Forecast, Detect, Intervene: Anomaly Detection for Time

Series.Deepak Agarwal

Yahoo! Research

Outline

• Approach– Forecast– Detect– Intervene

• Monitoring multiple series– Multiple testing, a Bayesian solution.

• Application

Issues• {yt} : univariate, regularly spaced time series to be

monitored for anomalies, “novel events” , surprises prospectively.– E.g. query volume, Hang-ups, ER admissions

• Goal: A semi-automated statistical approach – Forecast accurately : good baseline model.– Detect deviations from baseline:

• sensitivity/specificity/timeliness

– Baseline model adaptive: learn changes automatically • Important in applications : better forecasts →fewer false +ve

Approach• Three components : (West and Harrison, 1976)

–Forecast: Bayesian version of Kalman filter

–Detection: A new sequential algorithm

– Intervention: correct baseline model.

Forecasting

Kalman filter

• Observation Equation– Conditional distribution of data given parameters

• State Equation – Evolution of parameters (states) through time

• Posterior of states, predictive distribution– Estimated online by recursive algorithm

OBSERVATION EQUATIONy ~ (0, )

t

: observed value: mean (unknown and estimated from data): with variance

(usually unknown and estimated from data)

' independent conditional on 'i.e., if the truth

t

t

v N Vt t

ytTruet

v Noise Vt

y s st t

1

is known, ' provide no informationto predict the future.

This model severely overfits (more unknowns than knowns)Need to make simplifying assumptions to estimate the

true mean surface { }t T

t

y st

t

STATE EQUATION: What assumptions are appropriate for the “Truth”?

for all t

Too simple, performs poorly unless mean stationary.(Rarely works on real data)

position of particle at described by some differential equationt.

t

te g

Simple :

Determined by system dynamics :

. (Wikle,1998) studies ecological processes using reaction-diffusion equations.

:

Works well for empirical data analysis.Assumes function of true mean at previous time poit

Simple Markovian assumptions

0 gives back the constant model.

ntse.g. Simple Random Walk: ~ (0, )t 1

:

w N Wt tt

NB Wt

More general models

1

equationy ~ (0, )

equation~ (0, )

updated using Kalman filter

One can almost take any static model and make it dynamic.

e.g, is a 7-dim vector correspo

Tt t t t t

Tt t t

t t t t t

t

t

observationx Vx

StateG w W

nding to day of week effects.

Covariates whose coefficients evolve dynamically

1t

Yt-1 Yt

t

xtXt-1

Gt

Yt-1

Xt-1

Kalman Filter update at time t: t-1 t-1 1 1 1

t t 1

1 1

1

t

(a) Posterior for : ( | ) ~ ( , )

( ) Prior for : ( | ) ~ ( , )

Prior parameters derived by plugging posterior estimates in state equation

0

( )Likelihood of y :

t t t

t t t

t t t

t t t

D N m C

b D N a R

a m mR C W

c

t t

1 t t-1 t

t

t 1

( | ) ~ ( , )

( )Posterior for : ( | ) ~ ( , )

(y -m ); C A /( )

(1 A )

If , this gives the well known exponentially weighted mo

t t t t

t t t

t t t t t

t t t

t t t t

t

y N V

d D N m C

m m A AV

R R V

m A y m

A A

ving average (EWMA). Under mild conditions, this is true.

Estimating Variance componentst t-1

t 1 1

Assume W C

W (1 ) / / (0 1)

called "discount" factor, makes prior at vague to enablecurrent estimates being influenced more by recent past.

0:Prior too vague.1: Pri

t w w t t w w

w

w

w

C R C

t

2t 0 0

or too tight, close to a static model.

plays a role similar to window size in streaming algorithms.

In practice, it is better to be conservative and choose smaller .Reason: Var(e | ) (1 ( )

w

w

w wD Q

2/(1 ))

We select using initial data and re-adjust the value later if needed.

w

w

Detection

An existing method

errors. correlated-auto d)

changes variancec) shiftsmean b)a)outlier :Detects-

s.for factor Bayeson based algorithm sequentialA -

:

10 iid s null,Under -

)|(/))|(( 11

t

t

tttttt

u

),N(u

DyVDyEyu

'05) Salvador, and llowork(Garga Related

Gaussian ondistributi predictive when Residual

Pitfalls of GS

• What if predictive not Gaussian?– Mixtures of Gaussians, Poisson etc

• Bayes factor: specify alternative explicitly– Large number of unspecified parameters– Require explicit model for each alternative

Our approach

• Normal scores derived from p-values– Good for continuous, approximately good for

discrete, especially for large means.

• A sequential procedure with far less tweaking parameters.

• Our method has more power, we sacrifice on timeliness.

Sequential detection procedureAt time t, we are in one of these regions:• Acceptance region (A): The null model is true,

the system is behaving as expected, no anomalies, start a new run.

• Rejection region (R) : The null model is not true, an anomaly is generated which is reported to the user and/or the forecasting model is reset. Start a new run.

• Continue (C): Don’t have enough data to reach a decision, keep accumulating evidence by taking another sample.

Detecting outliers and mean shifts

r

r t-i+1i=0

inc

: | |

All other changes based on most recent run of ( 1) points in

: : ' are iid ( ,1) (|h|>0).

Test based on u u / ~ (0,1/ ) under null.

Test statistics: Pmean =

t

h i

u a

r

M u s N h

r N r

Outlier

C

Mean shifts

h=0 r r

dec h=0 r r

P (u u ) (to detect mean increase.) Pmean = P (u <= u ) (to detect mean decrease.)

Large shifts, would be identified as outliers. Moderateshifts when identified can

obs

obs

provide important information about the system.

Detecting variance shifts

k

2 2 2- 1

02 2( )

1

1

M : ' are iid (0, )

Test based on ~ with under null.

Test statistics: var ( ) (to detect variance increase) var (

tr

r t i ri

obsinc k r r

dec k r

u s N k

U u df r

P P U UP P U

2 2( ) ) (to detect variance decrease)

Large variance increase will be identified as an outlier, moderateincrease when detected can be important in applications.

Variance decrease indicates the s

obsrU

ystem getting stable or the discountparameters being set too low. Helps in improving the forecasting model,may not be that important to the user.

Gradual changes, auto correlated errors.

g i 1

Local patches of auto-correlated errors (Gargallo and Salvador, 2003).

Roughly, the residuals follow an AR(1) process.

M : u ~ (0,1)(0 1)

Residuals tend to be positive(negative) more often th

i igu N g

an negative(positive).Gradual ramp up(down) in the residuals, short run of consecutive moderatechanges.

Report the change, relax the discount factors to learn the gradual shifts.

Use two test statistics2 2

2- 1 1

0 0

1

:

g= / ~ (0,1/ ) under null for large r

For small r, exact distribution not available in closed form.

(1 ) ~ (0, (12

r r

t i t i t ii i

t t t

u u u N r

u u u N

(obs) (obs)1 g=0 2 g=0

(1 ) )) under null.

Test statistics: AC =P (g > g ); AC =min(P (g > g ), (| | | |)

r

obsnull t tP u u

2 : Cauchy with scale 1 and center 0. 2, analytical form not known, well

approximated by mixture of two normals.

( ) (0,1) (1 ) (0,1 )

( , ) estimated using si

r

rr r g

f r g PN P N

P

Distribution of g under g = 0

mulationsfrom the distribution.

For 7, the normal approximation is good.r

r P τ

3 .84 14.91

4 .87 4.40

5 .96 3.77

6 .98 3.98

7 .99 3.66

8 .98 0.10

9 .98 0.10

10 .85 0.09

11 .85 0.09

The sequential algorithm at time t

-1max 2

-1max 2

max 1 max

1: | | outlier else continue.

1: | | outlier compute max( ( , , var , var , ))

arg max( ( , , var , var , ))

accept null;

t

t

inc dec inc dec

inc dec inc dec

r u a

r u a elseS Pmean Pmean P P ACi Pmean Pmean P P AC

S c S

2 max

1 max 2

1 2

declare anomaly continue.

3.5, .25, 1.2, 3.0 for 5% false positive rate.

c ic S c

a c c

Blue:ours; red: Gargallo and Salvador(GS)

Intervention

Intervention to adjust the baseline.

• Outlier → A tail or rare event has occurred– Ignore points → short tail; more false +ve– Use points→ elongated tail, more false -ve

• A robust solution: ignore points but elongate tail – retain same prior mean, increase prior variance.– system adapts, re-initializing the monitor.

• Use the above for mean shifts and variance increase.

• Variance decrease: System stable, make prior tight.

• Slow changes: System under-adaptive, make prior vague.

Intervention strategy

2t 1 1 1

w

Outlier, mean shift, variance increase:

| ~ ( , )( 2,3); max(.9,.95 ) .

Variance decrease:min(1,1.025 ); min(1,1.025 )

Slow change, positive autocorrelation, persistent bli

t t t t v t

new neww v v

D N m m R m n n

w

ps.

max(.7,.95 ); max(.9,.95 )new neww v v

No intervention, m=1

strong intervention, m=3

Example: Blue is data, yellow is forecast.

Multiple testing

Multiple testing: A Bayesian Approach.

• Monitoring large number of independent streams – testing multiple hypotheses at each time point – Need correction for multiple testing.

• Main idea: – Derive an empirical null based on observed deviations– Present analyst with interesting cases adjusting for global

characteristics of the system.– We use a Bayesian approach to derive shrinkage

estimates of deviations– the “shrunk” deviations automatically build in penalty for

conducting multiple tests.

Bayesian procedure.

series). timeofnumber (moderate Hermite-Gauss :BayesianFully

series) timeofnumber (large EB using estimated : etersHyperparam

residuals. large of shrinkage-over prevents :component Two

)./(

),;()1/(),;()1/(

))1(()1(),,,;|(

. of meansposterior monitoringby anomaliesDetect

).,()(1)1(~);,(~|

on.distributi predictive of value-pon based score normal:

12

121

21

tttst

ttstttttsttstst

stststststtstttttstst

st

ttttsttsttststst

st

B

uNDPeNDPqq

BuBqquE

NPPNu

u

Experiment comparing multiple testing versus naïve procedure (threshold raw standardized

residuals)• Simulate K noise points N(0,1)

(K=500,1000,..), 100 signal points from [2,11]U[-2,-11].

• Adjust threshold of Bayesian residuals to match sensitivity of naive procedure.

• Compute False Discovery Rate (FDR) for both procedures.

FDR of naive and Bayesian procedures. The Bayesianmethod gets better with increase in number of time series.

Calculations based on 100 replications.The differences are statistically significant.

Application

• Goal: To find leading indicators of social disruption events

in China before it gets reported in the mainstream media.• Approach: Monitor the occurrence of a set of pre-defined

patterns on a collection of Chinese websites (mainly news sites, government sites and portals similar to yahoo located in eastern China).

Motivating Application (bio-surveillance).

English translation of some Chinese patterns being monitored

Notations and transformation.

./1)(

)/)1(1000)/(1000(5.

Tukey -Freeman

./)(Var ;)/(

model,Poisson aUnder

rate. Occurence

.downloaded pages ofnumber

.day on websiteon pattern of freq

rates. ofon distributi thesymmetrize

;dependence ncemean varia remove tion totransforma

ijt

ijtijtijtijt

ijtijtijtijtijtijtijt

ijt

ijt

ththijt

nijt

ZVar

nSnSijt

Z

nrnSrE

n

tjiS

Dotted solid lines: Days when reports appeared in mainstream media

Dotted gray lines: Days when our system found spikes related to the reports that appeared later.

Rough validation using actual media reports.

• July 24th : mystery illness kills 17 people in China, we noticed several spikes on July 17th and 18th alerting us on this.

• Sept 29th and Dec 7th : On Sept 29th , news reports of China carving out emergency plans to fight bird flu and prevent it from spreading to humans. On Dec 7th , a confirmed case of bird flu in humans reported.

• We reported several spikes on Sept 12th and 14th, Nov 2nd, 7th, 11th, and 16th mostly for the pattern influenza, flu, pneumonia, meningitis. On Nov 21st , four big spikes on bf3.syd.com.cn on influenza, flu, pneumonia, meningitis;

emergency, disaster, crisis; prevention and quarantine.

Questions?