STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2)

Size: px

Start display at page:

Download "STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2)"

Baldwin Gilmore
5 years ago
Views:

1 STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2)

2 Outline 1 Signal Extraction and Optimal Filtering 2 Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 2/ 17

3 Outline 1 Signal Extraction and Optimal Filtering 2 Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 3/ 17

4 Signal Extraction Problem Consider again the lagged regression model y t = r= β r x t r + v t But this time we know the coefficients β r, and we want to estimate the input signal x t ; i.e., we seek the filter a r which optimizes the estimate x t = r= a r y t r We shall assume x t and y t are jointly stationary. We wish to minimize the following mean square error ( ) 2 MSE = E x t a r y t r r= Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 4/ 17

5 Orthogonality Principle Revisited Again, we start with the normal equations: [( ) ] E x t a r y t r y t s = 0 r= for s = 0, ±1, ±2,... which leads to r= Proceeding just as before, we deduce a r γ y (s r) = γ xy (s). A(ω)f y (ω) = f xy (ω) where A(ω) and a r are Fourier transform pairs. However, we don t know x t, hence we don t know f x y(ω)! Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 5/ 17

6 Signal-To-Noise Ratio (SNR) Using the additional observation (exercise) leads to another formulation of A(ω) as f xy (ω) = B(ω)f x (ω) A(ω) = ( B(ω) ) B(ω) 2 + fv(ω) f x(ω) Although we don t know f x (ω), the quantity f x (ω)/f v (ω) is referred to as the signal-to-noise ratio which can be estimated (cf. Chapter 7). Therefore, given the SNR, the optimal filter a r can be estimated from the inverse Fourier transform of Â(ω) where Â(ω) = B(ω) ( ) B(ω) 2 + ŜNR(ω) Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 6/ 17

7 Outline 1 Signal Extraction and Optimal Filtering 2 Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 7/ 17

8 Introduction Long memory time series provide an intermediate comprimize between the ARMA models and the nonstationary ARIMA models. Long memory processes occur in: hydrology (e.g. long term storage capacity of reservois Hurst 1951) environmental series (e.g. varve data) econometrics (e.g. certain squared returns and interest rates) Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 8/ 17

9 Why ARMA won t do The causal ARMA(p, q) model φ(b)x t = θ(b)w t with the representation x t = ψ j w t j j=0 has coefficients ψ j that decay exponentially fast. Hence the acf, ρ(h), decays exponentially fast. However there are many stationary time series that have slowly (i.e. non-exponentially) decaying autocovariance functions! Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 9/ 17

10 ACF Plot Originally, when we see this: We would first difference the time series. Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 10/ 17

11 ACF Plot Originally, when we see this: We would first difference the time series. Not this time! Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 10/ 17

12 Fractional Differencing Levels of differencing: No differencing: (1 B) 0 x t Full differencing: (1 B) 1 x t Fractional differencing: (1 B) d x t where d may be between 0 and 1! This is how: (1 B) d x t = (1 db + (Taylor series expansion) d(d 1) B 2 2! d(d 1)(d 2) B 3 + )x t 3! Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 11/ 17

13 Stationary Fractionally Differenced Series The fractionally differenced series (1 B) d x t = w t is stationary only when d < 1/2. For d in this range, we write the Taylor series expansion of w t as w t = (1 B) d x t = π j B j x t = π j x t j where and also where π j = x t = (1 B) d w t = ψ j = j=0 Γ(j d) Γ(j + 1)Γ( d) ψ j B j w t = j=0 Γ(j + d) Γ(j + 1)Γ(d) j=0 ψ j w t j Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 12/ 17 j=0

14 ACF of Fractionally Differenced Series Using the causal representation of x t, we have ρ(h) = Therefore, for 0 < d <.5, Γ(h + d)γ(1 d) Γ(h d + 1)Γ(d) h2d 1 h= This indicates x t has a long memory! ρ(h) = Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 13/ 17

15 Using Fractional Differencing Consider the random walk model x t = x t 1 + w t which is non-stationary, but stationary after differencing. But maybe the time series on hand looks like a random walk but is actually stationary with a long memory. Then incorporating fractional differencing may be in order! Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 14/ 17

16 ARFIMA Models Definition (ARFIMA Model) The model φ(b) d (x t µ) = θ(b)w t where d may be any real number (integer or not!) defines the ARFIMA model which is stationary for 1/2 < d < 1/2. Two useful packages in R: longmemo fracdiff Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 15/ 17

17 Examples Example Here s how you can simulate data from an ARFIMA(1,.4,1) model: > library(longmemo) > arima.sim(100, model = list(ar=.9, ma = 1), + innov= simarma0(150,h=.4+1/2), n.start = 50) Example Here s a time series plot of a simple fractional Gaussian noise: > plot(simfgn0(2000,.8), lwd=3) Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 16/ 17

18 d estimation in R Example Here s an example estimation of d using the fracdiff package: > library(fracdiff) > varve = scan("mydata/varve.dat") Read 634 items > lvarve = log(varve)-mean(log(varve)) > varve.fd = fracdiff(lvarve, nar=0, nma=0, M=30) > varve.fd$d [1] > varve.fd$stderror.dpq [1] e-06 Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 17/ 17

γ 0 = Var(X i ) = Var(φ 1 X i 1 +W i ) = φ 2 1γ 0 +σ 2, which implies that we must have φ 1 < 1, and γ 0 = σ2 . 1 φ 2 1 We may also calculate for j 1

γ 0 = Var(X i ) = Var(φ 1 X i 1 +W i ) = φ 2 1γ 0 +σ 2, which implies that we must have φ 1 < 1, and γ 0 = σ2 . 1 φ 2 1 We may also calculate for j 1 4.2 Autoregressive (AR) Moving average models are causal linear processes by definition. There is another class of models, based on a recursive formulation similar to the exponentially weighted moving