VIEWS: 0 PAGES: 34 CATEGORY: Education POSTED ON: 1/4/2010
Motivation Parabolic scheme for stochastic inference Summary Inverse Langevin approach to time-series data analysis Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Universidade de Brasília Saratoga Springs, MaxEnt 2007 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference Newton’s method Implementation Examples A very naive application to ﬁnance 2 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference Newton’s method Implementation Examples A very naive application to ﬁnance 2 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise 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 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise 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) Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference Newton’s method Implementation Examples A very naive application to ﬁnance 2 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise Limit from discrete process 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) Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise Questions What the drift and the noise? Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise Questions What the drift and the noise? Is it Fokker-Planck or Kramer-Moyal? Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise Questions What the drift and the noise? Is it Fokker-Planck or Kramer-Moyal? Can we compare Fokker-Planck to Kramer-Moyal models? Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference Newton’s method Implementation Examples A very naive application to ﬁnance 2 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise Markov Chain Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Brownian Motion Limit from discrete process Inference about forces/noise Use Bayes theorem... Likelihood (µ: measurement noise, w: hidden state, η, θ: parameters of interest) p(η|y) = p(y|η) = 1 p(η)p(y|η) p(y) dw dµ dθ p(η, θ)p(µ)p(y|w, µ)p(w|η, θ) Similar thing for p(θ|y) Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference Newton’s method Implementation Examples A very naive application to ﬁnance 2 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance Exactly soluble propagators A Markov process is characterized by its propagator p(w0 , w1 , . . . , wN ) = p(w0 )p(w1 |w0 ) . . . p(wN |wN−1 ) We know analytical (Gaussians!) results only for simplest cases F(x, v; t) = −γv − ω 2 x + a0 m Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance 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 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference Newton’s method Implementation Examples A very naive application to ﬁnance 2 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance 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= dv p(y0 , v0 , y1 , v1 , . . . , yN , vN, |θ, η) Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance Main loop At each integration step, collect a bunch of coefﬁcients η αi − η (ai v2 +2bi vi +2ci vi vi−1 +di v2 +2ei vi−1 +fi ) i−1 e 2 i Gi After each step, some coefﬁcients must be updated before we start with the next integration Φ= η 2 − η f (θ,y) dv p(y, v|θ, η) = e 2 G(θ) N−1 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance 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 +σ+ d 2 2 p(η|y) is calculated analytically Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance 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 +σ+ d 2 2 p(η|y) is calculated analytically We use a ﬂat prior p(θ) (I’m ashamed!) Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference Newton’s method Implementation Examples A very naive application to ﬁnance 2 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance Oscillatory brownian motion Force parameters vs. data length constant linear quadratic 0.05 0.00 Coefficient -0.05 -0.10 -0.15 -0.20 103 104 105 Data length Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance 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 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance Some results Simple Harmonic Dissipative D-H η(1) 0.96(0.03) 0.78(0.03) 1.06(0.03) 1.07(0.03) θ1 (0) −0.07(0.02) −0.04(0.08) 0.08(0.02) −0.04(0.08) θ2 (0.1) −9 × 10−7 (5 × 10−4 ) −0.099(0.003) −5 × 10−4 (9 × 10−4 ) −0.096(0.004) Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference Newton’s method Implementation Examples A very naive application to ﬁnance 2 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance You will not win any money! There is no force. This will NOT make you win money! Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance 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 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Newton’s method Implementation Examples A very naive application to ﬁnance The time series data GE closing values 1.0 0.5 log2-return 0.0 -0.5 -1.0 -1.5 1970 1980 1990 2000 year Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Summary Given f (x, θ), it is possible to infer θ and the intensity of the noise which governs a Langevin system Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Summary 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. Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Summary 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!) Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Summary 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 Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis Motivation Parabolic scheme for stochastic inference Summary Summary 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) Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis