Time Series Analysis

Transcription

1 Time Series Analysis Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1

2 Outline of the lecture Input-Output systems The z-transform important issues from Sec. 4.4 Cross Correlation Functions from Sec Transfer function models; identification, estimation, validation, prediction, Chap. 8 2

3 The z-transform A way to describe dynamical systems in discrete time Z({x t }) = X(z) = t= x t z t (z complex) The z-transform of a time delay: Z({x t τ }) = z τ X(z) The transfer function of the system is called H(z) = h t z t t= y t = k= h k x t k Y (z) = H(z)X(z) Relation to the frequency response function: H(ω) = H(e iω ) 3

4 Cross covariance and cross correlation functions Estimate of the cross covariance function: C XY (k) = C XY ( k) = 1 N 1 N (X t X)(Y t+k Y ) N k t=1 (X t+k X)(Y t Y ) N k t=1 Estimate of the cross correlation function: ρ XY (k) = C XY (k)/ C XX (0)C Y Y (0) If at least one of the processes is white noise and if the processes are uncorrelated then ρ XY (k) is approximately normally distributed with mean 0 and variance 1/N 4

5 Systems without measurement noise Xt Y t System input output Y t = h i X t i i= Given γ XX and the system description we obtain γ Y Y (k) = γ XY (k) = h i h j γ XX (k j + i) (1) i= j= i= h i γ XX (k i). (2) 5

6 Systems with measurement noise N t X t input System Σ Y t output Y t = i= h i X t i + N t. 6

7 Time domain relations Given γ XX and γ NN we obtain γ Y Y (k) = γ XY (k) = h i h j γ XX (k j + i) + γ NN (k) (3) i= j= i= h i γ XX (k i). (4) IMPORTANT ASSUMPTION: No feedback in the system. 7

8 Spectral relations f Y Y (ω) = H(e iω )H(e iω )f XX (ω) + f NN (ω) = G 2 (ω)f XX (ω) + f NN (ω), f XY (ω) = H(e iω )f XX (ω) = H(ω)f XX (ω). The frequency response function, which is a complex function, is usually split into a modulus and argument H(ω) = H(ω) e iarg{h(ω)} = G(ω)e iφ(ω), where G(ω) and φ(ω) are the gain and phase, respectively, of the system at the frequency ω from the input {X t } to the output {Y t }. 8

9 Estimating the impulse response The poles and zeros characterize the impulse response (Appendix A and Chapter 8) If we can estimate the impulse response from recordings of input an output we can get information that allows us to suggest a structure for the transfer function True Impulse Response Lag SCCF Lag SCCF after pre whitening Lag 9

10 Estimating the impulse response On the previous slide we saw that we got a good picture of the true impulse response when pre-whitening the data The reason is γ XY (k) = i= h i γ XX (k i) and only if {X t } is white noise we get γ XY (k) = h k σ 2 X Therefore if {X t } is white noise the SCCF ˆρ XY (k) is proportional to ĥk Normally {X t } is not white noise we fix this using pre-whitening 10

11 Pre-whitening a) A suitable ARMA-model is applied to the input series: ϕ(b)x t = θ(b)α t. b) We perform a prewhitening of the input series α t = θ(b) 1 ϕ(b)x t c) The output series {Y t } is filtered with the same model, i.e. β t = θ(b) 1 ϕ(b)y t. d) Now the impulse response function is estimated by ĥ k = C αβ (k)/c αα (0) = C αβ (k)/sα. 2 11

12 Example using S-PLUS ## ARMA structure for x; AR(1) x.struct <- list(order=c(1,0,0)) ## Estimate the model (check for convergence): x.fit <- arima.mle(x - mean(x), model=x.struct) ## Extract the model: x.mod <- x.fit$model ## Filter x: x.start <- rep(mean(x), 1000) x.filt <- arima.sim(model=list(ma=x.mod$ar), innov=x, start.innov = x.start) ## Filter y: y.start <- rep(mean(y), 1000) y.filt <- arima.sim(model=list(ma=x.mod$ar), innov=y, start.innov = y.start) ## Estimate SCCF for the filtered series: acf(cbind(y.filt, x.filt)) 12

13 Graphical output Multivariate Series : cbind(y.filt, x.filt) y.filt y.filt and x.filt ACF x.filt and y.filt x.filt ACF Lag Lag 13

14 Systems with measurement noise N t X t input System Σ Y t output Y t = i= h i X t i + N t. γ XY (k) = i= h i γ XX (k i) 14

15 Transfer function models ε t θ(b) B b ϕ(b) N t Xt b ω(b) B δ(b) Σ Y t Y t = ω(b) δ(b) Bb X t + θ(b) ϕ(b) ε t Also called Box-Jenkins models Can be extended to include more inputs see the book. 15

16 Some names FIR: Finite Impulse Response ARX: Auto Regressive with external input ARMAX/CARMA: Auto Regressive Moving Average with external input / Controlled ARMA OE: Output Error Regression models with ARMA noise 16

17 Identification of transfer function models h(b) = ω(b)bb δ(b) = h 0 + h 1 B + h 2 B 2 + h 3 B 3 + h 4 B Using pre-whitening we estimate the impulse response and guess an appropriate structure of h(b) based on this (see page 197 for examples). It is a good idea to experiment with some structures. Matlab (use q 1 instead of B): A = 1; B = 1; C = 1; D = 1; F = [ ]; mod = idpoly(a, B, C, D, F, 1, 1) impulse(mod) PEZdemo (complex poles/zeros should be in pairs): 17

18 2 exponential 2 1.8B 1 1.8B+0.81B 2 Zeros Poles Im Im Re Re Impulse Response Step Response Lag Lag 18

19 2 real poles B+0.72B 2 Zeros Poles No Zeros Im Im Re Re Impulse Response Step Response Lag Lag 19

20 2 complex B+0.81B 2 Zeros Poles No Zeros Im Im Re Re Impulse Response Step Response Lag Lag 20

21 1 exp + 2 comp Zeros B+0.69B B+2.02B B 3 Poles Im Im Re Re Impulse Response Step Response Lag Lag 21

22 Identification of the transfer function for the noise After selection of the structure of the transfer function of the input we estimate the parameters of the model Y t = ω(b) δ(b) Bb X t + N t We extract the residuals {N t } and identifies a structure for an ARMA model of this series N t = θ(b) ϕ(b) ε t ϕ(b)n t = θ(b)ε t We then have the full structure of the model and we estimate all parameters simultaneously 22

23 Estimation Form 1-step predictions, treating the input {X t } as known in the future (if {X t } is really stochastic we condition on the observed values) Select the parameters so that the sum of squares of these errors is as small as possible If {ε t } is normal then the ML estimates are obtained For FIR and ARX models we can write the model as Y t = X T t θ + ε t and use LS-estimates Moment estimates: Based on the structure of the transfer function we find the theoretical impulse response and we make a match with the lowest lags in the estimated impulse response Output error estimates... 23

24 Model validation As for ARMA models with the additions: Test for cross correlation between the residuals and the input ˆρ εx (k) Norm(0,1/N) which is (approximately) correct when {ε t } is white noise and when there is no correlation between the input and the residuals A Portmanteau test can also be performed 24

25 Prediction Ŷt+k t We must consider two situations The input is controllable, i.e. we can decide it and we can predict under different input-scenarios. In this case the prediction error variance is originating from the ARMA-part only (N t ). The input is only known until the present time point t and to predict the output we must predict the input. In this case the prediction error variance depend also on the autocovariance of the input process. In the book the case where the input can be modelled as an ARMA-process is considered. 25

26 Prediction (cont nd) Ŷ t+k t = k 1 i=0 h i X t+k i t + h i X t+k i + N t+k t. i=k Y t+k Ŷt+k t = k 1 i=0 h i (X t+k i X t+k i t ) + N t+k N t+k t If the input is controllable then X t+k i t = X t+k i The book also considers the case where output is known until time t and input until time t + j 26

27 Prediction (cont nd) We have N t = ψ i ε t i i=0 And if we model the input as an ARMA-process we have X t = ψ i η t i i=0 An thereby we get: V [Y t+k Ŷt+k t] = σ 2 η k 1 l=0 i 1 +i 2 =l h i1 ψ i2 2 + σ 2 ε k 1 ψi 2 i=0 27

28 Y t = B X t ε t, σ 2 ε = y x h xf N y x h xf N NA NA NA NA To forecast y (9,10) we must filter x as in xf, calc. N for the historic data, forecast N and add that to xf (future values) > Nfc <- arima.forecast(n[1:8], model=list(ar=0.4), sigma2=0.036, n=2) > Nfc$mean: [1]

29 Intervention models I t = { 1 t = t0 0 t t 0 Y t = ω(b) δ(b) I t + θ(b) φ(b) ε t See a real life example in the book. 29