deepayan chakrabarticikm 20021 f4: large scale automated forecasting using fractals -deepayan...

Deepayan Chakrabarti

CIKM 2002 1

F4: Large Scale Automated Forecasting Using Fractals

-Deepayan Chakrabarti-Christos Faloutsos


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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General Problem Definition

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

Time

Value?


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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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Results Conclusions


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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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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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Q0: Interpolation

Using SVD (state of the art) [Sauer/1993]

Xt-1

xt


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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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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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Results Conclusions


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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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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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Results Conclusions


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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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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

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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Results Conclusions


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Fractal Dimensions FD = intrinsic dimensionality

“Embedding” dimensionality = 3

Intrinsic dimensionality = 1


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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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Results Conclusions


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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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Q2: Finding k(opt)

To find k(opt)

• Conjecture: k(opt) ~ O(f)

We choose k(opt) = 2*f + 1


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Results Conclusions


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Datasets Logistic Parabola:

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

Time

Value


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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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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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Logistic Parabola

• FD vs L plot flattens out

• L(opt) = 1

Timesteps

Value

Lag

FD


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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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Logistic Parabola

Our L(opt) = 1, which exactly minimizes NMSE

Lag

NM

SE

FD


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LORENZ

• L(opt) = 5

Timesteps

Value

Lag

FD


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LORENZ

Value

Timesteps

Our Prediction from here


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LORENZ

Timesteps

Value



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LORENZ

L(opt) = 5

Also NMSE is optimal at Lag = 5

Lag

NM

SE

FD


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Laser

• L(opt) = 7

Timesteps

Value

Lag

FD


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Laser

Timesteps

Value

Our Prediction starts here


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Laser

Timesteps

Value



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Laser

L(opt) = 7

Corresponding NMSE is close to optimal

Lag

NM

SE

FD


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Speed and Scalability Preprocessin

g is linear in N

Proportional to time taken to calculate FD


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Results Conclusions


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Conclusions

Our Method:

Automatically set parameters

L(opt) (answers Q1)

k(opt) (answers Q2)

In linear time on N


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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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Snapshothttp://snapdragon.cald.cs.cmu.edu/TSPExtra


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Future Work

Feature Selection Multi-sequence prediction

Extra


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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


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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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


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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Intuition

• The FD vs L plot does flatten out

• L(opt) = 1

Lag

Fractal dimension

Extra


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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


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


CIKM 2002 53

Speed and Scalability Preprocessin

g varies as L(opt)2

Extra

deepayan chakrabarticikm 20021 f4: large scale automated forecasting using fractals -deepayan...

Documents

lag plotsxt

lag plotscikm

problem ge

loptuse fractal dimensions

datasetslogistic p

logistic parabola xt

density of prey p

fractal dimensionsfd