deepayan chakrabarticikm 20021 f4: large scale automated forecasting using fractals -deepayan...
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![Page 1: Deepayan ChakrabartiCIKM 20021 F4: Large Scale Automated Forecasting Using Fractals -Deepayan Chakrabarti -Christos Faloutsos](https://reader035.vdocuments.net/reader035/viewer/2022062803/56649c7c5503460f94930e68/html5/thumbnails/1.jpg)
Deepayan Chakrabarti
CIKM 2002 1
F4: Large Scale Automated Forecasting Using Fractals
-Deepayan Chakrabarti-Christos Faloutsos
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Deepayan Chakrabarti
CIKM 2002 2
Outline Introduction/Motivation Survey and Lag Plots Exact Problem Formulation Proposed Method
Fractal Dimensions Background Our method
Results Conclusions
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Deepayan Chakrabarti
CIKM 2002 3
General Problem Definition
Given a time series {xt}, predict its future course, that is, xt+1, xt+2, ...
Time
Value?
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Deepayan Chakrabarti
CIKM 2002 4
Motivation
• Financial data analysis
• Physiological data, elderly care
• Weather, environmental studies
Traditional fields
Sensor Networks (MEMS, “SmartDust”)• Long / “infinite” series
• No human intervention “black box”
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Deepayan Chakrabarti
CIKM 2002 5
Outline Introduction/Motivation Survey and Lag Plots Exact Problem Formulation Proposed Method
Fractal Dimensions Background Our method
Results Conclusions
![Page 6: Deepayan ChakrabartiCIKM 20021 F4: Large Scale Automated Forecasting Using Fractals -Deepayan Chakrabarti -Christos Faloutsos](https://reader035.vdocuments.net/reader035/viewer/2022062803/56649c7c5503460f94930e68/html5/thumbnails/6.jpg)
Deepayan Chakrabarti
CIKM 2002 6
How to forecast? ARIMA but linearity assumption Neural Networks but large
number of parameters and long training times [Wan/1993, Mozer/1993]
Hidden Markov Models O(N2) in number of nodes N; also fixing N is a problem [Ge+/2000]
Lag Plots
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Deepayan Chakrabarti
CIKM 2002 7
Lag Plots
xt-1
xxtt
4-NNNew Point
Interpolate these…
To get the final prediction
Q0: Interpolation Method
Q1: Lag = ?
Q2: K = ?
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Deepayan Chakrabarti
CIKM 2002 8
Q0: Interpolation
Using SVD (state of the art) [Sauer/1993]
Xt-1
xt
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Deepayan Chakrabarti
CIKM 2002 9
Why Lag Plots?
Based on the “Takens’ Theorem” [Takens/1981]
which says that delay vectors can be used for predictive purposes
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Deepayan Chakrabarti
CIKM 2002 10
Inside TheoryExample: Lotka-Volterra equations
ΔH/Δt = rH – aH*P ΔP/Δt = bH*P – mP
H is density of preyP is density of predators
Suppose only H(t) is observed. Internal state is (H,P).
Extra
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Deepayan Chakrabarti
CIKM 2002 11
Outline Introduction/Motivation Survey and Lag Plots Exact Problem Formulation Proposed Method
Fractal Dimensions Background Our method
Results Conclusions
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Deepayan Chakrabarti
CIKM 2002 12
Problem at hand
Given {x1, x2, …, xN} Automatically set parameters
- L(opt) (from Q1) - k(opt) (from Q2)
in Linear time on N to minimise Normalized Mean
Squared Error (NMSE) of forecasting
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Deepayan Chakrabarti
CIKM 2002 13
Previous work/Alternatives
Manual Setting : BUT infeasible [Sauer/1992]
CrossValidation : BUT Slow; leave-one-out crossvalidation ~ O(N2logN) or more
“False Nearest Neighbors” : BUT Unstable [Abarbanel/1996]
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Deepayan Chakrabarti
CIKM 2002 14
Outline Introduction/Motivation Survey and Lag Plots Exact Problem Formulation Proposed Method
Fractal Dimensions Background Our method
Results Conclusions
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Deepayan Chakrabarti
CIKM 2002 15
Intuition
X(t-1)
X(t
)
The Logistic Parabola xt = axt-1(1-xt-1) + noise
time
x(t
) Intrinsic Dimensionality
≈ Degrees of Freedom
≈ Information about Xt given Xt-1
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CIKM 2002 16
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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Deepayan Chakrabarti
CIKM 2002 17
Intuition
To find L(opt): Go further back in time (ie., consider
Xt-2, Xt-3 and so on) Till there is no more information
gained about Xt
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Deepayan Chakrabarti
CIKM 2002 18
Outline Introduction/Motivation Survey and Lag Plots Exact Problem Formulation Proposed Method
Fractal Dimensions Background Our method
Results Conclusions
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Deepayan Chakrabarti
CIKM 2002 19
Fractal Dimensions FD = intrinsic dimensionality
“Embedding” dimensionality = 3
Intrinsic dimensionality = 1
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Deepayan Chakrabarti
CIKM 2002 20
Fractal Dimensions
FD = intrinsic dimensionality [Belussi/1995]
log(r)
log( # pairs)
Points to note:
• FD can be a non-integer
• There are fast methods to compute it
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Deepayan Chakrabarti
CIKM 2002 21
Outline Introduction/Motivation Survey and Lag Plots Exact Problem Formulation Proposed Method
Fractal Dimensions Background Our method
Results Conclusions
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Deepayan Chakrabarti
CIKM 2002 22
Q1: Finding L(opt) Use Fractal Dimensions
to find the optimal lag length L(opt)
Lag (L)
Fra
ctal
Dim
ensi
on
epsilon
L(opt)
f
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Deepayan Chakrabarti
CIKM 2002 23
Q2: Finding k(opt)
To find k(opt)
• Conjecture: k(opt) ~ O(f)
We choose k(opt) = 2*f + 1
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Deepayan Chakrabarti
CIKM 2002 24
Outline Introduction/Motivation Survey and Lag Plots Exact Problem Formulation Proposed Method
Fractal Dimensions Background Our method
Results Conclusions
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Deepayan Chakrabarti
CIKM 2002 25
Datasets Logistic Parabola:
xt = axt-1(1-xt-1) + noise Models population of flies [R. May/1976]
Time
Value
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Deepayan Chakrabarti
CIKM 2002 26
Datasets Logistic Parabola:
xt = axt-1(1-xt-1) + noise Models population of flies [R. May/1976]
LORENZ: Models convection currents in the air
Time
Value
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CIKM 2002 27
Datasets
Error NMSE = ∑(predicted-true)2/σ2
Logistic Parabola: xt = axt-1(1-xt-1) + noise Models population of flies [R. May/1976]
LORENZ: Models convection currents in the air
LASER: fluctuations in a Laser over time (from the Santa Fe Time Series Competition, 1992)
Time
Value
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Deepayan Chakrabarti
CIKM 2002 28
Logistic Parabola
• FD vs L plot flattens out
• L(opt) = 1
Timesteps
Value
Lag
FD
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Deepayan Chakrabarti
CIKM 2002 29
Logistic Parabola
Timesteps
Value
Our Prediction from here
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Deepayan Chakrabarti
CIKM 2002 30
Logistic Parabola
Timesteps
Value
Comparison of prediction to correct values
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Deepayan Chakrabarti
CIKM 2002 31
Logistic Parabola
Our L(opt) = 1, which exactly minimizes NMSE
Lag
NM
SE
FD
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Deepayan Chakrabarti
CIKM 2002 32
LORENZ
• L(opt) = 5
Timesteps
Value
Lag
FD
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Deepayan Chakrabarti
CIKM 2002 33
LORENZ
Value
Timesteps
Our Prediction from here
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Deepayan Chakrabarti
CIKM 2002 34
LORENZ
Timesteps
Value
Comparison of prediction to correct values
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Deepayan Chakrabarti
CIKM 2002 35
LORENZ
L(opt) = 5
Also NMSE is optimal at Lag = 5
Lag
NM
SE
FD
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Deepayan Chakrabarti
CIKM 2002 36
Laser
• L(opt) = 7
Timesteps
Value
Lag
FD
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Deepayan Chakrabarti
CIKM 2002 37
Laser
Timesteps
Value
Our Prediction starts here
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Deepayan Chakrabarti
CIKM 2002 38
Laser
Timesteps
Value
Comparison of prediction to correct values
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Deepayan Chakrabarti
CIKM 2002 39
Laser
L(opt) = 7
Corresponding NMSE is close to optimal
Lag
NM
SE
FD
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Deepayan Chakrabarti
CIKM 2002 40
Speed and Scalability Preprocessin
g is linear in N
Proportional to time taken to calculate FD
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Deepayan Chakrabarti
CIKM 2002 41
Outline Introduction/Motivation Survey and Lag Plots Exact Problem Formulation Proposed Method
Fractal Dimensions Background Our method
Results Conclusions
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Deepayan Chakrabarti
CIKM 2002 42
Conclusions
Our Method:
Automatically set parameters
L(opt) (answers Q1)
k(opt) (answers Q2)
In linear time on N
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Deepayan Chakrabarti
CIKM 2002 43
Conclusions Black-box non-linear time series
forecasting Fractal Dimensions give a fast,
automated method to set all parameters
So, given any time series, we can automatically build a prediction system
Useful in a sensor network setting
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Deepayan Chakrabarti
CIKM 2002 44
Snapshothttp://snapdragon.cald.cs.cmu.edu/TSPExtra
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Deepayan Chakrabarti
CIKM 2002 45
Future Work
Feature Selection Multi-sequence prediction
Extra
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Deepayan Chakrabarti
CIKM 2002 46
Discussion – Some other problems
How to forecast?
•x1, x2, …, xN
•L(opt)
•k(opt)How to find the k(opt) nearest neighbors quickly?
Given:
Extra
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Deepayan Chakrabarti
CIKM 2002 47
Motivation
Forecasting also allows us to
• Find outliers anything that doesn’t match our prediction!
• Find patterns if different circumstances lead to similar predictions, they may be related.
Extra
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Deepayan Chakrabarti
CIKM 2002 48
Motivation (Examples)
• EEGs : Patterns of electromagnetic impulses in the brain
• Intensity variations of white dwarf stars
• Highway usage over time
Traditional
Sensors• “Active Disks” for forecasting / prefetching / buffering
• “Smart House” sensors monitor situation in a house
• Volcano monitoring
Extra
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Deepayan Chakrabarti
CIKM 2002 49
General Method
• Store all the delay vectors {x{xt-1t-1, …, x, …, xt-L(opt)t-L(opt)} }
and corresponding prediction xand corresponding prediction x tt
Xt-1
xt• Find the latest delay vector
L(opt) = ?
• Find nearest neighbors
K(opt) = ?
Interpolate• Interpolate
Extra
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Deepayan Chakrabarti
CIKM 2002 50
Intuition
• The FD vs L plot does flatten out
• L(opt) = 1
Lag
Fractal dimension
Extra
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Deepayan Chakrabarti
CIKM 2002 51
Inside Theory
Internal state may be unobserved 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
Extra
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Deepayan Chakrabarti
CIKM 2002 52
Fractal Dimensions
Many real-world datasets have fractional intrinsic dimension
There exist fast (O(N)) methods to calculate the fractal dimension of a cloud of points [Belussi/1995]
Extra
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Deepayan Chakrabarti
CIKM 2002 53
Speed and Scalability Preprocessin
g varies as L(opt)2
Extra