Time series analysis - basic definitions and examples

Climate of the Earth system

Prof. Dr. Markus Meier
Leibniz Institute for Baltic Sea Research Warnemünde (IOW)
E-Mail: markus.meier@io-warnemuende.de

Time series analysis - basic definitions and examples#

  • decomposition of a time series into deterministic parts and noise

../_images/L13_1_timeseries.PNG

  • exmaples of the volatility of different temperature time series

../_images/L13_2_temperature_series.PNG

  • the El Nino time series spectrum:

../_images/L13_3_elnino.PNG

  • correlation of time series does depend on their time frame

../_images/L13_4_ts_correlation.PNG

  • definition of a stationary process: a stochastic process \(X_t: t \in \mathbb{Z}\) is said to be stationary if all stochastic properties are independent of \(t\) (index of time or space):

    1. \(\mathbf{X_t}\) has the same distribution function F for all \(t\), and

    2. for all \(t\) and \(s\) the parameters of the joint distribution function of \(\mathbf{X_t}\) and \(\mathbf{X_s}\) depend only on \(|t-s|\).

  • most climate variables are not stationary, mostly due to externally forced deterministic cycles such as he diurnal or annual cycles

  • cyclo-stationary processes:

    1. \(\varepsilon(\mathbf{X_t}) = \mu_{t|m}\). The mean is a function of the time within the external cycle, where \(t|m\) refers to the phase in the external cycle.

    2. \(\forall t,s\) is \(f_{X_t X_s} = f_{X_tX_s}(|t-s|,~t|m)\). For all \(t\) and \(s\) the parameters of the joint distribution function, \(f_{X_tX_s}\) of \(\mathbf{X_t}\) and \(\mathbf{X_s}\) depend only on \(|t-s|\) and the phase \(t|m\) in the external cycle.

  • stationarity does not apply to climate variables due to long-term trends in the boundary conditions (e.g. atmospheric carbon increase) -> remove trend and assume cycle-stationarity

  • ergodicity: a physical system is ergodic if the average in time:

\[ \widehat{\mu_t} = \frac{1}{n}\sum_{t=1}^n g(X_t)\]

  • and the ensemble mean

\[ \widehat{\mu_i} = \frac{1}{n}\sum_{i=1}^n g(X_i)\]

  • are equal to another

\[ \widehat{\mu_t} = \widehat{\mu_i}.\]

  • have to take into account that we cannot observe ensembles of the climate system

  • hypotheses testing: Does the climate chage?, insert explanation -> take home message?