time series analysis time series – collection of observations in time: x( t i ) x( t i ) discrete...
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TIME SERIES ANALYSIS
Time series – collection of observations in time: x( ti )
x( ti ) discrete time series with Δt
Deterministic process: Can be predicted exactly for all the values of the independent varriable ti
Stochastic process: Basically unpredictable – most geophysical phenomena
FOURIER ANALYSIS OF DETERMINISTIC PROCESS
Fourier Analysis is concerned with orthogonal functions:
ttT
nn
sin2
sin
tt
T
nn
cos2
cos
Tt 0
Any time series y(t) can be reproduced with a summation of cosines and sines:
n
nnnn tBtAtyty sincos Fourier series
Average Constants – Fourier Coefficients
n
nnnn tBtAtyty sincos Fourier series
Any time series y(t) can be reproduced with a summation of cosines and sines:
Collection of Fourier coefficients An and Bn forms a periodogram
defines contribution from each oscillatory component n to the total ‘energy’ of the observed signal – power spectral density
Both An and Bn need to be specified to build a power spectrum periodogram. Therefore, there are 2 dof per spectral estimate for the ‘raw’ periodogram.
1
0 sincos2
1
nnnnn tBtAAty
Construct y(t) through infinite Fourier series
yAn 20@ 0
An and Bn provide a measure of the relative importance of each frequency to the overall signal variability.
22
22
21
21 BABA e.g. if there is much more spectral energy at
frequency 1 than at 2
,2,1,0cos2
0
ndtttyT
AT
nn
To obtain coefficients:
tty ncos)(
,2,1,0sin2
0
ndtttyT
BT
nn tty nsin)(
1
0 cos2
1
nnnn tCCty
,2,1,0,22 nBAC nnn
,2,1,tan 1 nA
B
n
nn
Fourier series can also be expressed in compact form:
Tf 1
Tf 5
Tf 10
Tf 20
knn N
jnCtjy 2
cos
10100
2cos1
jtjy
520
2cos5.02
jtjy
210
2cos2.03
jtjy
85
2cos1.04
jtjy
(j)
2
10 sincos
2
1 N
njnnjnnj tBtAAty
Tnfnn 22
tjt j
2
10 2sin2cos
2
1 N
nnnj NjnBNjnAAty
2
10 2cos
2
1 N
nnn NjnCC
tNT
SUMMARY
2,,2,1,02cos2
1
NnNjnyN
AN
jjn
To obtain coefficients:
12,,2,1,02sin2
1
NnNjnyN
BN
jjn
0,2
01
0
ByN
AN
jj
0,cos1
21
2
N
N
jjN Bjy
NA
Multiplying data times sin and cos functions picks out frequency components specific to their trigonometric arguments
Orthogonality requires that arguments be integer multiples of total record length T = Nt, otherwise original series cannot be replicated correctly
Arguments2nj/N, are based on hierarchy of equally spaced frequencies n=2n/Nt and time increment j
Steps for computing Fourier coefficients:
1) Calculate arguments nj = 2nj/N, for each integer j and n = 1.
2) For each j = 1, 2, … , N evaluate the corresponding cos nj and sin nj ; effect sums of yj cos nj and yj sin nj
3) Increase n and repeat steps 1 and 2.
Requires ~N2 operations (multiplication & addition)
1601N ht 5.0
2,,2,1,02cos2
1
NnNjnyN
AN
jjn
12,,2,1,02sin2
1
NnNjnyN
BN
jjn
An Bn
Cn
,2,1,0,22 nBAC nnn
,2,1,tan 1 nA
B
n
nn
22nnn BAC
n
nn A
B1tan
70
10 2cos
2
1
nnnj NjnCCty
2
10 2cos
2
1 N
nnnj NjnCCty
22nnn BAC
n
nn A
B1tan
mra
dia
ns
Tnfnn 22
48/1601day per # NT