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Prediction Error Methods and Pseudo-Linear Regressions System Identification Lecture 10: Prediction error methods and pseudo-linear regressions Roy Smith 2016-11-22 10.1 Prediction e k p q H z p q v k y k p q u k p q G z p q ` p q Typical assumptions G z and H z are stable, p q p q H z is stably invertible (no zeros outside the unit disk) p q e k has known statistics: known pdf or known moments. p q One-step ahead prediction Given ZK u 0 , y 0 , . , u K 1 , y K 1 , “ t p q p q p ´ q p ´ qu what is the best estimate of y K ? p q 2016-11-22 10.2 Prediction v k e k p q H z p q p q Noise model invertibility Given, v k , k 0,...,K 1, can we determine e k , k 0,...,K 1? p q “ ´ p q “ ´ 8 Inverse filter: Hinv z : e k hinv i v k i p q p q “ i 0 p q p ´ q ÿ“ We also want the inverse filter to be causal and stable: 8 hinv k 0, k 0, and hinv k . p q “ ă k 0 | p q| ă 8 ÿ“ If H z has no zeros for z 1, then, p q | | ě 1 Hinv z . p q “ H z p q 2016-11-22 10.3 Prediction v k e k p q H z p q p q One step ahead prediction Given measurements of v k , k 0,...,K 1, can we predict v K ? p q “ ´ p q Assume that we know H z , how much can we say about v K ? p q p q Assume also that H z is monic (h 0 1). p q p q “ 8 v k h i e k i p q “ i 0 p q p ´ q ÿ“ 8 e k h i e k i “ p q ` i 1 p q p ´ q ÿ“ m k 1 “ p ´ q loooooooomoooooooon“observed” 2016-11-22 10.4 Prediction v k e k p q H z p q p q One-step ahead prediction The prediction of v k , based on measurements up to time k 1 is, p q ´ vˆ k k 1 . p | ´ q We will argue that a good choice in this case is, 8 vˆ k k 1 m k 1 h i e k i . p | ´ q “ p ´ q “ i 1 p q p ´ q ÿ“ The error in our prediction is e k — which we clearly can’t reduce. p q 2016-11-22 10.5 One-step prediction statistics General case Say e k is identically distributed with pdf: fe x , p q p q x δx ` Prob x e k x δx fe x dx fe x δx. t ď p q ď ` u “ p q « p q żx A posteriori distribution k 1 What are the statistics of v k given v ´ v , . , v k 1 ? p q ´8 “ t p´8q p ´ qu k 1 Prob x v k x δx v ´ t ď p q ď ` | ´8 u “ Prob x e k m k 1 x δx “ t ď p q ` p ´ q ď ` u Prob x m k 1 e k x m k 1 δx “ t ´ p ´ q ď p q ď ´ p ´ q ` u fe x m k 1 δx. « p ´ p ´ qq 2016-11-22 10.6 One-step ahead prediction statistics Maximum of the conditional (a posteriori) distribution Select the prediction estimate as the peak value of the conditional distribution: vˆ k k 1 argmax fe x m k 1 p | ´ q “ x p ´ p ´ qq m k 1 for the Gaussian case. “ p ´ q This is the most probable value of v k k 1 . p | ´ q Mean of the conditional distribution Select the prediction estimate as the mean value of the conditional distribution: k 1 vˆ k k 1 E v k v ´ E e k m k 1 p | ´ q “ t p q| ´8 u “ t p q ` p ´ qu m k 1 E e k m k 1 . “ p ´ q ` t p qu “ p ´ q This is the expected value of v k k 1 . p | ´ q 2016-11-22 10.7 One-step ahead prediction Calculation 8 vˆ k k 1 m k 1 h i e k i p | ´ q “ p ´ q “ i 1 p q p ´ q ÿ“ H z 1 e k (assuming H z is monic) “ p p q ´ q p q p q H z 1 p q ´ v k “ H z p q p q 1 Hinv z v k “ p ´ p qq p q 8 hinv i v k i “ ´ i 1 p q p ´ q ÿ“ Note that vˆ k k 1 depends only on values up to time k 1. p | ´ q ´ The best we can do is: k 8 vˆ k k 1 hinv i v k i hinv i v k i . p | ´ q “ ´ i 1 p q p ´ q « ´ i 1 p q p ´ q ÿ“ ÿ“ 2016-11-22 10.8 Example Moving average model 1 v k e k ce k 1 , H z 1 cz´ . p q “ p q ` p ´ q ùñ p q “ ` For H z to be stably invertible we require c 1. p q | | ă 1 8 i i H z c z´ . inv 1 cz 1 p q “ ´ “ i 0p´ q ` ÿ“ One-step ahead predictor 8 i vˆ k k 1 1 Hinv z v k c v k i p | ´ q “ p ´ p qq p q “ ´ i 1p´ q p ´ q ÿ“ k c iv k i « ´ i 1p´ q p ´ q ÿ“ cv k 1 c2v k 2 c3v k 3 c kv 0 . “ p ´ q ´ p ´ q ` p ´ q ` ¨ ¨ ¨ ´ p´ q p q 2016-11-22 10.9 Example Moving average model 1 v k e k ce k 1 , H z 1 cz´ . p q “ p q ` p ´ q ùñ p q “ ` Recursive formulation Note that, H z vˆ k k 1 H z 1 v k p q p | ´ q “ p p q ´ q p q So, vˆ k k 1 cvˆ k 1 k 2 cv k 1 p | ´ q ` p ´ | ´ q “ p ´ q vˆ k k 1 c v k 1 vˆ k 1 k 2 p | ´ q “ p p ´ q ´ p ´ | ´ qq k 1 (prediction error at k 1) p ´ q ´ c looooooooooooooooomooooooooooooooooonk 1 “ p ´ q 2016-11-22 10.10 Another example Autoregressive noise model Our noise model is: 8 v k aie k i a 1 for stability. p q “ i 0 p ´ q | | ă ÿ“ 8 i i 1 So, H z a z´ , 1 az 1 p q “ i 0 “ ´ ÿ“ ´ 1 and Hinv z 1 az´ (a moving average process) p q “ ´ Our one-step ahead predictor is, vˆ k k 1 1 Hinv z v k av k 1 . p | ´ q “ p ´ p qq p q “ p ´ q 2016-11-22 10.11 Output prediction e k p q H z y k G z u k v k p q p q “ p q p q ` p q v k y k p q u k p q G z p q ` p q One-step ahead prediction Maximise the expected value of the conditional distribution, yˆ k k 1 E y k ZK G z u k vˆ k k 1 p | ´ q “ t p q| u “ p q p q ` p | ´ q G z u k 1 Hinv z v k “ p q p q ` p ´ p qq p q Hinv z G z u k 1 Hinv z y k “ p q p q p q ` p ´ p qq p q 2016-11-22 10.12 Output prediction e k p q H z y k G z u k v k p q p q “ p q p q ` p q v k y k p q u k p q G z p q ` p q Prediction error y k yˆ k k 1 Hinv z G z u k Hinv z y k p q ´ p | ´ q “ ´ p q p q p q ` p q p q Hinv z y k G z u k Hinv z v k “ p qp p q ´ p q p qq “ p q p q e k “ p q The innovation is the part of the output prediction that cannot be estimated from past measurements. 2016-11-22 10.13 Prediction error based identification The one-step ahead predictor is parametrised by θ, yˆ k θ, ZK Hinv θ, z G θ, z u k 1 Hinv θ, z y k p | q “ p q p q p q ` p ´ p qq p q Define a parametrised prediction error, k, θ y k yˆ k, θ , p q “ p q ´ p q which we can optionally filter, F k, θ F z k, θ (weighted error). p q “ p q p q Define a cost function, K 1 1 ´ J θ, Z l k, θ typically l k, θ k, θ . K K F F F 2 p q “ k 0 p p qq p p qq “ } p q} ÿ“ θˆ argmin J θ, ZK . “ θ p q 2016-11-22 10.14 Prediction error methods: ARX models e k p q B θ, z G θ, z p q , p q “ A θ, z 1 p q A θ, z 1 p q H θ, z , A θ, z v k p q “ p q p q y k B θ, z u k p q p q p q ` A θ, z p q yˆ k θ Hinv θ, z G θ, z u k 1 Hinv θ, z y k p | q “ p q p q p q ` p ´ p qq p q B z u k 1 A z y k “ p q p q ` p ´ p qq p q θT φ k φT k θ.
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