An extension of the iterated moving average

Abstract

This work introduces an extension of the iterated moving average filter, called the Extended Kolmogorov-Zurbenko (EKZ) filter for time series and spatio-temporal analysis. The iterated application of a central simple moving average (SMA) filter, also known as a Kolmogorov-Zurbenko (KZ) filter, is a low-pass filter defined by the length of the moving average window and the number of iterations. These two arguments determine the filter properties such as the energy transfer function and cut-off frequency. However, the existing KZ filter is only defined for positive odd integer window lengths. Therefore, for any finite time series dataset there is only a relatively small selection of possible window lengths, determined by the length of the dataset, with which to apply a KZ filter. This inflexibility impedes use of KZ filters for a wide variety of applications such as time series component separation, filtration, signal reconstruction, energy transfer function design, modeling, and forecasting. The proposed EKZ filter extends the KZ and SMA filters by permitting a widened range of argument selection for the filter window length providing the choice of an infinite number of filters that may be applied to a dataset, affording enhanced control over the filter characteristics and greater practical application. Simulations and real data application examples are provided.