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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

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

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

Time

CIKM 2002

datasets1
ValueDatasets
  • 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
ValueDatasets
  • 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
ValueLogistic 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
FDLaser

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
FDLaser

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
ExtraSnapshot

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

CIKM 2002

future work
ExtraFuture Work
  • Feature Selection
  • Multi-sequence prediction

CIKM 2002

discussion some other problems
ExtraDiscussion – 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
ExtraMotivation
  • 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
ExtraMotivation (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
ExtraIntuition

Fractal dimension

  • The FD vs L plot does flatten out
  • L(opt) = 1

CIKM 2002

Lag

inside theory1
ExtraInside 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
ExtraFractal 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
ExtraSpeed and Scalability
  • Preprocessing varies as L(opt)2

CIKM 2002

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