Instructions
User has to input the end time of interval where g-renewal process parameters will be calculated, then input the number N of components observed. Each component should have a set of event times ending with integer number 0 or -1. Number 0 indicates that the component suspended at the time corresponding to the previous number. Number -1 corresponds to the case when observation finished at the last event which is the previous number. All N sets of event times should be stored in a text file (with .txt extension). If it is convenient, the user can prepare input data in Excel format and then copy and paste the data in a .txt file and then use it in the calculation.

Each number in the file should be separated by white space or new line ("Enter"). Note that two integer numbers without separator will be read as one number, for example "23" and "67" will be read as "2367". Two float numbers without separator could be corrupted, for example "0.34" and "3.5" typed as "0.343.5" can be read as "0.343" and "0.5". We check the format of input numbers and in the most cases one will get an error message if the format is not valid, but it is the responsibility of the user to prepare the input data properly. We also check for consistency of the input set of numbers. One can use "Browse" to select the required file and click "Upload" to submit.

In this calculation non-parametric estimation of expected number of events is performed first, then regression analysis is applied to obtained data for recurrent event prediction. To apply it, we have to calculate the cumulative intensity function (CIF) at each event time using non-parametric estimation (see General information). Note, that the accuracy of this calculation significantly depends on the number of observed components. The error is in-between 1/N and 1/rN where N is total number of components and rN is the number of operating components at the end of observation. Therefore, the formula is valid at time ti of a failure if corresponding value of CIF is much greater than 1/ri. We recommend to use this calculation only if N>10. Otherwise it is better to apply maximum likelihood estimator. However, in case of large enough number of components N, the suggested regression analysis allows better data approximation using Pade functions with many parameters and different shapes. For this reason it is also more efficient for data extrapolation (see Examples).

On the next page user should enter three parameters of approximation, which allow to govern the process of calculation and to reach required accuracy. The first one is the required accuracy of approximation. The second field is the maximum number of coefficients of Pade approximation (we recommend 3-5). The program calculates and chooses the best approximation by increasing this number from 2 until the maximum entered by user will be reached. All these values will be printed back with the result of calculation. The calculation will be completed if one of the following 2 conditions is reached:

  1. The error of approximation is equal to or less than that entered by user. The error of approximation is calculated as the ratio of the maximum error of approximation to the mean value of absolute values of ordinates of all given points.
  2. The current amount of coefficients of the approximation function is equal to the maximum entered by user.

At each calculation one will get and can print the approximation formula, the table of given data and their approximation, the calculated standard error, Weibull function parameters, and the accuracy of approximation. After analyzing the result, you can correct the input of governing parameters of approximation to get the desired result. We also show "The number of iterations without significant changing of error", which means that the change of standard deviation did not exceed 10% at last iterations. In this case user can try to reduce the number of coefficients of the approximation function without loss of accuracy.

In addition, the user has the option to discard the effect of censoring if "No" is selected in the field ""Select censoring option". In this case the lower bound of non-parametric estimation will be applied.

The software provides the possibility to extrapolate the process outside the time interval of observation because of a relatively large number of approximation parameters in the final formula. However this estimation should be used with caution because there is no reliable method for accuracy estimation of the extrapolation. We show the result of linear extrapolation of the data as well. Extrapolation formula should be considered as adjustment to linear extrapolation which cannot exceed about 30%.