Draft:Oscillatory process

Oscillatory processes are a class of non-stationary Gaussian processes introduced by M. B. Priestley in the 1960s as part of his framework for analyzing non-stationary stochastic processes through the concept of evolutionary spectra. These processes generalize stationary processes by allowing their spectral properties to vary smoothly over time while maintaining a meaningful physical interpretation of frequency.

Definition

Oscillatory Functions

A function φ_t(ω) is called an oscillatory function if, for some necessarily unique θ(ω), it can be written in the form

${\displaystyle \varphi _{t}(\omega )=A_{t}(\omega )e^{i\theta (\omega )t}}$

where A_t(ω) is the modulating amplitude of the form

${\displaystyle A_{t}(\omega )=\int _{-\infty }^{\infty }e^{itu}dK_{\omega }(u)}$

with |dK_ω(u)| having an absolute maximum at u = 0. This condition ensures that A_t(ω) varies extremely slowly with time t, which is essential for maintaining the physical interpretation of frequency.

Oscillatory Processes

A stochastic process {X(t)} is termed an oscillatory process if there exists a family of oscillatory functions {φ_t(ω)} such that the process has a representation of the form

${\displaystyle X(t)=\int _{-\infty }^{\infty }\varphi _{t}(\omega )dZ(\omega )}$

where Z(ω) is an orthogonal process with E[|dZ(ω)|²] = dμ(ω).

Evolutionary Power Spectrum

For an oscillatory process, the evolutionary power spectrum at time t is defined by

${\displaystyle dH_{t}(\omega )=|A_{t}(\omega )|^{2}d\mu (\omega )}$

When a smooth density function exists, the evolutionary spectral density function is given by

${\displaystyle h_{t}(\omega )=H'_{t}(\omega )}$

This spectrum describes the local power-frequency distribution at each instant of time, generalizing the classical power spectral density for stationary processes. The evolutionary spectrum satisfies

${\displaystyle {\text{Var}}(X(t))=\int _{-\infty }^{\infty }h_{t}(\omega )d\omega }$

for each time t.

Properties

Reduction to Stationary Case

When X(t) is stationary and θ(ω) = ω, the evolutionary spectrum dH_t(ω) reduces to the regular power spectrum of a stationary process, making oscillatory processes a proper generalization of stationary processes.

Semi-Stationary Processes

An important subclass is semi-stationary processes or uniformly modulated processes, where the oscillatory functions take the specific form

${\displaystyle \varphi _{t}(\omega )=A_{t}(\omega )e^{i\omega t}}$

with A_t(ω) = C(t) being a slowly varying function of time. This class is particularly tractable for statistical estimation and is related to modern concepts of locally stationary processes.

Heisenberg-Gabor Uncertainty Principle

A fundamental limitation in determining evolutionary spectra is expressed by an uncertainty principle: one cannot obtain simultaneously a high degree of resolution in both the time domain and frequency domain. More accurate determination of h_t(ω) as a function of time necessarily reduces the accuracy in the frequency domain, and vice versa.

Estimation

Priestley developed estimation methods based on the double-window technique. Given a realization x(t) of length T, the evolutionary spectrum at frequency ω₀ is estimated using

${\displaystyle y(t)=\int _{t-T}^{t}g(u)x(t-u)e^{-i\omega _{0}(t-u)}du}$

where g(u) is a linear filter with width much smaller than the characteristic bandwidth of the process. Under suitable conditions, E(|y(t)|²) provides an approximately unbiased estimate of h_t(ω₀).

To reduce sample fluctuations and obtain a consistent estimator, smoothing in time is applied through

${\displaystyle z(t)=\int _{-\infty }^{\infty }w_{T'}(u)|y(t-u)|^{2}du}$

where w_T′(t) is a weight function depending on a time window parameter T′.

Applications

Oscillatory processes have been applied to various fields including:

• Seismology: Analysis of earthquake accelerations and ground motion
• Meteorology and climatology: Study of non-stationary atmospheric and oceanic processes
• Engineering: Dynamic response analysis of structures to random excitations
• Signal processing: Time-frequency analysis of signals with time-varying spectral content

Historical Development

The theory of oscillatory processes was developed by M. B. Priestley through a series of papers in the 1960s, with comprehensive treatments appearing in his books Spectral Analysis and Time Series (1981) and Non-linear and Non-stationary Time Series Analysis (1988). The framework built upon earlier ideas by Emanuel Parzen (1959) regarding multiple representations of stochastic processes.

Priestley and T. Subba Rao (1969) developed statistical tests for stationarity based on the evolutionary spectrum framework, which were ahead of their time and not widely implemented in software until decades later.

Related Concepts

• Stationary process
• Spectral density estimation
• Time-frequency analysis
• Locally stationary process
• Wavelet

See Also

• Time series
• Stochastic process
• Fourier analysis
• Non-stationary signal

References

• Priestley, M. B. (1965). "Evolutionary spectra and non-stationary processes". Journal of the Royal Statistical Society: Series B (Methodological). 27 (2): 204–229.
• Priestley, M. B. (1967). "Power spectral analysis of non-stationary random processes". Journal of Sound and Vibration. 6 (1): 86–97.
• Priestley, M. B. (1981). Spectral Analysis and Time Series. Academic Press.
• Priestley, M. B. (1988). Non-linear and Non-stationary Time Series Analysis. Academic Press.
• Priestley, M. B.; Subba Rao, T. (1969). "A test for non-stationarity of time series". Journal of the Royal Statistical Society: Series B (Methodological). 31 (1): 140–149.

External Links

Category:Time series Category:Stochastic processes Category:Signal processing Category:Statistical theory