Ramaswamy, Rastogi & Shim - Efficient Algorithm for Mining Outliers in Large Data Sets

Ramaswamy, Rastogi & Shim - Efficient Algorithm for Mining Outliers in Large Data Sets
Efficient Algorithms for Mining Outliers from Large Data Sets
Sridhar Ramaswamy
Rajeev Rastogi
Kyuseok Shim
Epiphany Inc.
Palo Alto, CA 94403
[email protected]
Bell Laboratories
Murray Hill, NJ 07974
[email protected]
KAISTy and AITrcz
Taejon, KOREA
[email protected]
Abstract
In this paper, we propose a novel formulation for distance-based
outliers that is based on the distance of a point from its th nearest
neighbor. We rank each point on the basis of its distance to its
th nearest neighbor and declare the top points in this ranking
to be outliers. In addition to developing relatively straightforward
solutions to finding such outliers based on the classical nestedloop join and index join algorithms, we develop a highly efficient
partition-based algorithm for mining outliers. This algorithm first
partitions the input data set into disjoint subsets, and then prunes
entire partitions as soon as it is determined that they cannot contain
outliers. This results in substantial savings in computation. We
present the results of an extensive experimental study on real-life
and synthetic data sets. The results from a real-life NBA database
highlight and reveal several expected and unexpected aspects of
the database. The results from a study on synthetic data sets
demonstrate that the partition-based algorithm scales well with
respect to both data set size and data set dimensionality.
k
k
1
n
Introduction
Knowledge discovery in databases, commonly referred to
as data mining, is generating enormous interest in both the
research and software arenas. However, much of this recent
work has focused on finding “large patterns.” By the phrase
“large patterns”, we mean characteristics of the input data
that are exhibited by a (typically user-defined) significant
portion of the data. Examples of these large patterns
include association rules[AMS+ 95], classification[RS98]
and clustering[ZRL96, NH94, EKX95, GRS98].
In this paper, we focus on the converse problem of finding
“small patterns” or outliers. An outlier in a set of data
is an observation or a point that is considerably dissimilar
or inconsistent with the remainder of the data. From the
The work was done while the author was with Bell Laboratories.
y Korea Advanced Institute of Science and Technology
z Advanced Information Technology Research Center at KAIST
above description of outliers, it may seem that outliers are
a nuisance—impeding the inference process—and must be
quickly identified and eliminated so that they do not interfere
with the data analysis. However, this viewpoint is often too
narrow since outliers contain useful information. Mining for
outliers has a number of useful applications in telecom and
credit card fraud, loan approval, pharmaceutical research,
weather prediction, financial applications, marketing and
customer segmentation.
For instance, consider the problem of detecting credit card
fraud. A major problem that credit card companies face is
the illegal use of lost or stolen credit cards. Detecting and
preventing such use is critical since credit card companies
assume liability for unauthorized expenses on lost or stolen
cards. Since the usage pattern for a stolen card is unlikely
to be similar to its usage prior to being stolen, the new
usage points are probably outliers (in an intuitive sense) with
respect to the old usage pattern. Detecting these outliers is
clearly an important task.
The problem of detecting outliers has been extensively
studied in the statistics community (see [BL94] for a good
survey of statistical techniques). Typically, the user has
to model the data points using a statistical distribution,
and points are determined to be outliers depending on
how they appear in relation to the postulated model. The
main problem with these approaches is that in a number
of situations, the user might simply not have enough
knowledge about the underlying data distribution. In order
to overcome this problem, Knorr and Ng [KN98] propose
the following distance-based definition for outliers that is
both simple and intuitive: A point p in a data set is an outlier
with respect to parameters k and d if no more than k points
in the data set are at a distance of d or less from p1 . The
distance function can be any metric distance function 2.
The main benefit of the approach in [KN98] is that it does
not require any apriori knowledge of data distributions that
the statistical methods do. Additionally, the definition of
outliers considered is general enough to model statistical
1 The precise definition used in [KN98] is slightly different from, but
equivalent to, this definition.
2 The algorithms proposed assume that the distance between two points
is the euclidean distance between the points.
outlier tests for normal, poisson and other distributions. The
authors go on to propose a number of efficient algorithms
for finding distance-based outliers. One algorithm is a block
nested-loop algorithm that has running time quadratic in the
input size. Another algorithm is based on dividing the space
into a uniform grid of cells and then using these cells to
compute outliers. This algorithm is linear in the size of the
database but exponential in the number of dimensions. (The
algorithms are discussed in detail in Section 2.)
The definition of outliers from [KN98] has the advantages
of being both intuitive and simple, as well as being computationally feasible for large sets of data points. However, it
also has certain shortcomings:
the underlying data set, thus making it easier for the user to
specify compared to d.
The contributions of this paper are as follows:
1. It requires the user to specify a distance d which could be
difficult to determine (the authors suggest trial and error
which could require several iterations).
2. It does not provide a ranking for the outliers—for
instance a point with very few neighboring points within
a distance d can be regarded in some sense as being a
stronger outlier than a point with more neighbors within
distance d.
3. The cell-based algorithm whose complexity is linear in
the size of the database does not scale for higher number
of dimensions (e.g., 5) since the number of cells needed
grows exponentially with dimension.
In this paper, we focus on presenting a new definition for
outliers and developing algorithms for mining outliers that
address the above-mentioned drawbacks of the approach
from [KN98]. Specifically, our definition of an outlier
does not require users to specify the distance parameter
d. Instead, it is based on the distance of the k th nearest
neighbor of a point. For a k and point p, let D k (p) denote
the distance of the k th nearest neighbor of p. Intuitively,
Dk (p) is a measure of how much of an outlier point p is.
For example, points with larger values for D k (p) have more
sparse neighborhoods and are thus typically stronger outliers
than points belonging to dense clusters which will tend to
have lower values of D k (p). Since, in general, the user is
interested in the top n outliers, we define outliers as follows:
Given a k and n, a point p is an outlier if no more than n ; 1
other points in the data set have a higher value for D k than
p. In other words, the top n points with the maximum D k
values are considered outliers. We refer to these outliers as
the Dnk (pronounced “dee-kay-en”) outliers of a dataset.
The above definition has intuitive appeal since in essence,
it ranks each point based on its distance from its k th nearest
neighbor. With our new definition, the user is no longer
required to specify the distance d to define the neighborhood
of a point. Instead, he/she has to specify the number of
outliers n that he/she is in interested in—our definition
basically uses the distance of the k th neighbor of the nth
outlier to define the neighborhood distance d. Usually, n can
be expected to be very small and is relatively independent of
We propose a novel definition for distance-based outliers
that has great intuitive appeal. This definition is based on
the distance of a point from its k th nearest neighbor.
The main contribution of this paper is a partition-based
outlier detection algorithm that first partitions the input
points using a clustering algorithm, and computes lower
and upper bounds on D k for points in each partition.
It then uses this information to identify the partitions
that cannot possibly contain the top n outliers and
prunes them. Outliers are then computed from the
remaining points (belonging to unpruned partitions) in
a final phase. Since n is typically small, our algorithm
prunes a significant number of points, and thus results in
substantial savings in the amount of computation.
We present the results of a detailed experimental study
of these algorithms on real-life and synthetic data sets.
The results from a real-life NBA database highlight
and reveal several expected and unexpected aspects of
the database. The results from a study on synthetic
data sets demonstrate that the partition-based algorithm
scales well with respect to both data set size and data set
dimensionality. It also performs more than an order of
magnitude better than the nested-loop and index-based
algorithms.
The rest of this paper is organized as follows. Section 2 discusses related research in the area of finding outliers. Section 3 presents the problem definition and the
notation that is used in the rest of the paper. Section 4
presents the nested loop and index-based algorithms for outlier detection. Section 5 discusses our partition-based algorithm for outlier detection. Section 6 contains the results
from our experimental analysis of the algorithms. We analyzed the performance of the algorithms on real-life and
synthetic databases. Section 7 concludes the paper. The
work reported in this paper has been done in the context
of the Serendip data mining project at Bell Laboratories
(www.bell-labs.com/projects/serendip).
2
Related Work
Clustering algorithms like CLARANS [NH94], DBSCAN
[EKX95], BIRCH [ZRL96] and CURE [GRS98] consider
outliers, but only to the point of ensuring that they do not
interfere with the clustering process. Further, the definition
of outliers used is in a sense subjective and related to the
clusters that are detected by these algorithms. This is in
contrast to our definition of distance-based outliers which is
more objective and independent of how clusters in the input
data set are identified. In [AAR96], the authors address the
problem of detecting deviations – after seeing a series of
Symbol
k
Dk
n
N
M
dist
MINDIST
MAXDIST
Description
Number of neighbors of a point that we are interested in
Distance of point p to its kth nearest neighbor
Total number of outliers we are interested in
Total number of input points
Dimensionality of the input
Amount of memory available
Distance between a pair of points
Minimum distance between a point/MBR and MBR
Maximum distance between a point/MBR and MBR
Table 1: Notation Used in the Paper
similar data, an element disturbing the series is considered
an exception. Table analysis methods from the statistics
literature are employed in [SAM98] to attack the problem
of finding exceptions in OLAP data cubes. A detailed value
of the data cube is called an exception if it is found to differ
significantly from the anticipated value calculated using a
model that takes into account all aggregates (group-bys) in
which the value participates.
As mentioned in the introduction, the concept of distancebased outliers was developed and studied by Knorr and
Ng in [KN98]. In this paper, for a k and d, the authors
define a point to be an outlier if at most k points are within
distance d of the point. They present two algorithms for
computing outliers. One is a simple nested-loop algorithm
with worst-case complexity O(N 2 ) where is the number
of dimensions and N is the number of points in the dataset.
In order to overcome the quadratic time complexity of
the nested-loop algorithm, the authors propose a cell-based
approach for computing outliers in which the dimensional
d . The
space is partitioned into cells with sides of length 2p
time complexity of this cell-based algorithm is O(c + N )
where c is a number that is inversely proportional to d. This
complexity is linear is N but exponential in the number of
dimensions. As a result, due to the exponential growth in the
number of cells as the number of dimensions is increased,
the nested loop outperforms the cell-based algorithm for
dimensions 4 and higher.
While existing work on outliers focuses only on the identification aspect, the work in [KN99] also attempts to provide intensional knowledge, which is basically an explanation of why an identified outlier is exceptional. Recently, in
[BKNS00], the notion of local outliers is introduced, which
like Dnk outliers, depend on their local neighborhoods. However, unlike Dnk outliers, local outliers are defined with respect to the densities of the neighborhoods.
3
Problem Definition and Notation
In this section, we first present a precise statement of the
problem of mining outliers from point data sets. We then
present some definitions that are used in describing our
algorithms. Table 1 describes the notation that we use in
the remainder of the paper.
3.1 Problem Statement
Recall from the introduction that we use D k (p) to denote the
distance of point p from its k th nearest neighbor. We rank
points on the basis of their D k (p) distance, leading to the
following definition for Dnk outliers:
Definition 3.1 : Given an input data set with N points,
parameters n and k , a point p is a Dnk outlier if there are no
more than n ; 1 other points p0 such that Dk (p0 ) > Dk (p).3
In other words, if we rank points according to their
Dk (p) distance, the top n points in this ranking are
considered to be outliers. We can use any of the L p metrics
like the L1 (“manhattan”) or L2 (“euclidean”) metrics for
measuring the distance between a pair of points. Alternately,
for certain application domains (e.g., text documents),
nonmetric distance functions can also be used, making our
definition of outliers very general.
With the above definition for outliers, it is possible to rank
outliers based on their D k (p) distances—outliers with larger
Dk (p) distances have fewer points close to them and are thus
intuitively stronger outliers. Finally, we note that for a given
k and d, if the distance-based definition from [KN98] results
in n0 outliers, then each of them is a Dnk 0 outlier according
to our definition.
3.2 Distances between Points and MBRs
One of the key technical tools we use in this paper is
the approximation of a set of points using their minimum
bounding rectangle (MBR). Then, by computing lower and
upper bounds on D k (p) for points in each MBR, we are
able to identify and prune entire MBRs that cannot possibly
contain Dnk outliers. The computation of bounds for MBRs
requires us to define the minimum and maximum distance
between two MBRs. Outlier detection is also aided by
the computation of the minimum and maximum possible
distance between a point and an MBR, which we define
below.
In this paper, we use the square of the euclidean distance (instead of the euclidean distance itself) as the distance
metric since it involves fewer and less expensive computations. We denote the distance between two points p and q by
dist(p q ). Let us denote a point p in -dimensional space by
p1 p2 : : : p ] and a -dimensional rectangle R by the two
endpoints of its major diagonal: r = r1 r2 : : : r ] and
r0 = r10 r20 : : : r0 ] such that ri ri0 for 1 i n. Let us
denote the minimum distance between point p and rectangle
R by MINDIST(p R). Every point in R is at a distance of
at least MINDIST(p R) from p. The following definition of
MINDIST is from [RKV95]:
Definition 3.2: MINDIST(p R) =
P
2
i=1 xi , where
3 Note that more than n points may satisfy our definition of Dk
n
outliers—in this case, any n of them satisfying our definition are considered
k outliers.
Dn
8 r ;p
< i i
xi =
: p0i ; ri
0
if pi < ri
if ri0 < pi
otherwise
We denote the maximum distance between point p and
rectangle R by MAXDIST(p R). That is, no point in R is
at a distance that exceeds MAXDIST(p, R) from point p.
MAXDIST(p R) is calculated as follows:
xi
=
P
2
i=1 xi , where
r +r
ri ; pi if pi < i 2 i
pi ; ri otherwise
Definition 3.3: MAXDIST(p R) =
0
0
We next define the minimum and maximum distance
between two MBRs. Let R and S be two MBRs defined
by the endpoints of their major diagonal (r r 0 and s s0
respectively) as before. We denote the minimum distance
between R and S by MINDIST(R S ). Every point in R
is at a distance of at least MINDIST(R S ) from any point
in S (and vice-versa). Similarly, the maximum distance
between R and S , denoted by MAXDIST(R S ) is defined.
The distances can be calculated using the following two
formulae:
Definition 3.4: MINDIST(R S ) =
8 r ;s
< i i
xi =
: s0i ; ri
0
0
P
if s0i < ri
if ri0 < si
otherwise
Definition 3.5: MAXDIST(R S ) =
maxfjs0i ; ri j jri0 ; si jg.
4
2
i=1 xi , where
P
2
i=1 xi , where xi =
Nested-Loop and Index-Based
Algorithms
In this section, we describe two relatively straightforward
solutions to the problem of computing D nk outliers.
Block Nested-Loop Join: The nested-loop algorithm for
computing outliers simply computes, for each input point
p, Dk (p), the distance of its k th nearest neighbor. It then
selects the top n points with the maximum D k values. In
order to compute D k for points, the algorithm scans the
database for each point p. For a point p, a list of the k
nearest points for p is maintained, and for each point q from
the database which is considered, a check is made to see
if dist(p q ) is smaller than the distance of the k th nearest
neighbor found so far. If the check succeeds, q is included
in the list of the k nearest neighbors for p (if the list contains
more than k neighbors, then the point that is furthest away
from p is deleted from the list). The nested-loop algorithm
can be made I/O efficient by computing D k for a block of
points together.
Index-Based Join:
Even with the I/O optimization,
the nested-loop approach still requires O(N 2 ) distance
computations. This is expensive computationally, especially
if the dimensionality of points is high. The number of
distance computations can be substantially reduced by using
a spatial index like an R -tree [BKSS90].
If we have all the points stored in a spatial index like
the R -tree, the following pruning optimization, which was
pointed out in [RKV95], can be applied to reduce the
number of distance computations: Suppose that we have
computed D k (p) for p by looking at a subset of the input
points. The value that we have is clearly an upper bound for
the actual Dk (p) for p. If the minimum distance between p
and the MBR of a node in the R -tree exceeds the D k (p)
value that we have currently, none of the points in the subtree rooted under the node will be among the k nearest
neighbors of p. This optimization lets us prune entire subtrees containing points irrelevant to the k -nearest neighbor
search for p.4
In addition, since we are interested in computing only
the top n outliers, we can apply the following pruning
optimization for discontinuing the computation of D k (p)
for a point p. Assume that during each step of the indexbased algorithm, we store the top n outliers computed. Let
Dnmin be the minimum D k among these top outliers. If
during the computation of D k (p) for a point p, we find
that the value for D k (p) computed so far has fallen below
Dnmin , we are guaranteed that point p cannot be an outlier.
Therefore, it can be safely discarded. This is because
Dk (p) monotonically decreases as we examine more points.
Therefore, p is guaranteed to not be one of the top n outliers.
Note that this optimization can also be applied to the nestedloop algorithm.
Procedure computeOutliersIndex for computing D nk outliers is shown in Figure 1. It uses Procedure getKthNeighborDist in Figure 2 as a subroutine. In computeOutliersIndex, points are first inserted into an R -tree index (any other
spatial index structure can be used instead of the R -tree) in
steps 1 and 2. The R -tree is used to compute the k th nearest neighbor for each point. In addition, the procedure keeps
track of the n points with the maximum value for D k at any
point during its execution in a heap outHeap. The points are
stored in the heap in increasing order of D k , such that the
point with the smallest value for D k is at the top. This Dk
value is also stored in the variable minDkDist and passed to
the getKthNeighborDist routine. Initially, outHeap is empty
and minDkDist is 0.
The for loop spanning steps 5-13 calls getKthNeighborDist for each point in the input, inserting the point into outHeap if the point’s D k value is among the top n values seen
4 Note that the work in [RKV95] uses a tighter bound called MINMAXDIST in order to prune nodes. This is because they want to find the
maximum possible distance for the nearest neighbor point of p, not the k
nearest neighbors as we are doing. When looking for the nearest neighbor
of a point, we can have a tighter bound for the maximum distance to this
neighbor.
kn
Procedure computeOutliersIndex( , )
begin
1. for each point in input data set do
2.
insertIntoIndex(Tree, )
3. outHeap := 4. minDkDist := 0
5. for each point in input data set do f
6.
getKthNeighborDist(Tree.Root, , , minDkDist)
7.
if ( .DkDist minDkDist) f
8.
outHeap.insert( )
9.
if (outHeap.numPoints()
) outHeap.deleteTop()
)
10.
if (outHeap.numPoints()
11.
minDkDist := outHeap.top().DkDist
12.
g
13. g
14. return outHeap
end
p
p
p
p
>
pk
p
>n
=n
Figure 1: Index-Based Algorithm for Computing Outliers
so far (p.DkDist stores the Dk value for point p). If the
heap’s size exceeds n, the point with the lowest D k value is
removed from the heap and minDkDist updated.
Procedure getKthNeighborDist computes D k (p) for point
p by examining nodes in the R -tree. It does this using a
linked list nodeList. Initially, nodeList contains the root of
the R -tree. Elements in nodeList are sorted, in ascending
order of their MINDIST from p. 5 During each iteration
of the while loop spanning lines 4–23, the first node from
nodeList is examined.
If the node is a leaf node, points in the leaf node are
processed. In order to aid this processing, the k nearest
neighbors of p among the points examined so far are
stored in the heap nearHeap. nearHeap stores points in the
decreasing order of their distance from p. p.Dkdist stores
Dk for p from the points examined. (It is 1 until k points
are examined.) If at any time, a point q is found whose
distance to p is less than p.Dkdist, q is inserted into nearHeap
(steps 8–9). If nearHeap contains more than k points,
the point at the top of nearHeap discarded, and p.Dkdist
updated (steps 10–12). If at any time, the value for p.Dkdist
falls below minDkDist (recall that p.Dkdist monotonically
decreases as we examine more points), point p cannot
be an outlier. Therefore, procedure getKthNeighborDist
immediately terminates further computation of D k for p
and returns (step 13). This way, getKthNeighborDist avoids
unnecessary computation for a point the moment it is
determined that it is not an outlier candidate.
On the other hand, if the node at the head of nodeList
is an interior node, the node is expanded by appending its
children to nodeList. Then nodeList is sorted according to
MINDIST (steps 17–18). In the final steps 20–22, nodes
whose minimum distance from p exceed p.DkDist, are
pruned. Points contained in these nodes obviously cannot
qualify to be amongst p’s k nearest neighbors and can be
5 Distances for nodes are actually computed using their MBRs.
pk
Procedure getKthNeighborDist(Root, , , minDkDist)
begin
1. nodeList := f Root g
2. .Dkdist := 1
3. nearHeap := 4. while nodeList is not empty do f
5.
delete the first element, Node, from nodeList
6.
if (Node is a leaf) f
7.
for each point in Node do
8.
if (
.DkDist) f
9.
nearHeap.insert( )
) nearHeap.deleteTop()
10.
if (nearHeap.numPoints()
11.
if (nearHeap.numPoints()
)
12.
.DkDist :=
( , nearHeap.top())
13.
if ( .Dkdist minDkDist) return
g
14.
15.
g
16.
else f
17.
append Node’s children to nodeList
18.
sort nodeList by MINDIST
19.
g
20.
for each Node in nodeList do
21.
if ( .DkDist MINDIST( ,Node))
22.
delete Node from nodeList
23. g
end
p
q
dist(pq) < p
p
p
p
q
dist p
>k
=k
p
Figure 2: Computation of Distance for k th Nearest Neighbor
safely ignored.
5
Partition-Based Algorithm
The fundamental shortcoming with the algorithms presented
in the previous section is that they are computationally
expensive. This is because for each point p in the database
we initiate the computation of D k (p), its distance from its
k th nearest neighbor. Since we are only interested in the
top n outliers, and typically n is very small, the distance
computations for most of the remaining points are of little
use and can be altogether avoided.
The partition-based algorithm proposed in this section
prunes out points whose distances from their k th nearest
neighbors are so small that they cannot possibly make it
to the top n outliers. Furthermore, by partitioning the
data set, it is able to make this determination for a point p
without actually computing the precise value of D k (p). Our
experimental results in Section 6 indicate that this pruning
strategy can result in substantial performance speedups due
to savings in both computation and I/O.
5.1 Overview
The key idea underlying the partition-based algorithm is
to first partition the data space, and then prune partitions
as soon as it can be determined that they cannot contain
outliers. Since n will typically be very small, this additional
preprocessing step performed at the granularity of partitions
rather than points eliminates a significant number of points
as outlier candidates. Consequently, k th nearest neighbor
computations need to be performed for very few points, thus
speeding up the computation of outliers. Furthermore, since
the number of partitions in the preprocessing step is usually
much smaller compared to the number of points, and the
preprocessing is performed at the granularity of partitions
rather than points, the overhead of preprocessing is low.
We briefly describe the steps performed by the partitionbased algorithm below, and defer the presentation of details
to subsequent sections.
1. Generate partitions: In the first step, we use a clustering algorithm to cluster the data and treat each cluster as
a separate partition.
2. Compute bounds on Dk for points in each partition:
For each partition P , we compute lower and upper
bounds (stored in P .lower and P .upper, respectively)
on D k for points in the partition. Thus, for every point
p 2 P , Dk (p) P .lower and D k (p) P .upper.
3. Identify candidate partitions containing outliers: In
this step, we identify the candidate partitions, that is,
the partitions containing points which are candidates
for outliers. Suppose we could compute minDkDist,
the lower bound on D k for the n outliers. Then, if
P .upper for a partition P is less than minDkDist, none
of the points in P can possibly be outliers. Thus,
only partitions P for which P .upper minDkDist are
candidate partitions.
minDkDist can be computed from P .lower for the partitions as follows. Consider the partitions in decreasing
order of P .lower. Let P1 : : : Pl be the partitions with
the maximum values for P .lower such that the number of
points in the partitions is at least n. Then, a lower bound
on D k for an outlier is minfPi :lower : 1 i lg.
4. Compute outliers from points in candidate partitions: In the final step, the outliers are computed from
among the points in the candidate partitions. For each
candidate partition P , let P .neighbors denote the neighboring partitions of P , which are all the partitions within
distance P .upper from P . Points belonging to neighboring partitions of P are the only points that need to be examined when computing D k for each point in P . Since
the number of points in the candidate partitions and their
neighboring partitions could become quite large, we process the points in the candidate partitions in batches,
each batch involving a subset of the candidate partitions.
5.2 Generating Partitions
Partitioning the data space into cells and then treating each
cell as a partition is impractical for higher dimensional
spaces. This approach was found to be ineffective for more
than 4 dimensions in [KN98] due to the exponential growth
in the number of cells as the number of dimensions increase.
For effective pruning, we would like to partition the data
such that points which are close together are assigned to a
single partition. Thus, employing a clustering algorithm for
partitioning the data points is a good choice. A number of
clustering algorithms have been proposed in the literature,
most of which have at least quadratic time complexity
[JD88]. Since N could be quite large, we are more
interested in clustering algorithms that can handle large data
sets. Among algorithms with lower complexities is the
pre-clustering phase of BIRCH [ZRL96], a state-of-the-art
clustering algorithm that can handle large data sets. The
pre-clustering phase has time complexity that is linear in
the input size and performs a single scan of the database.
It stores a compact summarization for each cluster in a CFtree which is a balanced tree structure similar to an R-tree
[Sam89]. For each successive point, it traverses the CFtree to find the closest cluster, and if the point is within a
threshold distance of the cluster, it is absorbed into it; else,
it starts a new cluster. In case the size of the CF-tree exceeds
the main memory size M , the threshold is increased and
clusters in the CF-tree that are within (the new increased) distance of each other are merged.
The main memory size M and the points in the data set
are given as inputs to BIRCH’s pre-clustering algorithm.
BIRCH generates a set of clusters with generally uniform
sizes and that fit in M . We treat each cluster as a separate
partition – the points in the partition are simply the points
that were assigned to its cluster during the pre-clustering
phase. Thus, by controlling the memory size M input to
BIRCH, we can control the number of partitions generated.
We represent each partition by the MBR for its points. Note
that the MBRs for partitions may overlap.
We must emphasize that we use clustering here simply
as a heuristic for efficiently generating desirable partitions,
and not for computing outliers. Most clustering algorithms,
including BIRCH, perform outlier detection; however unlike
our notion of outliers, their definition of outliers is not
mathematically precise and is more a consequence of
operational considerations that need to be addressed during
the clustering process.
5.3 Computing Bounds for Partitions
For the purpose of identifying the candidate partitions, we
need to first compute the bounds P .lower and P .upper,
which have the following property: for all points p 2 P ,
P .lower D k (p) P .upper. The bounds P .lower/P .upper
for a partition P can be determined by finding the l partitions
closest to P with respect to MINDIST/MAXDIST such that
the number of points in P 1 : : : Pl is at least k . Since the
partitions fit in main memory, a main memory index can be
used to find the l partitions closest to P (for each partition,
its MBR is stored in the index).
Procedure computeLowerUpper for computing P .lower
and P .upper for partition P is shown in Figure 3. Among its
input parameters are the root of the index containing all the
Procedure computeLowerUpper(Root, P , k , minDkDist)
begin
1. nodeList := f Root g
2. P .lower := P .upper := 1
3. lowerHeap := upperHeap := 4. while nodeList is not empty do f
5.
delete the first element, Node, from nodeList
6.
if (Node is a leaf) f
7.
for each partition Q in Nodef
8.
if (MINDIST(P ,Q) < P .lower) f
9.
lowerHeap.insert(Q)
10.
while lowerHeap.numPoints() ;
11.
lowerHeap.top().numPoints() k do
12.
lowerHeap.deleteTop()
13.
if (lowerHeap.numPoints() k )
P .lower := MINDIST(P , lowerHeap.top())
14.
15.
g
16.
if (MAXDIST(P ,Q) < P .upper)f
17.
upperHeap.insert(Q)
18.
while upperHeap.numPoints() ;
19.
upperHeap.top().numPoints() k do
20.
upperHeap.deleteTop()
21.
if (upperHeap.numPoints() k )
22.
P .upper := MAXDIST(P , upperHeap.top())
23.
if (P .upper minDkDist) return
g
24.
25.
g
26.
g
27.
else f
28.
append Node’s children to nodeList
29.
sort nodeList by MINDIST
30.
g
31.
for each Node in nodeList do
32.
if (P .upper MAXDIST(P ,Node) and
P .lower MINDIST(P ,Node))
33.
34.
delete Node from nodeList
35. g
end
Figure 3: Computation of Lower and Upper Bounds for
Partitions
partitions and minDkDist, which is a lower bound on D k for
an outlier. The procedure is invoked by the procedure which
computes the candidate partitions, computeCandidatePartitions, shown in Figure 4 that we will describe in the next
subsection. Procedure computeCandidatePartitions keeps
track of minDkDist and passes this to computeLowerUpper
so that computation of the bounds for a partition P can be
optimized. The idea is that if P .upper for partition P becomes less than minDkDist, then it cannot contain outliers.
Computation of bounds for it can cease immediately.
computeLowerUpper is similar to procedure getKthNeighborDist described in the previous section (see Figure 2). It stores partitions in two heaps, lowerHeap
and upperHeap, in the decreasing order of MINDIST and
MAXDIST from P , respectively – thus, partitions with the
largest values of MINDIST and MAXDIST appear at the top
of the heaps.
5.4 Computing Candidate Partitions
This is the crucial step in our partition-based algorithm in
which we identify the candidate partitions that can potentially contain outliers, and prune the remaining partitions.
The idea is to use the bounds computed in the previous section to first estimate minDkDist, which is a lower bound on
Dk for an outlier. Then a partition P is a candidate only
if P .upper minDkDist. The lower bound minDkDist can
be computed using the P .lower values for the partitions as
follows. Let P1 : : : Pl be the partitions with the maximum
values for P .lower and containing at least n points. Then
minDkDist = minfPi :lower : 1 i lg is a lower bound
on D k for an outlier.
The procedure for computing the candidate partitions
from among the set of partitions PSet is illustrated in
Figure 4. The partitions are stored in a main memory
index and computeLowerUpper is invoked to compute the
lower and upper bounds for each partition. However,
instead of computing minDkDist after the bounds for all the
partitions have been computed, computeCandidatePartitions
stores, in the heap partHeap, the partitions with the largest
P .lower values and containing at least n points among
them. The partitions are stored in increasing order of
P .lower in partHeap and minDkDist is thus equal to P .lower
for the partition P at the top of partHeap. The benefit
of maintaining minDkDist is that it can be passed as
a parameter to computeLowerUpper (in Step 6) and the
computation of bounds for a partition P can be halted early
if P .upper for it falls below minDkDist. If, for a partition
P , P .lower is greater than the current value of minDkDist,
then it is inserted into partHeap and the value of minDkDist
is appropriately adjusted (steps 8–13).
Procedure computeCandidatePartitions(PSet, k , n)
begin
1. for each partition P in PSet do
2.
insertIntoIndex(Tree, P )
3. partHeap := 4. minDkDist := 0
5. for each partition P in PSet do f
6.
computeLowerUpper(Tree.Root, P , k , minDkDist)
7.
if (P .lower > minDkDist) f
8.
partHeap.insert(P )
9.
while partHeap.numPoints() ;
10.
partHeap.top().numPoints() n do
11.
partHeap.deleteTop()
12.
if (partHeap.numPoints() n)
13.
minDkDist := partHeap.top().lower
14.
g
15. g
16. candSet := 17. for each partition P in PSet do
18.
if (P .upper minDkDist) f
19.
candSet := candSet fP g
20.
P .neighbors :=
21.
fQ: Q 2 PSet and MINDIST(P ,Q) P .upperg
22.
g
23. return candSet
end
Figure 4: Computation of Candidate Partitions
In the for loop over steps 17–22, the set of candidate
partitions candSet is computed, and for each candidate
partition P , partitions Q that can potentially contain the k th
nearest neighbor for a point in P are added to P .neighbors
(note that P .neighbors contains P ).
5.5 Computing Outliers from Candidate Partitions
In the final step, we compute the top n outliers from the
candidate partitions in candSet. If points in all the candidate
partitions and their neighbors fit in memory, then we can
simply load all the points into a main memory spatial index.
The index-based algorithm (see Figure 1) can then be used to
compute the n outliers by probing the index to compute D k
values only for points belonging to the candidate partitions.
Since both the size of the index as well as the number of
candidate points will in general be small compared to the
total number of points in the data set, this can be expected to
be much faster than executing the index-based algorithm on
the entire data set of points.
In the case that all the candidate partitions and their
neighbors exceed the size of main memory, then we need to
process the candidate partitions in batches. In each batch,
a subset of the remaining candidate partitions that along
with their neighbors fit in memory, is chosen for processing.
Due to space constraints, we refer the reader to [RRS98] for
details of the batch processing algorithm.
6
Experimental Results
We empirically compared the performance of our partitionbased algorithm with the block nested-loop and index-based
algorithms. In our experiments, we found that the partitionbased algorithm scales well with both the data set size as
well as data set dimensionality. In addition, in a number of
cases, it is more than an order of magnitude faster than the
block nested-loop and index-based algorithms.
We begin by describing in Section 6.1 our experience with
mining a real-life NBA (National Basketball Association)
database using our notion of outliers. The results indicate
the efficacy of our approach in finding “interesting” and
sometimes unexpected facts buried in the data. We then
evaluate the performance of the algorithms on a class of
synthetic datasets in Section 6.2. The experiments were
performed on a Sun Ultra-2/200 workstation with 512 MB
of main memory, and running Solaris 2.5. The data sets were
stored on a local disk.
6.1 Analyzing NBA Statistics
We analyzed the statistics for the 1998 NBA season with
our outlier programs to see if it could discover interesting
nuggets in those statistics. We had information about all
471 NBA players who played in the NBA during the 19971998 season. In order to restrict our attention to significant
players, we removed all players who scored less then 100
points over the course of the entire season. This left us
with 335 players. We then wanted to ensure that all the
columns were given equal weight. We accomplished this
by transforming the value c in a column to c;cc where
c is the average value of the column and c its standard
deviation. This transformation normalizes the column to
have an average of 0 and a standard deviation of 1.
We then ran our outlier program on the transformed data.
We used a value of 10 for k and looked for the top 5
outliers. The results from some of the runs are shown in
Figure 5. (findOuts.pl is a perl front end to the outliers
program that understands the names of the columns in the
NBA database. It simply processes its arguments and calls
the outlier program.) In addition to giving the actual value
for a column, the output also prints the normalized value
used in the outlier calculation. The outliers are ranked based
on their D k values which are listed under the DIST column.
The first experiment in Figure 5 focuses on the three
most commonly used average statistics in the NBA: average points per game, average assists per game and average
rebounds per game. What stands out is the extent to which
players having a large value in one dimension tend to dominate in the outlier list. For instance, Dennis Rodman, not
known to excel in either assisting or scoring, is nevertheless the top outlier because of his huge (nearly 4.4 sigmas)
deviation from the average on rebounds. Furthermore, his
DIST value is much higher than that for any of the other
outliers, thus making him an extremely strong outlier. Two
other players in this outlier list also tend to dominate in one
or two columns. An interesting case is that of Shaquille
O’ Neal who made it to the outlier list due to his excellent
record in both scoring and rebounds, though he is quite average on assists. (Recall that the average of every normalized
column is 0.) The first “well-rounded” player to appear in
this list is Karl Malone, at position 5. (Michael Jordan is at
position 7.) In fact, in the list of the top 25 outliers, there are
only two players, Karl Malone and Grant Hill (at positions
5 and 6) that have normalized values of more than 1 in all
three columns.
When we look at more defensive statistics, the outliers are
once again dominated by players having large normalized
values for a single column. When we consider average steals
and blocks, the outliers are dominated by shot blockers like
Marcus Camby. Hakeem Olajuwon, at position 5, shows up
as the first “balanced” player due to his above average record
with respect to both steals and blocks.
In conclusion, we were somewhat surprise by the outcome
of our experiments on the NBA data. First, we found that
very few “balanced” players (that is, players who are above
average in every aspect of the game) are labeled as outliers.
Instead, the outlier lists are dominated by players who excel
by a wide margin in particular aspects of the game (e.g.,
Dennis Rodman on rebounds).
Another interesting observation we made was that the
outliers found tended to be more interesting when we
considered fewer attributes (e.g., 2 or 3). This is not entirely
surprising since it is a well-known fact that as the number
of dimensions increases, points spread out more uniformly
in the data space and distances between them are a poor
measure of their similarity/dissimilarity.
->findOuts.pl -n
NAME
Dennis Rodman
Rod Strickland
Shaquille Oneal
Jayson Williams
Karl Malone
5 -k 10 reb assists pts
DIST avgReb (norm) avgAssts (norm)
7.26 15.000 (4.376)
2.900 (0.670)
3.95
5.300 (0.750) 10.500 (4.922)
3.61 11.400 (3.030)
2.400 (0.391)
3.33 13.600 (3.852)
1.000 (-0.393)
2.96 10.300 (2.619)
3.900 (1.230)
->findOuts.pl -n
NAME
Marcus Camby
Dikembe Mutombo
Shawn Bradley
Theo Ratliff
Hakeem Olajuwon
5 -k 10 steal blocks
DIST
avgSteals (norm)
8.44
1.100
(0.838)
5.35
0.400
(-0.550)
4.36
0.800
(0.243)
3.51
0.600
(-0.153)
3.47
1.800
(2.225)
avgBlocks
3.700
3.400
3.300
3.200
2.000
avgPts (norm)
4.700 (-0.459)
17.800 (1.740)
28.300 (3.503)
12.900 (0.918)
27.000 (3.285)
(norm)
(6.139)
(5.580)
(5.394)
(5.208)
(2.972)
Figure 5: Finding Outliers from a 1998 NBA Statistics Database
Finally, while we were conducting our experiments on the
NBA database, we realized that specifying actual distances,
as is required in [KN98], is fairly difficult in practice.
Instead, our notion of outliers, which only requires us
to specify the k -value used in calculating k ’th neighbor
distance, is much simpler to work with. (The results are
fairly insensitive to minor changes in k , making the job of
specifying it easy.) Note also the ranking for players that we
provide in Figure 5 based on distance —this enables us to
determine how strong an outlier really is.
6.2 Performance Results on Synthetic Data
We begin this section by briefly describing our implementation of the three algorithms that we used. We then move
onto describing the synthetic datasets that we used.
6.2.1 Algorithms Implemented
Block Nested-Loop Algorithm:
This algorithm was
described in Section 4. In order to optimize the performance
of this algorithm, we implemented our own buffer manager
and performed reads in large blocks. We allocated as much
buffer space as possible to the outer loop.
Index-Based Algorithm: To speed up execution, an R tree was used to find the k nearest neighbors for a point,
as described in Section 4. The R -tree code was developed
at the University of Maryland.6 The R -tree we used was
a main memory-based version. The page size for the R tree was set to 1024 bytes. In our experiments, the R tree always fit in memory. Furthermore, for the index-based
algorithm, we did not include the time to build the tree (that
is, insert data points into the tree) in our measurements of
execution time. Thus, our measurement for the running time
of the index-based algorithm only includes the CPU time for
main memory search. Note that this gives the index-based
algorithm an advantage over the other algorithms.
6 Our thanks to Christos Faloutsos for providing us with this code.
Partition-Based algorithm: We implemented our partitionbased algorithm as described in Section 5. Thus, we used
BIRCH’s pre-clustering algorithm for generating partitions,
the main memory R -tree to determine the candidate partitions and the block nested-loop algorithm for computing
outliers from the candidate partitions in the final step. We
found that for the final step, the performance of the block
nested-loop algorithm was competitive with the index-based
algorithm since the previous pruning steps did a very good
job in identifying the candidate partitions and their neighbors.
We configured BIRCH to provide a bounding rectangle
for each cluster it generated. We used this as the MBR
for the corresponding partition. We stored the MBR and
number of points in each partition in an R -tree. We used
the resulting index to identify candidate and neighboring
partitions. Since we needed to identify the partition to which
BIRCH assigned a point, we modified BIRCH to generate
this information.
Recall from Section 5.2 that an important parameter to
BIRCH is the amount of memory it is allowed to use. In
the experiments, we specify this parameter in terms of the
number of clusters or partitions that BIRCH is allowed to
create.
6.2.2 Synthetic Data Sets
For our experiments, we used the grid synthetic data set
that was employed in [ZRL96] to study the sensitivity of
BIRCH. The data set contains 100 hyper-spherical clusters
arranged as a 10 10 grid. The center of each cluster is
located at (10i 10j ) for 1 i 10 and 1 j 10.
Furthermore, each cluster has a radius of 4. Data points for
a cluster are uniformly distributed in the hyper-sphere that
defines the cluster. We also uniformly scattered 1000 outlier
points in the space spanning 0 to 110 along each dimension.
Table 2 shows the parameters for the data set, along with
their default values and the range of values for which we
conducted experiments.
Parameter
Number of Points (N )
Number of Clusters
Number of Points per Cluster
Number of Outliers in Data Set
Number of Outliers to be Computed (n)
Number of Neighbors (k )
Number of Dimensions ( )
Maximum number of Partitions
Distance Metric
Default Value
101000
100
1000
1000
100
100
2
6000
euclidean
Range of Values
11000 to 1 million
100 to 10000
100 to 500
100 to 500
2 to 10
5000 to 15000
Table 2: Synthetic Data Parameters
Block Nested-Loop
Index-Based
Partition-Based (6k)
Total Execution Time
Clustering Time Only
1000
10000
Execution Time (sec.)
Execution Time (sec.)
100000
1000
100
100
10
10
1
11000 26000
51000
N
76000
101000
1
101000 251000
(a)
501000
N
751000
1.001e+06
(b)
Figure 6: Performance Results for N
6.2.3 Performance Results
Number of Points: To study how the three algorithms
scale with dataset size, we varied the number of points per
cluster from 100 to 10,000. This varies the size of the dataset
from 11000 to approximately 1 million. Both n and k were
set to their default values of 100. The limit on the number
of partitions for the partition-based algorithm was set to
6000. The execution times for the three algorithms as N
is varied from 11000 to 101000 are shown using a log scale
in Figure 6(a).
As the figure illustrates, the block nested-loop algorithm
is the worst performer. Since the number of computations
it performs is proportional to the square of the number of
points, it exhibits a quadratic dependency on the input size.
The index-based algorithm is a lot better than block nestedloop, but it is still 2 to 6 times slower than the partitionbased algorithm. For 101000 points, the block nestedloop algorithm takes about 5 hours to compute 100 outliers,
the index-based algorithm less than 5 minutes while the
partition-based algorithm takes about half a minute. In order
to explain why the partition-based algorithm performs so
well, we present in Table 3, the number of candidate and
neighbor partitions as well as points processed in the final
step. From the table, it follows that for N = 101000,
out of the approximately 6000 initial partitions, only about
160 candidate partitions and 1500 neighbor partitions are
processed in the final phase. Thus, about 75% of partitions
are entirely pruned from the data set, and only about 0.25%
of the points in the data set are candidates for outliers (230
out of 101000 points). This results in tremendous savings in
both I/O and computation, and enables the partition-based
scheme to outperform the other two algorithms by almost an
order of magnitude.
In Figure 6(b), we plot the execution time of only
the partition-based algorithm as the number of points is
increased from 100,000 to 1 million to see how it scales for
much larger data sets. We also plot the time spent by BIRCH
for generating partitions— from the graph, it follows that
this increases about linearly with input size. However, the
overhead of the final step increases substantially as the data
set size is increased. The reason for this is that since we
generate the same number, 6000, of partitions even for a
million points, the average number of points per partition
exceeds k , which is 100. As a result, computed lower
bounds for partitions are close to 0 and minDkDist, the lower
bound on the D k value for an outlier is low, too. Thus,
our pruning is less effective if the data set size is increased
without a corresponding increase in the number of partitions.
Specifically, in order to ensure a high degree pruning, a good
rule of thumb is to choose the number of partitions such that
the average number of points per partition is fairly small (but
not too small) compared to k . For example, N=(k=5) is a
good value. This makes the clusters generated by BIRCH to
have an average size of k=5.
N
11000
26000
51000
76000
101000
Avg. # of Points
per Partition
2.46
4.70
8.16
11.73
16.39
# of Candidate
Partitions
115
131
141
143
160
# of Neighbor
Partitions
1334
1123
1088
963
1505
# of Candidate
Points
130
144
159
160
230
# of Neighbor
Points
3266
5256
8850
11273
24605
Table 3: Statistics for N
k th Nearest Neighbor:
Figure 7(a) shows the result of
increasing the value of k from 100 to 500. We considered
the index-based algorithm and the partition-based algorithm
with 3 different settings for the number of partitions—
5000, 6000 and 15000. We did not explore the behavior
of the block-nested loop algorithm because it is very slow
compared to the other two algorithms. The value of n was
set to 100 and the number of points in the data set was
101000. The execution times are shown using a log scale.
As the graph confirms, the performance of the partitionbased algorithms do not degrade as k is increased. This
is because we found that as k is increased, the number
of candidate partitions decreases slightly since a larger k
implies a higher value for minDkDist which results in more
pruning. However, a larger k also implies more neighboring
partitions for each candidate partition. These two opposing
effects cancel each other to leave the performance of the
partition-based algorithm relatively unchanged.
On the other hand, due to the overhead associated with
finding the k nearest neighbors, the performance of the
index-based algorithm suffers significantly as the value of k
increases. Since the partition-based algorithms prune more
than 75% of points and only 0.25% of the data set are
candidates for outliers, they are generally 10 to 70 times
faster than the index-based algorithm.
Also, note that as the number of partitions is increased,
the performance of the partition-based algorithm becomes
worse. The reason for this is that when each partition contains too few points, the cost of computing lower and upper
bounds for each partition is no longer low. For instance,
in case each partition contains a single point, then computing lower and upper bounds for a partition is equivalent to
computing D k for every point p in the data set and so the
partition-based algorithm degenerates to the index-based algorithm. Therefore, as we increase the number of partitions, the execution time of the partition-based algorithm
converges to that of the index-based algorithm.
Number of outliers:
When the number of outliers n,
is varied from 100 to 500 with default settings for other
parameters, we found that the the execution time of all
algorithms increase gradually. Due to space constraints, we
do not present the graphs for these experiments in this paper.
These can be found in [RRS98].
Number of Dimensions: Figure 7(b) plots the execution
times of the three algorithms as the number of dimensions is
increased from 2 to 10 (the remaining parameters are set to
their default values). While we set the cluster radius to 4 for
the 2-dimensional dataset, we reduced the radii of clusters
for higher dimensions. We did this because the volume of
the hyper-spheres of clusters tends to grow exponentially
with dimension, and thus the points in higher dimensional
space become very sparse. Therefore, we had to reduce
the cluster radius to ensure that points in each cluster are
relatively close compared to points in other clusters. We
used radius values of 2, 1.4, 1.2 and 1.2, respectively, for
dimensions from 4 to 10.
For 10 dimensions, the partition-based algorithm is about
30 times faster than the index-based algorithm and about
180 times faster than the block nested-loop algorithm. Note
that this was without including the building time for the R tree in the index-based algorithm. The execution time of
the partition-based algorithm increases sub-linearly as the
dimensionality of the data set is increased. In contrast,
running times for the index-based algorithm increase very
rapidly due to the increased overhead of performing search
in higher dimensions using the R -tree. Thus, the partitionbased algorithm scales better than the other algorithms for
higher dimensions.
7
Conclusions
In this paper, we proposed a novel formulation for distancebased outliers that is based on the distance of a point
from its k th nearest neighbor. We rank each point on
the basis of its distance to its kth nearest neighbor and
declare the top n points in this ranking to be outliers. In
addition to developing relatively straightforward solutions
to finding such outliers based on the classical nested-loop
join and index join algorithms, we developed a highly
efficient partition-based algorithm for mining outliers. This
algorithm first partitions the input data set into disjoint
subsets, and then prunes entire partitions as soon as it can be
determined that they cannot contain outliers. Since people
are usually interested in only a small number of outliers, our
algorithm is able to determine very quickly that a significant
number of the input points cannot be outliers. This results in
substantial savings in computation.
We presented the results of an extensive experimental
study on real-life and synthetic data sets. The results from a
Index-Based
Partition-Based (15k)
Partition-Based (6k)
Partition-Based (5k)
1000
100
10
1
100
Block Nested-Loop
Index-Based
Partition-Based (6k)
100000
Execution Time (sec.)
Execution Time (sec.)
10000
10000
1000
100
10
1
200
300
k
400
500
2
4
(a)
6
8
Number of Dimensions
10
(b)
Figure 7: Performance Results for k and real-life NBA database highlight and reveal several expected
and unexpected aspects of the database. The results from a
study on synthetic data sets demonstrate that the partitionbased algorithm scales well with respect to both data set size
and data set dimensionality. Furthermore, it outperforms
the nested-loop and index-based algorithms by more than an
order of magnitude for a wide range of parameter settings.
Acknowledgments: Without the support of Seema Bansal
and Yesook Shim, it would have been impossible to complete this work.
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