Statistical Modeling, Inference, and Analysis of Nanoscopic Phenomena

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With the current state-of-the-art in nanoscopic measurement technology, researchers are able to observe the dynamics of molecules and atoms with unprecedented accuracy and reproducibility. These nanoscopic experiments present formidable challenges and opportunities for statistical modeling. On one hand, randomness naturally emerges from deterministic physical systems in which countless degrees of freedom are unobserved. On the other, departures from ideal experimental conditions require a detailed analysis of measurement error. Ultimately, the scientific goal is to describe a dynamical phenomenon with an explanatory model. This research program will address two important challenges to achieving this goal: efficient statistical inference for stochastic physical models, and effective evaluation of the agreement between theoretical models and experimental data. Specifically, the statistical models for nanoscopic phenomena are given in continuous time, whereas recorded data is almost always discrete. This poses a major hurdle to parametric inference, as the likelihood function induced by these data is generally unavailable. A common strategy known as "data augmentation" is to impute the missing continuous path of the data in order to calculate the likelihood, and then integrate it back out. Efficient imputation of continuous data in this context is one of the primary objectives of this research program, and will be performed by Gaussian Process Regression (GPR). This constitutes a novel application of this long-standing stochastic interpolation technique. Most of the statistical literature on inference for continuous-time models operates under a Markov assumption. While this is appropriate when dynamical phenomena have rapid decorrelation times, for many cutting-edge experiments with high frequency measurements, this is not the case. A particularly important non-Markov stochastic model is known as a Generalized Langevin Equation (GLE). This model has received considerable attention from the scientific community with applications in physics, chemistry, molecular biology, and even quantum mechanics. This research program will, for the very first time, address parametric inference for the GLE in the non-Gaussian setting. The key is an approximation to the continuous-time likelihood by a novel numerical discretization scheme, through which the path imputation framework described above can be directly applied. While the primary focus of the proposed research is on statistical inference, it is of critical importance to determine whether a given model -- possibly selected from competing alternatives -- in fact agrees with the experimental data. The validity of a scientific model lies, in part, is its ability to make predictions. In many nanoscopic experiments, however, the predictions of interest are not merely univariate quantities but entire functions, such as mean-squared displacement curves or frequency spectra. This research program will leverage the principles of functional data analysis to meaningfully reduce these functions to low-dimensional statistics, which can readily be used for both goodness-of-fit testing and model selection. In a broader statistical context, the inferential and analytical challenges encountered here arise when almost any continuous-time stochastic model is discretely observed. This is the case for a wide range of non-nanoscopic phenomena occurring in areas such as geology, climatology, neurology, and finance. As such, the proposed research program is highly relevant to a multitude of problems under active investigation by researchers in Canada and abroad.