ST414 – Spectral Analysis of
Time Series Data
Lecture 1
28 January 2014
Examples
2
Brockwell and Davis, 2002
Examples
3
Shumway & Stoffer, 2004
Examples
4
Ombao et al, 2001
Examples
5
Kakizawa et al, 1998
Examples
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Ombao et al, 2001
Examples
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Examples
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Shumway & Stoffer, 2004
Objective
Develop a working knowledge of statistical
theories and methodologies for spectral
analysis of time series data.
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Today’s Objectives
• Discuss periodicity
• Discuss basic time series models
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Periodicity
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Shumway & Stoffer, 2004
Periodicity
where
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Periodicity
More generally,
where
What is Var(X(t))?
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Periodicity
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Shumway & Stoffer, 2004
Preliminaries
X(t) is said to be strictly stationary if the
probabilistic behaviour of every collection of
values
is identical to that of the shifted set
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Preliminaries
X(t) is said to be weakly stationary if 1)
E(X(t)) is invariant with respect to t and 2)
𝐶𝑜𝑣 𝑋 𝑡 , 𝑋 𝑠 = 𝛾(ℎ), where h = |s-t|.
𝛾(ℎ) is called the autocovariance function
of X(t).
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Preliminaries
1. 𝛾 0 ≥ 0
2. |𝛾 ℎ | ≤ 𝛾 0
3. 𝛾 ℎ =𝛾 −ℎ
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Preliminaries
The autocorrelation function of a weakly
stationary time series X(t) is
𝛾(ℎ)
𝜌 ℎ =
𝛾(0)
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White Noise
X(t) is white noise if it is a collection of
uncorrelated random variables identically
distributed with mean 0 and finite variance
𝜎 2.
Is X(t) weakly stationary?
What is 𝛾(ℎ)?
What is 𝜌 ℎ ?
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White Noise
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White Noise
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MA(1)
𝑋 𝑡 = 𝑍 𝑡 + 𝜃𝑍(𝑡 − 1)
𝑍 𝑡 is white noise
Is X(t) weakly stationary?
What is 𝛾(ℎ)?
What is 𝜌 ℎ ?
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MA(1)
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MA(1)
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AR(1)
𝑋 𝑡 = 𝜙𝑋 𝑡 − 1 + 𝑍 𝑡
𝑍 𝑡 is white noise
Is X(t) weakly stationary?
What is 𝛾(ℎ)?
What is 𝜌 ℎ ?
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AR(1)
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AR(1)
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AR(1)
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More Preliminaries
X(t) is a linear process if it has the
representation
∞
𝑋 𝑡 =
𝜓𝑗 𝑍(𝑡 − 𝑗),
𝑗=−∞
for all t, where Z(t) is white noise and {𝜓𝑗 } is
a sequence of constants with
∞
𝑗=−∞ |𝜓𝑗 | < ∞.
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More Preliminaries
If X(t) ~ AR(1), then X(t) has a MA ∞
representation:
∞
𝜙 𝑗 𝑍(𝑡 − 𝑗),
𝑋 𝑡 =
𝑗=0
Is X(t) weakly stationary?
What is 𝛾(ℎ)?
What is 𝜌 ℎ ?
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MA(2)
Consider the MA(2) model:
𝑋 𝑡 = 𝑍 𝑡 + 𝜃1 𝑍 𝑡 − 1 + 𝜃2 𝑍(𝑡 − 2)
𝑍 𝑡 is white noise
2−|ℎ|
𝛾 ℎ =
𝜎2
𝜃𝑗 𝜃𝑗+ ℎ , if |ℎ| ≤ 2
𝑗=0
0,
where 𝜃0 = 1.
if ℎ > 2
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ACF of MA(2)
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ACF of AR(1)
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The PACF
Heuristically, take the correlation between
X(t) and X(s) with the linear effect of
everything in between removed.
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ACF of AR(1)
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PACF of AR(1)
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PACF of AR(2)
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Example
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Example
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Example
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An Exercise
Let
𝑋 𝑡 = 𝑈1 cos 2𝜋𝜔𝑡 + 𝑈2 sin 2𝜋𝜔𝑡
where
𝑈1 , 𝑈2 iid 0, 𝜎 2 .
Is X(t) weakly stationary? If so, what is 𝛾(ℎ)?
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