cmu scs sensor data mining and forecasting christos faloutsos cmu [email protected]
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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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Motivation - Applications (cont’d)
• 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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Motivation - Applications (cont’d)
• 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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Motivation - Applications (cont’d)
• Computer systems
– ‘Active Disks’ (buffering, prefetching)
– web servers (ditto)
– network traffic monitoring
– ...
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Settings & Applications
• One or more sensors, collecting time-series data
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Settings & Applications
Each sensor collects data (x1, x2, …, xt, …)
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Settings & Applications
Problem #1:Finding patternsin a single time sequence
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Settings & Applications
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
1 time seq. AR,AWSOM
F4
Many t.s. MUSCLES
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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
0102030405060708090
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:
0102030405060708090
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:
• Q1: Can it work with window w>1?
• A1: YES! (we’ll fit a hyper-plane, then!)
xt-2
xt
xt-1
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More details:
• Q1: Can it work with window w>1?
• 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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How to choose ‘w’?
• goal: capture arbitrary periodicities
• with NO human intervention
• on a semi-infinite stream
noisexaxax wtwtt 11
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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
1 time seq. AR,AWSOM
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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Outline• Problem definition - motivation• Linear forecasting
– AR – AWSOM
• Coevolving series - MUSCLES• Fractal forecasting - F4• Other projects
– graph modeling, outliers etc
Linear Non-linear
1 time seq. AR,AWSOM
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:
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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Outline• Problem definition - motivation• Linear forecasting
– AR – AWSOM
• Coevolving series - MUSCLES• Fractal forecasting - F4• Other projects
– graph modeling, outliers etc
Linear Non-linear
1 time seq. AR,AWSOM
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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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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Theoretical foundation
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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Theoretical foundation
• 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
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
Comparison of prediction to correct values
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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
Comparison of prediction to correct values
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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?
• avg degree is, say 3.3• pick a node at random
- what is the degree you expect it to have?
• A: 1!!
degree
count
avg: 3.3
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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?
• A: 1!!
degree
count
avg: 3.3
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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’
• 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)