![Approach III: Modeling High-Dimensional Outliers
Ex. Angle-based outliers: Kriegel, Schubert, and Zimek [KSZ08]
For each point o, examine the angle ∆xoy for every pair of points x, y.
Point in the center (e.g., a), the angles formed differ widely
An outlier (e.g., c), angle variable is substantially smaller
Use the variance of angles for a point to determine outlier
Combine angles and distance to model outliers
Use the distance-weighted angle variance as the outlier score
Angle-based outlier factor (ABOF):
Efficient approximation computation method is developed
It can be generalized to handle arbitrary types of data
43
Develop new models for high-
dimensional outliers directly
Avoid proximity measures and adopt
new heuristics that do not deteriorate
in high-dimensional data
A set of points
form a cluster
except c
(outlier)](https://image.slidesharecdn.com/12outlier-250117184545-665f387e/85/12Outlier-for-software-introductionalism-43-320.jpg)
![51
Outlier Discovery: Distance-Based Approach
Introduced to counter the main limitations imposed by
statistical methods
We need multi-dimensional analysis without knowing
data distribution
Distance-based outlier: A DB(p, D)-outlier is an object O in
a dataset T such that at least a fraction p of the objects in T
lies at a distance greater than D from O
Algorithms for mining distance-based outliers [Knorr & Ng,
VLDB’98]
Index-based algorithm
Nested-loop algorithm
Cell-based algorithm](https://image.slidesharecdn.com/12outlier-250117184545-665f387e/85/12Outlier-for-software-introductionalism-51-320.jpg)
12Outlier.for software introductionalism
- 1. 1 Data Mining: Concepts and Techniques (3rd ed.) — Chapter 12 — Jiawei Han, Micheline Kamber, and Jian Pei University of Illinois at Urbana-Champaign & Simon Fraser University ©2011 Han, Kamber & Pei. All rights reserved.
- 2. 2 Chapter 12. Outlier Analysis Outlier and Outlier Analysis Outlier Detection Methods Statistical Approaches Proximity-Base Approaches Clustering-Base Approaches Classification Approaches Mining Contextual and Collective Outliers Outlier Detection in High Dimensional Data Summary
- 3. 3 What Are Outliers? Outlier: A data object that deviates significantly from the normal objects as if it were generated by a different mechanism Ex.: Unusual credit card purchase, sports: Michael Jordon, Wayne Gretzky, ... Outliers are different from the noise data Noise is random error or variance in a measured variable Noise should be removed before outlier detection Outliers are interesting: It violates the mechanism that generates the normal data Outlier detection vs. novelty detection: early stage, outlier; but later merged into the model Applications: Credit card fraud detection Telecom fraud detection Customer segmentation Medical analysis
- 4. 4 Types of Outliers (I) Three kinds: global, contextual and collective outliers Global outlier (or point anomaly) Object is Og if it significantly deviates from the rest of the data set Ex. Intrusion detection in computer networks Issue: Find an appropriate measurement of deviation Contextual outlier (or conditional outlier) Object is Oc if it deviates significantly based on a selected context Ex. 80o F in Urbana: outlier? (depending on summer or winter?) Attributes of data objects should be divided into two groups Contextual attributes: defines the context, e.g., time & location Behavioral attributes: characteristics of the object, used in outlier evaluation, e.g., temperature Can be viewed as a generalization of local outliers—whose density significantly deviates from its local area Issue: How to define or formulate meaningful context? Global Outlier
- 5. 5 Types of Outliers (II) Collective Outliers A subset of data objects collectively deviate significantly from the whole data set, even if the individual data objects may not be outliers Applications: E.g., intrusion detection: When a number of computers keep sending denial-of-service packages to each other Collective Outlier Detection of collective outliers Consider not only behavior of individual objects, but also that of groups of objects Need to have the background knowledge on the relationship among data objects, such as a distance or similarity measure on objects. A data set may have multiple types of outlier One object may belong to more than one type of outlier
- 6. 6 Challenges of Outlier Detection Modeling normal objects and outliers properly Hard to enumerate all possible normal behaviors in an application The border between normal and outlier objects is often a gray area Application-specific outlier detection Choice of distance measure among objects and the model of relationship among objects are often application-dependent E.g., clinic data: a small deviation could be an outlier; while in marketing analysis, larger fluctuations Handling noise in outlier detection Noise may distort the normal objects and blur the distinction between normal objects and outliers. It may help hide outliers and reduce the effectiveness of outlier detection Understandability Understand why these are outliers: Justification of the detection Specify the degree of an outlier: the unlikelihood of the object being generated by a normal mechanism
- 7. 7 Chapter 12. Outlier Analysis Outlier and Outlier Analysis Outlier Detection Methods Statistical Approaches Proximity-Base Approaches Clustering-Base Approaches Classification Approaches Mining Contextual and Collective Outliers Outlier Detection in High Dimensional Data Summary
- 8. Outlier Detection I: Supervised Methods Two ways to categorize outlier detection methods: Based on whether user-labeled examples of outliers can be obtained: Supervised, semi-supervised vs. unsupervised methods Based on assumptions about normal data and outliers: Statistical, proximity-based, and clustering-based methods Outlier Detection I: Supervised Methods Modeling outlier detection as a classification problem Samples examined by domain experts used for training & testing Methods for Learning a classifier for outlier detection effectively: Model normal objects & report those not matching the model as outliers, or Model outliers and treat those not matching the model as normal Challenges Imbalanced classes, i.e., outliers are rare: Boost the outlier class and make up some artificial outliers Catch as many outliers as possible, i.e., recall is more important than accuracy (i.e., not mislabeling normal objects as outliers) 8
- 9. Outlier Detection II: Unsupervised Methods Assume the normal objects are somewhat ``clustered'‘ into multiple groups, each having some distinct features An outlier is expected to be far away from any groups of normal objects Weakness: Cannot detect collective outlier effectively Normal objects may not share any strong patterns, but the collective outliers may share high similarity in a small area Ex. In some intrusion or virus detection, normal activities are diverse Unsupervised methods may have a high false positive rate but still miss many real outliers. Supervised methods can be more effective, e.g., identify attacking some key resources Many clustering methods can be adapted for unsupervised methods Find clusters, then outliers: not belonging to any cluster Problem 1: Hard to distinguish noise from outliers Problem 2: Costly since first clustering: but far less outliers than normal objects Newer methods: tackle outliers directly 9
- 10. Outlier Detection III: Semi-Supervised Methods Situation: In many applications, the number of labeled data is often small: Labels could be on outliers only, normal objects only, or both Semi-supervised outlier detection: Regarded as applications of semi- supervised learning If some labeled normal objects are available Use the labeled examples and the proximate unlabeled objects to train a model for normal objects Those not fitting the model of normal objects are detected as outliers If only some labeled outliers are available, a small number of labeled outliers many not cover the possible outliers well To improve the quality of outlier detection, one can get help from models for normal objects learned from unsupervised methods 10
- 11. Outlier Detection (1): Statistical Methods Statistical methods (also known as model-based methods) assume that the normal data follow some statistical model (a stochastic model) The data not following the model are outliers. 11 Effectiveness of statistical methods: highly depends on whether the assumption of statistical model holds in the real data There are rich alternatives to use various statistical models E.g., parametric vs. non-parametric Example (right figure): First use Gaussian distribution to model the normal data For each object y in region R, estimate gD(y), the probability of y fits the Gaussian distribution If gD(y) is very low, y is unlikely generated by the Gaussian model, thus an outlier
- 12. Outlier Detection (2): Proximity-Based Methods An object is an outlier if the nearest neighbors of the object are far away, i.e., the proximity of the object is significantly deviates from the proximity of most of the other objects in the same data set 12 The effectiveness of proximity-based methods highly relies on the proximity measure. In some applications, proximity or distance measures cannot be obtained easily. Often have a difficulty in finding a group of outliers which stay close to each other Two major types of proximity-based outlier detection Distance-based vs. density-based Example (right figure): Model the proximity of an object using its 3 nearest neighbors Objects in region R are substantially different from other objects in the data set. Thus the objects in R are outliers
- 13. Outlier Detection (3): Clustering-Based Methods Normal data belong to large and dense clusters, whereas outliers belong to small or sparse clusters, or do not belong to any clusters 13 Since there are many clustering methods, there are many clustering-based outlier detection methods as well Clustering is expensive: straightforward adaption of a clustering method for outlier detection can be costly and does not scale up well for large data sets Example (right figure): two clusters All points not in R form a large cluster The two points in R form a tiny cluster, thus are outliers
- 14. 14 Chapter 12. Outlier Analysis Outlier and Outlier Analysis Outlier Detection Methods Statistical Approaches Proximity-Base Approaches Clustering-Base Approaches Classification Approaches Mining Contextual and Collective Outliers Outlier Detection in High Dimensional Data Summary
- 15. Statistical Approaches Statistical approaches assume that the objects in a data set are generated by a stochastic process (a generative model) Idea: learn a generative model fitting the given data set, and then identify the objects in low probability regions of the model as outliers Methods are divided into two categories: parametric vs. non-parametric Parametric method Assumes that the normal data is generated by a parametric distribution with parameter θ The probability density function of the parametric distribution f(x, θ) gives the probability that object x is generated by the distribution The smaller this value, the more likely x is an outlier Non-parametric method Not assume an a-priori statistical model and determine the model from the input data Not completely parameter free but consider the number and nature of the parameters are flexible and not fixed in advance Examples: histogram and kernel density estimation 15
- 16. Parametric Methods I: Detection Univariate Outliers Based on Normal Distribution Univariate data: A data set involving only one attribute or variable Often assume that data are generated from a normal distribution, learn the parameters from the input data, and identify the points with low probability as outliers Ex: Avg. temp.: {24.0, 28.9, 28.9, 29.0, 29.1, 29.1, 29.2, 29.2, 29.3, 29.4} Use the maximum likelihood method to estimate μ and σ 16 Taking derivatives with respect to μ and σ2 , we derive the following maximum likelihood estimates For the above data with n = 10, we have Then (24 – 28.61) /1.51 = – 3.04 < –3, 24 is an outlier since
- 17. Parametric Methods I: The Grubb’s Test Univariate outlier detection: The Grubb's test (maximum normed residual test) ─ another statistical method under normal distribution For each object x in a data set, compute its z-score: x is an outlier if where is the value taken by a t-distribution at a significance level of α/(2N), and N is the # of objects in the data set 17
- 18. Parametric Methods II: Detection of Multivariate Outliers Multivariate data: A data set involving two or more attributes or variables Transform the multivariate outlier detection task into a univariate outlier detection problem Method 1. Compute Mahalaobis distance Let ō be the mean vector for a multivariate data set. Mahalaobis distance for an object o to ō is MDist(o, ō) = (o – ō )T S –1 (o – ō) where S is the covariance matrix Use the Grubb's test on this measure to detect outliers Method 2. Use χ2 –statistic: where Ei is the mean of the i-dimension among all objects, and n is the dimensionality If χ2 –statistic is large, then object oi is an outlier 18
- 19. Parametric Methods III: Using Mixture of Parametric Distributions Assuming data generated by a normal distribution could be sometimes overly simplified Example (right figure): The objects between the two clusters cannot be captured as outliers since they are close to the estimated mean 19 To overcome this problem, assume the normal data is generated by two normal distributions. For any object o in the data set, the probability that o is generated by the mixture of the two distributions is given by where fθ1 and fθ2 are the probability density functions of θ1 and θ2 Then use EM algorithm to learn the parameters μ1, σ1, μ2, σ2 from data An object o is an outlier if it does not belong to any cluster
- 20. Non-Parametric Methods: Detection Using Histogram The model of normal data is learned from the input data without any a priori structure. Often makes fewer assumptions about the data, and thus can be applicable in more scenarios Outlier detection using histogram: 20 Figure shows the histogram of purchase amounts in transactions A transaction in the amount of $7,500 is an outlier, since only 0.2% transactions have an amount higher than $5,000 Problem: Hard to choose an appropriate bin size for histogram Too small bin size → normal objects in empty/rare bins, false positive Too big bin size → outliers in some frequent bins, false negative Solution: Adopt kernel density estimation to estimate the probability density distribution of the data. If the estimated density function is high, the object is likely normal. Otherwise, it is likely an outlier.
- 21. 21 Chapter 12. Outlier Analysis Outlier and Outlier Analysis Outlier Detection Methods Statistical Approaches Proximity-Base Approaches Clustering-Base Approaches Classification Approaches Mining Contextual and Collective Outliers Outlier Detection in High Dimensional Data Summary
- 22. Proximity-Based Approaches: Distance-Based vs. Density-Based Outlier Detection Intuition: Objects that are far away from the others are outliers Assumption of proximity-based approach: The proximity of an outlier deviates significantly from that of most of the others in the data set Two types of proximity-based outlier detection methods Distance-based outlier detection: An object o is an outlier if its neighborhood does not have enough other points Density-based outlier detection: An object o is an outlier if its density is relatively much lower than that of its neighbors 22
- 23. Distance-Based Outlier Detection For each object o, examine the # of other objects in the r-neighborhood of o, where r is a user-specified distance threshold An object o is an outlier if most (taking π as a fraction threshold) of the objects in D are far away from o, i.e., not in the r-neighborhood of o An object o is a DB(r, π) outlier if Equivalently, one can check the distance between o and its k-th nearest neighbor ok, where . o is an outlier if dist(o, ok) > r Efficient computation: Nested loop algorithm For any object oi, calculate its distance from other objects, and count the # of other objects in the r-neighborhood. If π∙n other objects are within r distance, terminate the inner loop Otherwise, oi is a DB(r, π) outlier Efficiency: Actually CPU time is not O(n2 ) but linear to the data set size since for most non-outlier objects, the inner loop terminates early 23
- 24. Distance-Based Outlier Detection: A Grid-Based Method Why efficiency is still a concern? When the complete set of objects cannot be held into main memory, cost I/O swapping The major cost: (1) each object tests against the whole data set, why not only its close neighbor? (2) check objects one by one, why not group by group? Grid-based method (CELL): Data space is partitioned into a multi-D grid. Each cell is a hyper cube with diagonal length r/2 24 Pruning using the level-1 & level 2 cell properties: For any possible point x in cell C and any possible point y in a level-1 cell, dist(x,y) ≤ r For any possible point x in cell C and any point y such that dist(x,y) ≥ r, y is in a level-2 cell Thus we only need to check the objects that cannot be pruned, and even for such an object o, only need to compute the distance between o and the objects in the level-2 cells (since beyond level-2, the distance from o is more than r)
- 25. Density-Based Outlier Detection Local outliers: Outliers comparing to their local neighborhoods, instead of the global data distribution In Fig., o1 and o2 are local outliers to C1, o3 is a global outlier, but o4 is not an outlier. However, proximity-based clustering cannot find o1 and o2 are outlier (e.g., comparing with O4). 25 Intuition (density-based outlier detection): The density around an outlier object is significantly different from the density around its neighbors Method: Use the relative density of an object against its neighbors as the indicator of the degree of the object being outliers k-distance of an object o, distk(o): distance between o and its k-th NN k-distance neighborhood of o, Nk(o) = {o’| o’ in D, dist(o, o’) ≤ distk(o)} Nk(o) could be bigger than k since multiple objects may have identical distance to o
- 26. Local Outlier Factor: LOF Reachability distance from o’ to o: where k is a user-specified parameter Local reachability density of o: 26 LOF (Local outlier factor) of an object o is the average of the ratio of local reachability of o and those of o’s k-nearest neighbors The lower the local reachability density of o, and the higher the local reachability density of the kNN of o, the higher LOF This captures a local outlier whose local density is relatively low comparing to the local densities of its kNN
- 27. 27 Chapter 12. Outlier Analysis Outlier and Outlier Analysis Outlier Detection Methods Statistical Approaches Proximity-Base Approaches Clustering-Base Approaches Classification Approaches Mining Contextual and Collective Outliers Outlier Detection in High Dimensional Data Summary
- 28. Clustering-Based Outlier Detection (1 & 2): Not belong to any cluster, or far from the closest one An object is an outlier if (1) it does not belong to any cluster, (2) there is a large distance between the object and its closest cluster , or (3) it belongs to a small or sparse cluster Case I: Not belong to any cluster Identify animals not part of a flock: Using a density- based clustering method such as DBSCAN Case 2: Far from its closest cluster Using k-means, partition data points of into clusters For each object o, assign an outlier score based on its distance from its closest center If dist(o, co)/avg_dist(co) is large, likely an outlier Ex. Intrusion detection: Consider the similarity between data points and the clusters in a training data set Use a training set to find patterns of “normal” data, e.g., frequent itemsets in each segment, and cluster similar connections into groups Compare new data points with the clusters mined—Outliers are possible attacks 28
- 29. FindCBLOF: Detect outliers in small clusters Find clusters, and sort them in decreasing size To each data point, assign a cluster-based local outlier factor (CBLOF): If obj p belongs to a large cluster, CBLOF = cluster_size X similarity between p and cluster If p belongs to a small one, CBLOF = cluster size X similarity betw. p and the closest large cluster 29 Clustering-Based Outlier Detection (3): Detecting Outliers in Small Clusters Ex. In the figure, o is outlier since its closest large cluster is C1, but the similarity between o and C1 is small. For any point in C3, its closest large cluster is C2 but its similarity from C2 is low, plus |C3| = 3 is small
- 30. Clustering-Based Method: Strength and Weakness Strength Detect outliers without requiring any labeled data Work for many types of data Clusters can be regarded as summaries of the data Once the cluster are obtained, need only compare any object against the clusters to determine whether it is an outlier (fast) Weakness Effectiveness depends highly on the clustering method used—they may not be optimized for outlier detection High computational cost: Need to first find clusters A method to reduce the cost: Fixed-width clustering A point is assigned to a cluster if the center of the cluster is within a pre-defined distance threshold from the point If a point cannot be assigned to any existing cluster, a new cluster is created and the distance threshold may be learned from the training data under certain conditions
- 31. 31 Chapter 12. Outlier Analysis Outlier and Outlier Analysis Outlier Detection Methods Statistical Approaches Proximity-Base Approaches Clustering-Base Approaches Classification Approaches Mining Contextual and Collective Outliers Outlier Detection in High Dimensional Data Summary
- 32. Classification-Based Method I: One-Class Model Idea: Train a classification model that can distinguish “normal” data from outliers A brute-force approach: Consider a training set that contains samples labeled as “normal” and others labeled as “outlier” But, the training set is typically heavily biased: # of “normal” samples likely far exceeds # of outlier samples Cannot detect unseen anomaly 32 One-class model: A classifier is built to describe only the normal class. Learn the decision boundary of the normal class using classification methods such as SVM Any samples that do not belong to the normal class (not within the decision boundary) are declared as outliers Adv: can detect new outliers that may not appear close to any outlier objects in the training set Extension: Normal objects may belong to multiple classes
- 33. Classification-Based Method II: Semi-Supervised Learning Semi-supervised learning: Combining classification- based and clustering-based methods Method Using a clustering-based approach, find a large cluster, C, and a small cluster, C1 Since some objects in C carry the label “normal”, treat all objects in C as normal Use the one-class model of this cluster to identify normal objects in outlier detection Since some objects in cluster C1 carry the label “outlier”, declare all objects in C1 as outliers Any object that does not fall into the model for C (such as a) is considered an outlier as well 33 Comments on classification-based outlier detection methods Strength: Outlier detection is fast Bottleneck: Quality heavily depends on the availability and quality of the training set, but often difficult to obtain representative and high- quality training data
- 34. 34 Chapter 12. Outlier Analysis Outlier and Outlier Analysis Outlier Detection Methods Statistical Approaches Proximity-Base Approaches Clustering-Base Approaches Classification Approaches Mining Contextual and Collective Outliers Outlier Detection in High Dimensional Data Summary
- 35. Mining Contextual Outliers I: Transform into Conventional Outlier Detection If the contexts can be clearly identified, transform it to conventional outlier detection 1. Identify the context of the object using the contextual attributes 2. Calculate the outlier score for the object in the context using a conventional outlier detection method Ex. Detect outlier customers in the context of customer groups Contextual attributes: age group, postal code Behavioral attributes: # of trans/yr, annual total trans. amount Steps: (1) locate c’s context, (2) compare c with the other customers in the same group, and (3) use a conventional outlier detection method If the context contains very few customers, generalize contexts Ex. Learn a mixture model U on the contextual attributes, and another mixture model V of the data on the behavior attributes Learn a mapping p(Vi|Uj): the probability that a data object o belonging to cluster Uj on the contextual attributes is generated by cluster Vi on the behavior attributes Outlier score: 35
- 36. Mining Contextual Outliers II: Modeling Normal Behavior with Respect to Contexts In some applications, one cannot clearly partition the data into contexts Ex. if a customer suddenly purchased a product that is unrelated to those she recently browsed, it is unclear how many products browsed earlier should be considered as the context Model the “normal” behavior with respect to contexts Using a training data set, train a model that predicts the expected behavior attribute values with respect to the contextual attribute values An object is a contextual outlier if its behavior attribute values significantly deviate from the values predicted by the model Using a prediction model that links the contexts and behavior, these methods avoid the explicit identification of specific contexts Methods: A number of classification and prediction techniques can be used to build such models, such as regression, Markov Models, and Finite State Automaton 36
- 37. Mining Collective Outliers I: On the Set of “Structured Objects” Collective outlier if objects as a group deviate significantly from the entire data Need to examine the structure of the data set, i.e, the relationships between multiple data objects 37 Each of these structures is inherent to its respective type of data For temporal data (such as time series and sequences), we explore the structures formed by time, which occur in segments of the time series or subsequences For spatial data, explore local areas For graph and network data, we explore subgraphs Difference from the contextual outlier detection: the structures are often not explicitly defined, and have to be discovered as part of the outlier detection process. Collective outlier detection methods: two categories Reduce the problem to conventional outlier detection Identify structure units, treat each structure unit (e.g., subsequence, time series segment, local area, or subgraph) as a data object, and extract features Then outlier detection on the set of “structured objects” constructed as such using the extracted features
- 38. Mining Collective Outliers II: Direct Modeling of the Expected Behavior of Structure Units Models the expected behavior of structure units directly Ex. 1. Detect collective outliers in online social network of customers Treat each possible subgraph of the network as a structure unit Collective outlier: An outlier subgraph in the social network Small subgraphs that are of very low frequency Large subgraphs that are surprisingly frequent Ex. 2. Detect collective outliers in temporal sequences Learn a Markov model from the sequences A subsequence can then be declared as a collective outlier if it significantly deviates from the model Collective outlier detection is subtle due to the challenge of exploring the structures in data The exploration typically uses heuristics, and thus may be application dependent The computational cost is often high due to the sophisticated mining process 38
- 39. 39 Chapter 12. Outlier Analysis Outlier and Outlier Analysis Outlier Detection Methods Statistical Approaches Proximity-Base Approaches Clustering-Base Approaches Classification Approaches Mining Contextual and Collective Outliers Outlier Detection in High Dimensional Data Summary
- 40. Challenges for Outlier Detection in High- Dimensional Data Interpretation of outliers Detecting outliers without saying why they are outliers is not very useful in high-D due to many features (or dimensions) are involved in a high-dimensional data set E.g., which subspaces that manifest the outliers or an assessment regarding the “outlier-ness” of the objects Data sparsity Data in high-D spaces are often sparse The distance between objects becomes heavily dominated by noise as the dimensionality increases Data subspaces Adaptive to the subspaces signifying the outliers Capturing the local behavior of data Scalable with respect to dimensionality # of subspaces increases exponentially 40
- 41. Approach I: Extending Conventional Outlier Detection Method 1: Detect outliers in the full space, e.g., HilOut Algorithm Find distance-based outliers, but use the ranks of distance instead of the absolute distance in outlier detection For each object o, find its k-nearest neighbors: nn1(o), . . . , nnk(o) The weight of object o: All objects are ranked in weight-descending order Top-l objects in weight are output as outliers (l: user-specified parm) Employ space-filling curves for approximation: scalable in both time and space w.r.t. data size and dimensionality Method 2: Dimensionality reduction Works only when in lower-dimensionality, normal instances can still be distinguished from outliers PCA: Heuristically, the principal components with low variance are preferred because, on such dimensions, normal objects are likely close to each other and outliers often deviate from the majority 41
- 42. Approach II: Finding Outliers in Subspaces Extending conventional outlier detection: Hard for outlier interpretation Find outliers in much lower dimensional subspaces: easy to interpret why and to what extent the object is an outlier E.g., find outlier customers in certain subspace: average transaction amount >> avg. and purchase frequency << avg. Ex. A grid-based subspace outlier detection method Project data onto various subspaces to find an area whose density is much lower than average Discretize the data into a grid with φ equi-depth (why?) regions Search for regions that are significantly sparse Consider a k-d cube: k ranges on k dimensions, with n objects If objects are independently distributed, the expected number of objects falling into a k-dimensional region is (1/ φ)k n = fk n,the standard deviation is The sparsity coefficient of cube C: If S(C) < 0, C contains less objects than expected The more negative, the sparser C is and the more likely the objects in C are outliers in the subspace 42
- 43. Approach III: Modeling High-Dimensional Outliers Ex. Angle-based outliers: Kriegel, Schubert, and Zimek [KSZ08] For each point o, examine the angle ∆xoy for every pair of points x, y. Point in the center (e.g., a), the angles formed differ widely An outlier (e.g., c), angle variable is substantially smaller Use the variance of angles for a point to determine outlier Combine angles and distance to model outliers Use the distance-weighted angle variance as the outlier score Angle-based outlier factor (ABOF): Efficient approximation computation method is developed It can be generalized to handle arbitrary types of data 43 Develop new models for high- dimensional outliers directly Avoid proximity measures and adopt new heuristics that do not deteriorate in high-dimensional data A set of points form a cluster except c (outlier)
- 44. 44 Chapter 12. Outlier Analysis Outlier and Outlier Analysis Outlier Detection Methods Statistical Approaches Proximity-Base Approaches Clustering-Base Approaches Classification Approaches Mining Contextual and Collective Outliers Outlier Detection in High Dimensional Data Summary
- 45. Summary Types of outliers global, contextual & collective outliers Outlier detection supervised, semi-supervised, or unsupervised Statistical (or model-based) approaches Proximity-base approaches Clustering-base approaches Classification approaches Mining contextual and collective outliers Outlier detection in high dimensional data 45
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- 50. 50 Outlier Discovery: Statistical Approaches Assume a model underlying distribution that generates data set (e.g. normal distribution) Use discordancy tests depending on data distribution distribution parameter (e.g., mean, variance) number of expected outliers Drawbacks most tests are for single attribute In many cases, data distribution may not be known
- 51. 51 Outlier Discovery: Distance-Based Approach Introduced to counter the main limitations imposed by statistical methods We need multi-dimensional analysis without knowing data distribution Distance-based outlier: A DB(p, D)-outlier is an object O in a dataset T such that at least a fraction p of the objects in T lies at a distance greater than D from O Algorithms for mining distance-based outliers [Knorr & Ng, VLDB’98] Index-based algorithm Nested-loop algorithm Cell-based algorithm
- 52. 52 Density-Based Local Outlier Detection M. M. Breunig, H.-P. Kriegel, R. Ng, J. Sander. LOF: Identifying Density-Based Local Outliers. SIGMOD 2000. Distance-based outlier detection is based on global distance distribution It encounters difficulties to identify outliers if data is not uniformly distributed Ex. C1 contains 400 loosely distributed points, C2 has 100 tightly condensed points, 2 outlier points o1, o2 Distance-based method cannot identify o2 as an outlier Need the concept of local outlier Local outlier factor (LOF) Assume outlier is not crisp Each point has a LOF
- 53. 53 Outlier Discovery: Deviation-Based Approach Identifies outliers by examining the main characteristics of objects in a group Objects that “deviate” from this description are considered outliers Sequential exception technique simulates the way in which humans can distinguish unusual objects from among a series of supposedly like objects OLAP data cube technique uses data cubes to identify regions of anomalies in large multidimensional data
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