Time Series Spectral Representation

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

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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

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

• Lowest frequency resolved
• Highest frequency resolved, corresponds to n/2
  (Nyquist frequency)
• General case, time interval ?t

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Aliasing

• Due to discretization, a sparsely sampled high
  frequency process may be erroneously attributed
  to a lower frequency

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Example. Fourier representation of streamflow
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Equivalent complex number representation
Note Integer truncation is used in the sum
limits. For example if n5 (odd) the limits are
-2, 2. If n6 (even) the limits are -2, 3.
(Same number of fourier coefficients as data
points)
Complex conjugate pair
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Fourier representation of an infinite (non
periodic) discrete series
Discrete transform pair
Lowest frequency Highest frequency Spacing As
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Fourier representation of an infinite (non
periodic) discrete series
Discrete transform pair
Re-center on (-n-1)/2, (n-1)/2
for
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Fourier transforms for discrete function on
infinite domain (aperiodic)
Wei section 10.5
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Fourier transforms for continuous function on
finite domain (periodic)
Wei section 10.6.1
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Fourier transforms for continuous function on
infinite domain (aperiodic)
Wei section 10.6.2
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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))

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Spectral decomposition of any function
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Spectral representation of a stationary random
process
Autocorrelation
Autocorrelation function
Time Series
Fourier Transform
Fourier Transform
Spectral density function
Fourier coefficients
Smoothing
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Time Domain Frequency Domain
C1, C2, C3,
Z1, Z2, Z3,
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The Periodogram
Ck2
k/n
Wei section 12.1
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Time Domain
r0.9
r0.2
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Frequency Domain
r0.9
r0.2
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The Spectral Density Function

• The spectral density function is defined as

S(w)dwE(C(w)2)
Wei section 12.2, 12.3
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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

n50
n200
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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

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Different degrees of smoothing
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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