Estimation of uncertainties


Period04 provides several tools to calculate the uncertainties of the parameters of a fit:
  1. Calculation of uncertainties from the error matrix of a least-squares calculation
  2. Monte Carlo Simulation
  3. Uncertainties calculated from analytically derived formulae assuming an ideal case.

Please note:
The errors of frequency and phase are correlated. However, by an appropriate choice of a zero point in time the uncertainties for frequency and phase can be decoupled (Montgomery & Odonoghue, 1999, DSSN, 13, 28). This is the case when
It is very likely that your data set does not fulfill this condition. Therefore, Period04 provides the possibility to shift the data set by the required value in time, for the purpose of determining the uncorrelated parameter uncertainties when the standard fitting formula is being used.

1. Calculation of uncertainties from the error matrix of a least-squares calculation


Period04 applies the curfit routine from Bevington, this is a Levenberg-Marquardt non-linear least-squares fitting procedure. As a by-product of least-squares fits an error matrix is available from which parameter uncertainties can be calculated.
In some cases though, i.e. when the error matrix is ill-conditioned, this method does not provide a good estimation of the uncertainties. In order to ensure that the calculated uncertainties are reliable, Period04 performs checks for these cases.


The output of common least-squares fits are correlated uncertainties. When the standard fitting formula is being used, Period04 additionally offers the possibility to calculate uncorrelated uncertainties.

To calculate the uncertainties of the fit parameters, press Calculate LS uncertainties in the "Goodness of Fit" tab. If you improved frequencies and phases simultaneously, a dialog will ask you whether you want to uncouple the uncertainties of frequency and phase (in other words: whether you want to calculate correlated or uncorrelated uncertainties).

After you made your choice, the uncertainties will be displayed in the text box.

2. Monte Carlo Simulation


Monte Carlo simulations are a very reliable way to determine parameter uncertainties. The principle idea is to repeat an experiment (in our case the optimization routine) on almost identical samples.
For the Monte Carlo simulation Period04 generates a set of time strings. Each data set is created as follows: For every data set a least-squares calculation will be done. Based on the distribution of fit parameters the program calculates the uncertainties of the parameters.

A short step-by-step guide for making Monte Carlo simulations:

3. Uncertainties calculated from analytically derived formulae assuming an ideal case


Based on some assumptions one can derive a formula for the uncertainties in frequency, amplitude and phase. See Breger M., Handler G., Garrido R., et al., 1999 A&A, 349, 225 for the derivation based on a monoperiodic fit. If cross terms can be neglected then the following equations can also be applied for each pulsation frequency separately:

N is the number of time points, T is the time length of the data set, σ(m) denotes the residuals from the fit and 'a' refers to the amplitude of the frequency.

To show these parameter uncertainties, select Show analytical uncertainties in the "Special" menu.
Please note:
This option is only available when the standard fitting formula is being used.