Theoretical and Practical Limits of Kolmogorov-Zurbenko Periodograms with Dynamic Smoothing in Estimating Signal Frequencies

This investigation establishes the theoretical and practical limits of Kolmogorov-Zurbenko periodograms with dynamic smoothing in their estimation of signal frequencies in terms of their sensitivity, accuracy, resolution, and robustness. While the DiRienzo-Zurbenko algorithm performs dynamic smoothing based on local variation in a periodogram, the Neagu-Zurbenko algorithm performs dynamic smoothing based on local departure from linearity in a periodogram. This article begins with a summary of the statistical foundations for both the DiRienzo-Zurbenko algorithm and the Neagu-Zurbenko algorithm, followed by instructions for accessing and utilizing these approaches within the R statistical program platform. Brief definitions, importance, statistical bases, theoretical and practical limits, and demonstrations are provided for their sensitivity, accuracy, resolution, and robustness in estimating signal frequencies. 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 these approaches are compared to the autoregressive integral moving average (ARIMA) approach, with support again garnered for their being robust when data is missing. Throughout, the article contrasts the limits of Kolmogorov-Zurbenko periodograms with dynamic smoothing to those of log-periodograms with static smoothing, while also comparing the performance of the DiRienzo-Zurbenko algorithm to that of the Neagu-Zurbenko algorithm. It concludes by delineating next steps to establish the precision with which Kolmogorov-Zurbenko periodograms with dynamic smoothing estimate signal strength.