One of the most intriguing facets of the climate system is that it exhibits variability across all temporal and spatial scales; pronounced examples are temperature and precipitation. The structure of this variability, however, is not arbitrary. Over certain spatial and temporal ranges, it can be described by scaling relationships in the form of power laws in probability density distributions and autocorrelation functions. These scaling relationships can be quantified by scaling exponents which measure how the variability changes across scales and how the intensity changes with frequency of occurrence. Scaling determines the relative magnitudes and persistence of natural climate fluctuations. Here, we review various scaling mechanisms and their relevance for the climate system. We show observational evidence of scaling and discuss the application of scaling properties and methods in trend detection, climate sensitivity analyses, and climate prediction.

This article aims at providing a comprehensive survey of recent developments in the field of integer-valued time series modelling, paying particular attention to models obtained as discrete counterparts of conventional autoregressive moving average and bilinear models, and based on the concept of thinning. Such models have proven to be useful in the analysis of many real-world applications ranging from economy and finance to medicine. We review the literature of the most relevant thinning operators proposed in the analysis of univariate and multivariate integer-valued time series with either finite or infinite support. Finally, we also outline and discuss possible directions of future research.

In this article, we introduce a class of self-exciting threshold integer-valued autoregressive models driven by independent Poisson-distributed random variables. Basic probabilistic and statistical properties of this class of models are discussed. Moreover, parameter estimation is also addressed. Specifically, the methods of estimation under analysis are the least squares-type and likelihood-based ones. Their performance is compared through a simulation study.