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CSE 881: Data Mining
Lecture 22: Anomaly Detection
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Anomaly/Outlier Detection

What are anomalies/outliers?
– Data points whose characteristics are considerably
different than the remainder of the data

Applications:
–
–
–
–
Credit card fraud detection
telecommunication fraud detection
network intrusion detection
fault detection
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Examples of Anomalies

Data from different classes
– An object may be different from other objects because
it is of a different type or class

Natural (random) variation in data
– Many data sets can be modeled by statistical
distributions (e.g., Gaussian distribution)
– Probability of an object decreases rapidly as its
distance from the center of the distribution increases
– Chebyshev inequality:
2
P(| X   | c) 

c2
Data measurement or collection errors
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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
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Anomalies

General characteristics
– Rare occurrence
– Deviant behavior compared to
the majority of the data

Distribution
– Natural variation

uniform distribution
– Data from different classes

distribution may be clustered
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Anomaly Detection

Challenges
– Method is (mostly) unsupervised

Validation can be quite challenging (just like for clustering)
– Small number of anomalies

Finding needle in a haystack
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Anomaly Detection Schemes

General Steps
– Build a profile of the “normal” behavior

Profile can be patterns or summary statistics for the normal 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
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Graphical Approaches

Boxplot (1-D), Scatter plot (2-D), Spin plot (3-D)

Limitations
– Time consuming
– Subjective
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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?
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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)
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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:
( N  1)
G
N
G
max X  X
s
t (2 / N , N 2 )
N  2  t (2 / N , N 2 )
‹#›
Statistical-based – Likelihood Approach

Assume the data set D consists of 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
– Choose a point xt that belongs to M and 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 an anomaly and
is moved permanently from M to A

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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 often assumed to be uniform distribution

Likelihood at time t:

 |At |

|M t |
Lt ( D )   PD ( xi )   (1   )  PM t ( xi )    PAt ( xi ) 
i 1
xi M t

 xiAt

LLt ( D )  M t log( 1   )   log PM t ( xi )  At log    log PAt ( xi )
N
xi M t
xi At
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Limitations of Statistical Approaches

Most of the tests are for a single attribute

In many cases, the data distribution may not be
known

For high dimensional data, it may be difficult to
estimate the true distribution
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Distance-based Approaches

Data is represented as a vector of features

Three approaches
– Nearest-neighbor based
– Density based
– Clustering based
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Nearest-Neighbor Based Approach

Approach:
– Compute the distance between every pair of data
points
– There are various ways to define outliers:
 Data
points with fewer than p points within a neighborhood of
radius D
 Data
points whose distance to the kth nearest neighbor is
among the highest
 Data
points whose average distance to the k nearest
neighbors is among the highest
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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
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Example

N=100,  = 5, f = 1/5 = 0.2, N  f2 = 4
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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

p2

p1
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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
‹#›
One-Class SVM

Based on support vector clustering
– Extension of SVM approach to clustering
– 2 key ideas in SVM:
It uses the maximal margin principle to find the linear
separating hyperplane

For nonlinearly separable data, it uses a kernel function to
project the data into higher dimensional space

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Support Vector Machine (Idea 1)

Maximal margin principle
2
Margin  
|| w ||
Objective function to minimize:
 2
|| w ||
 N k
L( w) 
 C   i 
2
 i 1 
subject to the following constraints:
 
if w  x i  b  1 - i
1
yi  
 
 1 if w  x i  b  1  i
 
wx b  0
‹#›
Support Vector Machine (Idea 2)
Original Space
High-dimensional Feature Space
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Support Vector Clustering
What is the corresponding
maximum margin
principle?
?
Original Space
High-dimensional Feature Space
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Support Vector Clustering

In SVM
– Start with the simplest case first, then make the
problem more complex
– Simplest case: linearly separable data

Apply same idea to clustering
– What is the simplest case?

All the points belong to a single cluster

The cluster is globular (spherical)
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Support Vector Clustering
B2
SVM
Choose the hyperplane with
largest margin
SVC
Choose the sphere with
smallest radius
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Support Vector Clustering
Let R be the radius of the sphere
 Goal is to:

min R 2
R
subject to:
a
i : xi  a  R 2
2
x
where:
– a is the center of the sphere
‹#›
Support Vector Clustering


Objective function: min R   i R  xi  a
2
R , a ,{ }
2
i
– where I’s are the Lagrange multipliers
– Subject to:

i  0

i :  i R  xi  a

2
2
 0
(KKT condition)
L
 2 R  2R  i  0   i  1
R
i
i
L
  2 i ( xi  a)  0    i xi  a
a
i
i
‹#›
2

Support Vector Clustering

Objective function (dual form):
LR
2
   R
2
i
 xi  a
2

i
2
 R 2    i R 2    i xi    j x j
i
i
j
 2

 R  R    i  xi  2  j xi  x j    j k x j  xk 
i
j
j ,k


2
2
   i xi    i j xi  x j
2
i

i
j
Find the I’s that maximizes the expression s.t.

i :  i R 2  xi  a
2
 0
(KKT condition)
‹#›
Support Vector Clustering


Since i :  i R  xi  a
2
2
 0
– If xi is located in the interior of the sphere, then i = 0
– If xi is located on the surface of the sphere then i  0

Support vectors are the data points located on
the cluster boundary
‹#›
Outliers

Outliers are considered data points located
outside the sphere

Let i be the error for xi

Goal is to:
min R 2  C   i
R ,{ }
a
i
– subject to:

i : xi  a  R 2   i ,  i  0
2
‹#›

x
Outliers

Lagrangian:
min

R  C i   i R  i  xi  a
2
R , a ,{ },{ }
i
i
– Subject to:
i :  R
i
2
 xi  a
2
2
2
   
i i
i
  0,    0
i i
(KKT conditions )
L
 2R  2R  i  0    i  1
R
i
i
L
  2 i ( xi  a )  0    i xi  a
a
i
i
L
 C   i  i  0   i  C  i
 i
‹#›
Outliers

Dual form:

L  R  C   i    i R   i  xi  a
2
i
2
i
2
   
i i
i

 R 2   ( i  i ) i    i  R 2   i  xi    i x j

i
i
j

2

 
i i
 
 i
2
   i xi    i x j
i
j
   i xi   i j xi x j
2
i
i
j
– Same as the previous (no outlier) case
‹#›
Outliers


Since i :  i R  xi  a
2
2
  0,    0
i i
– If xi is located in the interior of the sphere, then i = 0
– If xi is located on the surface of the sphere then i  0

Such points are called the support vectors
– If xi is located outside of the sphere then i = 0

Such points are called the bounded support vectors
‹#›
Irregular Shaped Clusters

What if the cluster have irregular shaped in the
original space?
– Instead of using a very large sphere, or a sphere with
large errors ( i), project the data into higherdimensional space (kernel trick)
(xi)
xi

‹#›
Irregular Shaped Clusters

Objective function (dual form):
  ( x )  ( x )    ( x ) ( x )
i
i
i

i
i
i
j
i
j
j
Kernel trick:
– Use kernel function in place of (xi) (xj)
– Typical kernel function:

Gaussian:
K ( xi , x j )  e
 q xi  x j
2
‹#›
References

Support Vector Clustering
By Ben-Hur, Horn, Siegelmann, and Vapnik
(Journal of Machine Learning Research, 2001)
http://citeseer.ist.psu.edu/hur01support.html

Cone Cluster Labeling for Support Vector
Clustering
By Lee and Daniels
(in Proc. of SIAM Int’l Conf on Data Mining, 2006)
http://www.siam.org/meetings/sdm06/proceedings/046lees.pdf
‹#›
Graph-based Method

Represent the data as a graph
– Objects  nodes
– Similarity  edges
Objects
Object Graph

Apply graph-based method to determine outliers
‹#›
Graph-based Method
Find the most outlying node in the graph
=> Opposite of finding the most “central” node
‹#›
Graph-based Method

Many measures of node centrality
– Degree
– Closeness: c(n)  
uV

1
d (u, n)
where d(u,n) is the geodesic distance between u and n
– Geodesic distance is the shortest path distance
– Betweenness: c(n)  
j k
g jk (n)
g jk
where gjk(n) is the number of geodesic paths from j to k that
pass through n

– Random walk method
‹#›
Random Walk Method

Random walk model
– Randomly pick a starting node, s
– Randomly choose a neighboring node linked to s. Set
current node s to be the neighboring node.
– Repeat step 2

Compute the probability that you will reach a
particular node in the graph
– The higher the probability, the more “central” the node
is.
‹#›
Random Walk Method

Goal: Find the stationary distribution c
– Vector c represents probability value for each object
– Initially, set c(i) = 1/N (for all i=1,…,N)

Let S be the adjacency matrix of the graph
– Normalized the rows so that S(i,j) becomes a
transition probability

Iteratively compute:
cS c
T
– Until c converges to a stationary distribution
– To ensure convergence, use a damping factor, d:
c  d  (1  d )S c
T
‹#›
Random Walk Method

Applications
– Web search (PageRank algorithm used by Google)
– Text summarization
– Keyword extraction
‹#›
Random Walk for Anomaly Detection

Assess the centrality or importance of individual
objects
Highly relevant
web pages
Anomalies
For closely related data (e.g.,
documents returned by PageRank)
For data containing anomalies
‹#›
Example

Sample dataset
6
5
4
3
2
O1
1
O2
0
0

1
2
3
4
5
Model parameter tuning
– damping factor=0.1
– Converge after 112 steps
Object
1
2
3
4
5
6
7
8
9
10
11
Connectivity
0.0835
0.0764
0.0930
0.0922
0.0914
0.0940
0.0936
0.0930
0.0942
0.0942
0.0939
Rank
2
1
5
4
3
9
7
6
10
11
8
‹#›