The cyclic variation of data points in a process is known as cyclostationarity. This process is mainly used for identifying random noise from periodic components of the signal, which are called as non-stationary random processes.
If any linear combination of random variables follows the statistical stationary system then it is termed to be a long-term dependency or memory on the system. In this case, both autocorrelation function and power spectral density functions are not able to identify these patterns properly making them more complex than white noise. It also shows how traditional methods fail in detecting signals with a long-term dependency.
What is Cyclostationary Process? – Definition and Application
A cyclostationary random process is a stochastic process with statistically stationary and independent, but the non-Gaussian short term statistical behavior which can be represented by a Fourier series having purely constant (or linearly time-variant) parts and periodically varying (cyclic) parts of frequencies not necessarily restricted to integer multiples of the fundamental frequency.
It is known as an anti-persistent process that has its mean square exponentially decaying. In this type of process, the first two moments do not have a desirable property as they fluctuate rather than being time-invariant or constant over different periods. Several statistical models including linear predictive coding, Wiener filtering, etc., have been proposed based on cyclostationarity.
The value of a function at multiple points in a time series is also known as a multivariate time series. A multivariate time series can be either stationary or non-stationary, but the parameters have to vary along with time for representing a non-stationary process.
The oscillating components that are part of the signal and do not have fixed amplitude and fixed frequency are called cyclic variations or Cyclical variations. In this context, it has been found that random noise from periodic components of signal is called non-stationary random processes. The level of statistical accuracy depends on these factors:
The time-domain analysis makes it easier to detect cyclostationary processes due to its better understanding capabilities when compared to the frequency domain. On the other hand, higher accuracy can be achieved from spectral analysis when compared to time series analysis.
For example, consider a data set consisting of multiple points at which wind speed is measured at different points in time. If the data fluctuates with fixed intervals between two consecutive measurements, it means that there is a cyclostationary component in the process and if this variation has an additional linear trend then it can be said to have a long-term dependency or memory on the system.
In general, many real world processes contain both cyclic variation and a linear trend, which makes them more complex than stationary processes. This concept enables us to identify random noise from periodic components using appropriate statistical measurements such as autocorrelation function and power spectral density functions.
The simplest example of a cyclostationary random process is a sinusoidal wave having period and frequency. The Fourier series for such a periodic function can be represented as:
Conclusion
Cyclostationary processes are random, but their behavior changes over time. The simplest example of a cyclostationary process is a sinusoidal wave having period and frequency. When we consider the -th partial sum: then: This shows that is an even function and has zero mean value which implies that statistically, its mean square variation or auto-variance decays exponentially with increasing lag. Hence, it possesses all the properties of noise with long term memory.
It also indicates that selecting data points at fixed intervals will not provide accurate statistical information about such processes as their behavior changes over time.
For many real-world processes, both cyclic variation and a trend can be observed. This makes them more complex than stationary processes which only contain linear trends. This concept enables us to identify random noise from periodic components of using appropriate statistical measurements such as autocorrelation function and power spectral density functions.
In another example, consider a time series containing data points collected from weather stations around the world over ten years. If the collected data is plotted as a histogram with different values of y-axis corresponding to different years, it will produce no obvious pattern in the resulting graph as each year contains its own set of values.
