Temporally correlated noise

This simulator module produces Gaussian noise the standard deviation of which may vary in time according to a polynomial trend. A temporal correlation coefficient $R_T$ between consecutive data points $t_{n-1}$, $t_n$ may be specified. In contrary to the serial correlation, the temporal correlation takes into account the width of the time interval between pairs of data points, which has implications on the noise behaviour of non-equidistantly sampled data. The serial correlation $R_S$ drops exponentially with the distance in time according to

$$R_S := R_T^{t_n - t_{n-1}}\: .$$ (25)

In this context, the temporal correlation coefficient may be interpreted as the serial correlation coefficient of two data points separated by one unit of time.

The keyword sim:temporal 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 temporal correlation coefficient $R_T$.

The standard deviation of the Gaussian noise follows the relation

$$\sigma\left( t\right) := \sigma _0\,\left( t-t_0\right) ^X\: .$$ (26)

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

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

Figure: Typical significance spectrum for temporally correlated noise, based on the sampling of the V photometry of IC4996#89. Temporal correlation produces systematically higher sigs in the low frequency region, which is quite comparable to serial correlation (Fig.32).

Example. The sample project sim-temporal contains the simulation and analysis of temporally 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-temporal.ini. The line

sim:temporal   0 0 1 0 0 0.01

specifies noise with a constant standard deviation of 1 and a temporal correlation coefficient of 0.01. Setting the first two parameters zero provides synthetic data for the entire time series. The resulting light curve is displayed in Fig.31. Comparing this light curve to the dataset generated in the project sim-serial (p.), the correlation between consecutive data points is obviously much stronger in the present example. Using Eq.25with a typical sampling interval width of 9 min for the dataset under consideration, the temporal correlation coefficient of 0.01 corresponds to a serial correlation coefficient of $\approx$ 0.97.

The significance spectrum displayed in Fig.33 shows the same overall characteristics as the corresponding spectrum for serially correlated noise (Fig.32, but the sigs at low frequencies are considerably higher, which is a consequence of the strong serial correlation associated to this setup.