Serially correlated noise

Figure 31: Time series generated by the simulator in the sample projects sim-serial (dots) and sim-temporal (open circles), respectively. The sampling represents the V photometry of IC4996#89. In both samples, the original observable values are replaced by the simulator.

This simulator module produces Gaussian noise the standard deviation of which may vary in time according to a polynomial trend. A serial correlation coefficient between consecutive data points may be specified additionally.

The keyword sim:serial is given with six floating-point parameters. They specify

1. the lower time limit,
2. the upper time limit,
3. the coefficient $\sigma _0$ for the standard deviation of the Gaussian noise,
4. the time zeropoint $t_0$ for the polynomial trend of the standard deviation,
5. the exponent $X$ for the polynomial trend of the standard deviation, and
6. the serial correlation coefficient.

The standard deviation of the Gaussian noise follows the relation

$$\sigma\left( t\right) := \sigma _0\,\left( t-t_0\right) ^X\: .$$ (24)

A full polynomial may be constructed by multiple keywords sim:serial with different parameters.

If the lower and upper time limits are both set zero, the noise is generated for the entire time base.

Figure 32: Typical significance spectrum for serially correlated noise, based on the sampling of the V photometry of IC4996#89. Serial correlation produces systematically higher sigs in the low frequency region.

Example. The sample project sim-serial contains the simulation and analysis of serially correlated noise. The sampling of the V photometry of IC4996#89 is used, and the simulator replaces the original observable values, according to the line

sim:replace

in the file sim-serial.ini. The line

sim:serial   0 0 1 0 0 0.8

specifies noise with a constant standard deviation of 1 and a serial correlation coefficient of 0.8. Setting the first two parameters zero provides synthetic data for the entire time series. The resulting light curve is displayed in Fig.31. The line

random number generator: file sim-serial.rnd

in the screen output indicates that a file sim-serial.rnd is found and used to initialise the random number generator. If such a file were not present, the system time would be used:

random number generator: system time initialisation

A significance spectrum is displayed in Fig.32. The overall shape of the spectrum is typical for serially correlated noise, characterised by higher amplitudes and sigs for low frequencies.