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Data Mining:
Concepts and Techniques
— Chapter 12 —

Chapter 12. Outlier Analysis
• Outlier and Outlier Analysis
• Outlier Detection Methods
• Statistical Approaches
• Proximity-Base Approaches
• Clustering-Base Approaches
• Classification Approaches
• Mining Contextual and Collective Outliers
• Outlier Detection in High Dimensional Data
• Summary 2

What Are Outliers?
• Outlier: A data object that deviates significantly from the normal objects as if it
were generated by a different mechanism
– Ex.: Unusual credit card purchase, sports: Michael Jordon, Wayne Gretzky, ...
• Outliers are different from the noise data
– Noise is random error or variance in a measured variable
– Noise should be removed before outlier detection
• Outliers are interesting: It violates the mechanism that generates the normal data
• Outlier detection vs. novelty detection: early stage, outlier; but later merged into
the model
• Applications:
– Credit card fraud detection
– Telecom fraud detection
– Customer segmentation
– Medical analysis
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Types of Outliers (I)
• Three kinds: global, contextual and collective outliers
• Global outlier (or point anomaly)
– Object is Og if it significantly deviates from the rest of the data set
– Ex. Intrusion detection in computer networks
– Issue: Find an appropriate measurement of deviation
• Contextual outlier (or conditional outlier)
– Object is Oc if it deviates significantly based on a selected context
– Ex. 80o
F in Urbana: outlier? (depending on summer or winter?)
– Attributes of data objects should be divided into two groups
• Contextual attributes: defines the context, e.g., time & location
• Behavioral attributes: characteristics of the object, used in outlier
evaluation, e.g., temperature
– Can be viewed as a generalization of local outliers—whose density
significantly deviates from its local area
– Issue: How to define or formulate meaningful context?
Global Outlier

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Types of Outliers (II)
• Collective Outliers
– A subset of data objects collectively deviate significantly
from the whole data set, even if the individual data objects
may not be outliers
– Applications: E.g., intrusion detection:
• When a number of computers keep sending denial-of-
service packages to each other
Collective Outlier
 Detection of collective outliers

Consider not only behavior of individual objects, but also that of
groups of objects

Need to have the background knowledge on the relationship
among data objects, such as a distance or similarity measure
on objects.
 A data set may have multiple types of outlier
 One object may belong to more than one type of outlier

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Challenges of Outlier Detection
 Modeling normal objects and outliers properly
 Hard to enumerate all possible normal behaviors in an application
 The border between normal and outlier objects is often a gray area
 Application-specific outlier detection
 Choice of distance measure among objects and the model of
relationship among objects are often application-dependent
 E.g., clinic data: a small deviation could be an outlier; while in
marketing analysis, larger fluctuations
 Handling noise in outlier detection
 Noise may distort the normal objects and blur the distinction
between normal objects and outliers. It may help hide outliers and
reduce the effectiveness of outlier detection
 Understandability
 Understand why these are outliers: Justification of the detection
 Specify the degree of an outlier: the unlikelihood of the object being
generated by a normal mechanism

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• Summary

Outlier Detection I: Supervised Methods
• Two ways to categorize outlier detection methods:
– Based on whether user-labeled examples of outliers can be obtained:
• Supervised, semi-supervised vs. unsupervised methods
– Based on assumptions about normal data and outliers:
• Statistical, proximity-based, and clustering-based methods
• Outlier Detection I: Supervised Methods
– Modeling outlier detection as a classification problem
• Samples examined by domain experts used for training & testing
– Methods for Learning a classifier for outlier detection effectively:
• Model normal objects & report those not matching the model as
outliers, or
• Model outliers and treat those not matching the model as normal
– Challenges
• Imbalanced classes, i.e., outliers are rare: Boost the outlier class and
make up some artificial outliers
• Catch as many outliers as possible, i.e., recall is more important than
accuracy (i.e., not mislabeling normal objects as outliers)
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Outlier Detection II: Unsupervised Methods
• Assume the normal objects are somewhat ``clustered'‘ into multiple groups,
each having some distinct features
• An outlier is expected to be far away from any groups of normal objects
• Weakness: Cannot detect collective outlier effectively
– Normal objects may not share any strong patterns, but the collective
outliers may share high similarity in a small area
• Ex. In some intrusion or virus detection, normal activities are diverse
– Unsupervised methods may have a high false positive rate but still miss
many real outliers.
– Supervised methods can be more effective, e.g., identify attacking some key
resources
• Many clustering methods can be adapted for unsupervised methods
– Find clusters, then outliers: not belonging to any cluster
– Problem 1: Hard to distinguish noise from outliers
– Problem 2: Costly since first clustering: but far less outliers than normal
objects
• Newer methods: tackle outliers directly
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Outlier Detection III: Semi-Supervised Methods
• Situation: In many applications, the number of labeled data is often small:
Labels could be on outliers only, normal objects only, or both
• Semi-supervised outlier detection: Regarded as applications of semi-
supervised learning
• If some labeled normal objects are available
– Use the labeled examples and the proximate unlabeled objects to train a
model for normal objects
– Those not fitting the model of normal objects are detected as outliers
• If only some labeled outliers are available, a small number of labeled outliers
many not cover the possible outliers well
– To improve the quality of outlier detection, one can get help from models
for normal objects learned from unsupervised methods
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Outlier Detection (1): Statistical Methods
• Statistical methods (also known as model-based methods) assume that the normal data
follow some statistical model (a stochastic model)
– The data not following the model are outliers.
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 Effectiveness of statistical methods: highly depends on whether the
assumption of statistical model holds in the real data
 There are rich alternatives to use various statistical models
 E.g., parametric vs. non-parametric
 Example (right figure): First use Gaussian distribution
to model the normal data
 For each object y in region R, estimate gD(y), the
probability of y fits the Gaussian distribution
 If gD(y) is very low, y is unlikely generated by the
Gaussian model, thus an outlier

Outlier Detection (2): Proximity-Based Methods
• An object is an outlier if the nearest neighbors of the object are far away, i.e., the proximity of
the object is significantly deviates from the proximity of most of the other objects in the same
data set
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 The effectiveness of proximity-based methods highly relies on the
proximity measure.
 In some applications, proximity or distance measures cannot be
obtained easily.
 Often have a difficulty in finding a group of outliers which stay close to
each other
 Two major types of proximity-based outlier detection
 Distance-based vs. density-based
 Example (right figure): Model the proximity of an
object using its 3 nearest neighbors
 Objects in region R are substantially different
from other objects in the data set.
 Thus the objects in R are outliers

Outlier Detection (3): Clustering-Based Methods
• Normal data belong to large and dense clusters, whereas outliers
belong to small or sparse clusters, or do not belong to any
clusters
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 Since there are many clustering methods, there are many
clustering-based outlier detection methods as well
 Clustering is expensive: straightforward adaption of a
clustering method for outlier detection can be costly and
does not scale up well for large data sets
 Example (right figure): two clusters
 All points not in R form a large cluster
 The two points in R form a tiny cluster,
thus are outliers

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• Summary

Statistical Approaches
• Statistical approaches assume that the objects in a data set are generated by
a stochastic process (a generative model)
• Idea: learn a generative model fitting the given data set, and then identify the
objects in low probability regions of the model as outliers
• Methods are divided into two categories: parametric vs. non-parametric
• Parametric method
– Assumes that the normal data is generated by a parametric distribution
with parameter θ
– The probability density function of the parametric distribution f(x, θ)
gives the probability that object x is generated by the distribution
– The smaller this value, the more likely x is an outlier
• Non-parametric method
– Not assume an a-priori statistical model and determine the model from
the input data
– Not completely parameter free but consider the number and nature of
the parameters are flexible and not fixed in advance
– Examples: histogram and kernel density estimation
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Parametric Methods I: Detection Univariate Outliers Based on
Normal Distribution
• Univariate data: A data set involving only one attribute or variable
• Often assume that data are generated from a normal distribution, learn the
parameters from the input data, and identify the points with low probability as
outliers
• Ex: Avg. temp.: {24.0, 28.9, 28.9, 29.0, 29.1, 29.1, 29.2, 29.2, 29.3, 29.4}
– Use the maximum likelihood method to estimate μ and σ
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 Taking derivatives with respect to μ and σ2
, we derive the following
maximum likelihood estimates
 For the above data with n = 10, we have
 Then (24 – 28.61) /1.51 = – 3.04 < –3, 24 is an outlier since

Parametric Methods I: The Grubb’s Test
• Univariate outlier detection: The Grubb's test (maximum normed residual test)
─ another statistical method under normal distribution
– For each object x in a data set, compute its z-score: x is an outlier if
where is the value taken by a t-distribution at a significance
level of α/(2N), and N is the # of objects in the data set
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Parametric Methods II: Detection of Multivariate
Outliers
• Multivariate data: A data set involving two or more attributes or variables
• Transform the multivariate outlier detection task into a univariate outlier
detection problem
• Method 1. Compute Mahalaobis distance
– Let ō be the mean vector for a multivariate data set. Mahalaobis
distance for an object o to ō is MDist(o, ō) = (o – ō )T
S –1
(o – ō) where S is
the covariance matrix
– Use the Grubb's test on this measure to detect outliers
• Method 2. Use χ2
–statistic:
– where Ei is the mean of the i-dimension among all objects, and n is the
dimensionality
– If χ2
–statistic is large, then object oi is an outlier
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Parametric Methods III: Using Mixture of Parametric
Distributions
• Assuming data generated by a normal distribution could
be sometimes overly simplified
• Example (right figure): The objects between the two
clusters cannot be captured as outliers since they are
close to the estimated mean
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 To overcome this problem, assume the normal data is generated by two
normal distributions. For any object o in the data set, the probability that
o is generated by the mixture of the two distributions is given by
where fθ1 and fθ2 are the probability density functions of θ1 and θ2
 Then use EM algorithm to learn the parameters μ1, σ1, μ2, σ2 from data
 An object o is an outlier if it does not belong to any cluster

Non-Parametric Methods: Detection Using Histogram
• The model of normal data is learned from the input
data without any a priori structure.
• Often makes fewer assumptions about the data, and
thus can be applicable in more scenarios
• Outlier detection using histogram:
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 Figure shows the histogram of purchase amounts in transactions
 A transaction in the amount of $7,500 is an outlier, since only 0.2%
transactions have an amount higher than $5,000
 Problem: Hard to choose an appropriate bin size for histogram
 Too small bin size → normal objects in empty/rare bins, false positive
 Too big bin size → outliers in some frequent bins, false negative
 Solution: Adopt kernel density estimation to estimate the probability
density distribution of the data. If the estimated density function is high,
the object is likely normal. Otherwise, it is likely an outlier.

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• Summary

Proximity-Based Approaches: Distance-Based vs. Density-
Based Outlier Detection
• Intuition: Objects that are far away from the others are outliers
• Assumption of proximity-based approach: The proximity of an
outlier deviates significantly from that of most of the others in
the data set
• Two types of proximity-based outlier detection methods
– Distance-based outlier detection: An object o is an outlier if
its neighborhood does not have enough other points
– Density-based outlier detection: An object o is an outlier if
its density is relatively much lower than that of its neighbors
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Distance-Based Outlier Detection
• For each object o, examine the # of other objects in the r-neighborhood of o, where r is a user-
specified distance threshold
• An object o is an outlier if most (taking π as a fraction threshold) of the objects in D are far away
from o, i.e., not in the r-neighborhood of o
• An object o is a DB(r, π) outlier if
• Equivalently, one can check the distance between o and its k-th nearest neighbor ok, where
. o is an outlier if dist(o, ok) > r
• Efficient computation: Nested loop algorithm
– For any object oi, calculate its distance from other objects, and count the # of other objects
in the r-neighborhood.
– If π n other objects are within r distance, terminate the inner loop
∙
– Otherwise, oi is a DB(r, π) outlier
• Efficiency: Actually CPU time is not O(n2
) but linear to the data set size since for most non-outlier
objects, the inner loop terminates early
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Distance-Based Outlier Detection: A Grid-Based Method
• Why efficiency is still a concern? When the complete set of objects cannot
be held into main memory, cost I/O swapping
• The major cost: (1) each object tests against the whole data set, why not only
its close neighbor? (2) check objects one by one, why not group by group?
• Grid-based method (CELL): Data space is partitioned into a multi-D grid. Each
cell is a hyper cube with diagonal length r/2
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
Pruning using the level-1 & level 2 cell properties:
 For any possible point x in cell C and any
possible point y in a level-1 cell, dist(x,y) ≤ r
 For any possible point x in cell C and any point y
such that dist(x,y) ≥ r, y is in a level-2 cell
 Thus we only need to check the objects that cannot be pruned, and
even for such an object o, only need to compute the distance between
o and the objects in the level-2 cells (since beyond level-2, the
distance from o is more than r)

Density-Based Outlier Detection
• Local outliers: Outliers comparing to their local
neighborhoods, instead of the global data
distribution
• In Fig., o1 and o2 are local outliers to C1, o3 is a global
outlier, but o4 is not an outlier. However, proximity-
based clustering cannot find o1 and o2 are outlier
(e.g., comparing with O4).
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 Intuition (density-based outlier detection): The density around an outlier
object is significantly different from the density around its neighbors
 Method: Use the relative density of an object against its neighbors as
the indicator of the degree of the object being outliers
 k-distance of an object o, distk(o): distance between o and its k-th NN
 k-distance neighborhood of o, Nk(o) = {o’| o’ in D, dist(o, o’) ≤ distk(o)}
 Nk(o) could be bigger than k since multiple objects may have
identical distance to o

Local Outlier Factor: LOF
• Reachability distance from o’ to o:
– where k is a user-specified parameter
• Local reachability density of o:
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 LOF (Local outlier factor) of an object o is the average of the ratio of
local reachability of o and those of o’s k-nearest neighbors
 The lower the local reachability density of o, and the higher the local
reachability density of the kNN of o, the higher LOF
 This captures a local outlier whose local density is relatively low
comparing to the local densities of its kNN

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• Summary

Clustering-Based Outlier Detection (1 & 2):
Not belong to any cluster, or far from the closest one
• An object is an outlier if (1) it does not belong to any cluster, (2) there is a
large distance between the object and its closest cluster , or (3) it belongs to a
small or sparse cluster
 Case I: Not belong to any cluster
 Identify animals not part of a flock: Using a density-
based clustering method such as DBSCAN
 Case 2: Far from its closest cluster
 Using k-means, partition data points of into clusters
 For each object o, assign an outlier score based on
its distance from its closest center
 If dist(o, co)/avg_dist(co) is large, likely an outlier
 Ex. Intrusion detection: Consider the similarity between
data points and the clusters in a training data set
 Use a training set to find patterns of “normal” data, e.g., frequent
itemsets in each segment, and cluster similar connections into groups
 Compare new data points with the clusters mined—Outliers are
possible attacks 28

• FindCBLOF: Detect outliers in small clusters
– Find clusters, and sort them in decreasing size
– To each data point, assign a cluster-based local outlier
factor (CBLOF):
– If obj p belongs to a large cluster, CBLOF =
cluster_size X similarity between p and cluster
– If p belongs to a small one, CBLOF = cluster size X
similarity betw. p and the closest large cluster
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Clustering-Based Outlier Detection (3):
Detecting Outliers in Small Clusters
 Ex. In the figure, o is outlier since its closest large cluster is C1, but the
similarity between o and C1 is small. For any point in C3, its closest
large cluster is C2 but its similarity from C2 is low, plus |C3| = 3 is small

Clustering-Based Method: Strength and Weakness
• Strength
– Detect outliers without requiring any labeled data
– Work for many types of data
– Clusters can be regarded as summaries of the data
– Once the cluster are obtained, need only compare any object against the clusters to
determine whether it is an outlier (fast)
• Weakness
– Effectiveness depends highly on the clustering method used—they may not be
optimized for outlier detection
– High computational cost: Need to first find clusters
– A method to reduce the cost: Fixed-width clustering
• A point is assigned to a cluster if the center of the cluster is within a pre-defined
distance threshold from the point
• If a point cannot be assigned to any existing cluster, a new cluster is created and
the distance threshold may be learned from the training data under certain
conditions

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• Summary

Classification-Based Method I: One-Class Model
• Idea: Train a classification model that can distinguish
“normal” data from outliers
• A brute-force approach: Consider a training set that
contains samples labeled as “normal” and others
labeled as “outlier”
– But, the training set is typically heavily biased: #
of “normal” samples likely far exceeds # of
outlier samples
– Cannot detect unseen anomaly
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 One-class model: A classifier is built to describe only the normal class.
 Learn the decision boundary of the normal class using classification
methods such as SVM
 Any samples that do not belong to the normal class (not within the
decision boundary) are declared as outliers
 Adv: can detect new outliers that may not appear close to any outlier
objects in the training set
 Extension: Normal objects may belong to multiple classes

Classification-Based Method II: Semi-Supervised Learning
• Semi-supervised learning: Combining classification-based and
clustering-based methods
• Method
– Using a clustering-based approach, find a large cluster, C,
and a small cluster, C1
– Since some objects in C carry the label “normal”, treat all
objects in C as normal
– Use the one-class model of this cluster to identify normal
objects in outlier detection
– Since some objects in cluster C1 carry the label “outlier”,
declare all objects in C1 as outliers
– Any object that does not fall into the model for C (such as
a) is considered an outlier as well
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 Comments on classification-based outlier detection methods
 Strength: Outlier detection is fast
 Bottleneck: Quality heavily depends on the availability and quality of
the training set, but often difficult to obtain representative and high-
quality training data

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• Summary

Mining Contextual Outliers I: Transform into
Conventional Outlier Detection
• If the contexts can be clearly identified, transform it to conventional outlier
detection
1. Identify the context of the object using the contextual attributes
2. Calculate the outlier score for the object in the context using a
conventional outlier detection method
• Ex. Detect outlier customers in the context of customer groups
– Contextual attributes: age group, postal code
– Behavioral attributes: # of trans/yr, annual total trans. amount
• Steps: (1) locate c’s context, (2) compare c with the other customers in the
same group, and (3) use a conventional outlier detection method
• If the context contains very few customers, generalize contexts
– Ex. Learn a mixture model U on the contextual attributes, and another
mixture model V of the data on the behavior attributes
– Learn a mapping p(Vi|Uj): the probability that a data object o belonging
to cluster Uj on the contextual attributes is generated by cluster Vi on the
behavior attributes
– Outlier score:
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Mining Contextual Outliers II: Modeling Normal Behavior with
Respect to Contexts
• In some applications, one cannot clearly partition the data into contexts
– Ex. if a customer suddenly purchased a product that is unrelated to those
she recently browsed, it is unclear how many products browsed earlier
should be considered as the context
• Model the “normal” behavior with respect to contexts
– Using a training data set, train a model that predicts the expected
behavior attribute values with respect to the contextual attribute values
– An object is a contextual outlier if its behavior attribute values
significantly deviate from the values predicted by the model
• Using a prediction model that links the contexts and behavior, these methods
avoid the explicit identification of specific contexts
• Methods: A number of classification and prediction techniques can be used
to build such models, such as regression, Markov Models, and Finite State
Automaton
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Mining Collective Outliers I: On the Set of
“Structured Objects”
• Collective outlier if objects as a group deviate significantly from
the entire data
• Need to examine the structure of the data set, i.e, the
relationships between multiple data objects
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 Each of these structures is inherent to its respective type of data

For temporal data (such as time series and sequences), we explore
the structures formed by time, which occur in segments of the time
series or subsequences

For spatial data, explore local areas

For graph and network data, we explore subgraphs
 Difference from the contextual outlier detection: the structures are
often not explicitly defined, and have to be discovered as part of the
outlier detection process.
 Collective outlier detection methods: two categories

Reduce the problem to conventional outlier detection

Identify structure units, treat each structure unit (e.g.,
subsequence, time series segment, local area, or subgraph) as
a data object, and extract features

Then outlier detection on the set of “structured objects”
constructed as such using the extracted features

Mining Collective Outliers II: Direct Modeling of the
Expected Behavior of Structure Units
• Models the expected behavior of structure units directly
• Ex. 1. Detect collective outliers in online social network of customers
– Treat each possible subgraph of the network as a structure unit
– Collective outlier: An outlier subgraph in the social network
• Small subgraphs that are of very low frequency
• Large subgraphs that are surprisingly frequent
• Ex. 2. Detect collective outliers in temporal sequences
– Learn a Markov model from the sequences
– A subsequence can then be declared as a collective outlier if it
significantly deviates from the model
• Collective outlier detection is subtle due to the challenge of exploring the
structures in data
– The exploration typically uses heuristics, and thus may be application
dependent
– The computational cost is often high due to the sophisticated mining
process
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39
• Summary

Challenges for Outlier Detection in High-Dimensional Data
• Interpretation of outliers
– Detecting outliers without saying why they are outliers is not very useful in high-D
due to many features (or dimensions) are involved in a high-dimensional data set
– E.g., which subspaces that manifest the outliers or an assessment regarding the
“outlier-ness” of the objects
• Data sparsity
– Data in high-D spaces are often sparse
– The distance between objects becomes heavily dominated by noise as the
dimensionality increases
• Data subspaces
– Adaptive to the subspaces signifying the outliers
– Capturing the local behavior of data
• Scalable with respect to dimensionality
– # of subspaces increases exponentially
40

Approach I: Extending Conventional Outlier Detection
• Method 1: Detect outliers in the full space, e.g., HilOut Algorithm
– Find distance-based outliers, but use the ranks of distance instead of the
absolute distance in outlier detection
– For each object o, find its k-nearest neighbors: nn1(o), . . . , nnk(o)
– The weight of object o:
– All objects are ranked in weight-descending order
– Top-l objects in weight are output as outliers (l: user-specified parm)
– Employ space-filling curves for approximation: scalable in both time and
space w.r.t. data size and dimensionality
• Method 2: Dimensionality reduction
– Works only when in lower-dimensionality, normal instances can still be
distinguished from outliers
– PCA: Heuristically, the principal components with low variance are preferred
because, on such dimensions, normal objects are likely close to each other
and outliers often deviate from the majority
41

Approach II: Finding Outliers in Subspaces
• Extending conventional outlier detection: Hard for outlier interpretation
• Find outliers in much lower dimensional subspaces: easy to interpret why and
to what extent the object is an outlier
– E.g., find outlier customers in certain subspace: average transaction
amount >> avg. and purchase frequency << avg.
• Ex. A grid-based subspace outlier detection method
– Project data onto various subspaces to find an area whose density is much
lower than average
– Discretize the data into a grid with φ equi-depth (why?) regions
– Search for regions that are significantly sparse
• Consider a k-d cube: k ranges on k dimensions, with n objects
• If objects are independently distributed, the expected number of
objects falling into a k-dimensional region is (1/ φ)k
n = fk
n,the standard
deviation is
• The sparsity coefficient of cube C:
• If S(C) < 0, C contains less objects than expected
• The more negative, the sparser C is and the more likely the objects in C
are outliers in the subspace
42

Approach III: Modeling High-Dimensional Outliers
• Ex. Angle-based outliers: Kriegel, Schubert, and Zimek [KSZ08]
• For each point o, examine the angle ∆xoy for every pair of points x, y.
– Point in the center (e.g., a), the angles formed differ widely
– An outlier (e.g., c), angle variable is substantially smaller
• Use the variance of angles for a point to determine outlier
• Combine angles and distance to model outliers
– Use the distance-weighted angle variance as the outlier score
– Angle-based outlier factor (ABOF):
– Efficient approximation computation method is developed
– It can be generalized to handle arbitrary types of data
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 Develop new models for high-
dimensional outliers directly
 Avoid proximity measures and adopt
new heuristics that do not deteriorate
in high-dimensional data
A set of points
form a cluster
except c (outlier)

Summary
• Types of outliers
– global, contextual & collective outliers
• Outlier detection
– supervised, semi-supervised, or unsupervised
• Statistical (or model-based) approaches
• Proximity-base approaches
• Clustering-base approaches
• Classification approaches
• Mining contextual and collective outliers
• Outlier detection in high dimensional data
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