Theoretical and Practical Limits of Kolmogorov-Zurbenko Periodograms with DiRienzo-Zurbenko Algorithm Smoothing in the Spectral Analysis of Time Series Data

This investigation establishes the theoretical and practical limits of the Kolmogorov-Zurbenko periodogram with DiRienzo-Zurbenko algorithm smoothing with respect to sensitivity (i.e., ability to detect weak signals), accuracy (i.e., ability to correctly identify signal frequencies), resolution (i.e., ability to separate signals with close frequencies), and robustness (i.e., sensitivity, accuracy, and resolution despite high levels of missing data). Compared to standard periodograms that utilize static smoothing with a fixed window width, Kolmogorov-Zurbenko periodograms with DiRienzo-Zurbenko algorithm smoothing utilize dynamic smoothing with a variable window width. This article begins with a summary of its statistical derivation and development followed by instructions for accessing and utilizing this approach within the R statistical program platform. Brief definitions, importance, statistical bases, theoretical and practical limits, and demonstrations are provided for its sensitivity, accuracy, resolution, and robustness. Next using a simulated time series in which two signals close in frequency are embedded in a significant level of random noise, the predictive power of this approach is compared to an autoregressive integral moving average (ARIMA), with support also garnered for its being robust even in the face of a high level of missing data. The article concludes with brief descriptions of studies across a range of scientific disciplines that have capitalized on the power of the Kolmogorov-Zurbenko periodogram with DiRienzo-Zurbenko algorithm smoothing.