Time series analysis based on running Mann-Whitney Z Statistics

Abstract

A time series analysis method based on the calculation of Mann-Whitney U statistics is described. This method samples data rankings over running time windows, converts those samples to Mann-Whitney U statistics, and then normalizes the U statistics to Z statistics using Monte-Carlo generated null parameters. Based on the Z statistics' magnitudes this algorithm can identify time windows containing significant incidences of low or high data rankings, where the window length is determined by the sample size. By repeating this process with sampling windows of varying duration ranking regimes of arbitrary onset and duration can be objectively identified in a time series. The simplicity of the procedure's output -a time series' most significant nonoverlapping ranking sequences -makes it possible to graphically identify common temporal breakpoints and patterns of variability in the analyses of multiple time series. This approach is demonstrated using United States annual temperature data during 1896-2008.

Snowpack variations since AD 1150 in the Andes of Chile and Argentina (30°–37°S) inferred from rainfall, tree‐ring and documentary records

“…The approach used for assessing the main IMD modes of variability in the snowpack reconstructions is novel in paleoclimatic studies and offers very promising perspectives for identifying the most extreme intervals in a given time series while testing their statistical significance in a simple and objective manner [ Mauget , 2011]. The tests performed using a 10‐year period as a reference revealed patterns that were very similar in the instrumental and reconstructed series (Figure 5).…”

“…A recently developed time series analysis technique [ Mauget , 2003, 2004, 2011] that is novel to paleoclimatic studies was used to identify and test the statistical significance of the most important intra‐ to multi‐decadal (IMD) ranking regimes in the snowpack reconstructions. Here, IMD time scales are considered to include regime periods of Ns = 5 to 30 years duration.…”

“…Here, null distributions for an Ns year sample size were calculated via Monte Carlo sampling of the rankings of autoregressive red noise time series with the same persistence characteristics of pre‐filtered snowpack series. As this pre‐filtering removed IMD variance and trends, the null hypothesis associated with the MWZ statistics holds that snowpack time series represent semi‐random climate variations with inter‐annual persistence, but are essentially stationary and trendless with no significant IMD ranking regimes (see Mauget [2011] for more details). As a result, MWZ statistics associated with a significant incidence of high or low rankings in a sample are defined by ±1.960, ±2.576 and ±3.291 MWZ values, at 95%, 99% and 99.9% confidence intervals respectively.…”

Streamflow variations across the Andes (18°–55°S) during the instrumental era

“…To characterize the most important low frequency temporal patterns and evaluate the severity of the most extreme periods in each region, we first annualized the regionally averaged mean monthly values using an April-March hydrological year, and then applied a nonparametric time series analysis procedure that uses a combination of Mann-Whitney U tests 26,27 and Monte Carlo simulations to provide significance levels for moving time windows of varying duration in hydroclimatic series 28–30 . Essentially this method samples data rankings over moving time windows (ranging here between five and 20 yrs in length), converts those samples to Mann Whitney U statistics, and then normalizes the U statistics to Z statistics using Monte Carlo generated null parameters.…”

“…Based on the magnitude of the Mann–Whitney Z values this algorithm can identify time windows containing significant incidences of low or high data rankings, where Z values of ±1.960, ±2.576 and ±3.291 represent the 2-sided 95%, 99% and 99.9% confidence levels, respectively. This technique offers an objective, relatively simple, and statistically based way of detecting the driest and wettest intervals in hydrological time series from a given region 28–30 .…”