Polynomial trend

The keyword sim:poly is given with five floating-point parameters. They specify

1. the lower time limit,
2. the upper time limit,
3. the coefficient $P_0$,
4. the time zeropoint $t_0$, and
5. the exponent $X$.

If the exponent is a non-integer number, the simulator evaluates

$$P\left( t\right) := P_0\left\vert t-t_0\right\vert ^X$$ (21)

instead and produces a power function.

For integer exponents, the trend is generated by the relation

$$P\left( t\right) := P_0\left( t-t_0\right) ^X\: .$$ (22)

Thus a full polynomial may be constructed by multiple keywords sim:poly with different parameters and integer exponents.

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

Figure 29: Time series generated by the simulator in the sample project sim-poly. The sampling represents the V photometry of IC4996#89. The simulator replaces the origninal observable by 16 different power functions.

Example. The sample project sim-poly contains the simulation and analysis of 16 individual power functions defined on different time intervals (Fig.29). 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-poly.ini. The specifications for the power functions are contained in the lines

sim:poly   2520.215 2521.088 4.298 2520.626  0.581
sim:poly   2521.088 2521.679 2.932 2521.443  1.195
sim:poly   2521.679 2522.442 1.092 2522.067  1.063
sim:poly   2522.442 2522.595 5.372 2522.466  0.676
sim:poly   2522.595 2523.351 2.495 2522.682  2.042
sim:poly   2523.351 2523.924 2.839 2523.607  0.221
sim:poly   2523.924 2524.478 8.357 2525.412 -0.899
sim:poly   2524.478 2525.399 2.304 2524.576  1.432
sim:poly   2525.399 2526.107 2.573 2525.721  1.205
sim:poly   2526.107 2526.550 6.350 2526.493  0.031
sim:poly   2526.550 2526.847 4.192 2526.589  2.893
sim:poly   2526.847 2527.616 0.345 2527.652 -0.472
sim:poly   2527.616 2528.264 3.583 2527.783  0.725
sim:poly   2528.264 2528.777 1.246 2528.704  0.610
sim:poly   2528.777 2529.606 3.534 2529.535  1.752
sim:poly   2529.606 2530.242 9.002 2529.694  1.119

The screen output contains the lines

*** simulator: replace *************************************

polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend
polynomial trend

to indicate that the simulator replaces the original observables by the synthetic values, and that 16 power functions are generated.

SIGSPEC detects 19 significant signal components, which are not discussed here.