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## 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}**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**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**Frequency and Wavenumber**cycles/time radians/time Due to orthogonality of Sin and Cos functions**Frequency Limits**Lowest frequency resolved Highest frequency resolved, corresponds to n/2 (Nyquist frequency) General case, time interval ∆t**Aliasing**Due to discretization, a sparsely sampled high frequency process may be erroneously attributed to a lower frequency**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**Fourier representation of an infinite (non periodic)**discrete series Discrete transform pair Lowest frequency Highest frequency Spacing As**Fourier representation of an infinite (non periodic)**discrete series Discrete transform pair Re-center on [(-n-1)/2, (n-1)/2] for**Fourier transforms for discrete function on infinite domain**(aperiodic) Wei section 10.5**Fourier transforms for continuous function on finite domain**(periodic) Wei section 10.6.1**Fourier transforms for continuous function on infinite**domain (aperiodic) Wei section 10.6.2**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))**Spectral representation of a stationary random process**Autocorrelation Autocorrelation function Time Series Fourier Transform Fourier Transform Spectral density function Fourier coefficients Smoothing**Time Domain Frequency Domain**C1, C2, C3, … Z1, Z2, Z3, …**|Ck|2 is random because Zt is random**Decomposition of variance Power at low frequency Persistence indicated by ACF The Periodogram |Ck|2 k/n Wei section 12.1**Time Domain**r=0.9 r=0.2**Frequency Domain**r=0.9 r=0.2**The Spectral Density Function**The spectral density function is defined as S(w)dw=E(|C(w)|2) Wei section 12.2, 12.3**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=200 n=50**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**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