The main goal of this paper is to propose the probabilistic description of cyclical (business) fluctuations. We generalize a fixed deterministic cycle model by incorporating the time-varying amplitude. More specifically, we assume that the mean function of cyclical fluctuations depends on unknown frequencies (related to the lengths of the cyclical fluctuations) in a similar way to the almost periodic mean function in a fixed deterministic cycle, while the assumption concerning constant amplitude is relaxed. We assume that the amplitude associated with a given frequency is time-varying and is a spline function. Finally, using a Bayesian approach and under standard prior assumptions, we obtain the explicit marginal posterior distribution for the vector of frequency parameters. In our empirical analysis, we consider the monthly industrial production in most European countries. Based on the highest marginal data density value, we choose the best model to describe the considered growth cycle. In most cases, data support the model with a time-varying amplitude. In addition, the expectation of the posterior distribution of the deterministic cycle for the considered growth cycles has similar dynamics to cycles extracted by standard bandpass filtration methods.
The aim of this study was to examine the changes in the chemical composition of shallow groundwater and its quality that have occurred in the last decade in an agriculturally used, heavily populated and characterized by a complex geological structure, catchment of the Stara Rzeka river, located in the ﬂysch part of the Outer Carpathians. Water samples were collected during 2013 from 19 still operating wells. Analyses of pH, electrolytic conductivity and chemical composition by ion chromatography were conducted. The obtained results were compared with the results of studies conducted in 2003 for the same wells. The quality of groundwater and its suitability for consumption was assessed based on the regulations currently existing in Poland. 21% of the wells still do not meet the requirements for drinking water in terms of at least one component. However, there was a decrease in the concentration of mineral forms of nitrogen and phosphorus in most of the wells and their mean concentration as compared to 2003 was reduced. In terms of physical and chemical characteristics groundwater of this region is typical of the hypergenic zone of the temperate climate. The highest concentrations were observed for Ca2+ and HCO3- ions, while K+ and Cl- were characterized by the largest variability. Principal Component Analysis (PCA) demonstrated that the factors determining the quality and chemical composition of the analyzed waters include the composition of bedrock (mineralogy of the rock environment) and human economic activity, and that they have not been signiﬁcantly changed over the past decade.