Practical Analysis of Time Series

Abstract

After a preliminary processing, the time series should be tested for stationarity. The test may fail if the time series contains a trend or if its mean value, variance, or spectrum are found to be time-dependent. Deleting the trend can be justified if it is caused by external factors or if it interferes with the higher-frequency part of the spectral density of interest to the researcher. A test for stationarity is suggested through splitting the time series in halves and estimating the mean values, variances and, if possible, spectral densities of the entire time series and its halves. The confidence bounds for estimates of statistical moments depend upon the number of independent observations in the time series. These numbers depend upon the correlation structure of the time series, and they can be much smaller than the total number of terms in the time series. A linear filtering is generally not recommended. The autoregressive approach allows one to determine frequencies of even strictly periodic oscillations contained in the time series (tides) with exceptionally high accuracy providing that the time series is long. A detailed example of autoregressive analysis is given.

Appendix

• 1. https://crudata.uea.ac.uk/cru/data/temperature/

• 2. https://climexp.knmi.nl/data/isunspots.dat

• 3. http://uhslc.soest.hawaii.edu/data/

• 4. https://tidesandcurrents.noaa.gov/harcon.html?id=9414317

• 5. https://www.esrl.noaa.gov/psd/data/timeseries/IPOTPI/.