Excerpts for Data Mining Anomaly Detection. Lecture Notes for Chapters 8 &10. Introduction to Data Mining

Transcription

1 Excerpts for Data Mining Anomaly Detection Lecture Notes for Chapters 8 &10 Introduction to Data Mining by Tan, Steinbach, Kumar Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1

2 Anomaly/Outlier Detection What are anomalies/outliers? The set of data points that are considerably different than the remainder of the data Variants of Anomaly/Outlier Detection Problems Given a database D, find all the data points x D with anomaly scores greater than some threshold t Given a database D, find all the data points x D having the topn largest anomaly scores f(x) Given a database D, containing mostly normal (but unlabeled) data points, and a test point x, compute the anomaly score of x with respect to D Applications: Credit card fraud detection, telecommunication fraud detection, network intrusion detection, fault detection

3 Importance of Anomaly Detection Ozone Depletion History In 1985 three researchers (Farman, Gardinar and Shanklin) were puzzled by data gathered by the British Antarctic Survey showing that ozone levels for Antarctica had dropped 10% below normal levels Why did the Nimbus 7 satellite, which had instruments aboard for recording ozone levels, not record similarly low ozone concentrations? The ozone concentrations recorded by the satellite were so low they were being treated as outliers by a computer program and discarded! Sources:

4 Anomaly Detection Challenges How many outliers are there in the data? Method is unsupervised u Validation can be quite challenging (just like for clustering) Finding needle in a haystack Working assumption: There are considerably more normal observations than abnormal observations (outliers/anomalies) in the data

5 Anomaly Detection Schemes General Steps Build a profile of the normal behavior u Profile can be patterns or summary statistics for the overall population Use the normal profile to detect anomalies u Anomalies are observations whose characteristics differ significantly from the normal profile Types of anomaly detection schemes Graphical & Statistical-based Distance-based Model-based

6 Graphical Approaches Boxplot (1-D), Scatter plot (2-D), Spin plot (3-D) Limitations Time consuming Subjective

7 Convex Hull Method Extreme points are assumed to be outliers Use convex hull method to detect extreme values What if the outlier occurs in the middle of the data?

8 Statistical Approaches Assume a parametric model describing the distribution of the data (e.g., normal distribution) Apply a statistical test that depends on Data distribution Parameter of distribution (e.g., mean, variance) Number of expected outliers (confidence limit)

9 Grubbs Test Detect outliers in univariate data Assume data comes from normal distribution Detects one outlier at a time, remove the outlier, and repeat H 0 : There is no outlier in data H A : There is at least one outlier Grubbs test statistic: Reject H 0 if: G > ( N 1) N G = max N t 2 X s ( α / N, N 2) 2 + t 2 X ( α / N, N 2)

10 Statistical-based Likelihood Approach Assume the data set D contains samples from a mixture of two probability distributions: M (majority distribution) A (anomalous distribution) General Approach: Initially, assume all the data points belong to M Let L t (D) be the log likelihood of D at time t For each point x t that belongs to M, move it to A u Let L t+1 (D) be the new log likelihood. u Compute the difference, Δ = L t (D) L t+1 (D) u If Δ > c (some threshold), then x t is declared as an anomaly and moved permanently from M to A

11 Statistical-based Likelihood Approach Data distribution, D = (1 λ) M + λ A M is a probability distribution estimated from data Can be based on any modeling method (naïve Bayes, maximum entropy, etc) A is initially assumed to be uniform distribution Likelihood at time t: = = = = t i t t i t t i t t t i t t A x i A t M x i M t t A x i A A M x i M M N i i D t x P A x P M D LL x P x P x P D L ) ( log log ) ( log ) log(1 ) ( ) ( ) ( ) (1 ) ( ) ( 1 λ λ λ λ

12 Limitations of Statistical Approaches Most of the tests are for a single attribute In many cases, data distribution may not be known For high dimensional data, it may be difficult to estimate the true distribution

13 Distance-based Approaches Data is represented as a vector of features Three major approaches Nearest-neighbor based Density based Clustering based

14 Nearest-Neighbor Based Approach Approach: Compute the distance between every pair of data points There are various ways to define outliers: u Data points for which there are fewer than p neighboring points within a distance D u The top n data points whose distance to the kth nearest neighbor is greatest u The top n data points whose average distance to the k nearest neighbors is greatest

15 Outliers in Lower Dimensional Projection In high-dimensional space, data is sparse and notion of proximity becomes meaningless Every point is an almost equally good outlier from the perspective of proximity-based definitions Lower-dimensional projection methods A point is an outlier if in some lower dimensional projection, it is present in a local region of abnormally low density

16 Outliers in Lower Dimensional Projection Divide each attribute into φ equal-depth intervals Each interval contains a fraction f = 1/φ of the records Consider a k-dimensional cube created by picking grid ranges from k different dimensions If attributes are independent, we expect region to contain a fraction f k of the records If there are N points, we can measure sparsity of a cube D as: Negative sparsity indicates cube contains smaller number of points than expected

17 Example N=100, φ = 5, f = 1/5 = 0.2, N f 2 = 4

18 Density-based: LOF approach For each point, compute the density of its local neighborhood Compute local outlier factor (LOF) of a sample p as the average of the ratios of the density of sample p and the density of its nearest neighbors Outliers are points with largest LOF value p 2 p 1 In the NN approach, p 2 is not considered as outlier, while LOF approach find both p 1 and p 2 as outliers

19 Clustering-Based Basic idea: Cluster the data into groups of different density Choose points in small cluster as candidate outliers Compute the distance between candidate points and non-candidate clusters. u If candidate points are far from all other non-candidate points, they are outliers

20 Model Evaluation Metrics for Performance Evaluation How to evaluate the performance of a model? Methods for Performance Evaluation How to obtain reliable estimates? Methods for Model Comparison How to compare the relative performance among competing models?

21 Model Evaluation Metrics for Performance Evaluation How to evaluate the performance of a model? Methods for Performance Evaluation How to obtain reliable estimates? Methods for Model Comparison How to compare the relative performance among competing models?

22 Metrics for Performance Evaluation Focus on the predictive capability of a model Rather than how fast it takes to classify or build models, scalability, etc. Confusion Matrix: PREDICTED CLASS ACTUAL CLASS Class=Yes Class=No Class=Yes a b Class=No c d a: TP (true positive) b: FN (false negative) c: FP (false positive) d: TN (true negative)

23 Metrics for Performance Evaluation PREDICTED CLASS Class=Yes Class=No ACTUAL CLASS Class=Yes Class=No a (TP) c (FP) b (FN) d (TN) Most widely-used metric: Accuracy = a a + b + + d c + d = TP TP + TN + TN + FP + FN

24 Limitation of Accuracy Consider a 2-class problem Number of Class 0 examples = 9990 Number of Class 1 examples = 10 If model predicts everything to be class 0, accuracy is 9990/10000 = 99.9 % Accuracy is misleading because model does not detect any class 1 example

25 Cost Matrix PREDICTED CLASS C(i j) Class=Yes Class=No ACTUAL CLASS Class=Yes C(Yes Yes) C(No Yes) Class=No C(Yes No) C(No No) C(i j): Cost of misclassifying class j example as class i

26 Computing Cost of Classification Cost Matrix ACTUAL CLASS PREDICTED CLASS C(i j) Model M 1 PREDICTED CLASS Model M 2 PREDICTED CLASS ACTUAL CLASS ACTUAL CLASS Accuracy = 80% Cost = 3910 Accuracy = 90% Cost = 4255

27 Cost vs Accuracy Count ACTUAL CLASS PREDICTED CLASS Class=Yes Class=No Class=Yes a b Class=No c d Accuracy is proportional to cost if 1. C(Yes No)=C(No Yes) = q 2. C(Yes Yes)=C(No No) = p N = a + b + c + d Accuracy = (a + d)/n Cost ACTUAL CLASS PREDICTED CLASS Class=Yes Class=No Class=Yes p q Class=No q p Cost = p (a + d) + q (b + c) = p (a + d) + q (N a d) = q N (q p)(a + d) = N [q (q-p) Accuracy]

28 Cost-Sensitive Measures a Precision (p) = a + c a Recall (r) = a + b 2rp F - measure (F) = r + p = 2a 2a + b + c Precision is biased towards C(Yes Yes) & C(Yes No) Recall is biased towards C(Yes Yes) & C(No Yes) F-measure is biased towards all except C(No No) Weighted Accuracy = wa 1 wa + w d w b + w c + w d 2 3 4

29 Model Evaluation Metrics for Performance Evaluation How to evaluate the performance of a model? Methods for Performance Evaluation How to obtain reliable estimates? Methods for Model Comparison How to compare the relative performance among competing models?

30 Methods for Performance Evaluation How to obtain a reliable estimate of performance? Performance of a model may depend on other factors besides the learning algorithm: Class distribution Cost of misclassification Size of training and test sets

31 Learning Curve Learning curve shows how accuracy changes with varying sample size Requires a sampling schedule for creating learning curve: Arithmetic sampling (Langley, et al) Geometric sampling (Provost et al) Effect of small sample size: - Bias in the estimate - Variance of estimate

32 Methods of Estimation Holdout Reserve 2/3 for training and 1/3 for testing Random subsampling Repeated holdout Cross validation Partition data into k disjoint subsets k-fold: train on k-1 partitions, test on the remaining one Leave-one-out: k=n Stratified sampling oversampling vs undersampling Bootstrap Sampling with replacement

33 Model Evaluation Metrics for Performance Evaluation How to evaluate the performance of a model? Methods for Performance Evaluation How to obtain reliable estimates? Methods for Model Comparison How to compare the relative performance among competing models?

34 ROC (Receiver Operating Characteristic) Developed in 1950s for signal detection theory to analyze noisy signals Characterize the trade-off between positive hits and false alarms ROC curve plots TP (on the y-axis) against FP (on the x-axis) Performance of each classifier represented as a point on the ROC curve changing the threshold of algorithm, sample distribution or cost matrix changes the location of the point

35 ROC Curve - 1-dimensional data set containing 2 classes (positive and negative) - any points located at x > t is classified as positive At threshold t: TP=0.5, FN=0.5, FP=0.12, FN=0.88

36 ROC Curve (TP,FP): (0,0): declare everything to be negative class (1,1): declare everything to be positive class (1,0): ideal Diagonal line: Random guessing Below diagonal line: u prediction is opposite of the true class

37 Using ROC for Model Comparison No model consistently outperform the other M 1 is better for small FPR M 2 is better for large FPR Area Under the ROC curve Ideal: Area = 1 Random guess: Area = 0.5

38 How to Construct an ROC curve Instance P(+ A) True Class Use classifier that produces posterior probability for each test instance P(+ A) Sort the instances according to P(+ A) in decreasing order Apply threshold at each unique value of P(+ A) Count the number of TP, FP, TN, FN at each threshold TP rate, TPR = TP/(TP+FN) FP rate, FPR = FP/(FP + TN)

39 How to construct an ROC curve Class Threshold >= TP FP TN FN TPR FPR ROC Curve:

40 Test of Significance Given two models: Model M1: accuracy = 85%, tested on 30 instances Model M2: accuracy = 75%, tested on 5000 instances Can we say M1 is better than M2? How much confidence can we place on accuracy of M1 and M2? Can the difference in performance measure be explained as a result of random fluctuations in the test set?

41 Confidence Interval for Accuracy Prediction can be regarded as a Bernoulli trial A Bernoulli trial has 2 possible outcomes Possible outcomes for prediction: correct or wrong Collection of Bernoulli trials has a Binomial distribution: u x Bin(N, p) x: number of correct predictions u e.g: Toss a fair coin 50 times, how many heads would turn up? Expected number of heads = N p = = 25 Given x (# of correct predictions) or equivalently, acc=x/n, and N (# of test instances), Can we predict p (true accuracy of model)?

42 Confidence Interval for Accuracy For large test sets (N > 30), P acc has a normal distribution with mean p and variance p(1-p)/n ( Z < < Z α / 2 1 α / 2 = 1 α acc p p(1 p) / N Confidence Interval for p: ) Area = 1 - α Z α/2 Z 1- α /2 p = 2 N acc + Z 2 α / 2 ± Z 2 α / 2 2( N + 4 N + Z 2 α / 2 ) acc 4 N acc 2

43 Confidence Interval for Accuracy Consider a model that produces an accuracy of 80% when evaluated on 100 test instances: N=100, acc = 0.8 Let 1-α = 0.95 (95% confidence) From probability table, Z α/2 = α Z N p(lower) p(upper)

44 Comparing Performance of 2 Models Given two models, say M1 and M2, which is better? M1 is tested on D1 (size=n1), found error rate = e 1 M2 is tested on D2 (size=n2), found error rate = e 2 Assume D1 and D2 are independent If n1 and n2 are sufficiently large, then Approximate: e e 1 2 ~ ~ ˆ σ = i N N ( µ, σ ) 1 ( µ, σ ) i 2 e (1 e ) i i n 1 2

45 Comparing Performance of 2 Models To test if performance difference is statistically significant: d = e1 e2 d ~ N(d t,σ t ) where d t is the true difference Since D1 and D2 are independent, their variance adds up: 2 σ t = σ + σ ˆ σ + ˆ σ 1 e1(1 e1) = + n1 2 1 e2(1 e2) n2 2 At (1-α) confidence level, d t = d ± Z α / 2 σ ˆ t

46 An Illustrative Example Given: M1: n1 = 30, e1 = 0.15 M2: n2 = 5000, e2 = 0.25 d = e2 e1 = 0.1 (2-sided test) 0.15(1 0.15) 0.25(1 0.25) ˆ = + = σ d At 95% confidence level, Z α/2 =1.96 d t = ± = ± => Interval contains 0 => difference may not be statistically significant

47 Comparing Performance of 2 Algorithms Each learning algorithm may produce k models: L1 may produce M11, M12,, M1k L2 may produce M21, M22,, M2k If models are generated on the same test sets D1,D2,, Dk (e.g., via cross-validation) For each set: compute d j = e 1j e 2j d j has mean d t and variance σ t k Estimate: 2 ( d d) 2 j= 1 j σ = ˆ d t t = d k( k ± t 1) 1 α, k 1 ˆ σ t

48 Base Rate Fallacy Bayes theorem: More generally:

49 Base Rate Fallacy (Axelsson, 1999)

50 Base Rate Fallacy Even though the test is 99% certain, your chance of having the disease is 1/100, because the population of healthy people is much larger than sick people

51 Base Rate Fallacy in Intrusion Detection I: intrusive behavior, I: non-intrusive behavior A: alarm A: no alarm Detection rate (true positive rate): P(A I) False alarm rate: P(A I) Goal is to maximize both Bayesian detection rate, P(I A) P( I A)

52 Detection Rate vs False Alarm Rate Suppose: Then: False alarm rate becomes more dominant if P(I) is very low

53 Detection Rate vs False Alarm Rate Axelsson: We need a very low false alarm rate to achieve a reasonable Bayesian detection rate