Fractal Forecasting of Financial Markets with Fraclet Algorithm

1. Fraclet Predictor Overview
Presented by
Quant Trade Technologies, Inc.

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 = ?

7. 7
Q0: Interpolation
Using SVD (state of the art)
Xt-1
xt

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

14. 14
Fractal Dimensions

FD = intrinsic dimensionality
“Embedding” dimensionality = 3
Intrinsic dimensionality = 1

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

17. 17
Q2: Finding k(opt)

To find k(opt)
•
Conjecture: k(opt) ~ O(f)
We choose k(opt) = 2*f + 1

18. 18
Logistic Parabola
00.511.522.5312345 Fractal Dimension LagFD vs LOur Choice
•
FD vs L plot flattens out
•
L(opt) = 1
Timesteps
Value
Lag
FD

19. 19
Prediction
Timesteps
Value
Our Prediction from here

20. 20
Logistic Parabola
Timesteps
Value
Comparison of prediction to correct values

21. 21
Logistic Parabola
00.511.522.5312345 Fractal Dimension LagFD vs LOur Choice00.050.10.150.20.250.30.35123456 NMSE LagNMSE vs LagOur Choice
Our L(opt) = 1, which exactly minimizes NMSE
Lag
NMSE
FD

22. 22
Lorenz Attractor
00.511.522.5312345678910 Fractal Dimension LagFD vs LOur Choice
•
L(opt) = 5
Timesteps
Value
Lag
FD

23. 23
Lorenz Attractor Prediction
Value
Timesteps
Our Prediction from here

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

26. 26
Laser
00.511.522.533.51234567891011121314151617 Fractal Dimension LagFD vs LOur Choice
•
L(opt) = 7
Timesteps
Value
Lag
FD

27. 27
Prediction
Timesteps
Value
Our Prediction starts here

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
Trading futures and options involves the risk of loss. You should consider carefully whether futures or options are appropriate to your financial situation. You must review the customer account agreement and risk disclosure prior to establishing an account. Only risk capital should be used when trading futures or options. Investors could lose more than their initial investment.
Past results are not necessarily indicative of futures results. The risk of loss in trading futures or options can be substantial, carefully consider the inherent risks of such an investment in light of your financial condition. Information contained, viewed, sent or attached is considered a solicitation for business.
Quant Trade, LLC has been a Commodity Futures Trading Commission (CFTC) registered Commodity Trading Advisor (CTA) since September 4, 2007 and a member of the National Futures Association (NFA).
Copyright @ 2012 Quant Trade, LLC. All rights reserved. No part of the materials including graphics or logos, available in this Web site may be copied, reproduced, translated or reduced to any electronic medium or machine- readable form, in whole or in part without written permission.
2 N Riverside Plaza
Suite 2325
Chicago, Illinois 60606
Quant Trade LLC
(872) 225-2110