CMU SCS
Sensor data mining and
forecasting
Christos Faloutsos
CMU
christos@cs.cmu.edu
CMU SCS
Outline
•
•
•
•
•
Problem definition - motivation
Linear forecasting - AR and AWSOM
Coevolving series - MUSCLES
Fractal forecasting - F4
Other projects
– graph modeling, outliers etc
Telcordia 2003
C. Faloutsos
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CMU SCS
Problem definition
• Given: one or more sequences
x1 , x2 , … , xt , …
(y1, y2, … , yt, …
…)
• Find
– forecasts; patterns
– clusters; outliers
Telcordia 2003
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Motivation - Applications
• Financial, sales, economic series
• Medical
– ECGs +; blood pressure etc monitoring
– reactions to new drugs
– elderly care
Telcordia 2003
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Motivation - Applications
(cont’d)
• ‘Smart house’
– sensors monitor temperature, humidity,
air quality
• video surveillance
Telcordia 2003
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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
# cars
Automobile traffic
2000
1800
1600
1400
1200
1000
800
600
400
200
0
time
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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
#sunspots per month
Sunspot Data
300
250
200
150
100
50
0
time
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Motivation - Applications
(cont’d)
• Computer systems
– ‘Active Disks’ (buffering, prefetching)
– web servers (ditto)
– network traffic monitoring
– ...
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Stream Data: Disk accesses
#bytes
time
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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
Sensors ‘report’ to a central site
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Settings & Applications
Problem #1:
Finding patterns
in a single time sequence
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Settings & Applications
Problem #2:
Finding patterns
in many time
sequences
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Problem #1:
Goal: given a signal (eg., #packets over time)
Find: patterns, periodicities, and/or compress
count
lynx caught per year
(packets per day;
temperature per day)
year
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Problem#1’: Forecast
Number of packets sent
Given xt, xt-1, …, forecast xt+1
90
80
70
60
50
40
30
20
10
0
??
1
3
5
7
9
11
Time Tick
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Problem #2:
Number of packets
• Given: A set of correlated time sequences
• Forecast ‘Sent(t)’
90
80
70
60
50
40
30
20
10
0
sent
lost
repeated
1
3
5
7
9
11
Time Tick
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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
Non-linear
Linear
F4
1 time seq. AR,
AWSOM
Many t.s. MUSCLES
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Outline
• Problem definition - motivation
Linear
• Linear forecasting
– AR
– AWSOM
1 time seq. AR,
AWSOM
Many t.s. MUSCLES
Non-linear
F4
• Coevolving series - MUSCLES
• Fractal forecasting - F4
• Other projects
– graph modeling, outliers etc
Telcordia 2003
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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/t
houghts.html
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Problem#1’: Forecast
Number of packets sent
• Example: give xt-1, xt-2, …, forecast xt
90
80
70
60
50
40
30
20
10
0
??
1
3
5
7
9
11
Time Tick
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Linear Regression: idea
patient
weight
height
Body height
85
80
75
1
27
43
70
2
43
54
65
3
54
72
60
…
N
…
25
55
…
50
45
??
40
15
25
35
45
Body weight
• express what we don’t know (= ‘dependent variable’)
• as a linear function of what we know (= ‘indep. variable(s)’)
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Linear Auto Regression:
Time
Packets
Sent (t-1)
Packets
Sent(t)
1
-
43
2
43
54
3
54
72
…
N
…
25
Telcordia 2003
…
??
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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:
xt  a1 xt 1    aw xt  w  noise
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80
70
60
50
40
30
20
10
0
1
??
3
5
7
9
Time Tick
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Time
Packets
Sent (t-1)
Packets
Sent(t)
1
-
43
2
43
54
3
54
72
…
N
…
25
…
??
Number of packets sent (t)
Linear Auto Regression:
85
‘lag-plot’
80
75
70
65
60
55
50
45
40
15
25
35
45
Number of packets sent (t-1)
• lag w=1
• Dependent variable = # of packets sent (S [t])
• Independent variable = # of packets sent (S[t-1])
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More details:
• Q1: Can it work with window w>1?
• A1: YES!
xt
xt-1
xt-2
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More details:
• Q1: Can it work with window w>1?
• A1: YES! (we’ll fit a hyper-plane, then!)
xt
xt-1
xt-2
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More details:
• Q1: Can it work with window w>1?
• A1: YES! (we’ll fit a hyper-plane, then!)
xt
xt-1
xt-2
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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
xt  a1 xt 1    aw xt  w  noise
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Outline
• Problem definition - motivation
• Linear forecasting
Linear
Non-linear
1 time seq. AR,
F4
AWSOM
Many t.s. MUSCLES
– AR
– AWSOM
• Coevolving series - MUSCLES
• Fractal forecasting - F4
• Other projects
– graph modeling, outliers etc
Telcordia 2003
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Problem:
• in a train of spikes (128 ticks apart)
• any AR with window w < 128 will fail
Impulse 128
250
What to do, then?
200
150
100
50
0
-50
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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?
freq
Impulse 128
250
200
150
100
50
0
-50
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w’
time
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wavelets
• main idea: variable-length window!
Impulse 128
250
200
150
f
100
50
t
0
-50
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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:
• Q: baritone/silence/
soprano - DWT?
f
t
value
time
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Wavelets - intuition:
• Q: baritone/soprano - DWT?
f
t
value
time
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AWSOM
xt
W1,1
W1,2
W1,3
W1,4
t
t
t
t
W2,2
=
t
t
frequency
W2,1
t
W3,1
t
V4,1
t
time
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AWSOM
xt
W1,1
W1,2
W1,3
W1,4
t
t
t
t
W2,2
t
t
frequency
W2,1
t
W3,1
t
V4,1
t
time
Telcordia 2003
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AWSOM - idea
Wl,t-2 Wl,t-1 Wl,t
Wl’,t’-2
Telcordia 2003
Wl’,t’-1
Wl’,t’
Wl,t 
Wl’,t’ 
C. Faloutsos
l,1Wl,t-1  l,2Wl,t-2  …
l’,1Wl’,t’-1  l’,2Wl’,t’-2  …
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More details…
• Update of wavelet coefficients (incremental)
• Update of linear models (incremental; RLS)
• Feature selection
(single-pass)
– Not all correlations are significant
– Throw away the insignificant ones (“noise”)
Telcordia 2003
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Results - Synthetic data
AWSOM
Telcordia 2003
AR
Seasonal AR
C. Faloutsos
• Triangle pulse
• Mix (sine +
square)
• AR captures
wrong trend (or
none)
• Seasonal AR
estimation fails
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Results - Real data
• Automobile traffic
– Daily periodicity
– Bursty “noise” at smaller scales
• AR fails to capture any trend
•Telcordia
Seasonal
AR
estimation
fails
2003
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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: OlgN + mk2  OlgN
Time: Ok2  O1
• Where
– N: number of points (so far)
– k: number of regression coefficients; fixed
– m:number of linear models; OlgN
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Conclusions
• AWSOM: Automatic, ‘hands-off’ traffic
modeling (first of its kind!)
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Outline
• Problem definition - motivation
Linear
• Linear forecasting
– AR
– AWSOM
1 time seq. AR,
AWSOM
Many t.s. MUSCLES
Non-linear
F4
• Coevolving series - MUSCLES
• Fractal forecasting - F4
• Other projects
– graph modeling, outliers etc
Telcordia 2003
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Co-Evolving Time Sequences
Number of packets
• Given: A set of correlated time sequences
• Forecast ‘Repeated(t)’
90
80
70
60
50
40
30
20
10
0
sent
lost
??
1
3
5
7
9
repeated
11
Time Tick
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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 timeticks; #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”
4
3.5
3
2.5
RMSE
2
AR
yesterday
MUSCLES
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14
Modems
MUSCLES outperforms AR & “yesterday”
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Accuracy - “Internet”
1.4
1.2
1
0.8
AR
RMSE
0.6
yesterday
0.4
MUSCLES
0.2
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Streams
MUSCLES consistently outperforms AR & “yesterday”
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Outline
• Problem definition - motivation
Linear
• Linear forecasting
– AR
– AWSOM
1 time seq. AR,
AWSOM
Many t.s. MUSCLES
Non-linear
F4
• Coevolving series - MUSCLES
• Fractal forecasting - F4
• Other projects
– graph modeling, outliers etc
Telcordia 2003
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Detailed Outline
• Non-linear forecasting
–
–
–
–
–
Problem
Idea
How-to
Experiments
Conclusions
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Recall: Problem #1
Value
Time
Given a time series {xt}, predict its future
course, that is, xt+1, xt+2, ...
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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)
Lag = 1,
k = 4 NN
xt
Interpolate
these…
To get the final
prediction
xt-1
4-NN
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New Point
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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( # pairs)
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log(r)
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x(t)
CMU SCS
Intuition
time
The Logistic
Parabola
xt = axt-1(1-xt-1) +
noise
X(t)
• Its lag plot for lag = 1
Telcordia 2003
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X(t-1)
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Intuition
x(t)
x(t)
x(t-1)
x(t)
x(t)
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Intuition
Fractal dimension
• The FD vs L plot
does flatten out
• L(opt) = 1
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Lag
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Proposed Method
Use Fractal Dimensions to find
the optimal lag length L(opt)
Fractal Dimension
•
epsilon
Choose this
Lag (L)
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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
Xt-1
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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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Skip
Theoretical foundation
Example: Lotka-Volterra equations
dH/dt = r H – a H*P
dP/dt = b H*P – m P
P
H is count of prey (e.g., hare)
P is count of predators (e.g., lynx)
H
Suppose only P(t) is observed (t=1, 2, …).
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Skip
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
P
P(t)
Telcordia 2003
H
C. Faloutsos
P(t-1)
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Detailed Outline
• Non-linear forecasting
–
–
–
–
–
Problem
Idea
How-to
Experiments
Conclusions
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x(t)
CMU SCS
Datasets
Logistic Parabola:
xt = axt-1(1-xt-1) + noise
Models population of flies [R. May/1976]
time
Lag-plot
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x(t)
CMU SCS
Datasets
Logistic Parabola:
xt = axt-1(1-xt-1) + noise
Models population of flies [R. May/1976]
time
Lag-plot
ARIMA: fails
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Logistic Parabola
Our Prediction from
here
Value
Timesteps
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Value
Logistic Parabola
Comparison of prediction
to correct values
Timesteps
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Value
Datasets
LORENZ: Models convection
currents in the air
dx / dt = a (y - x)
dy / dt = x (b - z) - y
dz / dt = xy - c z
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Value
LORENZ
Comparison of prediction
to correct values
Timesteps
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Value
Datasets
• LASER: fluctuations in
a Laser over time (used
in Santa Fe
competition)
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Time
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Value
Laser
Comparison of prediction
to correct values
Timesteps
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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?
count
avg: 3.3
Telcordia 2003
degree
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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!!
count
avg: 3.3
Telcordia 2003
degree
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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!!
count
avg: 3.3
Telcordia 2003
degree
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Patterns?
log(count)
• A: Power laws!
log {(out) degree}
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Other ‘laws’?
Effective Diameter
Count vs Outdegree
Eigenvalue vs Rank
Telcordia 2003
Count vs Indegree
“Network value”
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Hop-plot
Stress
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RMAT, to generate realistic graphs
Effective Diameter
Count vs Outdegree
Eigenvalue vs Rank
Telcordia 2003
Count vs Indegree
“Network value”
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Hop-plot
Stress
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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
Telcordia 2003
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/~christos
christos@cs.cmu.edu
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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 christos@cs.cmu.edu
• AWSOM & LOCI: spapadim@cs.cmu.edu
• F4, RMAT: deepay@cs.cmu.edu
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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 CoEvolving Time Sequences, ICDE 2000. (Describes
MUSCLES and Recursive Least Squares)
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