Fractal Forecasting of Financial Markets with Fraclet Algorithm

Fraclet Predictor Overview

Presented byQuant Trade Technologies, Inc.

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 = ?

7

Q0: Interpolation

Using SVD (state of the art)

Xt-1

xt

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)

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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

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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

17

Q2: Finding k(opt)

To find k(opt)

• Conjecture: k(opt) ~ O(f)

We choose k(opt) =

2*f + 1

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

19

Prediction

Timesteps

Value

Our Prediction from here

20

Logistic Parabola

Timesteps

Value

Comparison of prediction to correct values

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

23

Lorenz Attractor Prediction

Value

Timesteps

Our Prediction from here

24

Prediction Test

Timesteps

Value

Comparison of prediction to correct values

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

27

Prediction

Timesteps

Value

Our Prediction starts here

28

Prediction Test

Timesteps

Value

Comparison of prediction to correct values

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

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Pioneers in the fractal exploration of financial markets

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