AV I(U, V) = 2M I(U, V)−2E{M I(U, V)}
H(U) +H(V)−2E{M I(U, V)}
Like AMI, AVI takes a value of 1 when the two clusterings are identical and 0 when the mutual information between the two clusterings equals its expected value.
2.2
Outlier Detection
As illustrated in Section 1.1, outlier detection is a primary step in KDD. In data analysis, one of the preliminary steps is to detect the outlying observa- tions in the data sets. According to the definition of Hawkins [76], an outlier is defined as an observation that deviates so much from other observations as to arouse suspicion that it was generated by a different mechanism. Sim- ilarly, Grubbs [67] indicates that an outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs. Detecting such irregular observations in data sets is very curial for further data analysis. In addition, in this thesis, outliers also refer to noise objects. The terms “outlier” and “noise” are used in a synonymous way.
During last several decades, outlier detection has attracted huge attention in diverse applications, such as credit card fraud detection, clinical trials, voting irregularity analysis, network intrusion detection, fault detection, and athlete performance analysis. Many algorithms for outlier detection have thus been proposed [30, 91, 92, 24]. In the following, some widely-used outlier detection algorithms are generally described.
detection rely on the statistical analysis. For a given data set, it often assumes to follow a known distribution, such as Gaussian. If the data objects of the data set badly fit the data distribution and then they are considered as outliers. One of a well-known method is Mahalanobis distance. Given n
data objects in the d-dimensional data set, Mahalanobis distance for each data object xi is defined as [19]:
M ahalxi = v u u t n X i=1 (xi−x¯n)T ·Σ−1· n X i=1 (xi−x¯n) (2.2)
Where Σ is the covariance matrix of ndata objects and ¯xn denotes mean
vector.
Since Mahalanobis distance reflects the distribution of the data set, data objects with a large Mahalanobis distance are indicated as outliers.
CoCo: Based on information-theoretic principle, recently, B¨ohm, et al. proposed CoCo [24], a parameter-free outlier detection with coding cost. Based on the MDL principle, outliers are flagged as those objects which need more coding cost than regular objects. For coding each object, the optimal neighborhood size is heuristically determined. Independent Com- ponent Analysis and Exponential power distribution (EPD) are combined to estimate the probability and the corresponding coding cost. Like most distribution-based methods, CoCo tends to fail if the estimated distribution does not fit the data model well. It is also time consuming to find the opti- mal neighborhood to estimate the coding cost for each object by screening for suitable neighborhood sizes.
LOF: To handle the issue of varying densities in the data set, Breunig, et al. [30], introduce a notion of local outlier from a density-based perspective.
2.2 Outlier Detection 27
P1
P2 P3
Figure 2.3: LOF: local outlier vs. global outlier [30].
An object is regarded as an outlier if its local density does not fit well into the density of its neighboring objects. The local outlier factor (LOF) is then proposed to capture the degree to which the object is an outlier. It is defined as the average of the ratio of the local reachability density of the object and those of the objects in its neighborhood. A LOF value of approximately 1 indicates the object is located inside a cluster, while the objects with higher LOF values are more rather considered as outliers. In Figure 2.3, LOF shows the advantage of outlier detection for local outliers in the data set with varying densities. It can obtain the local outliers by computing the relative degree of isolation from its surrounding neighborhood.
LOCI: The Local Outlier Integral (LOCI) [120] flags outliers, based on probabilistic reasoning and motivated from the concept of a multi-granularity deviation factor (MDEF). Similar to LOF, the LOCI outlier model takes the local object density into account, but differently, the MDEF of LOCI uses
borhood in LOCI model is defined by two parameters: the counting and the sampling neighborhood. The counting neighborhood specifies some vol- ume of the feature space which is used to estimate the local object density. The sampling neighborhood is larger than the counting neighborhood and contains all points which are used to compute the average object density in the neighborhood. Objects which deviate in their local object density more than three times of the standard deviation are regarded as outliers. The flagging scheme of LOCI thus assumes the object densities follow a Gaussian distribution.
In the Chapter 8, the advantages and drawbacks will be further illus- trated and discussed by comparing with the new proposed outlier detection algorithm based on synchronization principle.