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

F4: Large Scale Automated Forecasting Using Fractals

-Deepayan Chakrabarti

-Christos Faloutsos

CIKM 2002


Outline

Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


General problem definition

?

General Problem Definition

Value

Time

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

CIKM 2002


Motivation

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


Outline1

Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


How to forecast

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


Lag plots

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


Q0 interpolation

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

xt

Xt-1

Q0: Interpolation

CIKM 2002


Why lag plots

Why Lag Plots?

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

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

CIKM 2002


Inside theory

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


Outline2

Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


Problem at hand

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

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


Outline3

Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


Intuition

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


Intuition1

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


Intuition2

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


Outline4

Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


Fractal dimensions

Fractal Dimensions

  • FD = intrinsic dimensionality

“Embedding” dimensionality = 3

Intrinsic dimensionality = 1

CIKM 2002


Fractal dimensions1

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)


Outline5

Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


Q1 finding l opt

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

Q2: Finding k(opt)

  • To find k(opt)

  • Conjecture: k(opt) ~ O(f)

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

CIKM 2002


Outline6

Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


Datasets

Value

Datasets

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

Time

CIKM 2002


Datasets1

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


Datasets2

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


Logistic parabola

Value

Timesteps

FD

Logistic Parabola

Lag

  • FD vs L plot flattens out

  • L(opt) = 1

CIKM 2002


Logistic parabola1

Logistic Parabola

Our Prediction from here

Value

Timesteps

CIKM 2002


Logistic parabola2

Value

Logistic Parabola

Comparison of prediction to correct values

Timesteps

CIKM 2002


Logistic parabola3

Logistic Parabola

FD

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

NMSE

CIKM 2002

Lag


Lorenz

FD

Value

Timesteps

LORENZ

Lag

  • L(opt) = 5

CIKM 2002


Lorenz1

LORENZ

Our Prediction from here

Value

Timesteps

CIKM 2002


Lorenz2

LORENZ

Value

Comparison of prediction to correct values

Timesteps

CIKM 2002


Lorenz3

LORENZ

FD

L(opt) = 5

Also NMSE is optimal at Lag = 5

NMSE

CIKM 2002

Lag


Laser

FD

Laser

Value

Lag

  • L(opt) = 7

Timesteps

CIKM 2002


Laser1

Laser

Our Prediction starts here

Value

Timesteps

CIKM 2002


Laser2

Laser

Value

Comparison of prediction to correct values

Timesteps

CIKM 2002


Laser3

FD

Laser

L(opt) = 7

Corresponding NMSE is close to optimal

NMSE

CIKM 2002

Lag


Speed and scalability

Speed and Scalability

  • Preprocessing is linear in N

  • Proportional to time taken to calculate FD

CIKM 2002


Outline7

Outline

  • Introduction/Motivation

  • Survey and Lag Plots

  • Exact Problem Formulation

  • Proposed Method

    • Fractal Dimensions Background

    • Our method

  • Results

  • Conclusions

CIKM 2002


Conclusions

Conclusions

Our Method:

  • Automatically set parameters

    • L(opt) (answers Q1)

    • k(opt) (answers Q2)

  • In linear time on N

  • CIKM 2002


    Conclusions1

    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


    Snapshot

    Extra

    Snapshot

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

    CIKM 2002


    Future work

    Extra

    Future Work

    • Feature Selection

    • Multi-sequence prediction

    CIKM 2002


    Discussion some other problems

    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


    Motivation1

    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


    Motivation examples

    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


    General method

    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


    Intuition3

    Extra

    Intuition

    Fractal dimension

    • The FD vs L plot does flatten out

    • L(opt) = 1

    CIKM 2002

    Lag


    Inside theory1

    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


    Fractal dimensions2

    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


    Speed and scalability1

    Extra

    Speed and Scalability

    • Preprocessing varies as L(opt)2

    CIKM 2002


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