A method to characterize climate, Earth or environmental vector random processes

We propose a general methodology to characterize a non-stationary random process that can be used for simulating random realizations that keep the probabilistic behavior of the original time series. The probability distribution of the process is assumed to be a piecewise function defined by several weighted parametric probability models. The weights are obtained analytically by ensuring that the probability density function is well defined and that it is continuous at the common endpoints. Any number of subintervals and continuous probability models can be chosen. The distribution is assumed to vary periodically in time over a predefined time interval by defining the model parameters and the common endpoints as truncated generalized Fourier series. The coefficients of the expansions are obtained with the maximum likelihood method. Different sets of orthogonal basis functions are tested. The method is applied to three time series with different particularities. Firstly, it is shown its good behavior to capture the high variability of the precipitation projected at a semiarid location of Spain for the present century. Secondly, for the Wolf sunspot number time series, the Schwabe cycle and time variations close to the 7.5 and 17 years are analyzed along a 22-year cycle. Finally, the method is applied to a bivariate time series that contains (1) freshwater discharges at the last regulation point of a dam located in a semiarid zone in Andalucia (Spain) which is influenced not only by the climate variability but also by management decisions and (2) the salinity at the mouth of the river. For this case, the analysis, that was combined with a vectorial autoregressive model, focus on the assessment of the goodness of the methodology to replicate the statistical features of the original series. In particular, it is found that it reproduces the marginal and joint distributions and the duration of sojourns above/below given thresholds.