7. Fits of the signals with abrupt changes of the first derivative

To fit sharp minima of eclipsing variables and asymmetric maxima of pulsating variables, one may use an extreme approximation by a broken line: where u(t)=t, if t>0 and u(t)=0 else. For comparison of the tested methods we used a model

Here asymmetry of the extrema is dependent on parameter b. Obviously, one may formally change the sign of time to make The calculated function coincides with a broken line, if in dimensionless units inside the interval (-1, 1)

where

For 4 methods of smoothing one may obtain U(t)=(t + 1)²/4 ("um"), (t + 1)²(t⁴-2t³-2t²+6t+5)/32 ("wm"), (t+1)²(-5t² + 10t+3)/32 ("up") and - (t+1)²(45t⁶-90t⁵-61t⁴+212t³-13t²-186t-35)/512 ("wp"), respectively. Dependence on parameter b of times of minima , smoothed value and second derivative (last is used in Eq. (47) for determination of the accuracy estimate of the moment of extremum) are presented in Table 1 (click here).

  
Figure 5: Approximations of a broken line (0) with b=0.5 by fits (1, 2, 3, 4). The crosses mark extrema of the smoothing functions

 

 

um wm up wp um wm up wp um wm up wp

0.1 0.8182 0.5247 0.3989 0.2856 0.0909 0.0654 0.0177 0.0155 0.5500 0.5415 0.9093 1.1499

0.2 0.6667 0.3946 0.3134 0.2203 0.1667 0.1128 0.0516 0.0398 0.6000 0.8019 1.1290 1.5229

0.3 0.5385 0.3057 0.2478 0.1724 0.2308 0.1513 0.0793 0.0595 0.6500 1.0016 1.3129 1.8288

0.4 0.4286 0.2373 0.1946 0.1346 0.2857 0.1838 0.1024 0.0758 0.7000 1.1688 1.4756 2.0940

0.5 0.3333 0.1817 0.1500 0.1034 0.3333 0.2119 0.1220 0.0897 0.7500 1.3149 1.6242 2.3313

0.6 0.2500 0.1350 0.1119 0.0769 0.3750 0.2366 0.1389 0.1017 0.8000 1.4459 1.7624 2.5479

0.7 0.1765 0.0947 0.0787 0.0540 0.4118 0.2585 0.1535 0.1122 0.8500 1.5653 1.8928 2.7485

0.8 0.1111 0.0594 0.0494 0.0339 0.4444 0.2783 0.1663 0.1214 0.9000 1.6756 2.0167 2.9362

0.9 0.0526 0.0281 0.0234 0.0160 0.4737 0.2962 0.1775 0.1295 0.9500 1.7784 2.1355 3.1132

1.0 0.0000 0.0000 0.0000 0.0000 0.5000 0.3125 0.1875 0.1367 1.0000 1.8750 2.2500 3.2813

As one may see, for a fixed value of , approximation becomes better according to a sequence "um-wm-up-wp". Systematic deviation of the minimum of the smoothing function from the true one is times smaller for running parabolae ("wp") than for classical running mean ("um"). Unweighted parabolae are shifted times more than "wp".