Novel Methods and Applications for Time Series Data

This paper develops a mathematical model and statistical methods to quantify trends in presence/absence observations of snow cover (not depths) and applies these in an analysis of Northern Hemispheric observations extracted from satellite flyovers during 1967-2021. A two state Markov chain model with periodic dynamics is introduced to analyze changes in the data in a cell by cell fashion. Trends, converted to the number of weeks of snow cover lost/gained per century, are estimated for each study cell. Uncertainty margins for these trends are developed from the model and used to assess the significance of the trend estimates. Cells with questionable data quality are explicitly identified. Among trustworthy cells, snow presence is seen to be declining in almost twice as many cells as it is advancing. While Arctic and southern latitude snow presence is found to be rapidly receding, other locations, such as Eastern Canada, are experiencing advancing snow cover.  

Sublevel set persistence diagrams from topological data analysis have found greater statistics and data science applications in recent years; e.g. in fields such as biomedical signal processing. In this talk, we detail two fundamental convergence results for such persistent diagrams for stationary time series and delineate their applications and efficacy in hypothesis testing for time series. Namely, we demonstrate both our strong law of large numbers and our central limit theorem for these topological statistics. We then discuss how these results apply to a wide variety of time series models and various functionals of the derived persistence diagrams. We will then discuss how these results can be applied to test for various characteristics of time series such as the presence of serial correlation and changepoints. 

In time series analysis, performing inference on parameters using Wald-type statistics requires adjustments for serial correlation or conditional heteroskedasticity. We use the spectral variance estimator for long-run variance, which makes minimal assumptions about the error term.

We examine two methods for obtaining critical values (CVs): adaptive-smoothing asymptotics, yielding standard chi-square CVs, and fixed-smoothing asymptotics, producing non-standard CVs that need approximation. The fixed-smoothing framework, including lugsail kernels, better captures long-run variance influences on test statistics, especially under strong correlation.

Our contributions include exploring methods for approximating fixed-smoothing CVs and comparing their performance across classical time series models in univariate and multivariate settings. We evaluate estimator corrections, mean squared error, bandwidth selection methods, and testing performance.

These findings extend the work of Kiefer and Vogelsang (2005), Lazarus et al. (2018), Sun (2014), and Kurtz-Garcia and Flegal (2024), offering practical support for researchers using fixed-smoothing frameworks and spectral variance-like estimators in various data contexts, such as spatial and longitudinal data. 

We study a popular tool in topological data analysis (TDA) called sublevel set persistent homology on discrete functions through the perspective of finite ordered sets of both linearly ordered and cyclically ordered domains. We prove duality of filtrations of sublevel sets that undergirds a range of duality results of sublevel set persistent homology without the need to invoke complications of continuous functions or classical Morse theory. We show that Morse-like behavior can be achieved for flat extrema without assuming genericity. Furthermore, we discuss aspects of barcode construction rules, surgery of circular and linearly ordered sets. We end by discussing ideas of future work that integrate this framework with more traditional statistical techniques for analyzing time series. 

In recent years there has been a large rise of battle royale tournaments. In this style tournament competitors are pitted against each and eliminated sequentially until a winner has been crowned. These competitions can be seen in video games, board games, endurance challenges, and reality competition series. We have developed a hierarchical Bayesian model to predict the results of a player-elimination tournament. In this talk we will use the reality competition series Project Runway as an illustrative example. We propose an adaptive model, which incorporates past results and player demographic information. We evaluate our model using various classification metrics.