ST414 – Spectral Analysis of
Time Series Data
Lecture 3
11 February 2014
Last Time
• Introduced the periodogram
• Introduced the spectral density function
• Derived the spectral density function for
time series models
2
Today’s Objectives
• Resume discussion on the periodogram
• The smoothed periodogram
• Asymptotic considerations
3
The Periodogram
The Discrete Fourier transform:
𝑗
𝑑
= 𝑇 −1/2
𝑇
𝑇
=𝑇
−1/2
𝑡=1
𝑇
𝑡=1
𝑗
𝑋 𝑡 exp −2𝜋𝑖𝑡
𝑇
𝑗
𝑋 𝑡 cos 2𝜋𝑡
𝑇
𝑇
−𝑖
𝑡=1
𝑗
𝑋 𝑡 sin 2𝜋𝑡
𝑇
4
The Periodogram
The periodogram:
𝐼 𝜔𝑗 = 𝑑 𝜔𝑗
2
5
The Spectral Density
Let
ℎ𝜖𝒁 |𝛾
ℎ | < ∞. Then
𝑓 𝜔 =
𝛾 ℎ exp(−𝑖2𝜋𝜔ℎ)
ℎ𝜖𝒁
is called the spectral density function.
Moreover,
1/2
𝛾 ℎ =
exp 𝑖2𝜋𝜔ℎ 𝑓 𝜔 𝑑𝜔
−1/2
6
White Noise
Let X(t) be white noise with variance 𝜎 2 .
What is 𝛾 ℎ ?
𝜎 2 if ℎ = 0
𝛾 ℎ =
0 if ℎ > 0
What is 𝑓 𝜔 ?
𝑓 𝜔 = 𝜎2
7
MA(1)
𝑋 𝑡 = 𝑍 𝑡 + 𝜃𝑍(𝑡 − 1)
𝑍 𝑡 is white noise (0, 𝜎 2 )
𝜎 2 1 + 𝜃 2 if ℎ = 0
𝛾 ℎ =
𝜎 2 𝜃 if |ℎ| = 1
0 if ℎ > 1
2
2
𝑓 𝜔 = 𝜎 𝜃 + 2𝜃 cos 2𝜋𝜔 + 1
8
AR(1)
𝑋 𝑡 = 𝜙𝑋 𝑡 − 1 + 𝑍 𝑡
𝑍 𝑡 is white noise (0, 𝜎 2 )
𝜎 2𝜙ℎ
γ ℎ =
1 − 𝜙2
𝜎2
𝑓 𝜔 = 2
𝜙 − 2𝜙 cos 2𝜋𝜔 + 1
9
AR(1)
10
AR(1) Periodogram
11
AR(1) Periodogram (Averaged)
12
AR(1)
13
AR(1) Periodogram
14
AR(1) Periodogram (Averaged)
15
AR(1), T = 512
16
AR(1), T = 2048
17
AR(1), T = 8192
18
The Periodogram
The periodogram is an (asymptotically)
unbiased estimator, but it is not consistent.
19
Asymptotic Distribution of the
Periodogram
Let X(1), …, X(T) be Gaussian white noise
with variance 𝜎 2 , and let
𝑇
𝑋𝑐 (𝜔𝑗 ) = 𝑇 −1/2
𝑋 𝑡 cos 2𝜋𝑡𝜔𝑗
𝑡=1
𝑇
𝑋𝑠 (𝜔𝑗 ) = 𝑇 −1/2
𝑋 𝑡 sin 2𝜋𝑡𝜔𝑗
𝑡=1
20
Asymptotic Distribution of the
Periodogram
Verify that
𝐸 𝑋𝑐 𝜔𝑗
= 𝐸 𝑋𝑠 𝜔𝑗
=0
and
𝑉𝑎𝑟 𝑋𝑐 𝜔𝑗
= 𝑉𝑎𝑟 𝑋𝑠 𝜔𝑗
= 𝜎 2 /2
21
Asymptotic Distribution of the
Periodogram
𝐶𝑜𝑣 𝑋𝑐 𝜔𝑗 , 𝑋𝑠 𝜔𝑗
=0
𝐶𝑜𝑣 𝑋𝑐 𝜔𝑗 , 𝑋𝑐 𝜔𝑘
=0
𝐶𝑜𝑣 𝑋𝑠 𝜔𝑗 , 𝑋𝑠 𝜔𝑘
=0
𝐶𝑜𝑣 𝑋𝑐 𝜔𝑗 , 𝑋𝑠 𝜔𝑘
=0
for any 𝑗 ≠ 𝑘.
22
Asymptotic Distribution of the
Periodogram
2(𝑋𝑐 2 𝜔𝑗 + 𝑋𝑠 2 𝜔𝑗 ) 2𝐼 𝜔𝑗
2
=
~𝜒
2
2
𝜎
𝑓(𝜔𝑗 )
This holds under more general conditions
(e.g., linearity, rapidly decaying
autocovariance function)
23
The Smoothed Periodogram
Idea: borrow information from neighbouring
frequencies.
24
The Smoothed Periodogram
𝑓 𝜔𝑗
1
=
2𝑀 + 1
𝑀
𝑘=−𝑀
𝑘
𝐼(𝜔𝑗 + )
𝑇
25
The Smoothed Periodogram
Under appropriate conditions,
2(2𝑀 + 1)𝑓 𝜔𝑗
~𝜒 2 2(2𝑀 + 1)
𝑓(𝜔𝑗 )
26
The Smoothed Periodogram
27
The Smoothed Periodogram
28
The Smoothed Periodogram
29
The Smoothed Periodogram
30
The Smoothed Periodogram
Recall
2(2𝑀 + 1)𝑓 𝜔𝑗
~𝜒 2 2(2𝑀 + 1)
𝑓(𝜔𝑗 )
and so
𝑉𝑎𝑟(𝑓 𝜔𝑗 ) ≈ 𝑓 2 (𝜔𝑗 )/(2𝑀 + 1)
31
The Smoothed Periodogram
Asymptotic framework for consistency:
1) 𝑇 → ∞
2) 𝑀 = 𝑀 𝑇 → ∞
3) 𝑀 𝑇 /𝑇 → 0
32
The Smoothed Periodogram
33
The Smoothed Periodogram
34
The Smoothed Periodogram
35
The Smoothed Periodogram
More generally,
𝑀
𝑓 𝜔𝑗 =
𝑘=−𝑀
where
ℎ𝑘 ≥ 0,
|𝑘|≤𝑀 ℎ𝑘
𝑘
ℎ𝑘 𝐼(𝜔𝑗 + ) ,
𝑇
= 1, and ℎ𝑘 = ℎ−𝑘
36
The Smoothed Periodogram
Asymptotic framework for consistency:
1) 𝑇 → ∞
2) 𝑀 = 𝑀 𝑇 → ∞
3) 𝑀 𝑇 /𝑇 → 0
4)
2
𝑀
𝑘=−𝑀 ℎ𝑘
→0
37
The Smoothed Periodogram
𝐸 𝑓 𝜔𝑗
→ 𝑓 𝜔𝑗
0 if 𝜔 ≠ 𝜆
𝐶𝑜𝑣 𝑓 𝜔 , 𝑓 𝜆
→
−1
𝑀
ℎ𝑘 2
𝑓 2 𝜔 if 𝜔 = 𝜆 ≠ 0, 0.5
𝑘=−𝑀
38
The Smoothed Periodogram
39
The Smoothed Periodogram
40