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2
Contents
Introduction/Motivation
Survey and Lag Plots
Exact Problem Formulation
Proposed Method› Fractal Dimensions Background› Our method
Results
Conclusions
3
General Problem Definition
Given a time series {xt
}, predict its future course, that is, xt+1
, xt+2
, ...
Time
Value?
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
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
Lag Plots
xt-1
xxtt
4-NNNew Point
Interpolate these…
To get the final prediction
Q0: Interpolation Method
Q1: Lag = ?
Q2: K = ?
8
Why Lag Plots?› Based on the “Takens’ Theorem”
[Takens/1981]› which says that delay vectors can be
used for predictive purposes
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
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
Transform Data
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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
CIKM 2002Your logo here 12
Cube the Data
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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
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
14
Fractal Dimensions
FD = intrinsic dimensionality
“Embedding”
dimensionality = 3
Intrinsic dimensionality = 1
15
Fractal Dimensions
FD = intrinsic dimensionality [Belussi/1995]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Y a
xis
X axis
Sierpinsky
7
8
9
10
11
12
13
14
15
16
-7 -6 -5 -4 -3 -2 -1 0 1 2
log
(# p
airs
with
in 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
Q1: Finding L(opt)
Use Fractal Dimensions to find the optimal lag length L(opt)
Lag (L)
Frac
tal D
imen
sion
epsilon
L(opt)
f
18
Logistic Parabola
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5
Fra
ctal D
imensi
on
Lag
FD vs L
Our Choice
• FD vs
L plot flattens out
• L(opt) = 1
Timesteps
ValueLag
FD
21
Logistic Parabola
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5
Fra
ctal
Dim
ensi
on
Lag
FD vs L
Our Choice
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 2 3 4 5 6
NM
SE
Lag
NMSE vs Lag
Our Choice
Our L(opt) = 1, which exactly minimizes NMSE
Lag
NM
SE
FD
22
Lorenz Attractor
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5 6 7 8 9 10
Fra
ctal
Dim
ensi
on
Lag
FD vs L
Our Choice
• L(opt) = 5
Timesteps
Value
Lag
FD
25
Optimal Prediction
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5 6 7 8 9 10
Fra
ctal
Dim
ensi
on
Lag
FD vs L
Our Choice
L(opt) = 5
Also NMSE is optimal at Lag = 5
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12
NM
SE
Lag
NMSE vs Lag
Our Choice
Lag
NM
SE
FD
26
Laser
0
0.5
1
1.5
2
2.5
3
3.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Fra
ctal
Dim
ensi
on
Lag
FD vs L
Our Choice
• L(opt) = 7
Timesteps
Value
Lag
FD
29
Optimal Prediction
0
0.5
1
1.5
2
2.5
3
3.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Frac
tal D
imen
sion
Lag
FD vs L
Our Choice
0
0.5
1
1.5
2
2.5
3
3.5
1 2 3 4 5 6 7 8 9 10 11 12 13
NM
SE
Lag
NMSE vs L
Our Choice
L(opt) = 7
Corresponding NMSE is close to optimal
Lag
NM
SE
FD
30
Speed and Scalability
Preprocessing is linear in N
Proportional to time taken to calculate FD
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
2000 4000 6000 8000 100001200014000160001800020000
Pre
pro
cess
ing
Tim
e
Number of points (N)
Time vs N
31
The Fraclet Way
Our Method:
Automatically set parameters
L(opt) (answers Q1)
k(opt) (answers Q2)
In linear time on N
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
Pioneers in the fractal exploration of financial markets
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