Inverse Langevin approach to time-series data analysis Fbio Macdo - PowerPoint PPT Presentation

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 2 Newton’s method Implementation Examples A very naive application to finance Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis

Motivation Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary Inference about forces/noise Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference 2 Newton’s method Implementation Examples A very naive application to finance Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis

Motivation Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary 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 Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary 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 Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary Inference about forces/noise Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference 2 Newton’s method Implementation Examples A very naive application to finance Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis

Motivation Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary 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 Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary 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 Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary 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 Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary 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 Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary Inference about forces/noise Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference 2 Newton’s method Implementation Examples A very naive application to finance Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis

Motivation Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary Inference about forces/noise Markov Chain Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis

Motivation Brownian Motion Parabolic scheme for stochastic inference Limit from discrete process Summary Inference about forces/noise Use Bayes theorem... Likelihood ( µ : measurement noise, � w : hidden state, η, θ : parameters of interest) 1 p ( η | � y ) = y ) p ( η ) p ( � y | η ) p ( � � p ( � y | η ) = d � w d µ d θ 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

Newton’s method Motivation Implementation Parabolic scheme for stochastic inference Examples Summary A very naive application to finance Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference 2 Newton’s method Implementation Examples A very naive application to finance Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis

Newton’s method Motivation Implementation Parabolic scheme for stochastic inference Examples Summary A very naive application to finance 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 ) = − γ v − ω 2 x + a 0 m Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis

Newton’s method Motivation Implementation Parabolic scheme for stochastic inference Examples Summary 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 δ t 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

Newton’s method Motivation Implementation Parabolic scheme for stochastic inference Examples Summary A very naive application to finance Outline 1 Motivation Brownian Motion Limit from discrete process Inference about forces/noise Parabolic scheme for stochastic inference 2 Newton’s method Implementation Examples A very naive application to finance Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis

Newton’s method Motivation Implementation Parabolic scheme for stochastic inference Examples Summary A very naive application to finance Delta function approximation Conjugate prior for p ( µ ) . At some level approximation for large data sets... � d µ 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 ) Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis

Newton’s method Motivation Implementation Parabolic scheme for stochastic inference Examples Summary A very naive application to finance Main loop At each integration step, collect a bunch of coefficients η α i 2 ( a i v 2 i − 1 + 2 e i v i − 1 + f i ) i + 2 b i v i + 2 c i v i v i − 1 + d i v 2 e − η G i After each step, some coefficients must be updated before we start with the next integration N − 1 � v | θ, η ) = η 2 G ( θ ) e − η 2 f ( θ,� y ) Φ = d � v p ( � y ,� Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis

Newton’s method Motivation Implementation Parabolic scheme for stochastic inference Examples Summary 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 ) p ( θ | � y ) ∝ � N + 1 2 + σ + d y ) − f (¯ � G ( θ ) f ( � y ) + f ( θ,� θ,� y ) + δ 2 p ( η | � y ) is calculated analytically Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Inverse Langevin approach to time-series data analysis