8: FORMULAS INVOLVING THE PERIODOGRAM
AND SAMPLE AUTOCOVARIANCES
Define the lag-r sample autocovariance by
1 n −1
ĉr = hh
xx
n t = er e t t−er e
Σ
for e r e < n . Note that ĉ −r = ĉr . The {ĉr } sequence is very important in the time-domain analysis of time
series (e.g., Box-Jenkins methods), and also plays an important role in frequency domain methods.
Interestingly, it can be shown that the periodogram is the Fourier transform of the {ĉr } sequence:
1
I (ω) = hhh
ĉr exp(−ir ω)
2π e r e < n
Σ
.
Proof:
1 n −1
1 n −1
I (ω) = hhhh e xt exp(−i ωt ) e 2 = hhhh
2πn t =0
2πn t =0
Σ
n −1
Σ uΣ=0xt xu exp{−i (t −u )ω}
.
Making the change of variables v = t −u , and rearranging terms (a diagram of the range of summation
will be helpful), we obtain
n −1 n −1−u
1
I (ω) = hhhh
2πn u =0
xv +u xu exp(−iv ω)
Σ vΣ
=−u
n −1 n −1−v
1
= hhhh
2πn v =0
1
n −1
−1
hhh
Σ uΣ=0 xv +u xu exp(−iv ω) + h2πn
Σ Σ xv +u xu exp(−iv ω)
v =−(n −1) u =−v
−1
1 n −1
1
= hhh [ ĉv exp(−iv ω)] + hhhh
2π v =0
2πn v =−(n −1)
Σ
Σ
n −1
Σ
u=ev e
1
= hhh
ĉr exp(−ir ω)
2π e r e < n
Σ
xu − e v e xu exp(−iv ω)
.
Another interesting formula is
π
ĉr =
∫ I (ω)exp(ir ω)
,
−π
showing that ĉr is the integral Fourier transform of the periodogram.
-2-
Proof:
π
π
∫ I (ω)exp(ir ω) =
−π
1
∫ hhh Σ ĉs exp(−is ω)exp (ir ω)
−π 2π e s e < n
π
1
ĉs hhh ∫ exp(i ω(r −s )) = ĉr
2π
es e <n
−π
Σ
=
.
1
Since I (ω) = hhh
ĉr exp (−ir ω), the {ĉr } sequence determines the periodogram. Furthermore,
2π e r e < n
Σ
the periodogram, if evaluated on a sufficiently fine grid, completely determines the covariance sequence.
We have
2π n ′−1
ĉr = hhh I (ω′ j ) exp(ir ω′ j )
n ′ j =0
Σ
,
2πj
ω′ j = hhhh
n′
,
n ′ = 2n
.
1
Proof: Append n zeros to x 0 , . . . , xn −1 to obtain y 0 , . . . , yn ′−1. We have ĉy , r = hh ĉx , r ( e r e < n ),
2
cˆy , r = 0
1
( e r e ≥ n ), and Iy (ω) = hh Ix (ω). Given r with e r e < n , the RHS of the formula to be proved is
2
2π n ′−1
2π n ′−1
2π n ′−1 1
hhh
Ix (ω′ j ) exp(ir ω′ j ) = hhh 2Iy (ω′ j ) exp(ir ω′ j ) = hhh [ hhh
ĉ exp(−is ω′ j )] exp(ir ω′ j )
n ′ j =0
2n j =0
n j =0 2π e s e < n ′ y , s
Σ
Σ
=
Σ ĉy , s
es e <n′
=
n ′−1
h1h exp(i (r −s )ω′ ) =
j
n j =0
Σ
Σ
Σ
1
n ′−1
Σ ĉx , s hnhh′ jΣ=0 exp(ij ω′r −s )
es e <n
Σ ĉx , s .Indicator(r −s =0 mod n ′) = ĉx , r
es e <n
= ĉr
.
As a consequence of the above identity, the {ĉr } sequence may be computed in O (n log n ) using the
FFT.