F4 large scale automated forecasting using fractals
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F4: Large Scale Automated Forecasting Using Fractals. -Deepayan Chakrabarti -Christos Faloutsos. Outline. Introduction/Motivation Survey and Lag Plots Exact Problem Formulation Proposed Method Fractal Dimensions Background Our method Results Conclusions. ?. General Problem Definition.

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F4: Large Scale Automated Forecasting Using Fractals

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F4: Large Scale Automated Forecasting Using Fractals

-Deepayan Chakrabarti

-Christos Faloutsos

CIKM 2002


Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


?

General Problem Definition

Value

Time

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

CIKM 2002


Motivation

Traditional fields

  • Financial data analysis

  • Physiological data, elderly care

  • Weather, environmental studies

Sensor Networks(MEMS, “SmartDust”)

  • Long / “infinite” series

  • No human intervention  “black box”

CIKM 2002


Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


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

CIKM 2002


Q0: Interpolation Method

Q1: Lag = ?

Q2: K = ?

Interpolate these…

To get the final prediction

4-NN

New Point

Lag Plots

xt

xt-1

CIKM 2002


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

xt

Xt-1

Q0: Interpolation

CIKM 2002


Why Lag Plots?

  • Based on the “Takens’ Theorem” [Takens/1981]

  • which says that delay vectors can be used for predictive purposes

CIKM 2002


Extra

Inside Theory

Example: 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).

CIKM 2002


Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


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

CIKM 2002


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]

CIKM 2002


Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


X(t)

Intrinsic Dimensionality

≈ Degrees of Freedom

≈ Information about Xt given Xt-1

X(t-1)

Intuition

x(t)

time

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

CIKM 2002


x(t)

x(t-1)

x(t-2)

x(t)

x(t)

x(t-1)

x(t-1)

x(t-2)

x(t-2)

Intuition

x(t)

x(t-1)

CIKM 2002


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

CIKM 2002


Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


Fractal Dimensions

  • FD = intrinsic dimensionality

“Embedding” dimensionality = 3

Intrinsic dimensionality = 1

CIKM 2002


Fractal Dimensions

FD = intrinsic dimensionality [Belussi/1995]

log( # pairs)

  • Points to note:

  • FD can be a non-integer

  • There are fast methods to compute it

CIKM 2002

log(r)


Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


epsilon

f

L(opt)

Q1: Finding L(opt)

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

Fractal Dimension

Lag (L)

CIKM 2002


Q2: Finding k(opt)

  • To find k(opt)

  • Conjecture: k(opt) ~ O(f)

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

CIKM 2002


Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


Value

Datasets

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

Time

CIKM 2002


Value

Datasets

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

Time

  • LORENZ: Models convection currents in the air

CIKM 2002


Value

Datasets

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

Error NMSE = ∑(predicted-true)2/σ2

Time

  • LORENZ: Models convection currents in the air

  • LASER: fluctuations in a Laser over time (from the Santa Fe Time Series Competition, 1992)

CIKM 2002


Value

Timesteps

FD

Logistic Parabola

Lag

  • FD vs L plot flattens out

  • L(opt) = 1

CIKM 2002


Logistic Parabola

Our Prediction from here

Value

Timesteps

CIKM 2002


Value

Logistic Parabola

Comparison of prediction to correct values

Timesteps

CIKM 2002


Logistic Parabola

FD

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

NMSE

CIKM 2002

Lag


FD

Value

Timesteps

LORENZ

Lag

  • L(opt) = 5

CIKM 2002


LORENZ

Our Prediction from here

Value

Timesteps

CIKM 2002


LORENZ

Value

Comparison of prediction to correct values

Timesteps

CIKM 2002


LORENZ

FD

L(opt) = 5

Also NMSE is optimal at Lag = 5

NMSE

CIKM 2002

Lag


FD

Laser

Value

Lag

  • L(opt) = 7

Timesteps

CIKM 2002


Laser

Our Prediction starts here

Value

Timesteps

CIKM 2002


Laser

Value

Comparison of prediction to correct values

Timesteps

CIKM 2002


FD

Laser

L(opt) = 7

Corresponding NMSE is close to optimal

NMSE

CIKM 2002

Lag


Speed and Scalability

  • Preprocessing is linear in N

  • Proportional to time taken to calculate FD

CIKM 2002


Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


Conclusions

Our Method:

  • Automatically set parameters

    • L(opt) (answers Q1)

    • k(opt) (answers Q2)

  • In linear time on N

  • CIKM 2002


    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

    CIKM 2002


    Extra

    Snapshot

    http://snapdragon.cald.cs.cmu.edu/TSP

    CIKM 2002


    Extra

    Future Work

    • Feature Selection

    • Multi-sequence prediction

    CIKM 2002


    Extra

    Discussion – Some other problems

    How to forecast?

    Given:

    • x1, x2, …, xN

    • L(opt)

    • k(opt)

    How to find the k(opt) nearest neighbors quickly?

    CIKM 2002


    Extra

    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.

    CIKM 2002


    Extra

    Motivation (Examples)

    Traditional

    • EEGs : Patterns of electromagnetic impulses in the brain

    • Intensity variations of white dwarf stars

    • Highway usage over time

    Sensors

    • “Active Disks” for forecasting / prefetching / buffering

    • “Smart House”  sensors monitor situation in a house

    • Volcano monitoring

    CIKM 2002


    Extra

    • Store all the delay vectors {xt-1, …, xt-L(opt)} and corresponding prediction xt

    • Find the latest delay vector

    xt

    • Find nearest neighbors

    Interpolate

    • Interpolate

    Xt-1

    General Method

    L(opt) = ?

    K(opt) = ?

    CIKM 2002


    Extra

    Intuition

    Fractal dimension

    • The FD vs L plot does flatten out

    • L(opt) = 1

    CIKM 2002

    Lag


    Extra

    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

    CIKM 2002


    Extra

    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]

    CIKM 2002


    Extra

    Speed and Scalability

    • Preprocessing varies as L(opt)2

    CIKM 2002


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