540 likes | 649 Views
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.
E N D
F4: Large Scale Automated Forecasting Using Fractals -Deepayan Chakrabarti -Christos Faloutsos CIKM 2002
Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002
? General Problem Definition Value Time Given a time series {xt}, predict its future course, that is, xt+1, xt+2, ... CIKM 2002
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
Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002
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
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
Using SVD (state of the art) [Sauer/1993] xt Xt-1 Q0: Interpolation CIKM 2002
Why Lag Plots? • Based on the “Takens’ Theorem” [Takens/1981] • which says that delay vectors can be used for predictive purposes CIKM 2002
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
Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002
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
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
Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002
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
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
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
Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002
Fractal Dimensions • FD = intrinsic dimensionality “Embedding” dimensionality = 3 Intrinsic dimensionality = 1 CIKM 2002
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)
Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002
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
Q2: Finding k(opt) • To find k(opt) • Conjecture: k(opt) ~ O(f) We choose k(opt) = 2*f + 1 CIKM 2002
Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002
Value Datasets • Logistic Parabola: xt = axt-1(1-xt-1) + noise Models population of flies [R. May/1976] Time CIKM 2002
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
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
Value Timesteps FD Logistic Parabola Lag • FD vs L plot flattens out • L(opt) = 1 CIKM 2002
Logistic Parabola Our Prediction from here Value Timesteps CIKM 2002
Value Logistic Parabola Comparison of prediction to correct values Timesteps CIKM 2002
Logistic Parabola FD Our L(opt) = 1, which exactly minimizes NMSE NMSE CIKM 2002 Lag
FD Value Timesteps LORENZ Lag • L(opt) = 5 CIKM 2002
LORENZ Our Prediction from here Value Timesteps CIKM 2002
LORENZ Value Comparison of prediction to correct values Timesteps CIKM 2002
LORENZ FD L(opt) = 5 Also NMSE is optimal at Lag = 5 NMSE CIKM 2002 Lag
FD Laser Value Lag • L(opt) = 7 Timesteps CIKM 2002
Laser Our Prediction starts here Value Timesteps CIKM 2002
Laser Value Comparison of prediction to correct values Timesteps CIKM 2002
FD Laser L(opt) = 7 Corresponding NMSE is close to optimal NMSE CIKM 2002 Lag
Speed and Scalability • Preprocessing is linear in N • Proportional to time taken to calculate FD CIKM 2002
Outline • Introduction/Motivation • Survey and Lag Plots • Exact Problem Formulation • Proposed Method • Fractal Dimensions Background • Our method • Results • Conclusions CIKM 2002
Conclusions Our Method: • Automatically set parameters • L(opt) (answers Q1) • k(opt) (answers Q2) • In linear time on N CIKM 2002
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
Extra Snapshot http://snapdragon.cald.cs.cmu.edu/TSP CIKM 2002
Extra Future Work • Feature Selection • Multi-sequence prediction CIKM 2002
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
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
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
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
Extra Intuition Fractal dimension • The FD vs L plot does flatten out • L(opt) = 1 CIKM 2002 Lag