Time Series and Spectral Methods for Imputation, Regression, and Environmental Health

统计方法的进步对于理解不断增长的数据流至关重要。时间序列是一种数据类型，由一个公共因子索引，由对物理现象的重复观测组成。该研究涉及此类时间序列中缺失样本的重构、谱估计以及两者在回归模型中的应用。这项研究属于两个不同的统计领域，都是处理数据的：第一个在分析前阶段(对原始时间序列进行插补)，第二个在分析阶段(开发分析模型)。 自然科学中的时间序列分析是解释现实世界现象的关键要素。随着数据序列长度的增加，这些分析具有越来越大的价值和预测能力。不幸的是，这样的系列经常被遗漏的唱片所困扰。插值法允许重建缺失记录的值，从而扩展整个序列的推理能力。拟议研究的一个方面是开发时间序列内插的算法和理论。这类算法的挑战在于真实世界时间序列的复杂性，而开发允许违反简单化假设的模型是这项研究的主要目标。 模型发展方面的研究包括将时间序列谱方法应用于两个问题：加性模型中有限时间尺度关联性的估计和滞后关联性的估计。第一个问题是混合数据的问题，预测因素和反应因素都有长期影响的驱动因素(例如，死亡率记录每年都不同)，但所需的推论是短期影响：需要仔细研究才能将两者分开。这些问题中的第二个是延迟效应问题，在这种情况下，预测者与反应具有时间延迟的联系。提出的研究为估计这些延迟效应开发了一个新的建模框架，消除了以前解决方案的可识别性问题。 这里提出的工作将解决这些与时间序列数据有关的问题，并导致将进一步科学地使用复杂时间序列的原创性研究。时间序列内插的工作对寻求分析长时间序列的研究人员将是有价值的，而关于加性模型的工作将对研究人员有价值，对他们来说，这些模型是估计时间尺度受限关联的主要工作工具。进一步的影响将是完善和提高人口健康风险估计的可靠性，这最终将影响我国政府制定的与加拿大人健康有关的政策。培训是这一研究计划的主要组成部分，该计划将支持1名博士和7名硕士研究生，以及8名本科生暑期研究学生。

Progress in statistical methods is vital for making sense of an ever-increasing flow of data. Time series are a type of data consisting of repeated observations of a physical phenomena, indexed by a common factor. The proposed research deals with the reconstruction of missing samples in such time series, the estimation of the spectra of them, and the use of both in regression models. The research falls within two distinct statistical areas, both dealing with data: the first in the pre-analysis stage (interpolating raw time series), and the second in the analysis stage (development of models for analysis). The analysis of time series in the natural sciences is a key element in the interpretation of real-world phenomena. These analyses have increasing value and predictive power as the length of the data series increases. Unfortunately, such series are often plagued by missing records. Interpolation allows for the reconstruction of values for missing records, extending the inferential power of the overall series. One prong of the proposed research centres on the development of algorithms and theory for the interpolation of time series. The challenge with such algorithms is the complex nature of real-world time series, and developing models which allow for violations of simplistic assumptions is the primary objective of the research. The proposed research on model development consists of application of time series spectral methods to two problems: the estimation of limited timescale associations in additive models, and the estimation of lagged associations. The first of these is a problem of mixed data, with both predictor and response having elements driven by long-term effects (e.g., mortality records vary annually), but with the inference desired being that of the short-term effects: careful work is required to separate the two. The second of these problems is that of delayed effects, where predictors have time delayed associations with responses. The research proposed develops a novel modelling framework for estimation of these delayed effects, eliminating an identifiability issue of previous solutions. The work proposed here will address these problems relating to time series data, and lead to original research that will further the scientific use of complex time series. The work on time series interpolation will be of value to researchers seeking to analyze long time series, while the work on additive models will be of value to researchers for whom these models are a primary working tool for estimation of timescale-limited associations. A further impact will be to refine and improve the reliability of risk estimation for population health, which ultimately will affect the policies made by our government that pertain to the health of Canadians. Training is a major component of this research program, which will support 1 Phd and 7 MSc graduate students, as well as 8 undergraduate summer research students.