  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.
