2. 2
Contents
Introduction/Motivation
Survey and Lag Plots
Exact Problem Formulation
Proposed Method
›
Fractal Dimensions Background
›
Our method
Results
Conclusions
3. 3
General Problem Definition
Given a time series {xt}, predict its future course, that is, xt+1, xt+2, ...
Time
Value
?
4. 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”
5. 5
Traditional Forecasting Methods
ARIMA but linearity assumption
Neural Networks but large number of parameters and long training times
Hidden Markov Models O(N2) in number of nodes N; also fixing N is a problem
Lag Plots
6. 6
Lag Plots
xt-1
xt
4-NN
New Point
Interpolate these…
To get the final prediction
Q0: Interpolation Method
Q1: Lag = ?
Q2: K = ?
8. 8
Why Lag Plots?
›
Based on the “Takens’ Theorem” [Takens/1981]
›
which says that delay vectors can be used for predictive purposes
9. 9
Inside Theory
Example: Lotka-Volterra equations
ΔH/Δt = rH – aH*P ΔP/Δt = bH*P – mP
H is density of prey P is density of predators
Suppose only H(t) is observed. Internal state is (H,P).
10. 10
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
11. 11
Transform Data
0.10.20.30.40.50.60.70.80.910.10.20.30.40.50.60.70.80.91 x(t) x(t-1) Logistic Parabola
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
12. CIKM 2002Your logo here 12
Cube the Data
0.10.20.30.40.50.60.70.80.910.10.20.30.40.50.60.70.80.91 x(t) x(t-1) Logistic Parabola
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)
13. 13
How Much Data is Enough?
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
15. 15
Fractal Dimensions
FD = intrinsic dimensionality [Belussi/1995]
00.10.20.30.40.50.60.70.80.9100.20.40.60.811.21.41.61.82 Y axis X axisSierpinsky78910111213141516-7-6-5-4-3-2-1012log(# pairs within r) log(r) FD plot= 1.56
log(r)
log( # pairs)
Points to note:
•
FD can be a non-integer
• There are fast methods to compute it
16. 16
Q1: Finding L(opt)
Use Fractal Dimensions to find the optimal lag length L(opt)
Lag (L)
Fractal Dimension
epsilon
L(opt)
f
24. 24
Prediction Test
Timesteps
Value
Comparison of prediction to correct values
25. 25
Optimal Prediction
00.511.522.5312345678910 Fractal Dimension LagFD vs LOur Choice
L(opt) = 5
Also NMSE is optimal at Lag = 5
00.20.40.60.81024681012 NMSE LagNMSE vs LagOur Choice
Lag
NMSE
FD
28. 28
Prediction Test
Timesteps
Value
Comparison of prediction to correct values
29. 29
Optimal Prediction
00.511.522.533.51234567891011121314151617 Fractal Dimension LagFD vs LOur Choice00.511.522.533.512345678910111213 NMSE LagNMSE vs LOur Choice
L(opt) = 7
Corresponding NMSE is close to optimal
Lag
NMSE
FD
30. 30
Speed and Scalability
Preprocessing is linear in N
Proportional to time taken to calculate FD
5001000150020002500300035004000450050002000400060008000100001200014000160001800020000 Preprocessing Time Number of points (N) Time vs N
31. 31
The Fraclet Way
Our Method:
Automatically set parameters
L(opt) (answers Q1)
k(opt) (answers Q2)
In linear time on N
32. 32
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
33. 33
Pioneers in the fractal exploration of financial markets
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