Inverse Langevin approach to time-series data analysis

Transcription

1 Inverse Langevin approach to time-series data analysis Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Universidade de Brasília Saratoga Springs, MaxEnt 2007

2 Outline 1 Motivation 2

3 Outline 1 Motivation 2

4 Classic Brownian motion Einstein: drunkard walk (Smoluchowski controversy) ) Lots of Brownian motions all around: physics, finance, communication etc. Random Gaussian increments (SDE s Wanier, Itô) dx = m(x; t) dt + σ(x; t) db

5 Langevin dynamics We must retain Newtonian physics: expand the state dv = a(x, v; t) dt + D(x, v; t) db dx = v dt rich set of solutions from simple equations: linear, exponential, oscillatory, etc Similar results (classic Brownian)

6 Outline 1 Motivation 2

7 What we tackle (work in progress) Wanier noise db = β dt: Fokker-Planck/forward Kolmogorov equation. (Brownian motion) Non-analytic noises, Levy, and others: Kramer-Moyal equation. (gas of rigid spheres)

8 Questions What the drift and the noise?

9 Questions What the drift and the noise? Is it Fokker-Planck or Kramer-Moyal?

10 Questions What the drift and the noise? Is it Fokker-Planck or Kramer-Moyal? Can we compare Fokker-Planck to Kramer-Moyal models?

11 Outline 1 Motivation 2

12 Markov Chain

13 Use Bayes theorem... Likelihood (µ: measurement noise, w: hidden state, η, θ: parameters of interest) p(η y) = 1 p( y) p(η)p( y η) p( y η) = d w dµ dθ p(η, θ)p(µ)p( y w, µ)p( w η, θ) Similar thing for p(θ y)

14 Outline 1 Motivation 2

15 Exactly soluble propagators A Markov process is characterized by its propagator p(w 0, w 1,..., w N ) = p(w 0 )p(w 1 w 0 )... p(w N w N 1 ) We know analytical (Gaussians!) results only for simplest cases F(x, v; t) m = γv ω 2 x + a 0

16 The parabolic method We don t know how to integrate every model Newton approximation The force at a small δt is almost constant (but depends on initial position + parameters) We can calculate the trajectory/propagator Repeat this procedure for the next step

17 Outline 1 Motivation 2

18 Delta function approximation Conjugate prior for p(µ). At some level approximation for large data sets... dµ p(µ)p( y w, µ) δ( y x) Now we have only Gaussian integrations over velocities. Φ = d v p(y 0, v 0, y 1, v 1,..., y N, v N, θ, η)

19 Main loop At each integration step, collect a bunch of coefficients η α i G i e η 2 (a i v 2 i +2b iv i +2c i v i v i 1 +d i v 2 i 1 +2e iv i 1 +f i) After each step, some coefficients must be updated before we start with the next integration Φ = d v p( y, v θ, η) = η N 1 2 η G(θ) e 2 f (θ, y)

20 Inference results Use Laplace approximation to normalize p(θ y, η), plug-in a Gamma prior p(η) p(θ y) p(θ) G(θ) [ f ( y) + f (θ, y) f ( θ, y) + δ ] N+1 2 +σ+ d 2 p(η y) is calculated analytically

21 Inference results Use Laplace approximation to normalize p(θ y, η), plug-in a Gamma prior p(η) p(θ y) p(θ) G(θ) [ f ( y) + f (θ, y) f ( θ, y) + δ ] N+1 2 +σ+ d 2 p(η y) is calculated analytically We use a flat prior p(θ) (I m ashamed!)

22 Outline 1 Motivation 2

23 Oscillatory brownian motion Force parameters vs. data length constant linear quadratic Coefficient Data length

24 Comments Works pretty well with artificially generated time series Still, it does not get the diffusion coefficient quite right in some cases It is also somewhat robust to the existence of dissipative forces

25 Some results η(1) θ 1 (0) θ 2 (0.1) Simple 0.96(0.03) 0.07(0.02) ( ) Harmonic 0.78(0.03) 0.04(0.08) 0.099(0.003) Dissipative 1.06(0.03) 0.08(0.02) ( ) D-H 1.07(0.03) 0.04(0.08) 0.096(0.004)

26 Outline 1 Motivation 2

27 You will not win any money! There is no force. This will NOT make you win money!

28 You will not win any money! There is no force. This will NOT make you win money! But this is the expected behavior, of course Things to explore (speculation-crash patterns): noise may not be Wanier volatility time we can introduce trends as a time-dependent force

29 The time series data GE closing values log2-return year

30 Given f (x, θ), it is possible to infer θ and the intensity of the noise which governs a Langevin system

31 Given f (x, θ), it is possible to infer θ and the intensity of the noise which governs a Langevin system Open possibilities: better linear approximations, time-dependent parameters, linear diffusion coefficient.

32 Given f (x, θ), it is possible to infer θ and the intensity of the noise which governs a Langevin system Open possibilities: better linear approximations, time-dependent parameters, linear diffusion coefficient. Open issues Work out a decent prior p(θ) (and calculate the evidence!)

33 Given f (x, θ), it is possible to infer θ and the intensity of the noise which governs a Langevin system Open possibilities: better linear approximations, time-dependent parameters, linear diffusion coefficient. Open issues Work out a decent prior p(θ) (and calculate the evidence!) Treatment of noise is not really satisfactory

34 Given f (x, θ), it is possible to infer θ and the intensity of the noise which governs a Langevin system Open possibilities: better linear approximations, time-dependent parameters, linear diffusion coefficient. Open issues Work out a decent prior p(θ) (and calculate the evidence!) Treatment of noise is not really satisfactory Dissipative forces (easy) and non-wanier and Levy noises (potentially very hard)