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