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F4: Large Scale Automated Forecasting Using Fractals

F4: Large Scale Automated Forecasting Using Fractals. -Deepayan Chakrabarti -Christos Faloutsos. Outline. Introduction/Motivation Survey and Lag Plots Exact Problem Formulation Proposed Method Fractal Dimensions Background Our method Results Conclusions. ?. General Problem Definition.

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F4: Large Scale Automated Forecasting Using Fractals

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  1. F4: Large Scale Automated Forecasting Using Fractals -Deepayan Chakrabarti -Christos Faloutsos CIKM 2002

  2. Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002

  3. ? General Problem Definition Value Time Given a time series {xt}, predict its future course, that is, xt+1, xt+2, ... CIKM 2002

  4. Motivation Traditional fields • Financial data analysis • Physiological data, elderly care • Weather, environmental studies Sensor Networks(MEMS, “SmartDust”) • Long / “infinite” series • No human intervention  “black box” CIKM 2002

  5. Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002

  6. How to forecast? • ARIMA but linearity assumption • Neural Networks  but large number of parameters and long training times [Wan/1993, Mozer/1993] • Hidden Markov Models  O(N2) in number of nodes N; also fixing N is a problem [Ge+/2000] • Lag Plots CIKM 2002

  7. Q0: Interpolation Method Q1: Lag = ? Q2: K = ? Interpolate these… To get the final prediction 4-NN New Point Lag Plots xt xt-1 CIKM 2002

  8. Using SVD (state of the art) [Sauer/1993] xt Xt-1 Q0: Interpolation CIKM 2002

  9. Why Lag Plots? • Based on the “Takens’ Theorem” [Takens/1981] • which says that delay vectors can be used for predictive purposes CIKM 2002

  10. Extra Inside Theory Example: Lotka-Volterra equations ΔH/Δt = rH – aH*P ΔP/Δt = bH*P – mP H is density of preyP is density of predators Suppose only H(t) is observed. Internal state is (H,P). CIKM 2002

  11. Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002

  12. 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 CIKM 2002

  13. Previous work/Alternatives • Manual Setting : BUT infeasible [Sauer/1992] • CrossValidation : BUT Slow; leave-one-out crossvalidation ~ O(N2logN) or more • “False Nearest Neighbors” : BUT Unstable [Abarbanel/1996] CIKM 2002

  14. Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002

  15. X(t) Intrinsic Dimensionality ≈ Degrees of Freedom ≈ Information about Xt given Xt-1 X(t-1) Intuition x(t) time The Logistic Parabola xt = axt-1(1-xt-1) + noise CIKM 2002

  16. x(t) x(t-1) x(t-2) x(t) x(t) x(t-1) x(t-1) x(t-2) x(t-2) Intuition x(t) x(t-1) CIKM 2002

  17. Intuition • 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 CIKM 2002

  18. Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002

  19. Fractal Dimensions • FD = intrinsic dimensionality “Embedding” dimensionality = 3 Intrinsic dimensionality = 1 CIKM 2002

  20. Fractal Dimensions FD = intrinsic dimensionality [Belussi/1995] log( # pairs) • Points to note: • FD can be a non-integer • There are fast methods to compute it CIKM 2002 log(r)

  21. Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002

  22. epsilon f L(opt) Q1: Finding L(opt) • Use Fractal Dimensions to find the optimal lag length L(opt) Fractal Dimension Lag (L) CIKM 2002

  23. Q2: Finding k(opt) • To find k(opt) • Conjecture: k(opt) ~ O(f) We choose k(opt) = 2*f + 1 CIKM 2002

  24. Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002

  25. Value Datasets • Logistic Parabola: xt = axt-1(1-xt-1) + noise Models population of flies [R. May/1976] Time CIKM 2002

  26. Value Datasets • Logistic Parabola: xt = axt-1(1-xt-1) + noise Models population of flies [R. May/1976] Time • LORENZ: Models convection currents in the air CIKM 2002

  27. Value Datasets • Logistic Parabola: xt = axt-1(1-xt-1) + noise Models population of flies [R. May/1976] Error NMSE = ∑(predicted-true)2/σ2 Time • LORENZ: Models convection currents in the air • LASER: fluctuations in a Laser over time (from the Santa Fe Time Series Competition, 1992) CIKM 2002

  28. Value Timesteps FD Logistic Parabola Lag • FD vs L plot flattens out • L(opt) = 1 CIKM 2002

  29. Logistic Parabola Our Prediction from here Value Timesteps CIKM 2002

  30. Value Logistic Parabola Comparison of prediction to correct values Timesteps CIKM 2002

  31. Logistic Parabola FD Our L(opt) = 1, which exactly minimizes NMSE NMSE CIKM 2002 Lag

  32. FD Value Timesteps LORENZ Lag • L(opt) = 5 CIKM 2002

  33. LORENZ Our Prediction from here Value Timesteps CIKM 2002

  34. LORENZ Value Comparison of prediction to correct values Timesteps CIKM 2002

  35. LORENZ FD L(opt) = 5 Also NMSE is optimal at Lag = 5 NMSE CIKM 2002 Lag

  36. FD Laser Value Lag • L(opt) = 7 Timesteps CIKM 2002

  37. Laser Our Prediction starts here Value Timesteps CIKM 2002

  38. Laser Value Comparison of prediction to correct values Timesteps CIKM 2002

  39. FD Laser L(opt) = 7 Corresponding NMSE is close to optimal NMSE CIKM 2002 Lag

  40. Speed and Scalability • Preprocessing is linear in N • Proportional to time taken to calculate FD CIKM 2002

  41. Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002

  42. Conclusions Our Method: • Automatically set parameters • L(opt) (answers Q1) • k(opt) (answers Q2) • In linear time on N CIKM 2002

  43. 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 CIKM 2002

  44. Extra Snapshot http://snapdragon.cald.cs.cmu.edu/TSP CIKM 2002

  45. Extra Future Work • Feature Selection • Multi-sequence prediction CIKM 2002

  46. Extra Discussion – Some other problems How to forecast? Given: • x1, x2, …, xN • L(opt) • k(opt) How to find the k(opt) nearest neighbors quickly? CIKM 2002

  47. Extra Motivation • Forecasting also allows us to • Find outliers  anything that doesn’t match our prediction!  • Find patterns  if different circumstances lead to similar predictions, they may be related. CIKM 2002

  48. Extra Motivation (Examples) Traditional • EEGs : Patterns of electromagnetic impulses in the brain • Intensity variations of white dwarf stars • Highway usage over time Sensors • “Active Disks” for forecasting / prefetching / buffering • “Smart House”  sensors monitor situation in a house • Volcano monitoring CIKM 2002

  49. Extra • Store all the delay vectors {xt-1, …, xt-L(opt)} and corresponding prediction xt • Find the latest delay vector xt • Find nearest neighbors Interpolate • Interpolate Xt-1 General Method L(opt) = ? K(opt) = ? CIKM 2002

  50. Extra Intuition Fractal dimension • The FD vs L plot does flatten out • L(opt) = 1 CIKM 2002 Lag

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