CMU SCS Sensor data mining and forecasting Christos Faloutsos CMU [email protected]

CMU SCS

Sensor data mining and forecasting

Christos Faloutsos

[email protected]

Telcordia 2003 C. Faloutsos 2

CMU SCS

Outline

• Problem definition - motivation

• Linear forecasting - AR and AWSOM

• Coevolving series - MUSCLES

• Fractal forecasting - F4

• Other projects– graph modeling, outliers etc


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Problem definition

• Given: one or more sequences x1 , x2 , … , xt , …

(y1, y2, … , yt, …

… )

• Find – forecasts; patterns– clusters; outliers


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Motivation - Applications• Financial, sales, economic series

• Medical

– ECGs +; blood pressure etc monitoring

– reactions to new drugs

– elderly care


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Motivation - Applications (cont’d)

• ‘Smart house’

– sensors monitor temperature, humidity, air quality

• video surveillance


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• civil/automobile infrastructure

– bridge vibrations [Oppenheim+02]

– road conditions / traffic monitoring


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Stream Data: automobile traffic

Automobile traffic

0200400600800

100012001400160018002000

time

# cars


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• Weather, environment/anti-pollution

– volcano monitoring

– air/water pollutant monitoring


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Stream Data: Sunspots

Sunspot Data

0

50

100

150

200

250

300

time

#sunspots per month


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• Computer systems

– ‘Active Disks’ (buffering, prefetching)

– web servers (ditto)

– network traffic monitoring

– ...


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Stream Data: Disk accesses

time

#bytes


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Settings & Applications

• One or more sensors, collecting time-series data


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Each sensor collects data (x1, x2, …, xt, …)


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Sensors ‘report’ to a central site


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Problem #1:Finding patternsin a single time sequence


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Problem #2:Finding patternsin many time sequences


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Problem #1:

Goal: given a signal (eg., #packets over time)

Find: patterns, periodicities, and/or compress

year

count lynx caught per year(packets per day;temperature per day)


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Problem#1’: ForecastGiven xt, xt-1, …, forecast xt+1

0102030405060708090

1 3 5 7 9 11

Time Tick

Nu

mb

er o

f p

ack

ets

sen

t

??


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Problem #2:• Given: A set of correlated time sequences

• Forecast ‘Sent(t)’

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1 3 5 7 9 11

Time Tick

Nu

mb

er o

f p

ack

ets

sent

lost

repeated


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Differences from DSP/Stat

• Semi-infinite streams – we need on-line, ‘any-time’ algorithms

• Can not afford human intervention– need automatic methods

• sensors have limited memory / processing / transmitting power– need for (lossy) compression


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Important observations

Patterns, rules, compression and forecasting are closely related:

• To do forecasting, we need– to find patterns/rules

• good rules help us compress• to find outliers, we need to have forecasts

– (outlier = too far away from our forecast)


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Pictorial outline of the talk

Linear Non-linear

1 time seq. AR,AWSOM

F4

Many t.s. MUSCLES


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Outline• Problem definition - motivation• Linear forecasting

– AR – AWSOM

• Coevolving series - MUSCLES• Fractal forecasting - F4• Other projects

– graph modeling, outliers etc

Linear Non-linear


F4

Many t.s. MUSCLES


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Mini intro to A.R.


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Forecasting

"Prediction is very difficult, especially about the future." - Nils Bohr

http://www.hfac.uh.edu/MediaFutures/thoughts.html


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Problem#1’: Forecast• Example: give xt-1, xt-2, …, forecast xt

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1 3 5 7 9 11

Time Tick

Nu

mb

er o

f p

ack

ets

sen

t

??


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Linear Regression: idea

40

45

50

55

60

65

70

75

80

85

15 25 35 45

Body weight

patient weight height

1 27 43

2 43 54

3 54 72

……

…

N 25 ??

• express what we don’t know (= ‘dependent variable’)• as a linear function of what we know (= ‘indep. variable(s)’)

Body height


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Linear Auto Regression:Time Packets

Sent (t-1)PacketsSent(t)

1 - 43

2 43 54

3 54 72

……

…

N 25 ??


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Problem#1’: Forecast• Solution: try to express

xt

as a linear function of the past: xt-2, xt-2, …,

(up to a window of w)

Formally:

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1 3 5 7 9 11Time Tick

??noisexaxax wtwtt 11


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Linear Auto Regression:

40

45

50

55

60

65

70

75

80

85

15 25 35 45

Number of packets sent (t-1)N

um

ber

of

pac

ket

s se

nt

(t)

Time PacketsSent (t-1)

PacketsSent(t)

1 - 43

2 43 54

3 54 72

……

…

N 25 ??

• lag w=1• Dependent variable = # of packets sent (S [t])• Independent variable = # of packets sent (S[t-1])

‘lag-plot’


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More details:

• Q1: Can it work with window w>1?

• A1: YES!

xt-2

xt

xt-1


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More details:


• A1: YES! (we’ll fit a hyper-plane, then!)

xt-2

xt

xt-1


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More details:


• A1: YES! (we’ll fit a hyper-plane, then!)

xt-2

xt-1

xt


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Even more details

• Q2: Can we estimate a incrementally?

• A2: Yes, with the brilliant, classic method of ‘Recursive Least Squares’ (RLS) (see, e.g., [Chen+94], or [Yi+00], for details)

• Q3: can we ‘down-weight’ older samples?

• A3: yes (RLS does that easily!)


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Mini intro to A.R.


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How to choose ‘w’?

• goal: capture arbitrary periodicities

• with NO human intervention

• on a semi-infinite stream

noisexaxax wtwtt 11


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– AR – AWSOM



Linear Non-linear


F4

Many t.s. MUSCLES


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Problem:

• in a train of spikes (128 ticks apart)

• any AR with window w < 128 will fail

What to do, then?

Impulse 128

-50

0

50

100

150

200

250


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Answer (intuition)

• Do a Wavelet transform (~ short window DFT)

• look for patterns in every frequency


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Intuition

• Why NOT use the short window Fourier transform (SWFT)?

• A: how short should be the window?

time

freq Impulse 128

-50

0

50

100

150

200

250

w’


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wavelets

Impulse 128

-50

0

50

100

150

200

250

t

f

• main idea: variable-length window!


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Advantages of Wavelets

• Better compression (better RMSE with same number of coefficients - used in JPEG-2000)

• fast to compute (usually: O(n)!)

• very good for ‘spikes’

• mammalian eye and ear: Gabor wavelets


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Wavelets - intuition:

t

f

• Q: baritone/silence/ soprano - DWT?

time

value


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Wavelets - intuition:

• Q: baritone/soprano - DWT?

t

f

time

value


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AWSOMxt

tt

W1,1

t

W1,2

t

W1,3

t

W1,4

t

W2,1

t

W2,2

t

W3,1

t

V4,1

time

frequ

ency=


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AWSOMxt

tt

W1,1

t

W1,2

t

W1,3

t

W1,4

t

W2,1

t

W2,2

t

W3,1

t

V4,1

time

frequ

ency


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AWSOM - idea

Wl,tWl,t-1Wl,t-2Wl,t l,1Wl,t-1 l,2Wl,t-2 …

Wl’,t’-1Wl’,t’-2Wl’,t’

Wl’,t’ l’,1Wl’,t’-1 l’,2Wl’,t’-2 …


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More details…

• Update of wavelet coefficients

• Update of linear models

• Feature selection– Not all correlations are significant– Throw away the insignificant ones (“noise”)

(incremental)

(incremental; RLS)

(single-pass)


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Results - Synthetic data• Triangle pulse

• Mix (sine + square)

• AR captures wrong trend (or none)

• Seasonal AR estimation fails

AWSOM AR Seasonal AR


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Results - Real data

• Automobile traffic– Daily periodicity– Bursty “noise” at smaller scales

• AR fails to capture any trend• Seasonal AR estimation fails


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Results - real data

• Sunspot intensity– Slightly time-varying “period”

• AR captures wrong trend• Seasonal ARIMA

– wrong downward trend, despite help by human!


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Complexity

• Model update

Space: OlgN + mk2 OlgNTime: Ok2 O1

• Where– N: number of points (so far)– k: number of regression coefficients; fixed– m: number of linear models; OlgN


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Conclusions

• AWSOM: Automatic, ‘hands-off’ traffic modeling (first of its kind!)


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– AR – AWSOM



Linear Non-linear


F4

Many t.s. MUSCLES


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Co-Evolving Time Sequences• Given: A set of correlated time sequences

• Forecast ‘Repeated(t)’

0102030405060708090

1 3 5 7 9 11

Time Tick

Nu

mb

er o

f p

ack

ets

sent

lost

repeated

??


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Solution:

Q: what should we do?


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Solution:

Least Squares, with

• Dep. Variable: Repeated(t)

• Indep. Variables: Sent(t-1) … Sent(t-w); Lost(t-1) …Lost(t-w); Repeated(t-1), ...

• (named: ‘MUSCLES’ [Yi+00])


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Examples - Experiments• Datasets

– Modem pool traffic (14 modems, 1500 time-ticks; #packets per time unit)

– AT&T WorldNet internet usage (several data streams; 980 time-ticks)

• Measures of success– Accuracy : Root Mean Square Error (RMSE)


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Accuracy - “Modem”

MUSCLES outperforms AR & “yesterday”

0

0.5

1

1.5

2

2.5

3

3.5

4

RMSE

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Modems

AR

yesterday

MUSCLES


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Accuracy - “Internet”

0

0.2

0.4

0.6

0.8

1

1.2

1.4

RMSE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Streams

AR

yesterday

MUSCLES

MUSCLES consistently outperforms AR & “yesterday”


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– AR – AWSOM



Linear Non-linear


F4

Many t.s. MUSCLES


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Detailed Outline

• Non-linear forecasting– Problem– Idea– How-to– Experiments– Conclusions


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Recall: Problem #1

Given a time series {xt}, predict its future course, that is, xt+1, xt+2, ...

Time

Value


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How to forecast?

• ARIMA - but: linearity assumption

• ANSWER: ‘Delayed Coordinate Embedding’ = Lag Plots [Sauer92]


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General Intuition (Lag Plot)

xt-1

xxtt

4-NNNew Point

Interpolate these…

To get the final prediction

Lag = 1,k = 4 NN


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Questions:

• Q1: How to choose lag L?• Q2: How to choose k (the # of NN)?• Q3: How to interpolate?• Q4: why should this work at all?


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Q1: Choosing lag L

• Manually (16, in award winning system by [Sauer94])

• Our proposal: choose L such that the ‘intrinsic dimension’ in the lag plot stabilizes [Chakrabarti+02]


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Fractal Dimensions

• FD = intrinsic dimensionality

Embedding dimensionality = 3

Intrinsic dimensionality = 1


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Fractal Dimensions

• FD = intrinsic dimensionality

log(r)

log( # pairs)


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Intuition

• Its lag plot for lag = 1

X(t-1)

X(t) The Logistic Parabola xt = axt-1(1-xt-1) + noise

time

x(t

)


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Intuition

x(t-1)

x(t)

x(t-2)

x(t)

x(t)

x(t-2)

x(t-2) x(t-1)

x(t-1)

x(t-1)

x(t)


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Intuition

• The FD vs L plot does flatten out

• L(opt) = 1

Lag

Fractal dimension


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Proposed Method

• Use Fractal Dimensions to find the optimal lag length L(opt)

Lag (L)

Fra

ctal

Dim

ensi

on

Choose this

epsilon


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Q2: Choosing number of neighbors k

• Manually (typically ~ 1-10)


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Q3: How to interpolate?

How do we interpolate between the k nearest neighbors?

A3.1: Average

A3.2: Weighted average (weights drop with distance - how?)


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Q3: How to interpolate?

A3.3: Using SVD - seems to perform best ([Sauer94] - first place in the Santa Fe forecasting competition)

Xt-1

xt


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Q4: Any theory behind it?

A4: YES!


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Theoretical foundation

• Based on the “Takens’ Theorem” [Takens81]

• which says that long enough delay vectors can do prediction, even if there are unobserved variables in the dynamical system (= diff. equations)


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Example: Lotka-Volterra equations

dH/dt = r H – a H*P dP/dt = b H*P – m P

H is count of prey (e.g., hare)P is count of predators (e.g., lynx)

Suppose only P(t) is observed (t=1, 2, …).

H

P

Skip


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• But the delay vector space is a faithful reconstruction of the internal system state

• So prediction in delay vector space is as good as prediction in state space

Skip

H

P

P(t-1)

P(t)


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Detailed Outline

• Non-linear forecasting– Problem– Idea– How-to– Experiments– Conclusions


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Datasets

Logistic Parabola: xt = axt-1(1-xt-1) + noise Models population of flies [R. May/1976]

time

x(t

)

Lag-plot


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Datasets

Logistic Parabola: xt = axt-1(1-xt-1) + noise Models population of flies [R. May/1976]

time

x(t

)

Lag-plot

ARIMA: fails


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Logistic Parabola

Timesteps

Value

Our Prediction from here


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Logistic Parabola

Timesteps

Value

Comparison of prediction to correct values


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Datasets

LORENZ: Models convection currents in the airdx / dt = a (y - x) dy / dt = x (b - z) - y dz / dt = xy - c z

Value


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LORENZ

Timesteps

Value



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Datasets

Time

Value

• LASER: fluctuations in a Laser over time (used in Santa Fe competition)


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Laser

Timesteps

Value



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Conclusions

• Lag plots for non-linear forecasting (Takens’ theorem)

• suitable for ‘chaotic’ signals


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Additional projects at CMU

• Graph/Network mining

• spatio-temporal mining - outliers


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Graph/network mining

• Internet; web; gnutella P2P networks

• Q: Any pattern?• Q: how to generate

‘realistic’ topologies?• Q: how to define/verify

realism?


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Patterns?

• avg degree is, say 3.3• pick a node at random

- what is the degree you expect it to have?

degree

count

avg: 3.3


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Patterns?



• A: 1!!

degree

count

avg: 3.3


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Patterns?



• A: 1!!

degree

count

avg: 3.3


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Patterns?

• A: Power laws!

log {(out) degree}

log(count)


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Other ‘laws’?

Count vs Outdegree Count vs Indegree Hop-plot

Eigenvalue vs Rank “Network value” Stress

Effective Diameter


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RMAT, to generate realistic graphs

Count vs Outdegree Count vs Indegree Hop-plot

Eigenvalue vs Rank “Network value” Stress

Effective Diameter


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Epidemic threshold?

• one a real graph, will a (computer / biological) virus die out? (given– beta: probability that an infected node will

infect its neighbor and– delta: probability that an infected node will

recover

NO MAYBE YES


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Epidemic threshold?

• one a real graph, will a (computer / biological) virus die out? (given– beta: probability that an infected node will

infect its neighbor and– delta: probability that an infected node will

recover

• A: depends on largest eigenvalue of adjacency matrix! [Wang+03]


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Additional projects

• Graph mining

• spatio-temporal mining - outliers


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Outliers - ‘LOCI’


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Outliers - ‘LOCI’

• finds outliers quickly,

• with no human intervention


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Conclusions

• AWSOM for automatic, linear forecasting

• MUSCLES for co-evolving sequences

• F4 for non-linear forecasting

• Graph/Network topology: power laws and generators; epidemic threshold

• LOCI for outlier detection


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Conclusions• Overarching theme: automatic discovery of

patterns (outliers/rules) in– time sequences (sensors/streams)– graphs (computer/social networks)– multimedia (video, motion capture data etc)

www.cs.cmu.edu/[email protected]


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Books

• William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery: Numerical Recipes in C, Cambridge University Press, 1992, 2nd Edition. (Great description, intuition and code for DFT, DWT)

• C. Faloutsos: Searching Multimedia Databases by Content, Kluwer Academic Press, 1996 (introduction to DFT, DWT)


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Books

• George E.P. Box and Gwilym M. Jenkins and Gregory C. Reinsel, Time Series Analysis: Forecasting and Control, Prentice Hall, 1994 (the classic book on ARIMA, 3rd ed.)

• Brockwell, P. J. and R. A. Davis (1987). Time Series: Theory and Methods. New York, Springer Verlag.


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Resources: software and urls

• MUSCLES: Prof. Byoung-Kee Yi:http://www.postech.ac.kr/~bkyi/or [email protected]

• AWSOM & LOCI: [email protected]• F4, RMAT: [email protected]


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Additional Reading

• [Chakrabarti+02] Deepay Chakrabarti and Christos Faloutsos F4: Large-Scale Automated Forecasting using Fractals CIKM 2002, Washington DC, Nov. 2002.

• [Chen+94] Chung-Min Chen, Nick Roussopoulos: Adaptive Selectivity Estimation Using Query Feedback. SIGMOD Conference 1994:161-172

• [Gilbert+01] Anna C. Gilbert, Yannis Kotidis and S. Muthukrishnan and Martin Strauss, Surfing Wavelets on Streams: One-Pass Summaries for Approximate Aggregate Queries, VLDB 2001


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Additional Reading

• Spiros Papadimitriou, Anthony Brockwell and Christos Faloutsos Adaptive, Hands-Off Stream Mining VLDB 2003, Berlin, Germany, Sept. 2003

• Spiros Papadimitriou, Hiroyuki Kitagawa, Phil Gibbons and Christos Faloutsos LOCI: Fast Outlier Detection Using the Local Correlation Integral ICDE 2003, Bangalore, India, March 5 - March 8, 2003.

• Sauer, T. (1994). Time series prediction using delay coordinate embedding. (in book by Weigend and Gershenfeld, below) Addison-Wesley.


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Additional Reading

• Takens, F. (1981). Detecting strange attractors in fluid turbulence. Dynamical Systems and Turbulence. Berlin: Springer-Verlag.

• Yang Wang, Deepayan Chakrabarti, Chenxi Wang and Christos Faloutsos Epidemic Spreading in Real Networks: An Eigenvalue Viewpoint 22nd Symposium on Reliable Distributed Computing (SRDS2003) Florence, Italy, Oct. 6-8, 2003


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Additional Reading

• Weigend, A. S. and N. A. Gerschenfeld (1994). Time Series Prediction: Forecasting the Future and Understanding the Past, Addison Wesley. (Excellent collection of papers on chaotic/non-linear forecasting, describing the algorithms behind the winners of the Santa Fe competition.)

• [Yi+00] Byoung-Kee Yi et al.: Online Data Mining for Co-Evolving Time Sequences, ICDE 2000. (Describes MUSCLES and Recursive Least Squares)

Documents

CMU SCS Sensor data mining and forecasting Christos Faloutsos CMU [email protected]