Stochastic Filtering by Projection - The Example of the Quadratic Sensor

28/08/2013
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Stochastic Filtering by Projection - The Example of the Quadratic Sensor

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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 dXt = ft(Xt) dt + σt(Xt) dWt, X0, dYt = bt(Xt) dt + dVt, Y0 = 0 . Xt is a process representing the state. Yt is a process representing the measurement. Wt and Vt are independent Wiener processes. Stochastic Filtering by Projection Stochastic Filtering Mathematical formulation dXt = ft(Xt) dt + σt(Xt) dWt, X0, dYt = bt(Xt) dt + dVt, Y0 = 0 . Xt is a process representing the state. Yt is a process representing the measurement. Wt and Vt are independent Wiener processes. Question What is the probability distribution for Xt given the values of Yt 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 pt satisfies: dpt = L∗ t ptdt + pt[bt − Ept {bt}][dYt − Ept {bt}dt] . where: L∗ = −ft ∂ ∂x + 1 2 at ∂ ∂x2 is the backward diffusion operator aT t a = σ and a is a square root of σ. Ept denotes expectation with respect to pt. Stochastic Filtering by Projection Stochastic Filtering The Kushner–Stratonovich equation With sufficient regularity and bounds, one can show that the probability density pt satisfies: dpt = L∗ t ptdt + pt[bt − Ept {bt}][dYt − Ept {bt}dt] . where: L∗ = −ft ∂ ∂x + 1 2 at ∂ ∂x2 is the backward diffusion operator aT t a = σ and a is a square root of σ. Ept denotes expectation with respect to pt. 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 pt 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 pt 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 pt 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 pt 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 p0, 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 p0, which represents the prior probability distribution for the state, is a Gaussian then pt is always a Gaussian The mean and standard deviation of pt 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 p0, which represents the prior probability distribution for the state, is a Gaussian then pt is always a Gaussian The mean and standard deviation of pt 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: pt(x) = i λi e(x−µi )/2σ2 i λi ≥ 0. i λi = 1. Gives rise to a 3m − 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: pt(x) = i λi e(x−µi )/2σ2 i λi ≥ 0. i λi = 1. Gives rise to a 3m − 1 dimensional family. The exponential family pt(x) = exp(a0 + a1x + a2x2 + . . . a2nx2n ) a2n < 0 Gives rise to a 2n 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 L2 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: dXt = ft(Xt) dt + σt(Xt) dWt is shorthand for an integral equation such as: XT = T 0 ft(Xt) dt + T 0 σt(Xt) dWt where the right hand integral is defined by the Ito integral: T 0 f (t) dWt = lim n→∞ ∞ i=1 f (ti )(Wti+1 − Wti ). Stochastic Filtering by Projection Projecting the equations Projecting SDE’s The Stratonovich integral Take the Ito integral: T 0 f (t) dWt = lim n→∞ ∞ i=1 f (ti )(Wti+1 − Wti ). and change the point where you evaluate the integrand T 0 f (t) ◦ dWt = lim n→∞ ∞ i=1 f ( ti + ti+1 2 )(Wti+1 − Wti ). 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 0 f (t) dWt = lim n→∞ ∞ i=1 f (ti )(Wti+1 − Wti ). and change the point where you evaluate the integrand T 0 f (t) ◦ dWt = lim n→∞ ∞ i=1 f ( ti + ti+1 2 )(Wti+1 − Wti ). 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. Stochastic Filtering by Projection Projecting the equations Projecting SDE’s A recipe for projecting SDE’s To project an SDE onto a submanifold parameterized by θ = (θ1, θ2, . . . , θn): Write the SDE as an SDE with vector coefficients in Stratonovich form. Project all the coefficients onto the tangent space. Equate both sides of the projected equations to get an SDE for the θi . Stochastic Filtering by Projection Projecting the equations Projecting SDE’s A recipe for projecting SDE’s To project an SDE onto a submanifold parameterized by θ = (θ1, θ2, . . . , θn): Write the SDE as an SDE with vector coefficients in Stratonovich form. Project all the coefficients onto the tangent space. Equate both sides of the projected equations to get an SDE for the θi . Since Stratonovich SDE’s transform like vector fields, this recipe is invariant of the parameterization. Stochastic Filtering by Projection Projecting the equations Projecting SDE’s The projected equations The end result for the case of L2 projection is: dθi = m j=1 hij p(θ), Lvj dt − γ0 (p(θ)), vj dt + γ1 (p(θ)), vj ◦ dY . Where: The vj = ∂p ∂θj give a basis for the tangent space hij and hij are the Riemannian metric tensor vi , vj . γ0 t (p) := 1 2 [|bt|2 − Ep{|bt|2}] γ1 t (p) := [bt − Ep{bt}]p ·, · is the L2 inner product. Note that the inner products and expectations give rise to integrals. We can compute these analytically for the normal mixture family. Stochastic Filtering by Projection Solving the SDE’s Solving the finite system of SDE’s Approximate the differential equation as a difference equation and solve numerically. This is more delicate for stochastic equations than ordinary ones. See Kloeden and Platen. We use the Stratonovich–Heun cheme. Note that the resulting difference equation will depend upon the choice of parameterization of the submanifold. Choose coordinates φ : Rn −→ M so that φ is defined on all of Rn. Stochastic Filtering by Projection Numerical example The quadratic sensor dXt = dWt dYt = X2 + dVt . Stochastic Filtering by Projection Numerical example The quadratic sensor dXt = dWt dYt = X2 + dVt . We do not receive any information on the sign of X. We expect that once X has hit the origin, p will be approximately symmetrical. We expect a bimodal distribution Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 0 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 1 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 2 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 3 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 4 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 5 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 6 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 7 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 8 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 9 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example Simulation for the Quadratic Sensor 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 6 8 X Distribution at time 10 Projection Exact Extended Kalman Exponential Stochastic Filtering by Projection Numerical example L2 residuals for the quadratic sensor 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 2 4 6 8 10 Time Residuals Projection Residual (L2 norm) Extended Kalman Residual (L2 norm) Hellinger Projection Residual (L2 norm) Stochastic Filtering by Projection Numerical example L´evy residuals for the quadratic sensor 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 1 2 3 4 5 6 7 8 9 10 Time ProkhorovResiduals Prokhorov Residual (L2NM) Prokhorov Residual (HE) Best possible residual (3Deltas) Stochastic Filtering by Projection Conclusions Conclusions Projection methods allow us to approximate the solution to nonlinear problems with surprising accuracy using only low dimensional manifolds. This conclusion holds for a variety of projection metrics and manifolds. L2 projection of normal mixtures is particularly promising since all integrals can be computed analytically.