Nuclear Science and Engineering / Volume 153 / Number 2 / June 2006 / Pages 184-191
The performances of autoregressive processes and the autoregressive moving average process of order two and one [ARMA(2,1)] have been investigated concerning the confidence interval estimation in Monte Carlo eigenvalue calculation. Two reasons exist for these model choices. First, the Wold decomposition states that any zero-mean stationary stochastic process can be expressed as the sum of a deterministic process and a moving average process of infinite order. This justifies the application of autoregressive fitting and autoregressive moving average fitting to a centered k-effective series from stationary iteration cycles. Second, ARMA(2,1) fitting is a logically natural refinement of first-order autoregressive fitting since the noise propagation in iterated source methods can be reduced to an autoregressive moving average model of orders p and p - 1 [ARMA(p, p - 1)]. Numerical results are presented for the "k-effective of the world" problem. The results indicate that ARMA(2,1) fitting performs much better than the autoregressive fitting of low orders. Also presented are some related theoretical results; MacMillan's formula to confidence limits can be derived from the ARMA(p, p - 1) representation of source distribution; and the multiplicity of higher eigenmodes can make the decay of the autocorrelation of source distribution much different than predicted by the sum of exponential terms. The latter result indicates poor performance that time series methods would exhibit for the confidence interval estimation of the fission rate distribution in the critical reactor with symmetric component placement.