Non-particle filters

Transcription

1 Non-particle filters Fred Daum & Misha Krichman Raytheon Company 225 Presidential Way Woburn, MA ABSTRACT We have developed a new nonlinear filter that is superior to particle filters in five ways: (1) it exploits smoothness; (2) it uses an exact solution of the Fokker-Planck equation in continuous time; (3) it uses a convolution to compute the effect of process noise at discrete times; (4) it uses the adjoint method to compute the optimal density of points in state space to represent the smooth conditional probability density, and (5) it uses Bayes rule exactly by exploiting the exponential family of probability densities. In contrast to particle filters, which do not exploit smoothness, the new filter avoids importance sampling and Monte Carlo methods. The new non-particle filter should be superior to particle filters for a broad class of practical problems. In particular, the new filter should dramatically reduce the curse of dimensionality for many (but not all) important real world nonlinear filter problems. KEY WORDS Nonlinear filters, particle filter, adjoint method, meshfree, Fokker-Planck equation, Kalman filter 1.0 INTRODUCTION We describe a new nonlinear filter that exploits smoothness to reduce the curse of dimensionality for a broad class of important practical problems. We use a hybrid model of nonlinear dynamics that allows us to solve the Fokker-Planck equation exactly; in particular, we use discrete time diffusion but continuous time drift. With this hybrid model, the Fokker- Planck equation is equivalent to an ODE for the unnormalized conditional density of the d- dimensional state vector. Therefore, we do not need an extremely fine quantization in time (or an implicit method or ADI method) to compensate for fine quantization in state space; that is, we do not need to worry about stability of the numerical solution of the Fokker-Planck equation, as defined by the Courant-Friedrich- Lewy stability criterion. Moreover, we implement Bayes rule exactly for updates of the unnormalized density with measurements using the exponential family of probability densities. The effect of diffusion (also called process noise by engineers) in the Fokker- Planck equation is computed using a fast convolution of two probability densities. The new filter is summarized in Tables 1, 2 & 3. The derivation in Table 3 assumes that the probability density is smooth and nowhere vanishing; this formula was known to Liouville. The new filter fully exploits the smoothness of the Fokker-Planck equation, and therefore it should be superior to particle filters, which do not exploit any smoothness and which do not exploit exact solutions or the exponential family. However, if we use a uniform grid in d- dimensional state space to represent the conditional probability density, then we would still suffer from the curse of dimensionality. Therefore, we represent the density using a sparse grid computed adaptively in real time 1

2 with the adjoint method [11]. The adjoint method for solving PDEs numerically is analogous to the adjoint (aka Lagrange multipliers) used in optimal control (see Table 2). The adjoint method is an industrial strength numerical algorithm that is widely used for solving PDEs. Intuitively, the reason that we can use this hybrid discrete-time/continuoustime model for nonlinear filtering is that engineers use the diffusion term (so-called process noise ) as a design parameter, unlike physics and chemistry, where the diffusion tensor is defined by Nature. In particular, engineers typically tune the process noise covariance matrix to get improved results with extended Kalman filters [4], but Nature does not allow such tuning of Avogadro s number or other physical constants. Engineers commonly increase or decrease process noise variance by a factor of two or three without any significant effect on filter performance, but changing the drift term by one percent can wreak havoc with performance in some applications. We exploit this insensitivity to model variation in diffusion, but we pay strict attention to the physics which is encoded in the drift term in the Fokker- Planck equation. This allows us to model process noise in discrete time and use a convolution to compute the effect of diffusion on the conditional density; this greatly reduces computational complexity. It would be a shame to lavish Gflops of computer throughput on carefully solving the Fokker-Planck equation with non-zero diffusion tensor, considering that an exact model of the diffusion tensor is both unknown and of little importance in practical engineering applications. In most practical applications the process noise covariance matrix is diagonal; if not, it can be diagonalized at the cost of d 3 computations; this means that we can use d one-dimensional convolutions of the two probability densities. The key issue in nonlinear filters is the curse of dimensionality, which is a phrase coined by Richard Bellman four decades ago. The curse of dimensionality means that the computational complexity of solving a problem increases extremely fast with the dimension of the problem. For nonlinear filters, the dimension refers to the dimension of the state vector of the dynamical system to be estimated. The term extremely fast is usually taken to mean that computer time increases exponentially with dimension. It is easy to see why the curse of dimensionality is relevant for nonlinear filters. As explained below, we need to solve a partial differential equation (PDE) in d-dimensional state space in order to solve the nonlinear filtering problem. Standard textbook methods for solving PDEs numerically use a fixed grid in d-dimensional space, and the computational complexity grows as N d where N is the number of grid points in each dimension. We can conclude from this that using a fixed grid results in computational complexity growing exponentially with dimension. Hence, using a fixed grid is an extremely bad idea, and that a non-uniform set of nodes computed adaptively is required to have any hope of mitigating the curse of dimensionality. That is the key idea of this paper, as well as particle filters, as well as all modern work on solving PDEs numerically. We emphasize that hardboiled engineers are only interested in good approximations rather than exact solutions. The question of what is good enough depends on the specific application. There are many different algorithms to solve the nonlinear filtering problem, including: extended Kalman filters, unscented Kalman filters, particle filters, explicit numerical solution of the Fokker-Planck equation, Daum filters, etc. A tutorial introduction to a wide range of state-of-the-art nonlinear filters is given in [4]. It has been asserted in engineering journals that particle filters beat the curse of dimensionality, but this is generally wrong. It turns out that particle filters depend on a good proposal density, and without such help the particle filters also suffer from the curse of dimensionality [4]. 2

3 Particle filters are extremely popular, owing to the ease of coding and the simple theory required. One can code a pretty good particle filter in one or two pages of MATLAB, and one does not need to understand the finer points of stochastic calculus or any fancy methods for solving partial differential equations. Also, particle filters are popular due to their generality and flexibility, as well as a certain amount of hype associated with the claim that they beat the curse of dimensionality. On the other hand, particle filters do not exploit the smoothness of the nonlinear filtering problem, and hence we expect that the new filter described here should be superior to particle filters for many practical applications. 2.0 THE VALUE OF SMOOTHNESS Smoothness can dramatically reduce computational complexity for high dimensional problems. In particular, for approximation of smooth functions, a well known theoretical bound [12] gives: c( d) σ T = d / s in which T = computation time to achieve an approximation error of σ d = dimension of independent variable s = smoothness of the functions being approximated c(d) = time for one function evaluation (e.g., d³ for typical engineering problems) We emphasize that the word smoothness in this context does not mean, for example twice continuously differentiable (for s = 2), but rather the word smoothness as used here defines a class of functions with mixed partial derivatives of order s that are bounded by unity [12]. To quote Nemirovsky & Yudin [13]: Smoothness does not, in itself, count for much; what is important is the values of the numerical parameters which characterize this smoothness (the values of the corresponding derivatives, and so on). This is intuitively obvious. For example, for d = 20, if the conditional density is in the class of functions with s = 2, we have reduced the computational complexity by an enormous factor, as if the dimension was only d = 10. We might be tempted to say that the effective dimension is d = 10 in this case. If the theoretical bound given above applies to our nonlinear filter problem, then we have not beaten the curse of dimensionality, but we have certainly improved the situation dramatically. Unfortunately, the simple bound given above is isotropic, whereas our problem might be much smoother in certain directions than others, and therefore it is difficult to quantify the reduction in computational complexity using a simple formula with just a few parameters. Nevertheless, the simple back-of-the-envelope formula above gives considerable insight into the benefit of smoothness. There are other bounds on computational complexity for multivariate integration of smooth functions in d-dimensions [12], as well as distinct formulas that apply for estimation of smooth probability densities in d-dimensions [10] and [14]-[15]. References (1) W. Bangerth and R. Rannacher, Adaptive finite element methods for differential equations, Birkhauser Inc., (2) M. B. Giles and E. Suli, Adjoint methods for PDEs, Acta Numerica, pages , Cambridge University Press, (3) R. Becker and R. Rannacher, An optimal control approach to a posteriori error 3

4 estimation in finite element methods, Acta Numerica, pages 1-102, Cambridge University Press, (4) F. E. Daum, Nonlinear filters: beyond the Kalman filter, special tutorial issue of IEEE AES Systems Magazine, August (5) F. E. Daum, Industrial Strength Nonlinear Filters, Proceedings of Workshop in honor of Yaakov Bar- Shalom, Monterey California, May (6) F. E. Daum, New Exact Nonlinear Filters, Chapter 8 in Bayesian Analysis of Time Series and Dynamic Models, edited by J. C. Spall, New York: Marcel Dekker, Inc (7) F. E. Daum, Exact Finite Dimensional Nonlinear Filters, IEEE Transactions on Automatic Control, July (8) M.-S. Oh, Monte Carlo integration via importance sampling: dimensionality effect and an adaptive algorithm, Contemporary Mathematics, volume 115, pages , (9) K. Kastella, Finite Difference Methods for Nonlinear Filtering and Automatic Target Recognition, in Multitarget/Multisensor Tracking Volume III, edited by Y. Bar-Shalom & W. D. Blair, Artech House, Inc., (10) Luc Devroye and Gabor Lugosi, Combinatorial Methods in Density Estimation, Springer-Verlag, (11) Fred Daum and Mikhail Krichman, Meshfree Adjoint Methods for Nonlinear Filtering, Proceedings of IEEE Aerospace Conference, Big Sky Montana, March (12) J. Traub and A. Werschultz, Complexity and Information, Cambridge University Press, (13) A. S. Nemirovsky and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization, translated by E. R. Dawson, John Wiley & Sons, Inc., (14) F. Cucker and S. Smale, On the mathematical foundations of learning, Bulletin of American Math. Society, volume 39, number 1, pages 1-49, (15) M. Griebel, Sparse grids and related approximation schemes for higher dimensional problems, Univ. Bonn, Table 1 New filter vs. Particle filter 1. Prediction of density (drift) 2. Prediction of density (diffusion) 3. Adaptive method to avoid uniform grid 4. Representation of density 5. Exploits smoothness NEW FILTER Exact solution of Fokker- Planck PDE convolution of two probability densities Adjoint method Hybrid of continuous & discrete in state-space Yes PARTICLE FILTER Monte Carlo Monte Carlo Importance sampling from proposal density Particles No 4

5 Table 2 Adjoint method for PDEs vs. optimal control PDEs Computes optimal density of points in state space: q(x,t) Uses feedback: residuals of both the primal & dual solutions Lp = f & L*v = g Functional to be minimized: error in numerical approximation of conditional mean Optimal control Computes optimal control: u(x,t) Uses feedback Euler-Lagrange equations Functional to be minimized: J = L(x, u,t) dt Table 3 Exact solution of Fokker- Planck equation for zero diffusion p/ t = - p/ x f p Tr( f/ x) + ½ Tr(Q ²p/ ²x) p/ t = - p/ x f p Tr( f/ x) for Q = 0 dp/dt = p/ t + p/ x f dp/dt = - p Tr( f/ x) dp/p = - Tr( f/ x) dt for p > 0. Hence, p(x, t) = p(x, 0) exp ( - Tr( f/ x) dt ) 5