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# Fractal Forecasting of Financial Markets with Fraclet Algorithm

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Fractal Forecasting of Financial Markets with Fraclet Algorithm

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### Fractal Forecasting of Financial Markets with Fraclet Algorithm

1. 1. Fraclet Predictor Overview Presented by Quant Trade Technologies, Inc.
2. 2. 2 Contents  Introduction/Motivation  Survey and Lag Plots  Exact Problem Formulation  Proposed Method › Fractal Dimensions Background › Our method  Results  Conclusions
3. 3. 3 General Problem Definition Given a time series {xt}, predict its future course, that is, xt+1, xt+2, ... Time Value ?
4. 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. 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. 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. 7 Q0: Interpolation Using SVD (state of the art) Xt-1 xt
8. 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. 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. 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. 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. 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. 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. 14 Fractal Dimensions  FD = intrinsic dimensionality “Embedding” dimensionality = 3 Intrinsic dimensionality = 1
15. 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. 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. 17 Q2: Finding k(opt)  To find k(opt) • Conjecture: k(opt) ~ O(f) We choose k(opt) = 2*f + 1
18. 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. 19 Prediction Timesteps Value Our Prediction from here
20. 20. 20 Logistic Parabola Timesteps Value Comparison of prediction to correct values
21. 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. 22 Lorenz Attractor 00.511.522.5312345678910 Fractal Dimension LagFD vs LOur Choice • L(opt) = 5 Timesteps Value Lag FD
23. 23. 23 Lorenz Attractor Prediction Value Timesteps Our Prediction from here
24. 24. 24 Prediction Test Timesteps Value Comparison of prediction to correct values
25. 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. 26 Laser 00.511.522.533.51234567891011121314151617 Fractal Dimension LagFD vs LOur Choice • L(opt) = 7 Timesteps Value Lag FD
27. 27. 27 Prediction Timesteps Value Our Prediction starts here
28. 28. 28 Prediction Test Timesteps Value Comparison of prediction to correct values
29. 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. 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. 31 The Fraclet Way Our Method:  Automatically set parameters  L(opt) (answers Q1)  k(opt) (answers Q2)  In linear time on N
32. 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