Autocovariance generating function and spectral density.... Introduction to Time Series Analysis.[r]
(1)Introduction to Time Series Analysis Lecture 16.
1 Review: Spectral density Examples
3 Spectral distribution function
(2)Review: Spectral density
If a time series {Xt} has autocovariance γ satisfying
P∞
h=−∞ |γ(h)| < ∞, then we define its spectral density as
f(ν) =
∞ X
h=−∞
γ(h)e−2πiνh
(3)Review: Spectral density
1 f(ν) is real f(ν) ≥
3 f is periodic, with period So we restrict the domain of f to
−1/2 ≤ ν ≤ 1/2
4 f is even (that is, f(ν) = f(−ν)) γ(h) =
Z 1/2 −1/2
(4)Examples
White noise: {Wt}, γ(0) = σw2 and γ(h) = for h 6=
f(ν) = γ(0) = σw2
AR(1): Xt = φ1Xt−1 + Wt, γ(h) = σw2 φ|h|1 /(1 − φ21)
f(ν) = σw2
1−2φ1cos(2πν)+φ2
If φ1 > (positive autocorrelation), spectrum is dominated by low frequency components—smooth in the time domain
If φ1 < (negative autocorrelation), spectrum is dominated by high
(5)Example: AR(1)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
10 20 30 40 50 60 70 80 90 100
ν
f(
ν
)
Spectral density of AR(1): X
(6)Example: AR(1)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
10 20 30 40 50 60 70 80 90 100
ν
f(
ν
)
Spectral density of AR(1): X
(7)Example: MA(1)
Xt = Wt + θ1Wt−1
γ(h) =
σw2 (1 + θ12) if h = 0,
σw2 θ1 if |h| = 1,
0 otherwise
f(ν) =
1
X
h=−1
γ(h)e−2πiνh
= γ(0) + 2γ(1) cos(2πν)
(8)Example: MA(1)
Xt = Wt + θ1Wt−1
f(ν) = σw2 + θ12 + 2θ1 cos(2πν)
If θ1 > (positive autocorrelation), spectrum is dominated by low frequency components—smooth in the time domain
If θ1 < (negative autocorrelation), spectrum is dominated by high
(9)Example: MA(1)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.5 1.5 2.5 3.5
ν
f(
ν
)
Spectral density of MA(1): X
(10)Example: MA(1)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.5 1.5 2.5 3.5
ν
f(
ν
)
Spectral density of MA(1): X
(11)Introduction to Time Series Analysis Lecture 16.
1 Review: Spectral density Examples
3 Spectral distribution function
(12)Recall: A periodic time series
Xt =
k X
j=1
(Aj sin(2πνjt) + Bj cos(2πνjt))
=
k X
j=1
(A2j + Bj2)1/2 sin(2πνjt + tan−1(Bj/Aj))
E[Xt] =
γ(h) =
k X
j=1
σj2 cos(2πνjh)
X
h
(13)Discrete spectral distribution function
For Xt = Asin(2πλt) + B cos(2πλt), we have γ(h) = σ2 cos(2πλh), and we can write
γ(h) =
Z 1/2 −1/2
e2πiνhdF(ν),
where F is the discrete distribution
F(ν) =
0 if ν < −λ,
σ2
2 if −λ ≤ ν < λ,
(14)The spectral distribution function
For any stationary {Xt} with autocovariance γ, we can write
γ(h) =
Z 1/2 −1/2
e2πiνhdF(ν),
where F is the spectral distribution function of {Xt}
We can split F into three components: discrete, continuous, and singular If γ is absolutely summable, F is continuous: dF(ν) = f(ν)dν
(15)The spectral distribution function
For Xt = Pkj=1 (Aj sin(2πνjt) + Bj cos(2πνjt)), the spectral distribution function is F(ν) = Pkj=1 σj2Fj(ν), where
Fj(ν) =
0 if ν < −νj,
1
2 if −νj ≤ ν < νj,
(16)Wold’s decomposition
Notice that Xt = Pkj=1 (Aj sin(2πνjt) + Bj cos(2πνjt)) is deterministic
(once we’ve seen the past, we can predict the future without error) Wold showed that every stationary process can be represented as
Xt = Xt(d) + Xt(n),
(17)Introduction to Time Series Analysis Lecture 16.
1 Review: Spectral density Examples
3 Spectral distribution function
(18)Autocovariance generating function and spectral density
Suppose Xt is a linear process, so it can be written
Xt = P∞i=0 ψiWt−i = ψ(B)Wt
Consider the autocovariance sequence,
γh = Cov(Xt, Xt+h)
= E ∞ X i=0
ψiWt−i
∞ X
j=0
ψjWt+h−j
= σw2
∞ X
i=0
(19)Autocovariance generating function and spectral density
Define the autocovariance generating function as
γ(B) =
∞ X
h=−∞
γhBh
Then, γ(B) = σw2
∞ X
h=−∞ ∞ X
i=0
ψiψi+hBh
= σw2
∞ X i=0 ∞ X j=0
ψiψjBj−i
= σw2
∞ X
i=0
ψiB−i
∞ X
j=0
(20)Autocovariance generating function and spectral density
Notice that
γ(B) =
∞ X
h=−∞
γhBh
f(ν) =
∞ X
h=−∞
γhe−2πiνh
= γ e−2πiν
= σw2 ψ e−2πiν ψ e2πiν
= σw2 ψ e2πiν
2
(21)Autocovariance generating function and spectral density
For example, for an MA(q), we have ψ(B) = θ(B), so
f(ν) = σw2 θ e−2πiνθ e2πiν
= σw2 θ e−2πiν
2
For MA(1),
f(ν) = σw2 + θ1e−2πiν
2
(22)Autocovariance generating function and spectral density
For an AR(p), we have ψ(B) = 1/φ(B), so
f(ν) = σ
2
w
φ (e−2πiν)φ (e2πiν)
= σ
2
w
|φ (e−2πiν)|2
For AR(1),
f(ν) = σ
2
w
|1 − φ1e−2πiν|2
= σ
2
w
(23)Spectral density of a linear process
If Xt is a linear process, it can be written Xt = P∞i=0 ψiWt−i = ψ(B)Wt Then
f(ν) = σw2 ψ e−2πiν
2
(24)
Introduction to Time Series Analysis Lecture 16.
1 Review: Spectral density Examples
3 Spectral distribution function