lecture notes for chapters 8 &10 introduction to data...

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

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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 top-

n 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


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:

http://exploringdata.cqu.edu.au/ozone.html http://www.epa.gov/ozone/science/hole/size.html


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


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


Graphical Approaches

  Boxplot (1-D), Scatter plot (2-D), Spin plot (3-D)

  Limitations –  Time consuming –  Subjective


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?


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)


Grubbs’ Test

 Detect outliers in univariate data  Assume data comes from normal distribution  Detects one outlier at a time, remove the outlier,

and repeat –  H0: There is no outlier in data –  HA: There is at least one outlier

 Grubbs’ test statistic:

 Reject H0 if: s

XXG

−=max

2

2

)2,/(

)2,/(

2)1(

−

−

+−−

>NN

NN

tNt

NNG

α

α


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 Lt(D) be the log likelihood of D at time t –  For each point xt that belongs to M, move it to A

u  Let Lt+1 (D) be the new log likelihood. u  Compute the difference, Δ = Lt(D) – Lt+1 (D) u  If Δ > c (some threshold), then xt is declared as an anomaly and moved permanently from M to A


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:

∑∑

∏∏∏

∈∈

∈∈=

+++−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−==

ti

t

ti

t

ti

t

t

ti

t

t

AxiAt

MxiMtt

AxiA

A

MxiM

MN

iiDt

xPAxPMDLL

xPxPxPDL

)(loglog)(log)1log()(

)()()1()()( ||||

1

λλ

λλ


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


Distance-based Approaches

  Data is represented as a vector of features

  Three major approaches –  Nearest-neighbor based –  Density based –  Clustering based


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


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


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 fk 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


Example

 N=100, φ = 5, f = 1/5 = 0.2, N × f2 = 4


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

p2 × p1

×

In the NN approach, p2 is not considered as outlier, while LOF approach find both p1 and p2 as outliers


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


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?


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)


Metrics for Performance Evaluation…

 Most widely-used metric:

PREDICTED CLASS

ACTUAL CLASS

Class=Yes Class=No

Class=Yes a (TP)

b (FN)

Class=No c (FP)

d (TN)

FNFPTNTPTNTP

dcbada

++++

=+++

+=Accuracy


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


Computing Cost of Classification

Cost Matrix

PREDICTED CLASS

ACTUAL CLASS

C(i|j) + - + -1 100 - 1 0

Model M1 PREDICTED CLASS

ACTUAL CLASS

+ - + 150 40 - 60 250

Model M2 PREDICTED CLASS

ACTUAL CLASS

+ - + 250 45 - 5 200

Accuracy = 80% Cost = 3910

Accuracy = 90% Cost = 4255


Cost vs Accuracy

Count PREDICTED CLASS

ACTUAL CLASS

Class=Yes Class=No

Class=Yes a b

Class=No c d

Cost PREDICTED CLASS

ACTUAL CLASS

Class=Yes Class=No

Class=Yes p q

Class=No q p

N = a + b + c + d

Accuracy = (a + d)/N

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]

Accuracy is proportional to cost if 1. C(Yes|No)=C(No|Yes) = q 2. C(Yes|Yes)=C(No|No) = p


Cost-Sensitive Measures

cbaa

prrp

baa

caa

++=

+=

+=

+=

222(F) measure-F

(r) Recall

(p)Precision

  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)

dwcwbwawdwaw

4321

41Accuracy Weighted+++

+=


Model Evaluation




competing models?


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


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


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


Model Evaluation




competing models?


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


ROC Curve

At threshold t:

TP=0.5, FN=0.5, FP=0.12, FN=0.88

- 1-dimensional data set containing 2 classes (positive and negative)

- any points located at x > t is classified as positive


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


Using ROC for Model Comparison

  No model consistently outperform the other   M1 is better for

small FPR   M2 is better for

large FPR

  Area Under the ROC curve   Ideal:

§  Area = 1   Random guess:

§  Area = 0.5


How to Construct an ROC curve

Instance P(+|A) True Class 1 0.95 + 2 0.93 + 3 0.87 - 4 0.85 - 5 0.85 - 6 0.85 + 7 0.76 - 8 0.53 + 9 0.43 -

10 0.25 +

•  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)


How to construct an ROC curve

Class + - + - - - + - + + P 0.25 0.43 0.53 0.76 0.85 0.85 0.85 0.87 0.93 0.95 1.00

TP 5 4 4 3 3 3 3 2 2 1 0

FP 5 5 4 4 3 2 1 1 0 0 0

TN 0 0 1 1 2 3 4 4 5 5 5

FN 0 1 1 2 2 2 2 3 3 4 5

TPR 1 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.2 0

FPR 1 1 0.8 0.8 0.6 0.4 0.2 0.2 0 0 0

Threshold >=

ROC Curve:


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?


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 = 50 × 0.5 = 25

 Given x (# of correct predictions) or equivalently, acc=x/N, and N (# of test instances),

Can we predict p (true accuracy of model)?



 For large test sets (N > 30), –  acc has a normal distribution

with mean p and variance p(1-p)/N

 Confidence Interval for p:

α

αα

−=

<−−

<−

1

)/)1(

(2/12/

ZNpp

paccZP

Area = 1 - α

Zα/2 Z1- α /2

)(2442

2

2/

22

2/

2

2/

α

αα

ZNaccNaccNZZaccNp

+

××−××+±+××=



 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, Zα/2=1.96

1-α Z

0.99 2.58

0.98 2.33

0.95 1.96

0.90 1.65

N 50 100 500 1000 5000

p(lower) 0.670 0.711 0.763 0.774 0.789

p(upper) 0.888 0.866 0.833 0.824 0.811


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 = e1

–  M2 is tested on D2 (size=n2), found error rate = e2

–  Assume D1 and D2 are independent –  If n1 and n2 are sufficiently large, then

–  Approximate:

( )( )222

111

,~,~σµ

σµ

NeNe

i

ii

i nee )1(ˆ −

=σ


Comparing Performance of 2 Models

 To test if performance difference is statistically significant: d = e1 – e2 –  d ~ N(dt,σt) where dt is the true difference –  Since D1 and D2 are independent, their variance adds

up:

–  At (1-α) confidence level,

2)21(2

1)11(1

ˆˆ 2

2

2

1

2

2

2

1

2

nee

nee

t

−+

−=

+≅+= σσσσσ

ttZdd σ

αˆ

2/±=


An Illustrative Example

 Given: M1: n1 = 30, e1 = 0.15 M2: n2 = 5000, e2 = 0.25

 d = |e2 – e1| = 0.1 (2-sided test)

 At 95% confidence level, Zα/2=1.96

=> Interval contains 0 => difference may not be

statistically significant

0043.05000

)25.01(25.030

)15.01(15.0ˆ =−

+−

=d

σ

128.0100.00043.096.1100.0 ±=×±=td


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 dj = e1j – e2j

–  dj has mean dt and variance σt –  Estimate:

tkt

k

j j

t

tddkkdd

σ

σ

αˆ)1()(

ˆ

1,1

1

2

2

−−

=

±=

−

−=∑


Base Rate Fallacy

  Bayes theorem:

  More generally:


Base Rate Fallacy (Axelsson, 1999)


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


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)


Detection Rate vs False Alarm Rate

 Suppose:

 Then:

 False alarm rate becomes more dominant if P(I) is very low


Detection Rate vs False Alarm Rate

  Axelsson: We need a very low false alarm rate to achieve a reasonable Bayesian detection rate

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