Stochastic Filtering by Projection The Example of the Quadratic - PowerPoint PPT Presentation

Stochastic Filtering by Projection Stochastic Filtering by Projection The Example of the Quadratic Sensor John Armstrong (King’s College London) collaboration with Damiano Brigo (Imperial College) GSI2013

Stochastic Filtering by Projection Stochastic Filtering Motivation Estimate the current state of a stochastic system from imperfect measurements

Stochastic Filtering by Projection Stochastic Filtering Motivation Estimate the current state of a stochastic system from imperfect measurements ◮ Estimate the position of a car

Stochastic Filtering by Projection Stochastic Filtering Motivation Estimate the current state of a stochastic system from imperfect measurements ◮ Estimate the position of a car ◮ Estimate the volatility of a stock from option prices

Stochastic Filtering by Projection Stochastic Filtering Motivation Estimate the current state of a stochastic system from imperfect measurements ◮ Estimate the position of a car ◮ Estimate the volatility of a stock from option prices ◮ Applications in weather forecasting, oil extraction ...

Stochastic Filtering by Projection Stochastic Filtering Motivation Estimate the current state of a stochastic system from imperfect measurements ◮ Estimate the position of a car ◮ Estimate the volatility of a stock from option prices ◮ Applications in weather forecasting, oil extraction ... The calculation should be performed online.

Stochastic Filtering by Projection Stochastic Filtering Mathematical formulation dX t = f t ( X t ) dt + σ t ( X t ) dW t , X 0 , dY t = b t ( X t ) dt + dV t , Y 0 = 0 . ◮ X t is a process representing the state. ◮ Y t is a process representing the measurement. ◮ W t and V t are independent Wiener processes.

Stochastic Filtering by Projection Stochastic Filtering Mathematical formulation dX t = f t ( X t ) dt + σ t ( X t ) dW t , X 0 , dY t = b t ( X t ) dt + dV t , Y 0 = 0 . ◮ X t is a process representing the state. ◮ Y t is a process representing the measurement. ◮ W t and V t are independent Wiener processes. Question What is the probability distribution for X t given the values of Y t up to time t?

Stochastic Filtering by Projection Stochastic Filtering The Kushner–Stratonovich equation With sufficient regularity and bounds, one can show that the probability density p t satisfies: d p t = L ∗ t p t d t + p t [ b t − E p t { b t } ][ d Y t − E p t { b t } d t ] . where: ◮ ∂ x + 1 ∂ ∂ L ∗ = − f t 2 a t ∂ x 2 is the backward diffusion operator ◮ a T t a = σ and a is a square root of σ . ◮ E p t denotes expectation with respect to p t .

Stochastic Filtering by Projection Stochastic Filtering The Kushner–Stratonovich equation With sufficient regularity and bounds, one can show that the probability density p t satisfies: d p t = L ∗ t p t d t + p t [ b t − E p t { b t } ][ d Y t − E p t { b t } d t ] . where: ◮ ∂ x + 1 ∂ ∂ L ∗ = − f t 2 a t ∂ x 2 is the backward diffusion operator ◮ a T t a = σ and a is a square root of σ . ◮ E p t denotes expectation with respect to p t . Question How can we efficiently approximate solutions to the infinite dimensional Kushner–Stratonovich equation?

Stochastic Filtering by Projection The geometric idea The geometric idea ◮ Choose a submanifold of the space of probability distributions so that points in the manifold can approximate p t well.

Stochastic Filtering by Projection The geometric idea The geometric idea ◮ Choose a submanifold of the space of probability distributions so that points in the manifold can approximate p t well. ◮ View the partial differential equation as defining a stochastic vector field.

Stochastic Filtering by Projection The geometric idea The geometric idea ◮ Choose a submanifold of the space of probability distributions so that points in the manifold can approximate p t well. ◮ View the partial differential equation as defining a stochastic vector field. ◮ Use projection to restrict the vector field to the tangent space.

Stochastic Filtering by Projection The geometric idea The geometric idea ◮ Choose a submanifold of the space of probability distributions so that points in the manifold can approximate p t well. ◮ View the partial differential equation as defining a stochastic vector field. ◮ Use projection to restrict the vector field to the tangent space. ◮ Solve the resulting finite dimensional stochastic differential equation.

Stochastic Filtering by Projection Choosing the submanifold The linear problem If: ◮ the coefficient functions a , b and f in the problem are all linear ◮ p 0 , which represents the prior probability distribution for the state, is a Gaussian then

Stochastic Filtering by Projection Choosing the submanifold The linear problem If: ◮ the coefficient functions a , b and f in the problem are all linear ◮ p 0 , which represents the prior probability distribution for the state, is a Gaussian then ◮ p t is always a Gaussian ◮ The mean and standard deviation of p t follow a finite dimensional SDE. This is called the Kalman filter .

Stochastic Filtering by Projection Choosing the submanifold The linear problem If: ◮ the coefficient functions a , b and f in the problem are all linear ◮ p 0 , which represents the prior probability distribution for the state, is a Gaussian then ◮ p t is always a Gaussian ◮ The mean and standard deviation of p t follow a finite dimensional SDE. This is called the Kalman filter . One can linearize any filtering problem at each point in time to obtain the Extended Kalman filter .

Stochastic Filtering by Projection Choosing the submanifold Two important families For multi modal problems, project onto one of the following families:

Stochastic Filtering by Projection Choosing the submanifold Two important families For multi modal problems, project onto one of the following families: ◮ A mixture of m Gaussian distributions: λ i e ( x − µ i ) / 2 σ 2 � p t ( x ) = i i ◮ λ i ≥ 0. � i λ i = 1. ◮ Gives rise to a 3 m − 1 dimensional family.

Stochastic Filtering by Projection Choosing the submanifold Two important families For multi modal problems, project onto one of the following families: ◮ A mixture of m Gaussian distributions: λ i e ( x − µ i ) / 2 σ 2 � p t ( x ) = i i ◮ λ i ≥ 0. � i λ i = 1. ◮ Gives rise to a 3 m − 1 dimensional family. ◮ The exponential family p t ( x ) = exp( a 0 + a 1 x + a 2 x 2 + . . . a 2 n x 2 n ) ◮ a 2 n < 0 ◮ Gives rise to a 2 n dimensional family.

Stochastic Filtering by Projection Projecting the equations The choice of metric Choice of metric for the projection Need to choose a Hilbert space structure on the space of probability distributions (more precisely some enveloping space).

Stochastic Filtering by Projection Projecting the equations The choice of metric Choice of metric for the projection Need to choose a Hilbert space structure on the space of probability distributions (more precisely some enveloping space). ◮ The Hellinger metric. ◮ Theoretical advantage of coordinate independence ◮ Works well with exponential families (Brigo) ◮ Meaningful for problems where density p does not exist. ◮ Requires numerical approximation of integrals to implement.

Stochastic Filtering by Projection Projecting the equations The choice of metric Choice of metric for the projection Need to choose a Hilbert space structure on the space of probability distributions (more precisely some enveloping space). ◮ The Hellinger metric. ◮ Theoretical advantage of coordinate independence ◮ Works well with exponential families (Brigo) ◮ Meaningful for problems where density p does not exist. ◮ Requires numerical approximation of integrals to implement. ◮ The direct L 2 metric. ◮ Works well with mixture families. ◮ All integrals that occur can be calculated analytically.

Stochastic Filtering by Projection Projecting the equations Projecting SDE’s Understanding stochastic differential equations A stochastic differential equation such as: dX t = f t ( X t ) dt + σ t ( X t ) dW t is shorthand for an integral equation such as: � T � T X T = f t ( X t ) dt + σ t ( X t ) dW t 0 0 where the right hand integral is defined by the Ito integral: � T ∞ � f ( t ) dW t = lim f ( t i )( W t i +1 − W t i ) . n →∞ 0 i =1

Stochastic Filtering by Projection Projecting the equations Projecting SDE’s The Stratonovich integral ◮ Take the Ito integral: � T ∞ � f ( t ) dW t = lim f ( t i )( W t i +1 − W t i ) . n →∞ 0 i =1 and change the point where you evaluate the integrand � T ∞ f ( t i + t i +1 � f ( t ) ◦ dW t = lim )( W t i +1 − W t i ) . 2 n →∞ 0 i =1 to get the Stratonvich integral . Hence you can define Stratonovich SDE’s.

Stochastic Filtering by Projection Projecting the equations Projecting SDE’s The Stratonovich integral ◮ Take the Ito integral: � T ∞ � f ( t ) dW t = lim f ( t i )( W t i +1 − W t i ) . n →∞ 0 i =1 and change the point where you evaluate the integrand � T ∞ f ( t i + t i +1 � f ( t ) ◦ dW t = lim )( W t i +1 − W t i ) . 2 n →∞ 0 i =1 to get the Stratonvich integral . Hence you can define Stratonovich SDE’s. ◮ The difference between the two integrals is an ordinary integral. This allows you to convert between the two formulations. ◮ Ito SDE’s model causality more naturally ◮ Stratonovich SDE’s transform like vector fields.