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Data Mining

Anomaly Detection

Lecture Notes for Chapter 9

Introduction to Data Mining

by

Tan, Steinbach, Kumar

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 *

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!

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

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

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

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 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)

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