1. Time Series Spectral Representation
Any mathematical function has a representation in terms of sin and cos functions.
Z(t) = {Z1, Z2, Z3, … Zn}

2. Spectral Analysis
Represent a time series in terms of the wavelengths associated with oscillations, rather than individual data values
The spectral density function describes the distribution of these wavelengths
Spectral analysis involves estimating the spectral density function.
Fourier analysis involves representing a function as a sum of sin and cos terms and is the basis for spectral analysis

3. Why Spectral Analysis
Yields insight regarding hidden periodicities and time scales involved
Provides the capability for more general simulation via sampling from the spectrum
Supports analytic representation of linear system response through its connection to the convolution integral
Is widely used in many fields of data analysis

4. Frequency and Wavenumber
cycles/time
radians/time
Due to orthogonality of Sin and Cos functions

5. Frequency Limits Lowest frequency resolved
Highest frequency resolved, corresponds to n/2 (Nyquist frequency)
General case, time interval ∆t

6. Aliasing
Due to discretization, a sparsely sampled high frequency process may be erroneously attributed to a lower frequency

7. Example. Fourier representation of streamflow

8. Equivalent complex number representation
Note: Integer truncation is used in the sum limits. For example if n=5 (odd) the limits are -2, 2. If n=6 (even) the limits are -2, 3. (Same number of fourier coefficients as data points)
Complex conjugate pair

9. Fourier representation of an infinite (non periodic) discrete series
Discrete transform pair
Lowest frequency
Highest frequency
Spacing As
wmax=pi
h=2
n=36
k=1:n
wk=k/n*wmax
Ck=exp(-(wk/h)^2)
plot(wk,Ck,type="l",lwd=3,xlim=c(0,pi),xaxs="i")
lines(wk,Ck)

10. Fourier representation of an infinite (non periodic) discrete series
Discrete transform pair
Re-center on [(-n-1)/2, (n-1)/2]
for

11. Fourier transforms for discrete function on infinite domain (aperiodic)
Wei section 10.5

12. Fourier transforms for continuous function on finite domain (periodic)
Wei section

13. Fourier transforms for continuous function on infinite domain (aperiodic)
Wei section

14. Frequency domain representation of a random process
Z(t) and C(w) are alternative equivalent representations of the data
If Z(t) is a random process C(w) is also random
ACF(Z(t))≡Spectral density function(C(w))

15. Spectral decomposition of any function

16. Spectral density function
Spectral representation of a stationary random process
Autocorrelation
Autocorrelation function
Time Series
Fourier Transform
Fourier Transform
Spectral density function
Fourier coefficients
Smoothing

17. Time Domain Frequency Domain
Z1, Z2, Z3, …
C1, C2, C3, …

18. The Periodogram |Ck|2 is random because Zt is random |Ck|2
Decomposition of variance
k/n
Power at low frequency Persistence indicated by ACF
Wei section 12.1

19. Time Domain
r=0.9
r=0.2

20. Frequency Domain
r=0.9
r=0.2

21. The Spectral Density Function
The spectral density function is defined as
S(w)dw=E(|C(w)|2)
Wei section 12.2, 12.3

22. Problem Estimating S(w)
Z(t) Stationary C(w) Independent
|C(w)|2 has 2 degrees of freedom (from real and imaginary parts
More data ∆w gets smaller, but still 2 degrees of freedom
n=50
n=200

23. S(w) estimated by smoothing the periodogram
Balance Spectral resolution versus precision
Tapering to minimize leakage to adjacent frequencies
Confidence bounds by 2 based on number of degrees of freedom involved with smoothing
Multitaper methods
A lot of lore

24. Different degrees of smoothing

25. Spectral analysis gives us
Decomposition of process into dominant frequencies
Diagnosis and detection of periodicities and repeatable patterns
Capability to, through sampling from the spectrum, simulate a process with any S(w) and hence any Cov()
By comparison of input and output spectra infer aspects of the process based on which frequencies are attenuated and which propagate through