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The spectra method for analyzing univariate and bivariate time series data using spectral density estimates, periodograms, and data windows such as hamming, tukey-hamming, parzen, bartlett, and user-specified windows. The document also provides references to books on fourier analysis and statistical time series.
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For all t, the series X (^) t can be represented by
X (^) t a x^ a (^) Kx^ f (^) K t b (^) Kx^ f (^) K t K
q = + − + − =
(^0) ∑ 1
4 cos^2 π^1 1 6 sin^2 π^1
where
t N
a X X X N
a (^) N X f t
b N
X f t
f K N
q
x (^) t t
N
Kx t K t
N
Kx t K t
N
K
!
" $
!
" $
= %& − '
=
=
=
0 1
1
1
cos
sin
if is even if is odd
π
π
1 0 56
1 0 56
0 5
The following statistics are calculated:
Frequency
f (^) K = K N , K = 1 , K,q
Period
1 f (^) K = N K , K = 1 , K,q
Fourier Cosine Coefficient
a (^) Kx^ , K = 1 , K,q
Fourier Sine Coefficient
b (^) Kx^ = 4 a (^) Kx^ − ib (^) Kx^94 a (^) Kx^ +ibKx 9
Periodogram
l (^) Kx^ = a (^) Kx^ +b (^) Kx N K q !
" $#^
4 9 4 9 =
2 2 2 , 1 , K,
spectral density estimate
s (^) Kx^ w lj K x j p m j p
p = (^) + + = =
∑ ,^ where^2 1 (number of spans)
Imaginary (^) 4 l (^) Kxy 9
1 6^ IC^ K =^ N^4 a (^) Kx^ b^ Ky^ −b^ Kx^ aKy 9 2
Cospectral Density Estimate
C (^) K w (^) j RCK j j p
p = (^) + =−
∑ 1 6
Quadrature Spectrum Estimate
Q (^) K w (^) j ICK j j p
p = (^) + =−
∑ 1 6
Cross-amplitude Values
1 2 4 9
Squared Coherency Values
K (^) s s K Kx^ Ky
2
Gain Values
G A^ s^ Y^ X^ f K (^) A s X Y f K Kx^ t t K K (^) Ky^ t t K
% &
K 'K^
gain of over at gain of over at
1 6 1 6
Phase Spectrum Estimate
K K
K K K K K K K K
K K K K
%
&
K K K
'
K K K
−
−
−
tan
tan ,
tan ,
1
1
1
1 6
1 6
1 6
if
if
if
π
π
1
The following spectral windows can be specified. Each formula defines the upper half of the window. The lower half is symmetric with the upper half. In all formulas, p is the integer part of the number of spans divided by 2. To be concise, the formulas are expressed in terms of the Fejer kernel:
q
q
q^ θ q
θ π π θ θ
1 6 1 6 1 6
^
% &
K
'
K
2
sin sin
otherwise
and the Dirichlet kernel:
q q^ θ q
θ π π θ θ
1 6 21 6 7 1 6
% &
K
'K
sin sin
otherwise
where q is any positive real number.
1 This algorithm applies to SPSS 6.0 and later releases.
Daniell window or rectangular window. The weights are
Wk = 1
for k = 0, K ,p.
No smoothing. If NONE is specified, the spectral density estimate is the same as the periodogram. It is also the case when the number of span is 1.
W− (^) p , K, W 0 , K,Wp
User-specified weights. If the number of weights is odd, the middle weight is applied to the periodogram value being smoothed and the weights on either side are applied to preceding and following values. If the number of weights are even (it is assumed that W (^) p is not supplied), the weight after the middle applies to the periodogram value being smoothed. It is required that the weight W 0 must be positive.
Bloomfield, P. 1976. Fourier analysis of time series, New York: John Wiley & Sons, Inc.
Fuller, W. A. 1976. Introduction to statistical time series. New York: John Wiley & Sons, Inc.
