Stochastic Processes. Stationarity

Stochastic processes concern sequences of events governed by probabilistic laws (Karlin and Taylor 1975). A stochastic process X={X(t), t є T} is a collection of random variables. That is, for each “t” in the domain T, X(t) is a random variable. We often interpret “t” as time and call X(t) the state of the process at time “t”. The domain T can be a discrete stochastic process, or a continuous one. Any set of X values is a sample.

We assume that the climate at a given site is fully described by a set of stochas­tic processes X1(t), X2(t), X3 (t)…, each of them representing the stochastic time evolution of a specific climatological quantity (solar irradiance, wind speed, wind direction, temperature, humidity, pressure, etc).

According to the goal, the continuous stochastic processes X1 (t), X2(t), X3 (t)… can be substituted by stochastic sequences (i. e. stochastic processes with discrete time parameter t), obtained by averaging, integrating or sampling them on an appro­priate time basis (for instance, hour or day).

Atmospheric observations separated by relatively short time intervals tend to cor­relate (Wilks 2006). The analysis of the nature of these correlations can be useful for both understanding atmospheric processes and forecasting future atmospheric events.

We do not expect the future values of a data series to be identical to past values of existing observations. However, in many cases, it may be very reasonable to assume that their statistical properties will be similar. The idea that past and future values of a time series will be similar in the statistical sense is an informal expression of what is called stationarity. Usually, this term refers to what is considered “weak” station – arity. In this sense, stationarity implies that the mean and autocorrelation function of the data series do not change in time. Different ensembles of a stationary time series can be regarded as having the same mean and variance.

Most methods of time series analysis assume stationarity of the data. However, many atmospheric processes are distinctly not stationary. Obvious examples of non­stationary atmospheric variables are those exhibiting annual or daily cycles. For instance, solar radiation exhibits inter-annual cycles.

When studying solar radiation, we expect stationarity conditions to be met only in its annual or diurnal cycles. If separated sub-series are compared each other for a period equal to a cycle we expect them to be consistent. Thus, for example, the daily global irradiation can be represented by a cycle-stationary stochastic sequence with a time step of one day and period of one year. Even with this simplification, a considerable number of years with data are needed in order to obtain a satisfactory statistical knowledge of the behaviour the parameters in concern. Since no secular changes in climatic conditions are considered, the set of processes corresponding to solar radiation is stochastically periodic (with period equal to one year), i. e. with probabilistic parameters periodically varying in time.

There are two approaches to deal with non-stationary variables. Both aim at processing the data in a way that will subsequently lead to stationarity. The first approach is to transform the non-stationary data to, approximately, stationary. For example, by subtracting a periodic function from the data subject to an annual cycle can derive a transformed data series with constant mean. In order to produce a series with both constant mean and variance, it might be necessary to parameterize these anomalies.

The alternative way is to stratify the data. That is, to conduct separate analyses for different subset of the data that are short enough to be regarded as nearly stationary, for instance monthly subsets of daily solar irradiation values (Wilks 2006).

Updated: July 31, 2015 — 11:46 am