TIME SERIES ANALYSIS Time series – collection of observations in time: x( t i ) x( t i ) discrete time series with Δt Deterministic process: Can be predicted exactly for all the values of the independent varriable t i Stochastic process: Basically unpredictable – most geophysical phenomena
FOURIER ANALYSIS OF DETERMINISTIC PROCESS Fourier Analysis is concerned with orthogonal functions: Any time series y(t) can be reproduced with a summation of cosines and sines: Fourier series AverageConstants – Fourier Coefficients
Fourier series Any time series y(t) can be reproduced with a summation of cosines and sines: Collection of Fourier coefficients A n and B n forms a periodogram power spectral density defines contribution from each oscillatory component n to the total ‘energy’ of the observed signal – power spectral density Both A n and B n need to be specified to build a power spectrum periodogram. Therefore, there are 2 dof per spectral estimate for the ‘raw’ periodogram.
Construct y(t) through infinite Fourier series A n and B n provide a measure of the relative importance of each frequency to the overall signal variability. e.g. if there is much more spectral energy at frequency 1 than at 2 To obtain coefficients:
Fourier series can also be expressed in compact form:
(j)
SUMMARY
To obtain coefficients: 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 = N t, otherwise original series cannot be replicated correctly Arguments2 nj/N, are based on hierarchy of equally spaced frequencies n =2 n/N t and time increment j
Steps for computing Fourier coefficients: 1) Calculate arguments nj = 2 nj/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 y j cos nj and y j sin nj 3) Increase n and repeat steps 1 and 2. Requires ~N 2 operations (multiplication & addition)
AnAn BnBn CnCn
m radians