Review: Spectral distribution function, spectral density... Introduction to Time Series Analysis.[r]
(1)Introduction to Time Series Analysis Lecture 17.
1 Review: Spectral distribution function, spectral density Rational spectra Poles and zeros
3 Examples
(2)Review: Spectral density and spectral distribution function
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
for −∞ < ν < ∞ We have
γ(h) =
Z 1/2
−1/2
e2πiνhf(ν)dν =
Z 1/2
−1/2
e2πiνh dF(ν),
(3)Review: 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
(4)
Spectral density of a linear process
For an ARMA(p,q), ψ(B) = θ(B)/φ(B), so
f(ν) = σw2 θ(e
−2πiν
)θ(e2πiν)
φ(e−2πiν)φ (e2πiν)
= σw2
θ(e−2πiν)
φ (e−2πiν)
2
(5)Rational spectra
Consider the factorization of θ and φ as
θ(z) = θq(z − z1)(z − z2)· · ·(z − zq)
φ(z) = φp(z − p1)(z − p2)· · ·(z − pp),
where z1, , zq and p1, , pp are called the zeros and poles.
f(ν) = σw2
θq Qq
j=1(e
−2πiν
− zj)
φp Qpj=1(e−2πiν − pj)
= σw2 θ q Qq j=1 e
−2πiν
− zj
φ2 p Qp
j=1 |e
−2πiν − p
j|2
(6)Rational spectra
f(ν) = σw2 θ
q
j=1 e
−2πiν
− zj φ2
p
Qp
j=1 |e
−2πiν − p
j|2
As ν varies from to 1/2, e−2πiν moves clockwise around the unit circle from to e−πi = −1
(7)Example: ARMA
Recall AR(1): φ(z) = − φ1z The pole is at 1/φ1 If φ1 > 0, the pole is
to the right of 1, so the spectral density decreases as ν moves away from If φ1 < 0, the pole is to the left of −1, so the spectral density is at its
maximum when ν = 0.5
Recall MA(1): θ(z) = + θ1z The zero is at −1/θ1 If θ1 > 0, the zero is to the left of −1, so the spectral density decreases as ν moves towards −1 If θ1 < 0, the zero is to the right of 1, so the spectral density is at its
(8)Example: AR(2)
Consider Xt = φ1Xt−1 + φ2Xt−2 + Wt Example 4.6 in the text considers
this model with φ1 = 1, φ2 = −0.9, and σw2 = In this case, the poles are
at p1, p2 ≈ 0.5555 ± i0.8958 ≈ 1.054e±i1.01567 ≈ 1.054e±2πi0.16165
Thus, we have
f(ν) = σ
2
w
φ22|e−2πiν − p
1|2|e−2πiν − p2|2 ,
(9)Example: AR(2)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
20 40 60 80 100 120 140
ν
f(
ν
)
Spectral density of AR(2): X
(10)Example: Seasonal ARMA
Consider Xt = Φ1Xt−12 + Wt
ψ(B) =
1 − Φ1B12 ,
f(ν) = σw2
(1 − Φ1e−2πi12ν)(1 − Φ
1e2πi12ν)
= σw2
1 − 2Φ1 cos(24πν) + Φ21
(11)Example: Seasonal ARMA
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.2 0.4 0.6 0.8 1.2 1.4 1.6
ν
f(
ν
)
Spectral density of AR(1)
(12)Example: Seasonal ARMA
Another view:
1 − Φ1z12 = ⇔ z = reiθ,
with r = |Φ1|−1/12, ei12θ = e−iarg(Φ1)
For Φ1 > 0, the twelve poles are at |Φ1|−1/12eikπ/6 for
k = 0,±1, ,±5,6
So the spectral density gets peaked as e−2πiν passes near
(13)Example: Multiplicative seasonal ARMA
Consider (1 − Φ1B12)(1 − φ1B)Xt = Wt
f(ν) = σw2
(1 − 2Φ1 cos(24πν) + Φ21)(1 − 2φ1 cos(2πν) + φ21)
This is a scaled product of the AR(1) spectrum and the (periodic) AR(1)12 spectrum
(14)Example: Multiplicative seasonal ARMA
1
f(
ν
)
Spectral density of AR(1)AR(1)
12: (1+0.5 B)(1+0.2 B 12
) X
(15)Introduction to Time Series Analysis Lecture 17.
1 Review: Spectral distribution function, spectral density Rational spectra Poles and zeros
3 Examples
(16)Time-invariant linear filters
A filter is an operator; given a time series {Xt}, it maps to a time series
{Yt} We can think of a linear process Xt = P∞
j=0 ψjWt−j as the output of
a causal linear filter with a white noise input.
A time series {Yt} is the output of a linear filter
A = {at,j : t, j ∈ Z} with input {Xt} if
Yt =
∞
X
j=−∞
at,jXj
If at,t−j is independent of t (at,t−j = ψj), then we say that the
filter is time-invariant.
(17)Time-invariant linear filters: Examples
1 Yt = X−t is linear, but not time-invariant
2 Yt = 13(Xt−1 + Xt + Xt+1) is linear, time-invariant, but not causal:
ψj =
3 if |j| ≤ 1,
0 otherwise
3 For polynomials φ(B), θ(B) with roots outside the unit circle,
(18)Time-invariant linear filters
The operation
∞
X
j=−∞
ψjXt−j
(19)Time-invariant linear filters
The sequence ψ is also called the impulse response, since the output {Yt} of the linear filter in response to a unit impulse,
Xt =
1 if t = 0,
0 otherwise, is
Yt = ψ(B)Xt =
∞
X
j=−∞
(20)Introduction to Time Series Analysis Lecture 17.
1 Review: Spectral distribution function, spectral density Rational spectra Poles and zeros
3 Examples
(21)Frequency response of a time-invariant linear filter
Suppose that {Xt} has spectral density fx(ν) and ψ is stable, that is,
P∞
j=−∞ |ψj| < ∞ Then Yt = ψ(B)Xt has spectral density
fy(ν) = ψ e2πiν
fx(ν)
The function ν 7→ ψ(e2πiν) (the polynomial ψ(z) evaluated on the unit circle) is known as the frequency response or transfer function of the linear filter
The squared modulus, ν 7→ |ψ(e2πiν)|2 is known as the power transfer
(22)Frequency response of a time-invariant linear filter
For stable ψ, Yt = ψ(B)Xt has spectral density
fy(ν) =
ψ e2πiν
fx(ν)
We have seen that a linear process, Yt = ψ(B)Wt, is a special case, since
fy(ν) = |ψ(e2πiν)|2σw2 = |ψ(e2πiν)|2fw(ν)
When we pass a time series {Xt} through a linear filter, the spectral density is multiplied, frequency-by-frequency, by the squared modulus of the
frequency response ν 7→ |ψ(e2πiν)|2
(23)Frequency response of a filter: Details
Why is fy(ν) = ψ e2πiν
fx(ν)? First,
γy(h) = E
∞
X
j=−∞
ψjXt−j
∞
X
k=−∞
ψkXt+h−k
=
∞
X
j=−∞
ψj
∞
X
k=−∞
ψkE[Xt+h−kXt−j]
=
∞
X
j=−∞
ψj
∞
X
k=−∞
ψkγx(h + j − k) =
∞
X
j=−∞
ψj
∞
X
l=−∞
ψh+j−lγx(l)
(24)Frequency response of a filter: Details
fy(ν) =
∞
X
h=−∞
γ(h)e−2πiνh
=
∞
X
h=−∞
∞
X
j=−∞
ψj
∞
X
l=−∞
ψh+j−lγx(l)e
−2πiνh
=
∞
X
j=−∞
ψje2πiνj
∞
X
l=−∞
γx(l)e−2πiνl
∞
X
h=−∞
ψh+j−le
−2πiν(h+j−l)
= ψ(e2πiνj)fx(ν)
∞
X
h=−∞
(25)Frequency response: Examples
For a linear process Yt = ψ(B)Wt, fy(ν) =
ψ e2πiν
σw2
For an ARMA model, ψ(B) = θ(B)/φ(B), so {Yt} has the rational
spectrum
fy(ν) = σw2
θ(e−2πiν)
φ (e−2πiν)
= σw2 θ q Qq j=1 e
−2πiν
− zj φ2
p
Qp
j=1 |e
−2πiν − p
j|2
,
where pj and zj are the poles and zeros of the rational function
(26)Frequency response: Examples
Consider the moving average
Yt = 2k +
k
X
j=−k
Xt−j
This is a time invariant linear filter (but it is not causal) Its transfer function is the Dirichlet kernel
ψ(e−2πiν) = Dk(2πν) = 2k +
k
X
j=−k
e−2πijν
=
(27)Example: Moving average
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −0.4
−0.2 0.2 0.4 0.6 0.8
ν
(28)Example: Moving average
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ν
Squared modulus of transfer function of moving average (k=5)
(29)Example: Differencing
Consider the first difference
Yt = (1 − B)Xt
This is a time invariant, causal, linear filter Its transfer function is
ψ(e−2πiν) = − e−2πiν,
(30)Example: Differencing
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
Transfer function of first difference
(31)Introduction to Time Series Analysis Lecture 17.
1 Review: Spectral distribution function, spectral density Rational spectra Poles and zeros
3 Examples