Extracting Common Time Trends from Concurrent Time Series: Maximum Autocorrelation Factors with Application to Tree Ring Time Series Data

Concurrent time series commonly arise in various applications, including when monitoring the environment such as in air quality measurement networks, weather stations, oceanographic buoys, or in paleo form such as lake sediments, tree rings, ice cores, or coral isotopes, with each monitoring or sampling site providing one of the time series. The goal in such applications is to extract a common time trend or signal in the observed data. Other examples where the goal is to extract a common time trend for multiple time series are in stock price time series, neurological time series, and quality control time series. For this purpose we develop properties of MAF [Maximum Autocorrelation Factors] that linearly combines time series in order to maximize the resulting SNR [signal-to-noise-ratio] where there are multiple smooth signals present in the data. Equivalence is established in a regression setting between MAF and CCA [Canonical Correlation Analysis] even though MAF does not require specific signal knowledge as opposed to CCA. We proceed to derive the theoretical properties of MAF and quantify the SNR advantages of MAF in comparison with PCA [Principal Components Analysis], a commonly used method for linearly combining time series, and compare their statistical sample properties. MAF and PCA are then applied to real and simulated data sets to illustrate MAFs efficacy.