1 Why Recursions for Estimation?

Transcription

1 Avd. Matematisk statistik TIDSSERIEANALYS SF2945 ON KALMAN RECURSIONS FOR PREDICTION Timo Koski 1 Why Recursions for Estimation? In the preceding lectures of sf2945 we have been dealing with the various of instances (e.g., linear prediction of a stationary time series) of the general problem of estimating X with a linear combination a 1 Y a N Y N selecting the parameters a 1,...,a N so that E [ (X (a 1 Y a N Y N )) 2] (1.1) is minimized (Minimal Mean Square Error). The following has been shown. Suppose Y 1,..., Y N and X are random variables, all with zero means, and γ mk = γ om = = E [Y m Y k ],m = 1,...,N; k = 1,...,N = E [Y m X], m = 1,...,N. (1.2) then [ E (X (a 1 Y a N Y N )) 2] (1.3) is minimized if the coefficients a 1,..., a N satisfy the system of N N equations N a k γ mk = γ om ; m = 1,...,N. (1.4) Let us now augment the notations by a dependence on the number of data points, N, as Ẑ N = a 1 (N)Y a N (N)Y N.

2 Suppose now that we get one more measurement, Y N+1. Then we need to find Ẑ N+1 = a 1 (N + 1)Y a N+1 (N + 1)Y N+1. In principle we can by the above find a 1 (N +1),..., a N+1 (N +1) by solving the system of (N + 1) (N + 1) equations N+1 a k (N + 1)r mk = r om ; m = 1,..., N + 1. (1.5) but if new data Y N+1, Y N+2,... is being obtained sequentially in real time, this will soon become practically unfeasible. Kalman filtering is a technique by which we calculate ẐN+1 recursively using ẐN, and the latest sample Y N+1. This requires a time series model (e.g., ARMA) for X. We consider the simplest special case. 2 Observing an AR(1) -Process State in Additive White Noise The state model is AR(1), i.e., X n = φx n 1 + Z n 1, n = 0, 1, 2,..., (2.1) where {Z n } is a white noise with expectation zero, and φ 1 (so that we may regard Z n 1 as non correlated with X n 1, X n 2,...). We have { σ R Z (k) = E [Z n Z n+k ] = σ 2 δ n,n+k = σ 2 2 k = 0 δ 0,k = (2.2) 0 k 0 where δ n,n+k is Kronecker s delta. We assume that E [X 0 ] = 0 and Var(X 0 ) = σ 2 0. The true state X n is not observed directly (X n is hidden from us), we see X n with added white measurement noise V n, or, as Y n in where Y n = cx n + V n, n = 0, 1, 2,..., (2.3) R V (k) = E [V n V n+k ] = σ 2 V δ n,n+k = σ 2 V δ 0,k = { σ 2 V k = 0 0 k 0 (2.4) The state and measurement noises {Z n } and {V n }, respectively, are independent.

3 3 The Recursive One-Step Prediction of an AR(1) -Process in Additive White Noise 3.1 Recursive Prediction We want to estimate X n in (2.1) using Y 0,..., Y n 1 accrued according to (2.3), so that E [ (X n (a 1 (n)y n a n (n)y 0 )) 2] (3.1) is minimized. We use a k (n) to denote that the best estimate depends on n. Next, we obtain Y n and want to estimate X n+1 using Y 0,...,Y n so that E [ (X n+1 (a 1 (n + 1)Y n a n+1 (n + 1)Y 0 )) 2] (3.2) is minimized. Let us set or and or X n+1 := a 1 (n + 1)Y n a n+1 (n + 1)Y 0 n+1 X n+1 := a k (n + 1)Y n+1 k, (3.3) X n := a 1 (n)y n a n (n)y 0 X n := a k (n)y n k. (3.4) By the general theory summarized above a 1 (n + 1),...,a n+1 (n + 1) satisfy n+1 a k (n + 1)E [Y m Y n+1 k ] = E [Y m X n+1 ];m = 0,..., n. (3.5) The course of action is as follows: we express a k (n + 1) for k = 2,...,n + 1 recursively as functions of a k (n)s. Then we show that X n+1 = φ X n + a 1 (n + 1) (Y n c X ) n, find a recursion for e n+1 = X n+1 X n+1, and determine finally a 1 (n+1) by minimizing E [ e 2 n+1]. We do this in steps of auxiliary statements (lemmas in the next subsection).

4 3.2 Lemmas for Kalman Recursions for One-Step Prediction of an AR(1) -Process in Additive White Noise Lemma 3.1 and for n 1 E [Y m X n+1 ] = φe [Y m X n ], m = 0,...,n 1. (3.6) E [Y m X n ] = E [Y my n ] ; m = 0,...,n 1. (3.7) c Proof: From the state equation (2.1) we get E [Y m X n+1 ] = E [Y m (φx n + Z n )] = φe [Y m X n ] + E [Y m Z n ]. Here E [Y m Z n ] =E [Y m ]E [Z n ] = 0, since Z n is a white noise independent of V m and X m for m n. Next, from (2.3) E [Y m X n ] = E [Y m (Y n V n ) /c] = E [Y m Y n ] /c, as claimed. The following lemma does the crucial task in our chosen effort. We start from (3.5), and use (3.6) and (3.7) to get an expression that will give us an opportunity to use the fact that a k (n) satisfy a k (n)e [Y m Y n k ] = E [Y m X n ] ; m = 0,...,n. (3.8) This lemma still leaves us with one free parameter, i.e., a 1 (n + 1), which will be determined in the sequel. Lemma 3.2 a k+1 (n + 1) = a k (n) (φ a 1 (n + 1)c);k = 1,...,n. (3.9)

5 Proof: Let m < n. From (3.3) and (3.5) we get n+1 E [Y m X n+1 ] = a k (n + 1)E [Y m Y n+1 k ] (3.10) n+1 = a 1 (n + 1)E [Y m Y n ] + a k (n + 1)E [Y m Y n+1 k ] (3.11) We change the index of summation from k to l = k 1. Then k=2 n+1 a k (n + 1)E [Y m Y n+1 k ] = k=2 a l+1 (n + 1)E [Y m Y n l ]. (3.12) l=1 On the other hand, from (3.6) we get in the left hand side of (3.10) that E [Y m X n+1 ] = φe [Y m X n ] and from (3.7) that a 1 (n + 1)E [Y m Y n ] = a 1 (n + 1)cE [Y m X n ]. Thus we can write (3.10), (3.11) by means of (3.12) as (φ a 1 (n + 1)c) E [Y m X n ] = a l+1 (n + 1)E [Y m Y n l ]. (3.13) l=1 But now, ladies and gentlemen, we compare with the system of equations in (3.8), i.e., a k (n)e [Y m Y n k ] = E [Y m X n ] ; m = 1,...,n. It must hold that or a k (n) = a k+1(n + 1) (φ a 1 (n + 1)c) a k+1 (n + 1) = a k (n) (φ a 1 (n + 1)c). We can keep a 1 (n + 1) undetermined and still get the following result.

6 Lemma 3.3 X n+1 = φ X n + a 1 (n + 1) (Y n c X ) n (3.14) Proof: By (3.3) and the same trick of changing the index of summation as done above we obtain n+1 X n+1 = a k (n + 1)Y n+1 k = a 1 (n + 1)Y n + so from (3.9) in the preceding lemma = a 1 (n + 1)Y n + = a 1 (n + 1)Y n + φ a k+1 (n + 1)Y n k a k (n) [φ a 1 (n + 1)c]Y n k a k (n)y n k ca 1 (n + 1) From (3.4) we obviously get in the right hand side = φ X n + a 1 (n + 1) (Y n c X ) n, a k (n)y n k. as was to be proved. We must choose the value for a 1 (n+1) which minimizes the second moment of the prediction error defined as e n+1 = X n+1 X n+1. First we find recursions for e n+1 and E [ e 2 n+1]. Lemma 3.4 and e n+1 = [φ a 1 (n + 1)c]e n + Z n a 1 (n + 1)V n, (3.15) E [ e 2 n+1] = [φ a1 (n + 1)c] 2 E [ e 2 n] + σ 2 + a 2 1(n + 1)σ 2 V. (3.16) Proof : We have from the equations above that e n+1 = X n+1 X n+1 = φx n + Z n φ X n a 1 (n + 1) (cx n + V n c X ) n = φ (X n X ) n a 1 (n + 1)c (X n X ) n + Z n a 1 (n + 1)V n,

7 and with e n = X n X n the result is (3.15). If we square both sides of (3.15) we get where e 2 n+1 = [φ a 1(n + 1)c] 2 e 2 n + (Z n a 1 (n + 1)V n ) 2 + 2τ, τ = [φ a 1 (n + 1)c] e n (Z n (a 1 (n + 1))V n ) By the properties stated in section 2 we get that E [e n Z n ] = E [Z n V n ] = E [e n V n ] = 0 (Note that X n uses X n 1, X n 2,... and is thus noncorrelated with V n.) Thus we get E [ e 2 n+1] as asserted. We can apply orthogonality to find the expression for a 1 (n + 1), but differentiation to be invoked in the next lemma is faster. Lemma 3.5 minimizes E [ e 2 n+1]. a 1 (n + 1) = φce [e2 n ] σv 2 + (3.17) c2 E [e 2 n ]. Proof: We differentiate (3.16) w.r.t. a 1 (n + 1) and set the derivative equal to zero. This entails or 2c [φ a 1 (n + 1)c] E [ e 2 n] + 2a1 (n + 1)σ 2 V = 0 a 1 (n + 1) ( c 2 E [ e 2 n] + σ 2 V ) = φce [ e 2 n ], which yields the solution as claimed. Next we summarize and comment the formulas obtained above. 4 The result: The Kalman Filter for One- Step Prediction of an AR(1) -Process in Additive White Noise 4.1 Summary of the Algorithm and Innovations Representation The final expressions may look messy, but this should not obstruct us from realizing the fact that the equations are readily implemented in software and/or hardware.

8 We change notation for a 1 (n + 1) determined in lemma 3.5, (3.17). The traditional way (in the engineering literature) is to write this quantity as the predictor gain, also known as the Kalman gain K(n) or The initial conditions are Then we have obtained above that and K(n) := φce [e2 n] σv 2 + (4.1) c2 E [e 2 n ]. X 0 = 0, E [ e 2 0] = σ 2 0. (4.2) E [ e 2 n+1] = [φ K(n)c] 2 E [ e 2 n] + σ 2 + K 2 (n)σ 2 V, (4.3) X n+1 = K(n) [Y n c X ] n + φ X n. (4.4) This shows that the filter processes one measurement Y n at a time, instead of all measurements. The data preceding Y n, i.e., Y 0,...,Y n 1 are summarized in X n. The equations (4.1) -(4.4) are an algorithm for recursive computation of Xn+1 for all n 0. This forms a simple case of an important statistical processor of time series data known as a Kalman filter. Most importantly the principles used above apply to recursive filtering with multivariate and timevariant state and noise statistics. A Kalman filter is in view of (4.4) based on predicting X n+1 using φ X n and correcting the prediction by the measured innovations ε n = [Y n c X ] n. Here ε n is the part of Y n which is not exhausted by c X n. The innovations are modulated by the filter gains K(n) that depend on the error variances E [e 2 n ]. We can write the filter also with an innovations representation X n+1 = φ X n + K(n)ε n (4.5) Y n = c X n + ε n, (4.6) which by comparison with (2.1) and (2.3) shows that the Kalman filter follows equations similar to the original ones, but is driven by the innovations ε n as noise.

9 4.2 Riccati Recursion for the Variance of the Prediction Error It turns out to be useful to find a further recursion (c.f., (3.16)) for E [ en+1] 2. We start with a general identity for Minimal Mean Square Error estimation. We have E [ ] [ ] [ ] e 2 n+1 = E X 2 n+1 E X2 n+1. (4.7) To see this, let us note that E [ [ ] ( e 2 n+1 = E X n+1 X ) ] 2 n+1 = E [ ] ] [ ] Xn+1 2 2E [X n+1 Xn+1 + E X2 n+1. Here ] [( ) ] E [X n+1 Xn+1 = E Xn+1 + e n+1 Xn+1 = E [ ] ] X2 n+1 + E [e n+1 Xn+1. But by orthogonality principle of Minimal Mean Square Error estimation we have ] E [e n+1 Xn+1 = 0, and this proves (4.7). Next we insert from the state equation (2.1) in the right hand side of (4.7) to get E [ X 2 n+1] = φ 2 E [ X 2 n] + σ 2. (4.8) We note next writing that ε n = Y n c X n = cx n + V n c X n = ce n + V n E [ ε 2 n] = σ 2 V + c 2 E [ e 2 n]. (4.9) When we use this formula we get from (4.4) or X n+1 = K(n)ε n + φ X n that [ ] [ ] E X2 n+1 = φ 2 E X2 n + K(n) ( 2 σv 2 + c2 E [ en]) 2. (4.10) But in view of our definition of the Kalman gain in (4.1) we have K(n) 2 ( σ 2 V + c2 E [ e 2 n ]) = φ 2 c 2 E [e 2 n] 2 σ 2 V + c2 E [e 2 n]

10 As in [1, p.273] we set and Hence we have from (4.8) and (4.10) or, again using (4.7), θ n = φce [ e 2 n], (4.11) n = σ 2 V + c2 E [ e 2 n], (4.12) E [ e 2 n+1] = φ 2 E [ X 2 n] + σ 2 φ 2 E E [ e 2 n+1] = φ 2 E [ e 2 n [ ] X2 n θ2 n n ] + σ 2 θ2 n n. (4.13) Hence we have shown the following proposition (c.f., [1, p.273]) about Kalman prediction. Proposition 4.1 and if then where and X n+1 = φ X n + θ n n ε n. (4.14) e n+1 = X n+1 X n+1 E [ e 2 n+1] = φ 2 E [ e 2 n ] + σ 2 θ2 n n, (4.15) θ n = φce [ e 2 n], (4.16) n = σ 2 V + c 2 E [ e 2 n]. (4.17) The recursion in (4.15) is a first order nonlinear difference equation known as the Riccati equation 1 for the prediction error variance. 1 named after Jacopo Francesco Riccati, , who was a mathematician born in Venice, who wrote on philosophy, physics and differential equations. history/biographies/riccati.html

11 4.3 Stationary Kalman Filter We say that the prediction filter has reached a steady state, if E [e 2 n] is a constant, say P = E [e 2 n ], that does not depend on n. Clearly this requires, e.g., that the coefficients in the model are not functions of n. Then the Riccati equation in (4.15) becomes a quadratic algebraic Riccati equation or and further by some algebra P = φ 2 P + σ 2 φ2 c 2 P 2 σ 2 V + c2 P, (4.18) P = σ 2 + σ2 V φ2 P σ 2 V + c2 P, c 2 P 2 + ( σ 2 V σ 2 V φ 2 σ 2 c 2) P σ 2 σ 2 V = 0. (4.19) This can be solved in a well known manner. We take only the non negative root into account. Given the stationary P we have the stationary Kalman gain as K := φcp σv 2 + c2 P. (4.20) 4.4 Examples on Kalman Prediction Example 4.2 Consider a discrete time Brownian motion (Z n is a Gaussian white noise) or a random walk observed in noise Then and X n+1 = X n+1 = X n + Z n, n = 0, 1, 2,..., Y n = X n + V n, n = 0, 1, 2,..., X n+1 = X n + E [e2 n] σ 2 V + E [e2 n ] ( Y n X n ). ( 1 E [e2 n ] ) σv 2 + E [e2 n ] X n + E [e2 n ] σ 2 V + E [e2 n ]Y n. It can be shown that there is convergence to the stationary filter ( X n+1 = 1 P ) σv 2 + P X n + P σv 2 + P Y n,

12 where P is found by solving (4.19). We have in this example φ = 1, c = 1. We select σ 2 = 1, σ 2 V = 1. Then (4.19) becomes P 2 P 1 = 0 P = = 1.618, and the stationary Kalman gain is K = In the first figure there is depicted a simulation of the state process and the computed trajectory of the Kalman predictor. (If you check this document in the course homepage, the Kalman predictor has the green path.) In the next figure we see the fast convergence of the Kalman gain

13 K(n) to K = Example 4.3 Consider an exponential decay, 0 < φ < 1, observed in noise X n+1 = φx n, n = 0.1, 2,..., Y n = X n + V n, n = 0, 1, 2,.... Then there is convergence to the stationary filter ( X n+1 = φ 1 P ) σv 2 + P X n + φp σv 2 + P Y n. Here c = 1 and let us take in this example φ = 0.9. We have σ 2 = 0, and select σv 2 = 1. Then (4.19) becomes P 2 + ( ) P = 0 P = 0, (where we neglect the negative root) and the stationary Kalman gain is K = 0. In the figures there is first depicted the state process and the computed trajectory of the Kalman predictor. (If you check this document in the course homepage, the Kalman predictor follows the green path.) In the next figure we see the fast convergence of the Kalman gain K(n) to

14 K = 0. We see the corresponding noisy measurements of the exponential decay in the final graph.

15 5 Notes and Literature on the Kalman Filter The recursive algorithm was invented by Rudolf E. Kalman 2. His original work is found in [3]. The Kalman filter was first applied to the problem of trajectory estimation for the Apollo space program of the NASA (in the 1960s), and incorporated in the Apollo space navigation computer. Perhaps the most commonly used type of Kalman filter is nowadays the phase-locked loop found everywhere in radios, computers, and nearly any other type of video or communications equipment. New applications of the Kalman Filter (and of its extensions like particle filtering) continue to be discovered, including for radar, global positioning systems (GPS), hydrological modelling, atmospheric observations and automated drug delivery systems (?). The presentation in this lecture above is to a large degree based on the treatment in [2]. 2 b in Budapest, but studied and graduated in electrical engineering in USA, Professor Emeritus in Mathematics at ETH, the Swiss Federal Institute of Technology in Zürich.

16 A Kalman filter webpage with lots of links to literature, software, and extensions (like particle filtering) is found at welch/kalman/ It has been understood only recently that the Danish mathematician and statistician Thorvald N. Thiele 3 discovered the principle (and a special case) of the Kalman filter in his book published in Copenhagen in 1889 (!): Forelæsningar over Almindelig Iagttagelseslære: Sandsynlighedsregning og mindste Kvadraters Methode. A translation of the book and an exposition of Thiele s work is found in [4]. Referenser [1] P.J. Brockwell and R.A. Davies: Introduction to Time Series and Forecasting. Second Edition Springer New York, [2] R.M. Gray and L.D. Davisson: An introduction to statistical signal processing. Cambridge University Press, Cambridge, U.K., [3] R.E. Kalman: A New Approach to Linear Filtering and Prediction Problems. Transactions of the ASME Journal of Basic Engineering, 1960, 82, (Series D): [4] S.L. Lauritzen: Thiele: pioneer in statistics. Oxford University Press, , a short biography is in history/biographies/thiele.html