) is very low

Detection Rate vs False Alarm Rate

Axelsson: We need a very low false alarm rate to achieve a reasonable Bayesian detection rate

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