A NEW NONLINEAR FILTER

Transcription

1 COMMUNICATIONS IN INFORMATION AND SYSTEMS c 006 International Press Vol 6, No 3, pp 03-0, A NEW NONLINEAR FILTER ROBERT J ELLIOTT AND SIMON HAYKIN Abstract A discrete time filter is constructed where both the observation and signal process have non-linear dynamics with additive white Gaussian noise Using the reference probably framework a convolution Zakai equation is obtained which updates the unnormalized conditional density Our work obtains approximate solutions of this equation in terms of Gaussian sum when second order expansions are introduced for the non-linear terms Acknowledgements The authors thank NSERC for its continued support 1 Introduction The most successful filter has without doubt been the Kalman filter This considers noisy observations of a signal process where the dynamics are linear and the noise is additive and Gaussian Extensions of the Kalman filter to cover non-linear dynamics were obtained by taking first order Taylor expansions of the non-linear terms about the current mean The resulting filter is the so-called extended Kalman filter, or EKF A popular method in recent years has been the so-called particle filter approach However, this is only based on Monte-Carlo simulation In this paper the reference probability framework is used to obtain a discrete time version of the Zakai equation This looks like a convolution equation and it provides an update for the unnormalized conditional density of the state process given the observations If first order Taylor series approximations are used for the non-linear terms in the signal and observation processes, the convolution equation can be solved explicitly and the extended Kalman filter re-derived Taylor expansions of the nonlinear terms to second order are then considered and approximate solutions in terms of Gaussian sums obtained Detailed proofs and numerical work will appear in [] Dynamics Consider non-linear signal and observation processes x, y where the noise is additive and Gaussian Suppose the processes are defined on Ω, F, P where w = {w k, k = 0, 1,, }, v = {v k, k = 0, 1,, } are sequences of independent N0, I n, resp N0, I m, random variables Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada, TN 1N4, relliott@ucalgaryca Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada, haykin@mcmasterca 03

2 04 ROBERT J ELLIOTT AND SIMON HAYKIN Suppose for each k = 0, 1,, are measurable functions A k : R n R n C k : R n R m We suppose the signal dynamics have the form: 1 x k+1 = A k x k + B k w k+1 and the observation dynamics have the form Observation y k = C k x k + D k v k To simplify notation we suppose the coefficients are time independent Further we suppose that B : R n R n and D : R m R m are symmetric and non-singular Measure Change We shall follow the methods of Elliott and Krishnamurthy [3] and show how the dynamics 1, can be modelled starting with a reference probability P Suppose on Ω, F, P we have two sequences of random variables x = {x k, k = 0, 1,, } y = {y k, k = 0, 1,, } We suppose that under P the x k are independent n-dimensional N0, I n random variables and the y k are independent N0, I m random variables For x R n, y R m write Here the prime ψx = π n/ exp x x φy = π n/ exp y y denotes transpose Define the σ-fields G = σ{x 0, x 1,, x k, y 0, y 1,, y k } Y = σ{y 0, y 1,,y k } Then G k represents the possible histories of x and y to time k and Y k represents the history of y to time k For any square matrix B write B for its determinant Write λ 0 = φ D 1 y 0 Cx 0 D φy 0

3 A NEW NONLINEAR FILTER 05 and for l 1 λ l = φ D 1 y l Cx l D φy l ψ B 1 x l Ax l 1 B ψx l For k 0 define Λ k = k λ l We can define a new probability measure P on Ω, V k G k by setting dp dp l=0 Gk = Λ k For l = 0, 1,, define v l := D 1 y l Cx l For l = 1,, define w l := B 1 x l Ax l 1 As in [3] we can then prove: Lemma 1 Under the measure P v = {v l, l = 0, 1,, } w = {w l, l = 1,, } are sequences of independent N0, I m and N0, I n random variables, respectively That is, under the measure P x l = Ax l 1 + Bw l y l = Cx l + Dv l However, P is a nicer measure under which to work 3 Recursive Densities The filtering problem is concerned with the estimation of x k, and its statistics, given the observations y 0, y 1,,y k, that is, given Y k The problem would be completely solved if we could determine the conditional density γ k x of x k given Y k That is, γ k xdx = Px k dx Y k = E[Ix k dx Y k ] Using a form of Bayes Theorem, see [1], for any measurable function g : R n R E[gx k Y k ] = E[Λ kgx k Y k ] E[Λ k Y k ] where the expectations on the right are taken under the measure P The numerator E[Λ k gx k Y k ] is an unnormalized conditional expectation of gx k given Y k The denominator is the special case of the numerator obtained by taking g = 1

4 06 ROBERT J ELLIOTT AND SIMON HAYKIN The map g E[Λ k gx k Y k ] is a continuous linear functional and so defines a measure We assume this measure has a density α k x so that E[Λ k gx k Y k ] = gxα k xdx R n and heuristically E[Λ k Ix k dx Y k ] = α k xdx Then E[gx k Y k ] = R gxα n k xdx α R n k xdx and α k x is an unnormalized conditional density of x k given Y k Modifying the proof in [3] we have: Theorem 31 α k satisfies the recursion: 31 α k x = φ D 1 y k Cx α k 1 zψ B 1 x Az dz B C φy k R n Proof See [3] Remark 3 This equation provides the recursion for the unnormalized conditional density of x k given Y k It is a discrete time version of the Zakai equation The problem is now to solve this equation As shown in the book [1] and the paper [3], when the dynamics are linear, Ax = Ax and Cx = Cx, for suitable matrices A and C, with linear Gaussian noise terms the integral in 31 can be evaluated and a Gaussian density obtained for α k x The extended Kalman filter is obtained below by linearizing A and C and solving 31 We then obtain approximate solutions when second and higher order terms are included in the approximations of Ax and Cx 4 Notation Recall our model is Signal Observation x k = Ax k 1 + Bw k R n y k = Cx k + Dv k R m where A and C can be non-linear functions: A : R n R n, C : R n R m Notation 41 The transpose of any matrix or vector M will be denoted by M, Write Ax = A 1 x,, A n x Cx = C 1 x,, C m x

5 A NEW NONLINEAR FILTER 07 If A resp C are twice differentiable we shall write A = A 1,, A n = A 1 x 1, A 1 x,, A n x 1, A 1 x n A n x n, C = C 1,, C m A will denote the matrix of Hessians A 1 A A = A n where Similarly, A k = A k x 1 A k x 1 x A k x n x 1 C = C 1 C m A k x 1 x n A k x n Approximations 4 For any µ R n we can consider the Taylor expansions of A and C to second order: 41 4 Az Aµ + Aµ z µ + 1 Aµ z µ Cx C Aµ + C Aµ x Aµ + 1 C Aµ x Aµ Here Aµ z µ R n and Aµ z µ denotes the vector z µ A 1 µz µ,, z µ A n µ z µ From the Gaussian densities involved in the recursion for α k we shall be interested in scalar quantities of the form: 1 B x Aµ Aµ z µ Aµz µ

6 08 ROBERT J ELLIOTT AND SIMON HAYKIN Write ξ = ξ 1, ξ,,ξ n for the vector B x Aµ Aµ z µ Then 1 B x Aµ Aµ z µ Aµ z µ = 1 ξ Aµ z µ z µ A 1 µz µ = 1 ξ 1,, ξ n z µ A n µz µ = z µ Hµz µ where Hµ is the symmetric matrix defined symbolically by A 1 µ Hµ = 1 ξ 43 A n µ = 1 ξ1 A 1 µ + ξ A µ + ξ n A n µ Similarly, for C C Aµ x Aµ will denote the vector x Aµ C 1 Aµ x Aµ,, x Aµ C m Aµ x Aµ form In the exponential involving C terms we shall consider scalar quantities of the This can be written 1 y C Aµ D C Aµ x Aµ x Aµ Z 1 µ x Aµ where Z 1 µ is the symmetric matrix 44 C 1 Aµ 1 ζ = 1 ζ 1 C 1 Aµ +ζ C Aµ + +ζm C m Aµ C m Aµ with ζ = D y C Aµ 5 The EKF We shall take the first two terms in the Taylor expansions of Ax and Cx and show how equation 31 can be solved to re-derive the EKF Extended Kalman Filter Of course, if the dynamics are linear the calculations below show how the Kalman filter can be derived from 31

7 A NEW NONLINEAR FILTER 09 Linear Approximations Signal The signal has dynamics x k = Ax k 1 + Bw k R n If µ k 1 is the conditional mean determined at time k 1 we consider the first two terms in the Taylor expansion of Ax k 1 about µ k 1 and write 51 x k Aµ k 1 + Aµ k 1 x k 1 µ k 1 + Bw k Similarly for the Observation y k = Cx k + Dv k R m we take the first two terms of the Taylor expansion about Aµ k 1 and write 5 y k C Aµ k 1 + C Aµ k 1 x k Aµ k 1 + Dv k R m We are supposing that the conditional density of x k 1 given Y k 1 is N µ k 1, Σ k 1, that is the normalized conditional density of xk 1 given Y k 1 is 53 Σk 1 1/ ψ Σ 1 k 1 x µ k 1 Therefore, α k 1 z ψ Σ 1 k 1 z µ k 1 Let us note that x k k 1 : = E[x k Y k 1 ] 54 = E[Ax k 1 + Bw k ]Y k 1 ] = E[Ax k 1 Y k 1 ] E[Aµ k 1 + Aµ k 1 x k 1 µ k 1 Y k 1 ] = Aµ k 1 as µ k 1 = E[x k 1 Y k 1 ] In the paper of Ito and Xiong [5] a more accurate approximation is given using quadrature for E[Ax k 1 Y k 1 ] This could be used rather than Aµ k 1 Also write 55 Σ k k 1 = E[x k x k k 1 x k x k k 1 Y k 1 ] E [ Ax k 1 + Bw k Aµ k 1 Ax k 1 + Bw k Aµ k 1 Yk 1 ] E [ Aµ k 1 x k 1 µ k 1 + Bw k Aµ k 1 x k 1 µ k 1 + Bw k Yk 1 ] = Aµ k 1 Σ k 1 Aµ k 1 + B Recall the matrix inversion Lemma MIL: Lemma 51 For suitable matrices B + AΣA 1 = B 1 B 1 AΣ 1 + A B 1 A 1 A B 1

8 10 ROBERT J ELLIOTT AND SIMON HAYKIN Corollary 5 Σ 1 k k 1 = B B Aµ k 1 Σ 1 k 1 + Aµ k 1 B Aµ k 1 1 Aµk 1 B Linear algebra and the MIL then enable us to obtain the following result Theorem 53 EKF Suppose x k 1 k 1 = E[x k 1 Y k 1 ] is Nµ k 1 Σ k 1 Then, approximately, x k k is Nµ k, Σ k where 56 Σ k = Σ k k 1 Σ k k 1 C Aµ k 1 D + C Aµ k 1 Σ k k 1 C Aµ k 1 1 C Aµk 1 Σ k k 1 and 57 µ k = Aµ k 1 + Σ k k 1 C Aµ k 1 D + C Aµ k 1 Σ k k 1 C Aµ k 1 1 y k C Aµ k 1 Proof See [] 6 A New Nonlinear Filter We extend the EKF by including second order terms in the approximations for Ax and Cx This has two consequences: 1 we can no longer explicitly evaluate the integral I We approximate this using Gauss-Hermite quadrature rules after some algebra when the integral is multiplied by φ D 1 y k Cx we would like to obtain Gaussian densities By modifying the algebra of Section 5 we are able to do this Expansions 61 We recall the second order expansions 41 and 4 61 Az Aµ + Aµ z µ + 1 Aµ z µ 6 Cx C Aµ + C Aµ x Aµ + 1 C Aµ x Aµ Theorem 6 Suppose α k 1 z is given by a weighted sum of Gaussian densities: α k 1 z = Nk 1 λ k 1,i exp 1 Σ x µk 1,i k 1,i x µk 1,i

9 A NEW NONLINEAR FILTER 11 Then α k x Nk 1 mk,i j=1 λ k,i,j exp 1 x µk,i,j Σ 1 k,i x µk,i,j where and 618 with Σ k,i,j = Γ k,i,j Γ k,i,j C Aµ k 1,i D + C Aµ k 1 Γ k,i,j C Aµ k 1,i 1 C Aµk 1,i Γ k,i µ k,i,j = Aµ k 1 + Γ k,i,j C Aµ k 1,i D + C Aµ k 1,i Γ k,i,j C Aµ k 1,i 1 y k C Aµ k 1,i Σ k,i,j γ k,i,j The values of Γ k,i,j, g k,i,j and λ k,i,j are given, respectively, by 619, 61 and Proof Suppose that α 0 z is a sum of Gaussian densities: α 0 z = N0 λ oi exp 1 z µ oi Σ 1 oi z µ o,i λ oi = ρ oi π n/ Σ oi 1/ > 0 We can, and shall, suppose α 0 z is a normalized density, that is R n α 0 zdz = 1, so N0 ρ oi = 1 Later unnormalized conditional densities are obtained from the recursion 31: α k x = φ D 1 y k Cx α k 1 zψ B 1 x Az dz B D φy k R n Suppose α k 1 z is approximated as a sum of Gaussian densities: α k 1 z Nk 1 λ k 1,i exp 1 z µ k 1,i Σ 1 k 1,i z µ k 1,i

10 1 ROBERT J ELLIOTT AND SIMON HAYKIN where again λ k 1,i = ρ k 1,i π n/ Σ k 1,i 1/ > 0 and from the recursion α k x φ D 1 y k Cx B D φy k exp 1 R n Nk 1 λ k 1,i Nk 1 [ z µk 1,i Σ 1 k 1,i z µ k 1,i + x Az B x Az ] dz ρ k 1,i = 1 Then Consider one integral term in the sum To simplify the notation we shall drop the suffices for the moment Using the second order Taylor expansion for Az the integral term is approximately: I k 1,i = exp 1 [ z µ Σ 1 z µ R n 64 Here, as in Section 5, and A is the matrix of Hessians + x A Az µ 1 Az µ B x A Az µ 1 Az µ ] dz A = Aµ k 1,i R n A = Aµ k 1,i R n n Dropping the z µ 4 term this is approximately exp 1 [ z µ Σ 1 z µ R n + x A Az µ B x A Az µ ] Gx, zdz where 1 B Gx, z = exp x A Az µ Az µ 1 = exp z µ Hx, µz µ in the notation of 43, where A 1 Hx, µ = 1 B x A Az µ A n Completing the square we have I k 1,i Kx exp 1 R z σ kiδ ki σ 1 ki z σ kiδ ki G k,i x, zdz n

11 A NEW NONLINEAR FILTER 13 where 65 σ 1 ki = Σ 1 k 1,i + Aµ k 1,i B Aµ k 1,i and 68 δ k,i = Σ 1 k 1,µ µ k 1,i + Aµ k 1,i B x Aµ k 1,i + Aµ k 1,i µ k 1,i 1 G k,i x, z = exp x Aµk 1,i Aµ k 1,i z µ k 1,i B Aµ k 1,i z µ k 1,i 1 Kx = exp δ ki σ ki δ ki µ k 1,iΣ 1 k 1,i µ k 1,i + x Aµ k 1,i + Aµ k 1,i µ k 1,i B x Aµ k 1,i + Aµ k 1,i µ k 1,i The problem is now to evaluate the integral exp 1 R z σ kiδ ki σ 1 ki z σ kiδ ki G ki x, zdz n As in Ito and Xiong, [5], one way is to use Gauss-Hermite quadrature of which the Julier-Uhlmann unscented filter is a special case That is, we assume and change coordinates writing The integral is then 69 S ki exp R n σ ki = S ki S ki z = S kit + σ ki δ ki, t R n t t Gx, S kit + σ ki δ ki dt Using the Gauss-Hermite formula this is approximately mk,i j=1 mk,i = where j=1 ω k,i,j Gx, S ki t j + σ ki δ ki x ω k,i,j exp Aµk 1,i Aµ k 1,i S kit j + σ ki δ ki µ k 1,i γk,i,j 610 A 1 µ k 1,i γ k,i,j = 1 B S kit j + σ ki δ ki µ k 1,i S ki t j + σ ki δ ki µ k 1,i A n µ k 1,i

12 14 ROBERT J ELLIOTT AND SIMON HAYKIN The order of approximation mk, i can be chosen Weights ω k,i,j are determined as in Golub [4] In fact if J is the symmetric tridiagonal matrix with a zero diagonal and J i,i+1 = i/, 1 i mk, i 1 then the t j are the eigenvalues of J and the ω k,i,j are given by v i 1, where v i 1 is the first element of the i th normalized eigenvector of J and As our integrand in 69 is not a N0, I n density in our case ω k,i,j = π n/ S ki ω k,i,j Write X i = x Aµ k 1,i and recall from 66 that Also write δ k,i = Σ 1 k 1,i µ k 1,i + Aµ k 1,i B X i + Aµ k 1,i µ k 1,i M kij = S ki t j + σ ki Σ 1 k 1,i µ k 1,i + Aµ k 1,i B Aµ k 1,i µ k 1,i µk 1,i Then F ki = σ ki Aµ k 1,i B 611 I k 1,i mk,i j=1 [ 1 ω k,i,j K ki exp Xi Aµ k 1,i M kij + F ki X i ] M kij + F ki X i Aµ k 1,i M kij + F ki X i Here again Aµ k 1,i denotes the matrix of Hessians Consider one term in this sum Retaining only terms of order x Aµ k 1,i = X i and dropping suffices this is: where: [ 1 ] ωkxexp X H 1 X + g X + κ 61 H 1 = H 1 kij = I AF B M AF B AM AF g = g kij = 1 I AF B M AM F AM B A M and κ = κ kij = M AB M AM

13 A NEW NONLINEAR FILTER 15 Therefore, I k 1,i mk,i j=1 ω kij K ki x exp [ 1 X i H 1 kij X i + g kij X i + κ kij ] where X i = x Aµ k 1,i Substituting in 63 α k x φ D 1 y k Cx B D φy k Nk 1 mk,i j=1 λ k 1,i ω kij K ki x Write exp [ 1 X ih 1 kij X i + g kijx i + κ kij ] 613 ρ kij = λ k 1,iω kij expκ kij B D φy k so α k x Nk 1 mk,i j=1 ρ kij φ D 1 y k Cx K ki xexp [ 1 X i H 1 kij X i + g kij X i] Consider one term only in the sum and replace Cx by its second order Taylor expansion about Aµ k 1,i Again, drop suffices to simplify the notation Then L kij x = Lx : = ρφ D 1 y k Cx Kx [ 1 exp x A H 1 x A + g x A] [ 1 ] ρkx exp X ih 1 X i + g X i [ exp 1 y C CXi 1 X i CX i D y C CX i 1 X i CX i Here again C = C Aµ k 1,i R m and C = C Aµ k 1,i R m n C 1 Aµk 1,i C = C m Aµk 1,i A = Aµ k 1,i R n X i = x Aµ k 1,i = x A

14 16 ROBERT J ELLIOTT AND SIMON HAYKIN Keeping only terms of order x A = X i this gives Lx ρkx exp 1 y C CXi D y C CX i 1 [ 1 ] exp y C D X i i CX exp X i H 1 X i + g X i As in 44 we can write as 1 exp y C D X i CX i 1 exp X i R 1 X i where R 1 = y k C Aµ k 1,i D C Aµ k 1,i Write Then dropping suffices and Z 1 := R 1 + H 1 [ 1 ] Lx ρkxexp X iz 1 X i + g X i exp 1 D y C CXi y C CX i α k x Nk 1 mi,j j=1 ρ k,i,j L k,i,j x Recalling X = X i = x Aµ k 1,i and substituting for Kx from 68 we have 1 [ L kij x = Lx ρ exp Σ 1 µ + AB X + Aµ σ k Σ 1 µ + AB X + Aµ µ Σ 1 µ + X + Aµ B X + Aµ ] 1 exp X Z 1 X + g X exp 1 D y C CX y C CX Collecting terms in X = X i = x Aµ k 1,i this is: = G exp 1 X Σ 1 X X

15 A NEW NONLINEAR FILTER 17 where 614 Σ 1 = Σ 1 kij = C Aµ k 1,i D C Aµ k 1,i + B B Aµ k 1,i σ k,i Aµ k 1,i B Z 1 kij and = kij 615 Further 616 = C Aµ k 1,i D y k C Aµ k 1,i B Aµ k 1,i µ k 1,i + g k,i,j + B Aµ k 1,i σ k,i Σ 1 k 1,i µ k 1,i + Aµ k 1,i B Aµ k 1,i µ k 1,i 1 G = G kij = ρ exp Σ 1 µ + A B A µ σ Σ 1 µ + AB A µ µ Σ 1 µ µ A B Aµ 1 y C D y C Completing the square we have 617 L k,i,j x = Lx λ k,i,j exp where 618 λ k,i,j = G kij exp Now as in 55 [ 1 ] X i Σ Σ 1 X i Σ 1 Σ Σ k k 1,i := Aµ k 1,i Σ k 1,i Aµ k 1,i + B is an approximate conditional variance of x k, given Y k 1, if Then, as in Corollary 5 so, from 614 x k 1 k 1 N µ k 1,i, Σ k 1,i Σ 1 k k 1,i = B B Aµ k 1,i σ k,i Aµ k 1,i B Σ 1 = Σ 1 k,i,j = C D C + Σ 1 k k 1,i Z 1 Write Γ 1 = Γ 1 k,i,j := Σ 1 k k 1,i Z 1 k,i,j so, using the MIL, Lemma Γ = Σ k k 1,i Σ k k 1,i Σk k 1,i Z k,i,j 1Σk k 1,i

16 18 ROBERT J ELLIOTT AND SIMON HAYKIN Also, as again using Lemma 51 Σ 1 = Σ 1 k,i,j = C D C + Γ 1 60 Σ = Σ k,i,j = Γ Γ C D + CΓ C 1 CΓ this is Finally we evaluate µ k,i,j = Σ k,i,j k,i,j Dropping suffices, from 614 and 615 this is [ Γ Γ C D + CΓ C 1 CΓ ] C D y C + Σ [ B Aσ Σ 1 µ + A B Aµ ] + Σg ΣB Aµ = [ Γ C Γ C D + CΓ C 1 CΓ C + D D ] D y C + Σ [ B A σ Σ 1 + A B A µ ] + Σg ΣB Aµ Recalling from 65 that Therefore, in terms of the variable the mean is σ = Σ 1 + A B A 1 = Γ C D + CΓ C 1 y C + Σg X = X i = x Aµ k 1,i Σ k,i,j k,i,j = Γ C D + CΓ C 1 y C + Σg so in terms of the original variable x the mean is = A + Γ C D + CΓ C 1 y C Σg That is, now writing in suffices, we have obtained a Gaussian density in the sum which determines α k x which has a variance Σ k,i,j = Γ k,i,j Γ k,i,j C Aµ k 1,i D + C Aµ k 1,i Γ k,i,j C Aµ k 1,i 1 C Aµk 1,i Γk,i,j,

17 A NEW NONLINEAR FILTER 19 where Γ 1 k,i,j = Σ 1 k k 1,i Z 1 k,i,j, and a mean µ k,i,j = Aµ k 1,i + Γ k,i,j C Aµ k 1,i D + C Aµ k 1,i Γ k,i,j C Aµ k 1,i 1 y k C Aµ k 1,i Σ k,i,j g k,i,j Here g k,i,j is given by 61 Also, the new weights λ k,i,j are given by 618 Therefore, α k x Nk 1 mk,i j=1 λ k,i,j exp 1 x µ k,i,j Σ 1 k,i,j x µ k,i,j Pruning can be effected by selecting only so many of the terms based on the size of the λ k,i,j 7 Conclusion A discrete time version of the Zakai equation has been obtained Using first order linearizations the equation can be solved and the extended Kalman filter derived Second order Taylor expansions of the non-linear terms were then considered and approximate solutions of the Zakai equation derived in terms of Gaussian sums Formulae for updating the means, variances and weights of the terms in the sums are given Detailed proofs will appear in [] REFERENCES [1] R J Elliott, L Aggoun, and J B Moore, Hidden Markov Models: Estimation and Control, Springer-Verlag, Berlin, Heidelberg, New York, Second edition 006 [] R J Elliott and S Haykin, An extended EKF, University of Calgary preprint 006, to appear [3] R J Elliott and V Krishnamurthy, New finite dimensional filters for parameter estimation of discrete time linear Gaussian models IEEE Trans Automatic Control, , pp [4] G H Golub, Some modified matrix eigenvalue problems, SIAM Review, , pp [5] K Ito and K Xiong,Gaussian filters for non-linear filtering problems, IEEE Trans Automatic Control, 45000, pp [6] H Kushner and A S Budhiraja, A nonlinear filtering algorithm based on an approximation of the conditional distribution, IEEE Trans Automatic Control, 45000, pp

18 0 ROBERT J ELLIOTT AND SIMON HAYKIN