ST414 – Spectral Analysis of
Time Series Data
Lecture 2
30 January 2014
Last Time
• Weak stationarity
• The autocovariance function
• MA & AR models
2
Today’s Objectives
• Introduce the periodogram
• Introduce the spectral density function
• Derive the spectral density function for
time domain models
3
Recall
where
𝑞
𝜎𝑘2 cos(2𝜋𝜔𝑘 ℎ)
𝛾 ℎ =
𝑘=1
4
Recall
5
Shumway & Stoffer, 2004
Two Problems
1. What is the amplitude for the oscillations?
2. How do you pick the frequencies that
drive the data?
6
Example
7
Problem 1
Regress the data on the sines and cosines
that oscillate at the frequency of interest.
8
Problem 1
True values: 𝛽1 = 2 cos 𝜙 = −.62,
𝛽2 = −2 sin 𝜙 = −1.90
9
,
Problem 1
10
Problem 2
Regress sine and cosine waves oscillating
at each frequency.
11
Problem 2
From the first part:
2𝜋𝑡
𝑡 cos(
) 2
50 =
𝛽1 =
2𝜋𝑡
𝑇
𝑇
2
𝑐𝑜𝑠
(
)
𝑡=1
50
2𝜋𝑡
𝑇
2
𝑡=1 𝑋 𝑡 sin( 50 )
𝛽2 =
=
2𝜋𝑡
𝑇
𝑇
2
𝑠𝑖𝑛
(
)
𝑡=1
50
𝑇
𝑡=1 𝑋
𝑇
2𝜋𝑡
𝑋 𝑡 cos(
)
50
𝑡=1
𝑇
2𝜋𝑡
𝑋 𝑡 𝑠𝑖𝑛(
)
50
𝑡=1
12
Problem 2
Consider instead
𝑇
𝑗
2
𝑗
𝛽1
=
𝑋 𝑡 cos(2𝜋𝑡 )
𝑇
𝑇
𝑇
𝑡=1
𝑇
𝑗
2
𝑗
𝛽2
=
𝑋 𝑡 𝑠𝑖𝑛(2𝜋𝑡 )
𝑇
𝑇
𝑇
𝑡=1
From here, look at the “squared correlations”:
2
2
𝑗
𝑗
𝛽1
+ 𝛽2
𝑛
𝑛
13
Problem 2
14
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
15
The Periodogram
𝑗
𝑑
𝑇
2
1
=
𝑇
𝑇
𝑡=1
Recall:
𝑗
2
𝛽1
=
𝑇
𝑇
𝑗
2
𝛽2
=
𝑇
𝑇
2
𝑗
𝑋 𝑡 cos 2𝜋𝑡
𝑇
1
+
𝑇
𝑇
𝑡=1
2
𝑗
𝑋 𝑡 sin 2𝜋𝑡
𝑇
𝑇
𝑗
𝑋 𝑡 cos(2𝜋𝑡 )
𝑇
𝑡=1
𝑇
𝑗
𝑋 𝑡 𝑠𝑖𝑛(2𝜋𝑡 )
𝑇
𝑡=1
16
The Periodogram
The periodogram can be easily (and
quickly!) computed using the Fast Fourier
transform.
17
The Periodogram
18
Shumway & Stoffer, 2004
The Periodogram
19
Example
20
Example
21
The Spectral Density
The periodogram is an estimator for a
population-level statistic called the spectral
density function.
22
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
23
The Spectral Density
𝛾 ℎ and 𝑓 𝜔 are Fourier transform pairs:
If 𝛾𝑓 ℎ =
0.5
exp(𝑖2𝜋𝜔ℎ)𝑓
−0.5
0.5
exp(𝑖2𝜋𝜔ℎ)𝑔
−0.5
𝜔 𝑑𝜔 =
𝜔 𝑑𝜔 = 𝛾𝑔 ℎ , then
𝑓 𝜔 =𝑔 𝜔 .
24
The Spectral Density
1. 𝑓 𝜔 ≥ 0
2. 𝑓 𝜔 = 𝑓(−𝜔)
3. 𝑓 𝜔 = 𝑓(1 − 𝜔)
25
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
26
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
27
MA(1)
28
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
29
AR(1)
30
AR(1)
31
AR(1)
32
AR(1)
33
AR(1)
34
AR(1)
35
The Periodogram
Recall the DFTs:
𝑇
𝑑 𝜔𝑗 = 𝑇 −1/2
𝑋 𝑡 exp −2𝜋𝑖𝜔𝑗 𝑡
𝑡=1
Their asymptotic distribution (under general
conditions) is complex Gaussian with mean
0 and variance 𝑓 𝜔 .
36
The Periodogram
The asymptotic distribution of the
periodogram 𝐼(𝜔) is
2𝐼 𝜔
𝑓 𝜔
𝑑
𝜒 2 (2)
37