Extracting common time trends from concurrent time series [electronic resource] : maximum autocorrelation factors with applications
Keywords: Earth System Science.

Concurrent time series commonly arise in applications such as environmental monitoring networks, with each monitoring or sampling site providing one of the time series. Examples are air quality measurement networks, weather stations, oceanographic buoys. Concurrent multiple time series also arise from proxy paleo climate data such as those from lake sediments, tree rings, ice cores, or coral isotopes, The goal that we address in this thesis is the efficient extraction of common time trends or time signals in the observed time series without any parametric representation. For this purpose we apply a non-parametric method referred to as MAF (Maximum Autocorrelation Factors) that linearly combines time series according to an optimality criterion that maximizes autocorrelation. We develop optimality properties of MAF in the context of a model where the time series differentially manifest the underlying signal time trend with additive random short-scale variability unrelated to the underlying trend. We establish optimality properties of MAF trend extraction where the time series are of this form, and further extend the optimality result to more complex data models that include the possibility of superposed multiple trend components. We proceed to compare MAF with the commonly used method of Principal Component Analysis (PCA) for combining time series. We quantify the advantages of MAF over PCA under time series models for the data that are of the form described above. We explore and quantify the statistical variability of both MAF and PCA computations and show the convergence of MAF computations with increasing time series length. As an illustrative application we apply the MAF methodology for combining concurrent tree ring time series to obtain regional representations of the shared time trend information. The extracted ``proxy'' tree ring time trends are then calibrated to the corresponding regional temperature record for the available instrumental period. The resulting calibration between the tree ring proxy trends and the regional temperature is used to backcast the pre-instrumental temperature time trend from the available earlier tree ring proxy time trends. We extend the MAF trend extraction methodology to accommodate time series with missing observations, in particular time series with assorted start and stop times. The conventional statistical approach to missing data is the Expectation-Maximization [EM] iterative algorithm that uses available data to impute missing data from which unknown parameters of a data model are estimated. However, EM does not exploit the time series nature of the data and requires specific parametric representations for the data. We extend EM in a way that exploits time structure and does not require parametric representations. Finally, we address computational difficulties that arise with MAF trend extraction when the number of concurrent time series is large in relation to the number of time steps within each time series. The difficulties arise in such situations because of the singularity of the matrix of cross-covariances between pairs of time series. To overcome these difficulties we introduce a multi-stage grouping strategy where separate MAF time trends are first extracted from small groups time series. The group-specific time trends are then themselves combined using the MAF method to obtain overall time trend estimates. We explore statistical properties of time-series grouping for common trend extraction using simulated data generated from a specified data model and quantify the loss of efficiency.