Comparison of two sampling theories to calculate the integration error for one-dimensional processes

The integration errors in some simulated and real time series were calculated as a function of the sampling interval. The errors were calculated from the variogram according to Gyʹs theory [1] and from the autocorrelation function according to Kateman and Müskensʹ theory [2]. The latter theory has been derived for first-order autoregressive processes. The calculated errors were compared with each other and with the true sampling error when this could be calculated. The true integration error and the errors calculated with the two theories were quite similar in the simulated first-order processes which were only slightly internally correlated. The processes with strong internal correlation were created with two simulation methods. In these processes Gyʹs integration error was smaller than the error by Kateman et al. and the true error. The normally distributed random noise added to the simulated processes increased Gyʹs integration error, but did not affect the error by Kateman et al. The error by Kateman et al. can be compared only with the non-periodic continuous term of Gyʹs integration error. The errors were calculated with two sample sizes in the simulated processes which were slightly internally correlated. The smaller the sample size, the larger the integration errors. The characteristics of the methods to calculate the errors with minimum sampling interval are discussed. Neither theories could calculate the error in a periodic process. Gyʹs integration error is much smaller than the error by Kateman et al. in a real periodic process.