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Statistical Selection of Relevant Subspace

Projections for Outlier Ranking

Emmanuel M¨ uller•, Matthias Schiffer◦, Thomas Seidl◦

•Karlsruhe Institute of Technology (KIT), Germany

emmanuel.mueller@kit.edu

◦RWTH Aachen University, Germany

{mschiffer, seidl}@cs.rwth-aachen.de

Abstract—Outlier mining is an important data analysis task to

distinguish exceptional outliers from regular objects. For outlier

mining in the full data space, there are well established methods

which are successful in measuring the degree of deviation for out-

lier ranking. However, in recent applications traditional outlier

mining approaches miss outliers as they are hidden in subspace

projections. Especially, outlier ranking approaches measuring

deviation on all available attributes miss outliers deviating from

their local neighborhood only in subsets of the attributes.

In this work, we propose a novel outlier ranking based on

the objects deviation in a statistically selected set of relevant

subspace projections. This ensures to find objects deviating in

multiple relevant subspaces, while it excludes irrelevant projec-

tions showing no clear contrast between outliers and the residual

objects. Thus, we tackle the general challenges of detecting

outliers hidden in subspaces of the data. We provide a selection

of subspaces with high contrast and propose a novel ranking

based on an adaptive degree of deviation in arbitrary subspaces.

In thorough experiments on real and synthetic data we show that

our approach outperforms competing outlier ranking approaches

by detecting outliers in arbitrary subspace projections.

I. INTRODUCTION

Outlier mining has become an important data mining task to

detect inconsistent or suspicious objects in large databases. For

recent applications, outlier mining as an unsupervised learning

task is important for consistency checks of sensor network

measurements, fraud detection in financial transactions, emer-

gency detection in health surveillance and many more. As

measuring and storing of data has become very cheap, in all

of these applications, objects are described by many attributes.

However, for each object only subsets of relevant attributes

provide the meaningful information, the residual attributes are

irrelevant for this object. For example in health surveillance,

for one patient attributes such as “age” and “skin humidity”

might be important to detect the abnormal “dehydration” status

of this patient. Other attributes such as “heart beat rate” are

irrelevant for the detection of this outlier, but are relevant for

the detection of abnormal patients with a heart disease. All

of these attributes are required for some outlier detection, but

each outlier occurs only in subsets of these attributes. Thus,

the distinction between outliers and regular objects is heavily

hindered by considering all available attributes as typically

done in traditional outlier mining methods.

Traditional techniques are well established for outlier min-

ing in the full space, but miss outliers which are hidden in

subspace projections. Thus, our general aim is to develop a

novel outlier ranking based on object deviation in subspace

projections. We focus on outlier ranking a special research

field of outlier mining which sorts objects according to their

local degree of deviation. Local outlier rankings use the local

neighborhood around each object to report an ordered list

presenting the most outlying object first. They provide more

information than just the binary decision about being an

outlier or not. Outlier rankings provide for each object the

extent of outlierness. However, traditional outlier rankings

using outlierness measures in full space are not appropriate

for outliers hidden in subspaces. In the full space all objects

appear to be alike so that traditional outlier rankings cannot

distinguish the outlierness of objects any more.

In this work we measure outlierness (the degree of devia-

tion) of an object in projections of the database taking only

subsets of attributes into account. Hence, we can successfully

detect an outlier in a set of relevant subspace projections in

which this outlier stands out from its surrounding objects. We

model various behaviors of one object in different projections

of the database. An object might show high deviation com-

pared to its neighborhood in one subspace. In addition, the

same object might cluster with some other objects in a second

subspace or might not show up as an outlier in a third scattered

subspace where all objects seem to be outliers. To illustrate

this, we have depicted several projections of a toy example

with two hidden outliers in Figure 1. Please note, that each

object requires the detection of individual subspace projections

to detect its outlier properties. This is in contrast to related

paradigms such as subspace clustering [14] or dimensionality

reduction techniques [10]. Global dimensionality reduction

techniques such as principal components analysis provide only

a single projection for all objects. In contrast, we aim at

detecting multiple relevant subspaces per object. Subspace

clustering detects multiple projections, however, focuses on

subspaces for groups of clustered objects. For outlier ranking,

the focus is on the individual objects and subspaces in which

an outlier is highly deviating from its local neighborhood.

Thus, outlier ranking in subspaces poses novel challenges not

tackled by these research topics.

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subspaces {1} {2} subspace {1,2,3}subspace {3,4} subspace {1,2}

dense regions

no outliers

dense regions

one outlier

dense regions

another outlier

scattered space

all seem outliers

Fig. 1.Example: Outliers in arbitrary subspaces

In this work, we focus on two key properties for outlier

ranking in subspace projections. First, for each object we

statistically select a set of projections for outlier ranking,

we call these relevant subspaces. In relevant subspaces the

neighborhood of an object is clustered and the object is an

outlier if it deviates from these clustered objects. In case

these relevant subspaces inhabit different clusters in terms

of objects, some objects might be members of a cluster in

one subspace while being outliers in others at the same time.

In contrast, in irrelevant subspaces the neighborhood of an

object is distributed uniformly random such that all objects

seem to be outliers. Overall, our outlier ranking is confronted

with arbitrary subspaces, while only very few are relevant

and contribute to a distinction between clustered objects and

outliers. Second, object deviation increases with the number of

attributes in a relevant subspace. As distances between objects

grow more and more alike due to the “curse of dimensionality”

[5], objects are clustered in dense regions in one dimensional

subspaces while objects are scattered in higher dimensional

spaces (cf. Figure 1). In general, the deviation of objects is

highly influenced by the number of attributes in the considered

subspaces. Thus, for outlier ranking in subspace projections,

we have to cope with two major challenges:

• Outliers appear only in relevant subspaces.

• Incomparable deviation in different subspaces.

To tackle both of these challenges, we propose OUTRES a

new method for outlier ranking in relevant subspace projec-

tions. For our outlier ranking we consider only a selection of

non-uniformly distributed projections. We exclude uniformly

random distributed subspaces by a statistical test, as they

hinder the distinction between outliers and regular objects.

Furthermore, we propose a novel adaptive outlier ranking

measuring comparable degrees of deviation for objects in

arbitrary subspaces. We define outliers to be objects highly

deviating from the estimated density in their local subspace

neighborhood. Overall in contrast to traditional outlier rank-

ing approaches, our outlier ranking considers deviation in

subspaces and adapts to the number of attributes in each

considered subspace.

II. RELATED WORK

In general, outliers are objects that deviate from the rest of

the data to a great extent. However, there have been various

outlier models proposed in the literature. We categorize these

models into two paradigms, traditional outlier mining methods

and subspace outlier mining techniques.

a) Traditional Methods:

Different models have been proposed modeling deviation glob-

ally e.g. in distance-based [12], cluster-based [9] or statistical-

based [4] outlier mining methods. However, such techniques

suffer from difficulties in parametrization, as the extent of

deviation is usually hard to quantify globally. This has led to

outlier ranking based on the local degree of deviation for each

object, as in the well established local outlier factor (LOF)

approach [6] or its extension (LOCI) based on local deviation

[24]. Further extensions have been proposed based on this

general local outlier factor idea. The most recent approach

proposes an angle based outlier factor (ABOF) [15]. Based on

the assumption, that angles between objects are more stable

than distances, ABOF computes for each object an angle range

to the residual objects. However, all of these outlier ranking

methods base on the full space, and thus, fail to separate

outliers from regular objects in subspace projections.

b) Subspace Methods:

In contrast, recent approaches consider subspace projections

for outlier ranking. The key property for all of these ap-

proaches is the appropriate choice of considered subspaces.

As the most basic approach a random choice of subspaces has

been proposed for outlier ranking in subspaces by RPLOF

[16] and Isolation Forest [17]. However, clearly such simple

heuristics might miss outliers due to the random selection

of subspaces. Recently, a more meaningful selection has

been proposed that selects only one subspace spanned as a

hyperplane by a set of reference points (SOD) [13]. Its general

hypothesis states that outliers deviate within this hyperplane.

However, SOD determines the outlierness of an object only

in this single subspace, if objects deviate in two or more

subspaces SOD is unable to distinguish between their outlier

factors.

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Recent approaches base on subspace clustering as a related

paradigm. As first approach one based on the subspace clus-

tering method CLIQUE [2] to derive objects deviating from

subspace clusters [1]. However, designed as a binary decision

this approach does not provide an outlier ranking. As first

outlier ranking based on subspace clusters a ranking function

using cluster properties as indicators for outliers has been

proposed (OutRank) [20]. However, based on the aggregated

information of clusters one ignores the actual deviation of each

object in the considered subspaces.

Overall, traditional full space outlier ranking methods sim-

ply compute object deviation in one fixed space and thus miss

outliers in subspace projections. In contrast, outlier ranking in

subspace projections take arbitrary projections into account.

However, none of the proposed subspace methods considers a

meaningful selection of relevant subspaces. Furthermore, they

all ignore the incomparable deviation of objects in different

subspaces for their outlierness measures.

III. OUTLIER RANKING IN SUBSPACES

Our general idea is to measure deviation of each object in a

set of relevant subspace projections. In contrast to traditional

outlier ranking approaches, we consider for each object its

deviation in multiple subspaces. This ensures to find objects

deviating in projections of the data, but, it also poses new

major challenges for outlier ranking. In the following we

start with some basic notions and provide a formalization

of these challenges before we propose our novel selection

of significantly non-uniformly distributed subspaces and our

novel adaptive outlierness measure.

A. Notions and Challenges

In general, the aim of outlier ranking is to provide a sorting

of all objects o given in a database DB. Technically, one ranks

according to the degree of deviation measured by a ranking

function r : DB → R. The ranking function provides a real

valued measure of the objects’ outlierness. Ranking functions

can be defined arbitrarily based on the object’s features o =

(o1,...,od). In contrast to traditional approaches that measure

the degree of deviation in the full d-dimensional space D =

{1,...,d}, we measure deviation in subspace projections S ⊆

D. Thus, we ensure to find outliers hidden in any possible

subspace projection.

The general challenge for outlier ranking approaches, is

to provide a meaningful ranking function which achieves to

distinguish between an outlier object o and a regular object

p by providing a clear distinction: r(o) ? r(p). However,

traditional outlier ranking functions fail for outliers hidden in

subspaces as they provide for all objects very similar ranking

values r(o) ≈ r(p) ∀o,p ∈ DB. This can be explained

by the scattered full space of such data sets. While each

object has only a subset of relevant attributes, the residual

attributes provide more or less random values. Considering

distances between objects using all of these attributes one

observes an effect termed the “curse of dimensionality”. As

traditional ranking functions consider all attributes for distance

computation distD(o,p) they cannot distinguish outliers from

regular objects. For the typically used Euclidean distance

??

, distances between o ∈ DB and any residual objects grow

more and more alike with increasing number of attributes

|D| → ∞:

distD(o,p) =

i∈D

(oi− pi)2

lim

|D|→∞

maxp∈DBdistD(o,p) − minp∈DBdistD(o,p)

minp∈DBdistD(o,p)

→ 0

As consequence, ranking values based on these full space

distances become meaningless:

lim

|D|→∞r(o) − r(p) → 0 ⇒ r(o) ≈ r(p) ∀o,p ∈ DB

Although outliers do not show up in full space, they deviate

in subspace projections. Thus, we cope with the curse of

dimensionality by considering the outlierness of each object

in a selection of relevant subspaces. This set of relevant

subspaces RS(o) is selected individually for each object o

such that these subspaces provide a high contrast between o

and its surrounding neighborhood. We measure the outlierness

score(o,S) by restricting distance functions distS(o,p) to the

subspace dimensions in S. The overall ranking value r(o) of an

object o is then simply computed by aggregating its outlierness

in all relevant subspaces:

Definition 1: Subspace Ranking Function

The overall ranking value r(o) of an object o ∈ DB w.r.t. a

set of relevant subspaces RS(o) and an outlierness measure

score(o,S) is defined as:

?

r(o) =

S∈RS(o)

score(o,S)

As aggregation of all outlierness measures in different

subspaces one could use several meaningful functions. Please

note, that we use scoring values in the range of 0...1

with outliers represented by low scores. Thus, the minimum

over all scorings would provide a meaningful aggregation.

However, this would highlight only the outlierness in one

single subspace. Using the sum of scores as aggregation would

lead to low contrast as objects found in clusters with high score

values would blur the overall ranking value. In contrast, we

use the product incorporating outlier properties from different

subspaces such that low scores in multiple subspaces highlight

an object as clear outlier providing high contrast between

outliers and regular objects.

While traditional ranking functions consider the outlierness

of an object only in the full space D, we aim at considering

outlierness in a set of subspaces RS(o) ⊆ P(D) out of the

powerset of possible subspace projections. This is meaningful

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as outliers might be hidden in multiple subspace projections.

However, two novel challenges arise:

• How to choose the set of relevant subspaces RS(o) for

meaningful outlier ranking

• How

score(o,S) over multiple subspaces S ∈ RS(o).

To tackle these challenges, our key hypothesis is that

outliers can be distinguished in local neighborhoods of non-

uniformly distributed subspaces. We base on the idea of local

outlier ranking as proposed by full space approaches [6], [24].

According to this outlier mining paradigm, we define outliers

as objects that are highly deviating from their local neighbor-

hood. Traditional methods already observed varying density

distributions and proposed local outlierness measures to tackle

outlier ranking in full space. For subspace outlier ranking

we observe an additional factor for varying densities not yet

addressed in the literature. Different subspace provide highly

varying densities ranging from densely clustered subspaces up

to uniformly distributed subspace. To tackle this variance in

densities we propose a statistical selection of subspaces and

our novel adaptive outlierness measure for outlier ranking in

relevant subspaces.

As a density-based approach OUTRES measures its out-

lierness score(o,S) according to the density den(o,S) of

an object in subspace S. Low density values on an object

indicate its outlierness and lead to low scores. However,

objects might be outliers in multiple subspaces, thus, a mean-

ingful outlierness measure has to be comparable over different

subspaces. We will propose an instantiation of our outlierness

function score(o,S) in Section III-C, where we will also give

details about the underlying adaptive density den(o,S) for a

comparable outlierness measurement. As a flexible approach

any density estimation method could be used in our outlierness

measure. Thus, for the general discussion in this section let

us assume the typically used density in a local neighborhood

defined as the number of objects in ε-distance to the object o

in subspace S:

to achieve comparable outliernessvalues

den(o,S) = |N(o,S)| = |{p | distS(o,p) ≤ ε}|

Based on this common density instantiation one can formally

derive two major challenges for an outlier ranking in sub-

spaces:

Challenge 1: Comparability of Outlierness

Outlierness measures are not comparable over multiple sub-

spaces if: For subspace S,T ⊆ D with T ⊂ S

due to curse of dimensionality

⇒

⇒

⇒

⇒

As density drops for increasing dimensionality, outlierness

measures based on density in subspace projections are biased

w.r.t. the dimensionality of the considered subspaces. As stated

in Challenge 1, considering a subspace T and one of its higher

∀p ∈ DB : distS(o,p) ≥ distT(o,p)

den(o,S) ≤ den(o,T)

score(o,S) ≤ score(o,T)

outlierness is biased w.r.t. dimensionality

dimensional projections S, density drops from T to S. Thus,

overall aggregation (cf. Def. 1) of outlierness is hindered by

incomparable measures. Please keep in mind that we set low

values in score(o,S) for highly deviating objects as we sort

our ranking in ascending order. With such an incomparable

measure, subspaces with many attributes would dominate the

ranking value and outliers in low dimensional projections

could not show up in the overall ranking. Thus, as we take

multiple subspaces into account we have to provide an adaptive

outlierness measure with comparable outlierness in arbitrary

subspace projections to achieve a fair ranking of objects in

any subspace.

Challenge 2: Relevance of Subspaces

A subspace S hinders the distinction of outliers if:

S is distributed uniformly random

∀o,p,q ∈ DB : distS(o,q) ≈ distS(p,q)

∀o,p ∈ DB : den(o,S) ≈ den(p,S)

∀o,p ∈ DB : score(o,S) ≈ score(p,S)

distinction of outliers is hindered

⇒

⇒

⇒

⇒

Obviously the full space D is such an irrelevant subspace for

increasing number of attributes |D| → ∞

With decreasing density, one reaches subspaces with uni-

formly distributed objects where outliers do not show up. In-

cluding such an irrelevant subspace projection S into a ranking

function yields very similar ranking values for all objects.

Thus, our key property for the set of relevant subspaces is

to exclude subspaces which are distributed uniformly random.

B. Selection of Relevant Subspaces

First, we propose a statistical selection of the set of relevant

subspaces RS(o) that can distinguish between the object o

and its local neighborhoods in the selected subspaces. As

motivated by Challenge 2, such a distinction based on the

objects density is not possible in scattered subspaces that show

uniformly random distributed data due to the low contrast

between outliers and regular objects. Thus, we propose to

exclude such scattered subspaces from outlier ranking by

testing the underlying distribution in the local neighborhood

N(o,S).

Our test is based on a statistical significance test aiming at

reducing the probability that a uniformly distributed subspace

passes into the set of relevant subspaces. We test against

the null hypothesis that data is uniformly distributed with

|N(o,S)| ∼ Binomial(|DB|,vol(N(o,S))). W.l.o.g, we

assume that data is normalized to 0...1 such that the volume

of N(o,S) provides us the probability of observing one object

in this neighborhood. As given for uniformly distributed data,

the expected number of objects is then |DB| · vol(N(o,S)).

Each object has equal probability of being in the neighborhood

depending only on the neighborhoods volume. Based on this

null hypothesis, we define H0(S is irrelevant) and H1(S is

a relevant subspace) for object o. As uniformly distributed

subspaces hinder the detection of meaningful outliers, this

definition ensures with a given significance level α that

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density

subspace dimensionality

1234

dense

subspaces

scattered

subspaces

subspace

subspace

subspace 2,3 2,3

1,2

1,4

11443322

1,21,2 1,41,42,3 2,3 3,43,4

1,2,3 1,2,3 2,3,42,3,4 1,3,41,3,41,2,41,2,4

1,2,3,4 1,2,3,4

1,3

2,4

scattered

subspaces

overall dense

subspaces

dimension 3

is irrelevant

relevant

subspaces

Fig. 2. Relevant subspace projections for outlier mining

uniformly distributed subspaces are only included into the

ranking with a very low probability of less than α. While for

uniformly distributed data one expects |DB| · vol(N(o,S))

many objects in the neighborhood, a relevant subspace should

contain significantly more objects.

Definition 2: Relevance Test

For subspace S and neighborhood N(o,S) we define hypothe-

ses H0and H1:

H0:

H1:

S is distributed uniformly random in N(o,S)

S is distributed non-uniformly in N(o,S)

ensuring significantly low first error:

P(H0is rejected |H0is true ) ≤ α

As statistical tool for testing uniform distribution we use the

Kolmogorov-Smirnov goodness of fit test for the uniform dis-

tribution [28]. In recent mining tasks this test has shown good

performance for subspace cluster detection [18]. In contrast to

subspace clustering where one is interested in sets of objects

grouped in a certain subspace, we select relevant subspaces for

each object to distinguish between outliers and regular objects.

For good ranking quality, we have to ensure that such uniform

subspace are only included in very rare cases by setting a low

α value. As significance level for the statistical hypothesis test

we set α = 0.01. Thus, the probability of wrongly rejecting

the hypothesis H0 (the subspace is uniformly distributed) is

only 1%, i.e. for one out of hundred uniform subspaces the

test will make an error and state that this subspace is relevant.

We will show the influence of the α parameter for the overall

outlier ranking quality in Section IV.

As illustrated on the left side of Figure 2, we exclude

uniformly distributed subspaces for each object o individually.

Starting considering 1d projections first, typically these low

dimensional projects are uniformly dense. The whole database

seems to be one dense region. Furthermore, outliers in 1d

projections could be easily detected as pre-processing. By

including more and more dimensions, due to correlations

of the data, the database diverts in multiple dense regions.

Density shows a high variance between dense clusters and

deviating outliers. For our outlier ranking we only take the

outlierness of objects in these relevant subspaces into account.

Adding even more dimensions the subspaces become scattered

like the full space. All objects seem to be outliers.

For a toy example we have depicted three subspaces where

a single hidden outlier is clearly deviating only in the relevant

subspace {1,2}. Measuring outlierness in this subspace yields

a clear distinction between this outlier and its local neigh-

borhood. Please note, it is crucial to exclude the scattered

subspace {1,4} as all objects seem outliers. Thus, outlierness

measures would lose their contrast as low score values are

provided for all objects. In contrast, subspace {2,3} is not

excluded by the relevance test as dense regions with high

scores do not affect the contrast of Def. 1.

Our general idea is to include only subspaces which are

distributed significantly different then the uniformly random

distribution. Hence, we exclude the subspaces that do not

provide any distinction between objects and hinder our outlier

detection. A key observation for relevant subspaces is that

with increasing number of attributes in a subspace S, one

reaches subspaces with uniformly distributed objects where

outliers do not show up any more. By including more and more

attributes distances between objects grow more and more alike

[5]. Thus, the selection of relevant subspaces can be reduced to

the selection of significant attributes to be included in a given

subspace projection S, as stated in the following corrollary:

Corrollary 1: Uniformly distributed subspaces

Let S = {d1,...,dk} be a subspace. Then it holds true:

S uniformly distributed ⇒

d1uniformly distributed

∧ ... ∧

dkuniformly distributed

Consequently, by testing each attribute diwe can assure that

no uniformly distributed subspace is included in the set of

relevant subspaces RS(o). Moreover, we discard a subspace

based on these insights as soon as at least one attribute is

distributed uniformly random.

We base on statistical tests to detect significant subspaces

by excluding uniformly distributed attributes from further

consideration. We perform an incremental processing of the

subspaces including in each step an additional attribute for the

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considered subspace S. By adding attribute dito S we check

if objects are uniformly distributed in di. We call an attribute

direlevant for outlier ranking, if objects are significantly non-

uniformly distributed. In contrast to other attributes, a relevant

attribute might be added to S while preserving the clustered

regions of subspace S also in subspace S∪di. Summing up, we

detect meaningful outliers by searching subspaces consisting

only of relevant attributes containing clustered regions from

which outliers can deviate.

Furthermore, as we aim at detecting outliers that deviate

from clustered objects in their local neighborhood we check

uniformly distribution according to this neighborhood N(o,S)

and not w.r.t. the entire subspace. Based on this, our out-

lier ranking is computed for each object w.r.t. its subspace

neighborhood, instead of the whole DB. Hence, the choice of

relevant subspaces occurs strictly on the basis of the object

locality. This is in contrast to subspace search approaches

[25], which provide global subspace estimations supporting

clustering with interesting projections. Compared to such

approaches, the main advantage of our subspace selection is

in the selection of locally significant subspaces for each object

taking local deviation into account.

To illustrate the effects of relevant subspace selection, we

depict a subspace lattice with all possible subspaces of a 4d

data space in Figure 2. Starting considering 1d projections

first, typically these low dimensional projects are uniformly

dense, and thus, they do not affect our outlier ranking. In

the relevant subspace projections highlighted in bold, we find

the detect outliers deviating from their local neighborhood.

As depicted we prune the higher-dimensional subspaces. For

example, adding dimension 3 to subspace {1,2} results in

such an irrelevant subspace. Incrementally using the statistical

test for each dimension we detect the irrelevant dimensions

and stop further processing of higher dimensional subspaces.

Formally, we define relevant subspaces RS(o) in Definition 3

to be the set of subspaces that are significantly non-uniformly

distributed.

Definition 3: Set of Relevant Subspaces

The set of relevant subspaces contains subspaces that are

significantly non-uniformly distributed:

RS(o) = {S ∈ P(D) | S passes H1}

Only these subspaces are considered for our outlier ranking

(cf. Def. 1).

Computing the outlierness of an object in its relevant sub-

spaces RS(o) yields a high contrast between the object and its

local neighborhood. Although we have excluded the irrelevant

subspaces a major challenge remains. Subspaces in RS(o)

have arbitrary dimensionality and show highly varying density

values as motivated in Challenge 1.

C. Adaptive Outlierness in Subspaces

For a meaningful outlier ranking based on outlierness in

multiple subspace projections the definition of score(o,S) has

to provide an adaptive outlierness measure as the overall rank-

ing combines object properties out of very different subspaces

S ∈ RS(o). We propose such an adaptive outlierness measure

by defining an adaptive density and a local deviation for each

object.

1) Adaptive object density: As formalized in Challenge 1,

measuring density in multiple subspaces leads to a challenging

task, namely the strong dependence of densities on the number

of attributes in the considered subspaces. For two subspaces

S,T ⊆ D with T ⊂ S a simple counting of objects in a

fixed neighborhood yields den(o,S) ≤ den(o,T). The main

problem for density-based mining of different subspaces is

the fixed neighborhood [3], [19], [23]. As distances between

objects grow with increasing number of attributes, a fixed

neighborhood N(o,S) = {p | distS(o,p) ≤ ε} becomes

empty. All objects tend to have higher distance than the fixed ε

parameter. To tackle this general problem of density estimation

in arbitrary subspaces, we propose an adaptive density using

a variable neighborhood. By increasing the neighborhood dis-

tance ε with increasing number of attributes, our density mea-

sure can automatically adapt to the expected data distribution.

Thus, different subspaces become comparable and outlierness

based on density estimation can automatically adapt to the

number of attributes.

In general, we propose an adaptive neighborhood AN(o,S)

based on a variable ε(|S|) range.

AN(o,S) = {p | distS(o,p) ≤ ε(|S|)}

The general idea is to derive the variable range out of a

common observation in subspace projections. While increasing

the number of attributes in a subspace projection the volume

of a fixed neighborhoods decreases significantly compared to

the overall volume of the subspace. For example, consider

the volume of the neighborhood covering the whole data

range 0...1 of one attribute with ε = 0.5. If one keeps the

neighborhood range fixed, the volume in subspace S is given

by

vol(N(o,S)) =

π|S|/2

Γ(|S|/2 + 1)· 0.5|S|

with the gamma function Γ(n + 1) = n · Γ(n),Γ(1) =

1,Γ(1/2) =

number of attributes.

√π. The volume decreases with increasing

vol(N(o,S)) ? vol(N(o,T)) for |T| < |S|

.

Thus, the expectation of detecting objects in such neigh-

borhoods is decreasing as well, resulting in very low density

estimations. Our variable neighborhood range adapts to this

phenomenon. While for 1-dimensional subspaces neighbor-

hood rages are typically set to lower values ε ≤ 0.5, in

higher-dimensional subspaces the range should be increased

with the number of dimensions. By increasing the range we

ensure that the expected number of objects remains constant.

Thus, we provide a comparable density estimation in arbitrary

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subspaces. In our prior work, we have shown that such an

adaptive neighborhood can be of general benefit for other

mining paradigms as well, e.g. for density-based subspace

clustering [19].

2) Instantiation of adaptive density: In the following we

instantiate the basic idea of adaptive neighborhoods to a

specific density estimation technique. As a flexible outlier

model, OUTRES could be used with any density measure such

as the simple counting of objects in the objects neighborhood

(cf. Section III-A). However, we base our density measure

on more enhanced and well established density estimation

techniques [26]. As the overall density distribution of the data

is not known in advance, density den(o,S) of an object o can

be estimated by using kernel density estimators.

Each object o contributes to the overall density by a local

impact defined by a kernel function K(x) with x =distS(o,p)

being the scaled distance of any other object p to the object

o. The bandwidth parameter h is used to scale the influence

of each object to a maximal distance of h. The overall density

for an object o is then simply the sum of kernel function

over all objects in the database. As kernel function we use

the Epanechnikov Kernel

h

Ke(x) = (1 − x2), x < 1

, providing optimal density estimation according to the mean

integrated squared error [26]. Concludingly, den(o,S) is cal-

culated by the formula:

?

Since objects being farther away than h from a certain object

do not contribute to its density, we obtain a local density on

which our outlier detection is based.

In contrast to the simple counting of objects in the neighbor-

hood given by |AN(o,S)|, kernel density estimation has major

benefits due to the weighted influence of each object. The

sum of Epanechnikov Kernels provides a theoretically sound

density definition. However, for density estimation in arbitrary

subspace projections the fixed bandwidth h shows similar

drawbacks to the fixed ε range. For comparable outlierness

over arbitrary subspaces, we propose to adapt the density by

a variable kernel bandwidth ε(|S|). As the true underlying

density distribution is unknown, we only use the dimension-

ality of the space to derive the bandwidth for adaptive density

estimation. For a fixed space with dimensionality d optimal

bandwidth hoptimal(d) is given by the following formula:

?

π

den(o,S) =

1

|DB|

p∈DB

Ke

?distS(o,p)

h

?

hoptimal(d) =

8 · Γ(d

2+ 1)

d

2

· (d + 4) · (2√π)d

?

· n

−1

d+4

where n = |DB| is the database size and the gamma function

as in the computation of the neighborhood volume [26].

As motivated in the previous paragraph, optimal bandwidth

is computed based on the expected number of objects in a

neighborhood. The formula for optimal bandwidth can simply

be seen as the optimal radius of an ε-sphere such that one yield

statistically optimal density estimation results. For density

estimation in subspaces this means that one has to choose

a bandwidth for each individual subspace. Assuming that

n is fixed in a static database, we observe hoptimal(d) to

be a monotonically increasing function. So intuitively, for

increasing dimensionality the influence (bandwidth) of each

object is increased as well in order to maintain optimal

density estimates, while the data space is becoming sparse.

For comparable outlierness we use the optimal bandwidth

to adapt density estimation in arbitrary subspaces. By the

user parameter ε we allow the user to quantify a notion of

locality and adjust this value for arbitrary subspaces based on

the optimal bandwidth. Formally, the bandwidth for a given

number of attributes in a subspace S is defined by Definition 4:

Definition 4: Adaptive neighborhood

For a subspaces dimensionality |S|, |S| ≥ 2, the adaptive

neighborhood ε(|S|) is defined by

ε(|S|) = ε ·hoptimal(|S|)

hoptimal(2)

Thus, we simply scale the given starting bandwidth ε from

2d space up to full data space and use these value for density

estimation. In contrast to the fixed bandwidth in kernel density

estimation we use our adaptive neighborhood as variable

bandwidth for each individual subspace. Consequently, our

automatic bandwidth adaption ensures comparable density

estimates for arbitrary dimensional subspaces.

3) Local object deviation: For our outlier ranking based

on deviations of density we first compute the density for

each object and compare it with the local (average) density

in a relevant subspace. By that, our approach is able to

detect objects highly deviating from the residual data in a

relevant subspace, i.e., objects having exceptionally low den-

sities. While our adaptive density ensures comparability over

multiple subspaces, our local deviation ensures meaningful

outlierness values inside one subspace. Hence, in addition to

the adaptive density, we ensure to highlight an outlier with

very low density compared to its local neighborhood in the

considered subspace.

Having such a comparable density estimation, an outlier can

be detected as an object showing significantly low density. As

we aim at a local outlierness we measure deviation based on

an adaptive threshold. As first filter step we select only objects

with significantly low density

den(o,S) < μ − 2 · σ

compared to μ and σ as the mean and standard deviation of

den(o,S) in the neighborhood of object o. From statistical

observations, only very rare objects deviate more than two

standard deviations from the mean value (cf. Chebyshev’s

inequality [8]). As statistical probability for such objects is

low (e.g. for normal distributed data it is less than 2.1%),

their outlierness has to be high. Using mean μ and standard

deviation σ of the estimated (local) density we ensure to be

Page 8

adaptive to varying density. Object deviation is then given by

the following definition.

Definition 5: Object deviation

The deviation of an object o with respect to mean and standard

deviation of the estimated density:

dev(o,S) =μ − den(o,S)

2 · σ

An object shows high deviation if its density compared

to the average density μ in its neighborhood AN(o,S) is

significantly low.

4) Adaptive Outlierness: Overall the outlierness of an ob-

ject o has to fulfill two major requirements. First, it has to be

adaptive to arbitrary dimensional subspaces. Thus, based on

our adaptive object density we propose an adaptive outlierness

which is comparable for different subspaces (cf. Sec. III-C2).

Second, our adaptive outlierness has to cope with object

deviation considering statistically deviation from the mean

value (cf. Sec. III-C3). Incorporating both aspects in our

adaptive outlierness measure we define score(o,S) as follows:

Definition 6: Adaptive Outlierness

The outlierness of an object o in subspace S is derived by its

density and its deviation in this subspace:

?

1

Our novel outlierness incorporates both aspects derived by

density and the deviation of each object: low density and

high deviation are both indicates for high outlierness. Highly

deviating objects show up by dev(o,S) ≥ 1 as density is

significantly low compared to mean and standard deviation.

Overall we cope with the different behaviors of objects

in different subspaces: Scattered irrelevant subspaces are ex-

cluded by our relevance testing (cf. Sec. III-B). Objects in

a dense subspace S result only in high density and almost

no deviation such that we set score(o,S) = 1 they do not

affect the ranking value (cf. Def. 1). Only if objects show up

with low density or high deviation in a relevant subspace they

contribute to the overall ranking value (cf. Def. 6).

score(o,S) =

den(o,S)

dev(o,S)

, if dev(o,S) ≥ 1

, else.

D. Computation of OUTRES

For the overall computation of our outlier ranking, first of

all one has to select the relevant subspaces for each object. In a

naive solution, one would test for each object o ∈ DB its local

neighborhood AN(o,S) in arbitrary subspaces S ∈ P(D).

As the number of possible subspaces increases exponentially

with the number of given attributes in D this selection is

obviously not practically feasible. For each object, one would

compute RS(o) out of 2|D|possible subspaces by using our

relevance test. Furthermore, we require for all objects a density

computation which yields a quadratic complexity w.r.t. the

number of objects. Overall the complexity of outlier ranking

based on relevant subspaces would be O(|DB|·(2|D|·|DB|)).

Thus, for an efficient processing we propose an approx-

imative selection of relevant subspaces based on a pruning

heuristic. We process subspaces bottom-up and prune based

on the observation that having reached a sparse subspace

with uniformly distributed data we may stop processing, as

data is scattered even more in higher dimensional projections.

As most important property, this pruning ensures to exclude

all irrelevant subspace. This provides a high contrast for our

outlier ranking as all considered subspaces are non-uniformly

distributed. As an approximative selection we cannot ensure

to include all non-uniformly distributed subspaces, but as

highlighted also by our experiments we achieve high quality

outlier ranking with this simple pruning heuristic. We even

observe some redundant subspaces that actually are selected

for RS(o), but do not contribute to the overall outlier ranking.

Thus, further enhancements for algorithmic solutions of our

novel outlier ranking model seem to be promising tasks for

future research. Especially, pruning of dense regions that

do not show any hidden outliers might lead to even better

runtimes. In addition to our pruning of irrelevant subspaces

one might add a second filter to prune some of the relevant

subspaces that might contribute only scores equal to 1 for the

outlierness measure. Such an enhanced filtering could safely

exclude further parts of the search space without any quality

losses.

Algorithm 1 OUTRES(o, S)

FOREACH i ∈ D \ S

S?= S ∪ {i};

IF S?is relevant

den(o,S?) =

dev(o,S?) =μ−den(o,S?)

IF dev(o,S?) ≥ 1

r(o) = r(o) ·den(o,S?)

OUTRES(o, S?);

ELSE

// break recursion for higher dimensional subspaces

// relevance test (cf. Def. 2)

p∈AN(o,S?)Ke(distS?(o,p)

;

// high deviation (cf. Def. 6)

dev(o,S?);

// recursively next subspace

1

|DB|

?

2·σ

ε(|S?|)

);

// aggregation of scoring

For the OUTRES algorithm, we test the relevance of sub-

spaces in a bottom-up processing. We base on the observation

that objects are dense in low dimensional spaces, while for

higher dimensional spaces they diverge until they form an

uniformly distributed scattered space. In Figure 2 we show

a box-plot for the varying distribution of density in various

dimensions as a toy example. Our bottom-up processing starts

with low dimensional subspaces and tests step-by-step each

subspace. The computation of the ranking r(o) based on the

relevant subspaces of the object o is given in Algorithm 1. We

start for each object o the recursive processing OUTRES(o,{}),

recursion stops if an irrelevant subspace has been detected

for object o. This ensures to exclude all irrelevant subspace

projections, which would hinder the distinction of outliers in

our ranking. In the worst case (if for all objects all subspaces

are relevant) complexity remains O(2|D|· |DB|2), but in

practical cases we yield efficient processing as we show in

Section IV in addition to the high quality outlier ranking

results.

Page 9

IV. EXPERIMENTS

We demonstrate the quality of our OUTRES approach on

both synthetic and real world data. We compare OUTRES to

the well established LOF [6] and its recent extensions ABOF

[15] as full space approaches. Furthermore, we compare

against OutRank [20] and SOD [13] as the most recent outlier

rankings based on subspace projections.

For comparability, we implemented all algorithms in our

open-source framework [21]. By extending the popular WEKA

framework we base our work on a widely used data input

format for repeatable and expandable experiments. We used

original implementations provided by the authors and best-

effort re-implementations based on the original papers. We en-

sure comparable evaluations and repeatability of experiments,

as we deploy all implemented algorithms on our website1.

With our SOREX system [22], we ensure that all of our results

will be reproducible, publicly available, and thus, might be

used for comparison in future publications.

For fair comparison we base on objective quality measures.

We believe that our evaluation setup provides a better quality

assessment than showing only some examples of the detected

outlier. In contrast to such a commonly used subjective eval-

uation, we highlight the achieved quality enhancement by

three different quality measures. We measure true positive

(TPR) and false positive (FPR) rates visualized in the well

established ROC plot. Both of these measures are useful to

derive if a ranking detects a high ratio of correct detected

outliers (TPR) while providing only few non-outlier as de-

tected outliers (FPR). However, they only take the ratio of

detected outliers and non-outliers into account ignoring more

or less the positioning of the objects in the ranking. Thus,

we additionally evaluate the results with a ranking coefficient

based on Spearmans Ranking Coefficient [27]. In contrast

to the ROC plot, ranking coefficients take also the ranking

positions of detected outliers into account. This leads to a

more fine grained quality measure.

To illustrate the quality of the rankings we use the quality

measures for the top-k ranked objects (cf. Definition 7). The

TPR measure is simply the fraction of found true outliers in

the first k objects

found true outliers(R,k) =

{or1...ork} ∩ DBhidden outliers

compared to the set of hidden outliers DBhidden outliersin

the database DB. Analogue, FPR is the fraction of found

non-outliers in the first k objects

found false outliers(R,k) =

{or1...ork} \ found true outliers(R,k)

compared to the overall set of non-outlier objects in the

database.

1http://dme.rwth-aachen.de/OpenSubspace/SOREX

Definition 7: TPR and FPR measures

The true positive rate for the first k objects of a ranking R =

{or1...orn} is defined as:

TPR(R,k) =|found true outliers(R,k)|

|DBhidden outliers|

The false positive rate is defined as:

FPR(R,k) =|found false outliers(R,k)|

|DBhidden non-outliers|

More detailed measures can be derived by ranking coefficients

[27]. Spearmans Ranking Coefficient SRC(R1,R2) computes

the correlation of two given rankings R1 and R2. We use

SRC to measure the quality for one ranking by comparing

it with the optimal ranking Rbest, ranking all outliers first.

Furthermore, we normalize with the ranking coefficient for the

worst ranking Rworsthaving all outliers in the last positions.

We define outlier ranking coefficient ORC(R,k) for the first

k objects in ranking R as given in Definition 8.

Definition 8: Ranking coefficient measure

The outlier ranking coefficient for the first k objects of a

ranking R = {or1...orn} is defined as:

ORC(R,k) =SRC({or1...ork},Rbest)

SRC(Rworst,Rbest)

For these measures the optimal ranking results in

TPR(Roptimal,k) = 1 ∧ FPR(Roptimal,k) = 0 and

ORC(Roptimal,k) = 1 for k = |DB|outliers

For non-optimal rankings TPR = 1 is reached for larger k

with FPR ? 0, while the ORC measure does not reach

the maximal value of 1 at all for non-optimal rankings. Thus,

the ORC measure is more appropriate for evaluation of outlier

rankings. By taking the actual positing of objects into account,

ORC is able to distinguish between two rankings having

found the same amount of outliers in the first k positions.

In such a case, TPR and FPR show same results as they

only consider the object ratio and cannot distinguish between

these two rankings. Taking also positioning information into

account ORC shows more fine grained differences in rank-

ings. Especially, one can compare ranking quality by taking the

overall ORC(R,|DB|) for comparison. Thus, after showing

all three measures in the first experiment we use only the

ranking coefficient measure for comparison in the following

experiments.

A. Synthetic Data

For scalability experiments, we generate synthetic data

following a method proposed in [11], [3] to generate density-

based clusters in arbitrary subspaces. In addition, our generator

adds outliers deviating from one of these subspace clusters. As

there are no global patterns hidden in data, the hidden outliers

do not appear in the scattered full space. In our first exper-

iment, we evaluate the quality of the competing approaches

Page 10

on a synthetic data set with 4765 objects represented by four

subspace clusters each using 4 out the 16 given attributes and

additionally 61 hidden outliers deviating from these clusters.

Figure 3(a) illustrates the quality with respect to ROC plot.

We observe that all approaches show high increase in true

positive rates of detected outliers with only very few false

positive. However, all hidden outliers (TPR = 1) are found

after thousands of considered objects, indicated by FPR ? 0.

Our novel OUTRES shows best performance compared to

LOF, ABOF, SOD and OutRank, as it archives to detect more

hidden outliers within the first ranked objects showing both

higher TPR and lower FPR than the competitors. Com-

paring ROC plot and ranking coefficient in Figure 3 for the

same experiment, we observe that OUTRES outperforms all

competing approaches in both quality measures independent

of the number of ranked objects. For ranking coefficient it

always shows highest correlation with the optimal ranking.

As ranking coefficient provides more information about the

positioning of the objects than the ROC plot, we use only this

measure in the following experiments.

0

0,2

0,4

0,6

0,8

1

0 0,2 0,40,60,81

True Positive Rate

False Positive Rate

(a) ROC plot

OUTRES

OutRank

LOF

ABOF

SOD

0

0,2

0,4

0,6

0,8

1

0 10002000 3000 4000

ranking coefficient

ranked objects

OUTRES

OutRank

LOF

ABOF

SOD

(b) ranking coefficient

Fig. 3.Ranking quality on synthetic data

In our second experiment, we evaluate the scalability of

outlier rankings with respect to the number of given attributes

in the database. As outlier ranking in subspace projections

aims to detect outliers hidden in any subset of the given at-

tributes, scalabiltiy w.r.t. number of given attributes is crucial.

We varied the number of attributes from 10 up to 50, while

keeping number of hidden subspace clusters, hidden outliers

and database size constant, as in the previous experiment.

Figure 4(a) shows the almost constantly high quality of our

approach. As we add more an more attributes, hidden outliers

disappear in the overall scattered full space. However, as

OUTRES investigates only relevant subspace projections it

scales w.r.t. number of given attributes. It outperforms all

competing approaches in terms of quality.

1

0,8

nt

oefficien

king?co

ran

0,6

0,4

OUTRES

OutRank OutRank

LOF

ABOF

ABOF

SOD

0,2

00

1020 30 40 50

data?space?dimensionality

(a) ranking quality

1

10

100

1000

10000

100000

1000000

10 2030 4050

runtime [sec.]

data space dimensionality

ABOF

OUTRES

SOD

OutRank

LOF

(b) runtime

Fig. 4. Scalability w.r.t. number of attributes

In Figure 4(b) we compare the runtimes. In conjunction

with the previous quality plot we observe that OUTRES

achieves to perform both efficient outlier ranking and a high

outlier ranking quality. Although OUTRES has to search for

outliers in arbitrary subsets of the attributes, the proposed

pruning heuristic by excluding uniformly distributed attributes

shows both high quality results but also efficient computation.

We skip further scalability experiments, as experiments have

shown that database size has less impact on both quality

and runtime. However, as an important issue we discuss

parametrization of our approach in the following experiments.

B. Parametrization

For the two main parameters α and ε we show the robust-

ness of the ranking quality of OUTRES. On the synthetic data

set from previous experiment, we varied the neighborhood

parameter ε from 5 to 45 (data ranges from 0 to 100).

Page 11

As depicted in Figure 5(a), OUTRES shows a quite robust

ranking quality only slightly decreasing for high ε values.

By increasing the neighborhood around each object density

is increasing for all objects. Especially for outliers, density is

becoming similar to clustered objects. Overall we achieve a

robust approach w.r.t. ε due to our automatic adaption of the

neighborhood range for the arbitrary subspace projections con-

sidered in OUTRES (cf. Def. 4). As default setting of ε in our

experiments we use ε = 15 showing best results. In general,

with our adaptive neighborhood one can set the usual low ε

neighborhood ranges in low dimensional subspace. These are

increased automatically for higher-dimensional subspaces and

provide a high quality density estimation for arbitrary subspace

projections.

0,5

0,4

0,6

0,7

0,8

0,9

1

coefficient

ranking?

OUTRES

0

0,1

0,2

0,3

5 10 1520 25 3035 4045

? ? parameter

(a) Variation of ε

0,5

0,4

0,6

0,7

0,8

0,9

1

coefficient

ranking?

OUTRES

0

0,1

0,2

0,3

0 0,10,20,30,4

? ? parameter

(b) Variation of α

Fig. 5. Robustness of OUTRES w.r.t parameters

For the second parameter α we observe similar effects. As

depicted in Figure 5(b) we observe best ranking quality for

low α settings. For higher α settings, OUTRES accepts more

and more uniform distributed subspaces as relevant subspaces

for outlier ranking. As one cannot distinguish between outliers

and regular objects in these scattered subspaces, the overall

ranking quality decreases. Keeping a low α setting (default

α = 0.01), thus, ensures to measure outlierness of objects

only in relevant subspace projections.

C. Real World Data

We analyzed the quality of outlier ranking on three real

world data sets (Ionosphere, Breast Cancer and Pendigits) from

the UCI repository [7]. All of these data sets provide scattered

full spaces, while subsets of the given attributes can be used to

distinguish between the hidden patterns and outlying objects.

For example, in the Pendigits data set objects are described by

(x,y) positions concatenated in a digit trajectory. Clearly not all

of the pen positions are important to detect outlying objects.

Some digits deviate significantly in the first position (first two

given attributes) from the residual objects starting typically at

similar positions (upper left area). Similarly, also in the other

data sets outliers can be distinguished from regular objects

using subspace projections of the database. For our evaluation

measures, we used one of the class labels reduced to 10% of its

size as ground truth for hidden outliers. In contrast to adding

artificial outliers into the database, such a reduction of the

orginal data distribution seems more natural. The remaining

objects of the reduced class show high deviation from other

classes and have low density due to the eliminated parts of

their own class. In our experiments, we show that outlier

ranking approaches successfully detects these very rare hidden

observations in subspaces of the given databases.

In Figure 6 we show the ranking coefficients for the real

world databases. For all data sets we observe a high ranking

quality of OUTRES, outperforming competing approaches by

detecting outliers as top ranked objects. Hidden outliers are

clearly distinguished by the selection of relevant subspaces.

For example, in the pendigits data one object representing the

digit four has been first ranked due to its high deviation in

the last positions of its trajectory. Overall, our novel OUTRES

approach provides highest quality results, while the competing

approaches show varying quality over multiple data sets.

V. CONCLUSION

In this work, we proposed a novel outlier ranking for objects

deviating in subspace projections. The OUTRES approach

computes local density deviation by looking at a selection of

relevant subspaces for each object. Relevance of subspaces is

measured by statistical significance tests. Thus, only relevant

subspaces that are not distributed uniformly random are used

for our outlier ranking. For comparable outlierness measures

in different subspaces, we derive an adaptive density mea-

sure which automatically adapts to the considered subspace.

OUTRES computes an overall high quality outlier ranking

by aggregating this adaptive outlierness of objects in relevant

subspaces. Our thorough evaluation on both synthetic and real

world data shows that OUTRES outperforms competing outlier

ranking approaches. Especially, OUTRES achieves to detect

outliers hidden in subspace projections.

ACKNOWLEDGMENT

This work has been supported in part by the UMIC Research

Centre, RWTH Aachen University, Germany.

Page 12

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0,4

0,3

0,5

0,6

0,7

0,8

0,9

1

nking?coefficient

ran

OUTRES

OutRank

LOF

ABOF

SOD

0

0,1

0,2

0 1000 20003000

ranked?objects

400050006000 7000

(a) Pendigits (digit 4)

0,4

0,3

0,5

0,6

0,7

0,8

0,9

1

nking?coefficient

ran

0

0,1

0,2

0 10002000 3000

ranked?objects

4000500060007000

OUTRES

OutRank

LOF

ABOF

SOD

(b) Pendigits (digit 6)

1

0,8

nt

oefficien

king?co

ran

0,6

0,4

OUTRES

OutRank

LOF

ABOF

SOD

0,2,

00

0 50100

ranked?objects

150200 250

(c) Ionosphere

1

0,8

nt

oefficien

king?co

ran

0,6

0,4

OUTRES

OutRank OutRank

LOF

ABOF ABOF

SOD

0,2

00

0 50 100150

ranked?objects

(d) Breast Cancer

Fig. 6. Ranking quality on real world data