<p style='text-indent:20px;'>We introduce a novel method for Additive Noise Analysis for Persistence Thresholding (ANAPT) which separates significant features in the sublevel set persistence diagram of a time series based on a statistics analysis of the persistence of a noise distribution. Specifically, we consider an additive noise model and leverage the statistical analysis to provide a noise cutoff or confidence interval in the persistence diagram for the observed time series. This analysis is done for several common noise models including Gaussian, uniform, exponential, and Rayleigh distributions. ANAPT is computationally efficient, does not require any signal pre-filtering, is widely applicable, and has open-source software available. We demonstrate the functionality of ANAPT with both numerically simulated examples and an experimental data set. Additionally, we provide an efficient <inline-formula><tex-math id="M1">\begin{document}$ \Theta(n\log(n)) $\end{document}</tex-math></inline-formula> algorithm for calculating the zero-dimensional sublevel set persistence homology.</p>