A cyclostationary random process is a stationary random process whose sample functions can be expressed as a linear function of an underlying autoregression. By this definition, cyclostationarity is a special case of the general concept of spatio temporal stationarity in time series analysis, and it encompasses classical nonstationary processes such as trend-cycle components and seasonal components.
What is cyclostationary random process?
In a purely mathematical sense, a signal could be considered “cyclostationary” if its covariance functions has no explicit time lags, but this usage is very uncommon. In practice, the term refers to processes for which there exist parameters so that can be expressed as. This includes all well-known classical stochastic processes with finite variance (i.e. normal random variables) and many discrete or continuous time stochastic processes that are not normally distributed (such as Poisson or Markov point processes), but excludes purely deterministic functions such as the trigonometric sine and cosine functions.
A real-valued cyclostationary process is equivalently defined by its autocovariance function, where is usually normalized so that. The concept can be extended to include complex-valued signals, in which case the autocovariance function is replaced with the autocorrelation function. This definition allows for stationary components at arbitrary frequencies, unlike classical stationarity. However, if all of these components are strictly zero mean (i.e., they have finite variance), then the process is strictly speaking cyclostationary.
AR(1) process with stationary white noise error term. The autocorrelation function of an AR(1) process can be written as: . Thus, it is cyclo-stationary if and only if .
If is equal to , then the series can also be called IID (independent and identically distributed), since each observation has the same statistical properties for all times except possibly for a constant offset.
This simplifies analysis by allowing one to use standard results about independent random variables. However, many authors reserve the term IID for purely stationary processes, in which case this terminology may be confusing.
A cyclostationary random process is irregular in the sense that its statistical properties do not remain constant over time. Consequently, it has neither finite mean nor variance even though the mean and variance of an underlying stochastic trend term may exist. Instead, the mean and variance are best described by means of a spectral decomposition. For instance, if is unit-mean cyclo-stationary with power spectral density , then .
The autocovariance function can be written as:
Where formula_3 for some deterministic . The last integral fails to converge because there are only a countable number of times at which events can happen (e.g., occurs at discrete points 0, formula_4, etc.).
This is a singularity which makes the AR(1) cyclostationary process non-stationary even though it is a special case of a stationary cyclo-stationary process.
Example: Let be an IID (independent and identically distributed) Gaussian random variable with zero mean and variance . Then the autocovariance function of becomes:
Where is the Kronecker delta function. By taking , we see that this process becomes purely periodic with period 1, as desired. If we let denote the unit-mean IID Gaussian random variable :
Note that the presence of the nonlinear term breaks time stationarity; however, if we define (a linear function of time), then this process becomes a stationary cyclostationary random process. The power spectral density for this process is given by:
Where . This example demonstrates that an AR(1) cyclostationary processes may be non-Gaussian even though the underlying Gaussian IID increments are asymptotically Gaussian as formula_8. In fact, if we define (a differentiable function of time), then the IID increments will be strictly zero mean and finite variance as desired, but they will no longer have the standard normal distribution.
The following theorem provides necessary and sufficient conditions for a stochastic process to be a cyclo-stationary random process:
Theorem: A real-valued function of stationary random variables is cyclo-stationary if and only if it is linear with respect to each random variable in turn.
Both properties are sufficient, but not necessary for the process to be cyclo-stationary. For example, a strictly stationary process is also required to be monotone (or piecewise constant), but neither property alone suffices.
This can be applied in practice via spectral analysis or through use of an appropriate moving average representation.
One way of constructing cycle stationary processes with finite mean and autocovariances which decay at infinity is to define . Then has the same qualitative characteristics as the original AR(1) model except that its covariance functions decay exponentially at infinity.
Cyclostationary Random Processes are irregular in the sense that their statistical properties do not remain constant over time. If you need to generate a cyclo-stationary process, it helps to know how spectral decomposition works and what conditions must be met for a stochastic process to be cyclo-stationary. When generating an AR(1) or MA(1) stationary random processes with finite mean and autocovariances which decay at infinity, use . A linear function of time can also work as long as it is strictly monotonic (or piecewise constant), but neither property alone suffices.
Read More : The complete Study Guide For Cyclostationarity